Sur La Description Des Formations De . . .

We are interested in nite-time blow-up phenomena for heat equations of the type: @u @t = u+ jujp 1u (1) where u : (x; t) 2 RN [0; T )! R, 1 < p, (N 2)p < N + 2. We rst construct for (1) a solution u which blows-up in nite time T at only one blow-up point x0 2 RN , and describe completely its blow-up pro le (or asymptotic behavior). This construction is based on a priori estimates' technique which reduces the problem to a nite-dimensional one, and on a Brouwer type lemma. This method allows us to derive a stability result of the behavior of u with respect to initial data or perturbation of the nonlinearity. In addition, we generalize the method to the case of vector-valued equations with a non gradient nonlinearity, as well as a vortex reconnection with the boundary in super-conductivity. In a second step, we consider the following equation derived form (1): @w @s = w 1 2y:rw w p 1 + wp; (2) and prove a Liouville Theorem which classi es all uniformly bounded globally (in space and time) de ned solutions of (2). We then obtain a localization property of equation (1) (if u 0) which allows a precise comparison with solutions of the associated ordinary di erential equation. In a third step, we use a consequence of the Liouville Theorem to prove the equivalence of di erent notions of blow-up pro le or asymptotic behavior near a blow-up point x0 of u, namely in variables x, y = x x0 pT t or z = x x0 p(T t)j log(T t)j . Key words: heat equation, singularity, nite time blow-up, nite time quenching, pro le, asymptotic behavior, vector-valued equations, super-conductivity. Table des mati eres 1 Introduction 15 1 Aper cu historique et directions fondamentales de l' etude . . . . . 17 2 Existence et stabilit e d'une solution de (2) avec les comportements (14) et (11) . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Estimations g en erales sur les solutions explosives positives de (2) 24 4 Existence de pro l a l'explosion pour les solutions de (2) . . . . . 26 I Existence et stabilit e de solutions explosives pour des equations de type chaleur et description pr ecise de leur pro l a l'explosion 33 1 Stabilit e du pro l a l'explosion pour les equations du type ut = u+ jujp 1u 35 2 Stability of the blow-up pro le for equations of the type ut = u+ jujp 1u 45 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . 51 2.1 Choice of variables . . . . . . . . . . . . . . . . . . . . . . 51 2.2 Decomposition of q . . . . . . . . . . . . . . . . . . . . . . 52 3 Existence of a blow-up solution with the given free-boundary pro le 54 3.1 Transformation of the problem . . . . . . . . . . . . . . . 55 3.2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . 56 3.3 Proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . . 62 4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1 Case N = 1: . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Case N 2: . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A Proof of lemma 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 77 B Proof of lemma 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3 Blow-up results for vector-valued nonlinear heat equations with no gradient structure 91 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . 95 2.1 Formal asymptotic analysis . . . . . . . . . . . . . . . . . 96 2.2 Transformation of the problem . . . . . . . . . . . . . . . 97 3 Existence of a blow-up solution for equation (2) . . . . . . . . . . 100 3.1 Geometrical property for q . . . . . . . . . . . . . . . . . 100 3.2 Proof of the geometrical property on q(s) . . . . . . . . . 102 3.3 Proof of proposition 3.2 . . . . . . . . . . . . . . . . . . . 105 4 Blow-up pro le of u(t) solution of (2) near blow-up point . . . . 110 5 Generalization and comments . . . . . . . . . . . . . . . . . . . . 114 A Appendix: A blow-up result for @u @t = u+ jujp 1u +ijujq 1u on bounded domain for q small . . . . . . . . . . . . . 116 B Appendix: Proof of lemma 3.3 . . . . . . . . . . . . . . . . . . . . 118 4 Reconnection of vortex with the boundary and nite time Quenching 127 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 1.1 The physical motivation and results . . . . . . . . . . . . 128 1.2 Mathematical setting and strategy of the proof . . . . . . 131 2 Existence of a blow-up solution for equation (16) . . . . . . . . . 136 3 A priori estimates of u(t) in the blow-up zone . . . . . . . . . . . 150 4 A priori estimates in P2 and P3 . . . . . . . . . . . . . . . . . . . 156 4.1 Estimates in P2 . . . . . . . . . . . . . . . . . . . . . . . . 156 4.2 Estimates in P3 . . . . . . . . . . . . . . . . . . . . . . . . 157 A Proof of lemma 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 158 B Proof of lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 164 II Estimations g en erales des solutions positives explosives de l' equation de la chaleur non lin eaire et notions de pro l a l'explosion 177 1 Estimations uniformes a l'explosion pour les equations de la chaleur non lin eaires et applications 179 1 Un th eor eme de Liouville pour l' equation (4) . . . . . . . . . . . 181 2 Estimations optimales a l'explosion . . . . . . . . . . . . . . . . . 182 3 Localisation a l'explosion . . . . . . . . . . . . . . . . . . . . . . 182 4 Notion de Pro l au voisinage d'un point d'explosion . . . . . . . 183 2 Optimal estimates for blow-up rate and behavior for nonlinear heat equations 187 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 2 Optimal blow-up estimates for equation (1) . . . . . . . . . . . . 193 2.1 L1 estimates for the solution of (1) . . . . . . . . . . . . 193 2.2 Global approximated behavior like an ODE . . . . . . . . 198 3 Classi cation of connections between critical points of equation (3) in L1loc . . . . . . . . . . . . . . . . . . . . 200 A Proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . . . . . . 210 B Proof of Proposition 3.4 . . . . . . . . . . . . . . . . . . . . . . . 215 C Proof of i) of Proposition 3.7 . . . . . . . . . . . . . . . . . . . . 219 3 Re ned uniform estimates at blow-up and applications for nonlinear heat equations 229 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 2 L1 estimates of order one for solutions of (3) . . . . . . . . . . . 235 2.1 Formulation and reduction of the problem . . . . . . . . . 235 2.2 Proof of the boundary estimates . . . . . . . . . . . . . . 238 3 Blow-up pro le notions for equation (1) . . . . . . . . . . . . . . 252 A Proof of lemma 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 261 B Proof of Proposition 2.3 . . . . . . . . . . . . . . . . . . . . . . . 262 C Proof of lemma 2.11 . . . . . . . . . . . . . . . . . . . . . . . . . 265 Chapitre 1 Introduction 16 Introduction L'objet de cette th ese est l' etude de la formation en temps ni de singularit es dans des syst emes de r eaction-di usion de type chaleur: 8<: @u @t = u+ F (u) dans [0; T ) u = 0 sur @ [0; T ) u(:; 0) = u0 dans (1) o u u : (x; t) 2 [0; T )! RM , u0 : ! RM , est un ouvert convexe born e et r egulier de RN ou = RN , T > 0, ( u)i = ui, F : RM ! RM est de classe C1, et N;M 2 N. (La condition de bord est a ignorer si = RN ). Ce syst eme constitue un mod ele simpli e pour beaucoup de ph enom enes physiques de r eaction-di usion. Il apparâ t notamment en combustion (voir Williams [58], Kapila [31], Kassoy et Poland [33], [34] (explosions thermiques), Bebernes et Eberly [2] (en particulier, un mod ele de combustion solide), Bebernes et Kassoy [3], Lacey [36], Galaktionov, Kurdyumov et Samarskii [19], Galaktionov et Vazquez [20]). On le retrouve aussi dans beaucoup de situations physiques, de la m ecanique des uides a l'optique, sous la forme de l' equation de Ginzburg-Landau complexe (voir Levermore et Oliver [37]). Le syst eme (1) a egalement un int erêt en neuro-biologie (voir Nagasawa [49], McKean [41]), et dans des mod eles g en etiques (voir Fisher [14]). Le probl eme de Cauchy (local en temps) pour (1) peut être r esolu dans une grande classe d'espaces fonctionnels. Citons par exemple l'espace des fonctions de C( ) nulles sur @ (si est born e) ainsi que l'espace H1 0 \L1( ) que nous consid erons dans la suite (voir Friedman [15], Henry [27], Pazy [50], Weissler [56]). On peut alors d e nir T > 0 comme etant le temps maximum d'existence de la solution de (1). D'apr es la th eorie locale, u 2 C [0; T ); H1 0 \ L1( ) . Deux cas se pr esentent: T = +1: existence globale. T < +1: dans ce cas, lim t!T ku(t)kL1( ) = lim t!T ku(t)kH1( ) = +1: On dit alors que u explose en temps ni T . Par la suite, on s'int eresse a l' etude de telles solutions explosives. Pour cela, on introduit la notion de point d'explosion (voir par exemple Friedman et McLeod [17]): D e nition 1 Un point a 2 est dit point d'explosion de u si u(x; t) n'est pas localement born ee au voisinage de (a; T ), autrement dit s'il existe (xn; tn) ! (a; T ) tel que ju(xn; tn)j ! +1 quand n! +1. Il est classique que Si u explose en temps ni T , alors elle admet (au moins) un point d'explosion. Plusieurs questions se posent autour de l' etude de l'explosion en temps ni dans (1): Aper cu historique et directions fondamentales de l' etude 17 Question 1: Existence. Existe-t-il des solutions de (1) qui explosent en temps ni? Existe-t-il des conditions su santes sur u0 et F qui entrâ nent l'explosion? Question 2: Normes de u. Peut-on avoir des estimations pr ecises des normes (spatiales) de u et de ses d eriv ees a l'explosion? Question 3: Comportement asymptotique. Comment se comporte u asymptotiquement au voisinage de (a; T ) o u a 2 est un point d'explosion? Estce que ce comportement est universel (i.e. ind ependant des donn ees initiales)? Est-il stable par rapport a des perturbations dans les donn ees initiales ou le terme non lin eaire? Question 4: Interaction de la di usion et de la r eaction. Peut-on comparer l' equation (1) a une equation de dynamique locale en espace (une equation di erentielle), ce qui permettrait de comprendre les rôles respectifs et les interactions entre le terme de di usion u et le terme de r eaction F (u), surtout au voisinage de l'explosion. 1 Aper cu historique et directions fondamentales de l' etude Dans la litt erature sur l'explosion en temps ni pour le syst eme (1), le cas suivant a constitu e un prototype int eressant retenu par plusieurs auteurs: @u @t = u+ jujp 1u (2) avec u : (x; t) 2 [0; T )! R, ouvert de RN , 1 < p et si N 3, p < N+2 N 2 . Dans le cas N =M = 1, d'autres auteurs se sont int eress es au cas de @u @t = u+ eu: (3) Nous nous int eressons essentiellement a l' equation prototype (2). Les premi eres conditions su santes d'existence d'une solution explosive pour les equations de type (2) sont dues en particulier a Kaplan [32], Friedman [16], Fujita [18], Levine [38], Ball [1] et Weissler [57]. Par exemple, dans le cas d'un domaine born e , Ball a obtenu, grâce a l' energie associ ee a (2) E(u) = 1 2 Z jruj2dx 1 p+ 1 Z jujp+1dx; et a des m ethodes d' equations di erentielles ordinaires, une obstruction a l'existence globale d'une solution de (2): Si u0 2 H1 0 ( ), u0 6= 0 et E(u0) 0, alors u(t) explose en temps ni. Dans un contexte plus g en eral ( = RN ou born e), Giga et Kohn [23] se sont appuy es sur l' energie locale pond er ee suivante: Ea;t(') = t 2 p 1 N2 +1 Z 12 jr'j2 1 p+ 1 j'jp+1 e jx aj2=4tdx (4) + t 2 p 1 N2 Z 1 2(p 1) j'j2e jx aj2=4tdx 18 Introduction pour trouver une condition n ecessaire de non explosion au voisinage d'un point donn e: Il existe une constante (N; p) > 0 telle que si u(t) explose en temps ni T et si est etoil e en un point a 2 v eri ant Ea;T (u0) < ; alors a n'est pas point d'explosion pour u(t). L' etude du comportement asymptotique de u(t) au voisinage de (a; T ) o u a est un point d'explosion s'est faite d'abord grâce a l'introduction de variables auto-similaires: y = x a pT t ; s = log(T t); wa(y; s) = (T t) 1 p 1u(x; t): (5) D'apr es (2), wa (ou simplement w) v eri e: 8s logT , 8y 2Wa;s, @w @s = w 12y:rw w p 1 + jwjp 1w (6) o u Wa;s = ( a)e s2 . Ainsi, l' etude de u(t) au voisinage de (a; T ) est equivalente a l' etude du comportement asymptotique de wa(s) quand s ! +1. D'ailleurs, l' energie locale Ea;t d e nie en (4) n'est autre que l' equivalent pour u de la fonctionnelle de Liapunov associ ee a (6). Giga et Kohn ont d emontr e dans [21] et [22] que si u0 0 ou p < 3N + 8 3N 4 ou N = 1; alors il existe 0 > 0 et C > 0 tels que 0 < 0 lim s!+1 kw(s)kL1(Wa;s) 1 0 (7) et krw(s)kL1 + k w(s)kL1 + kr w(s)kL1 C: (8) Ceci revient a dire en terme de u que 0 lim t!T(T t) 1 p 1 ku(t)kL1 1 0 et (T t) 1 p 1+ 1 2 kru(t)kL1 + (T t) 1 p 1+1k u(t)kL1 +(T t) 1 p 1+ 32 kr u(t)kL1 C: Une premi ere approche dans la recherche d'un d eveloppement asymptotique pour wa a consist e en une etude de (6) dans L2 o u (y) = e jyj2 4 (4 )N=2 ; (9) Aper cu historique et directions fondamentales de l' etude 19 ce qui a permis d'avoir des convergences de w(s) quand s ! +1 valables uniform ement sur tout compact. Cette etude a et e initi ee par Giga et Kohn qui ont d emontr e dans [22] et [23] qu'il existe la 2 f ; g tel que sup jyj R jwa(y; s) laj ! 0 quand s! +1 o u = (p 1) 1 p 1 : Notons que , et 0 sont les seules solutions de (6) ind ependantes du temps. Ce r esultat a et e pr ecis e dans le cas = RN par Filippas et Kohn [11], Filippas et Liu [13], et Herrero et Vel azquez [28], [53] (voir aussi les articles de revue [30] et [52]), grâce a une analyse dans des espaces de Sobolev avec poids Gaussien (9). Ces auteurs ont montr e que deux cas peuvent se produire: soit il existe k 2 f0; :::; N 1g et une matrice orthonorm ee Q tels que 8R > 0; sup jyj R wa(y; s) 2ps (N k) 12yTAky = o 1s (10) o u Ak = Q 1 IN k 0 0 0 Q et IN k est la matrice identit e (N k) (N k), soit il existe > 0 tel que 8R > 0; sup jyj R jwa(y; s) j C(R)e s: Herrero et Vel azquez ont a n e ce cas de convergence exponentielle a l'ordre 1 (voir [29] et [53]). D'un point de vue physique, ces d eveloppements asymptotiques sont insu sants. En e et, une fois traduite dans les variables d'origine (x; t), la convergence est uniforme uniquement a l'int erieur de paraboles du type jx aj RpT t, ce qui ne permet pas de d eduire un pro l asymptotique de u(t) dans la variable x. N eanmoins, le domaine de convergence a pu être etendu par Herrero et Vel azquez [28], [55] (voir aussi [54]) aux ensembles jzj C o u z = y ps en dimension N. Ils se sont appuy es sur une estimation lin eaire dans des espaces de Sobolev avec poids Gaussien de l'e et du terme convectif 1 2y:rw dans l' equation (6). Ce r esultat de Herrero et Vel azquez leur a permis de d egager dans le cas (10) une notion de pro l limite pour la fonction u au sens o u u(x; t) ! u (x) quand t! T si x 6= a et x est voisin de a, avec u (x) 8pj log jx ajj (p 1)2jx aj2 1 p 1 quand x! a: (11)Ce probl eme a et e egalement explor e d'un point de vue num erique. Citons en particulier une etude num erique de Berger et Kohn [4] qui a permis de d egager 20 Introduction l'existence d'un pro l asymptotique pour certaines solutions de (6) f(z) = p 1 + (p 1)2 4p jzj2 1 p 1 : (12) Il a et e observ e num eriquement dans [4] que w(y; s) f y ps quand s! +1: Bricmont et Kupiainen ont d emontr e dans [6] (voir aussi [5] et Bricmont, Kupiainen et Lin [7]) l'existence d'une donn ee initiale pour w telle que sup y2RN w(y; s) f y ps ! 0 quand s! +1: (13) Grâce a la transformation (5), ceci donne pour tout a 2 RN une solution explosive u(t) de (2) telle que sup x2RN (T t) 1 p 1u(x; t) f x a p(T t)j log(T t)j! ! 0 (14) quand t! T . Nous nous proposons de d evelopper trois directions dans cette th ese: Dans une premi ere direction, on propose une seconde m ethode de d emonstration du r esultat (13) de Bricmont et Kupiainen, bas ee sur la technique d'estimations a priori des solutions de (6) qui permet une r eduction en dimension nie du probl eme, et sur un lemme de type Brouwer. Cette m ethode permet de d egager un r esultat de stabilit e du comportement (13) par rapport a des perturbations dans les donn ees initiales ou dans le terme non lin eaire de r eaction. D'autre part, la m ethode se g en eralise a des equations vectorielles de type chaleur avec non-lin earit e sans structure de gradient, ainsi qu'au traitement d'un probl eme de reconnexion d'un vortex avec la paroi en supra-conductivit e. Dans une seconde direction, on a ne les estimations (7) et (8) de Giga et Kohn, grâce a un Th eor eme de Liouville qui donne une classi cation des solutions globales de (6). On obtient egalement une propri et e de localisation de l' equation (2) qui permet de la comparer de fa con pr ecise a la solution de l' equation di erentielle associ ee. En n, on s'int eresse de nouveau a la notion de pro l et on utilise les estimations qui d ecoulent du Th eor eme de Liouville pour prouver un r esultat d' equivalence des trois notions de pro ls d'explosion ou de d eveloppements asymptotiques en variable x, y, ou z = y ps . 2 Existence et stabilit e d'une solution de (2) avec les comportements (14) et (11) a) Equation de la chaleur avec une non-lin earit e en puissance Existence et stabilit e d'une solution de (2) 21 On consid ere le probl eme de construction d'une solution de l' equation 8<: @u @t = u+ jujp 1u u(x; t) 2 R; x 2 RN ; t 0 1 < p; et si N 3; p < N+2 N 2 (15) qui explose en temps ni donn e T > 0 en un point unique donn e a 2 RN , et telle que u v eri e (14) et (11). Dans [48] et [59] (voir aussi [47]), le r esultat suivant a et e obtenu (Th eor eme 1 page 48, Th eor eme 1 page 93 et Proposition 1 page 93): Th eor eme 1 (Existence) Il existe T0 > 0 tel que pour tous 0 < T̂ T0 et â 2 RN , il existe û0 2 L1 \W 1;p+1(RN ) telle que l' equation (15) avec donn ee initiale û0 admet une solution û(t) explosant en temps ni T̂ uniquement au point â 2 RN , et qui v eri e: i) sup x2RN (T̂ t) 1 p 1 û(x; t) f 0@ x â q(T̂ t)j log(T̂ t)j1A ! 0 (16) quand t! T̂ o u f est d e nie en (12), ii) 8x 2 RN nfâg, û(x; t)! û (x) quand t! T̂ et û (x) 8pj log jx âjj (p 1)2jx âj2 1 p 1 quand x! â: (17) Signalons d'abord que ii) est une cons equence directe de i) grâce a l'invariance de l' equation (15) sous la transformation ! u (x; t) = 2 p 1u( x; 2t); et a des estimations de r egularit e parabolique de (15) s'appuyant sur une condition su sante de non explosion de solutions de (15) due a Giga et Kohn [23] (voir section 4 dans [59]). L'objet du th eor eme se r eduit donc a la construction d'une donn ee initiale u0 pour (15) telle que i) soit satisfaite. La preuve de ceci s'appuie sur: 1) La transformation du probl eme grâce a (5) et a des estimations a priori sur les solutions de (6) au voisinage du pro l f d e ni en (12), ce qui permet de r eduire le probl eme de construction a un probl eme de dimension nie, 2) Une r esolution de ce probl eme de dimension nie a l'aide d'un argument topologique. La m ethode de r eduction en dimension nie initi ee pour la preuve du Th eor eme 1 dans [48] permet d'obtenir un r esultat de stabilit e du comportement (16) (et donc de (17)) par rapport a des perturbations L1 \W 1;p+1(RN ) de la donn ee initiale. Plus pr ecis ement (Th eor eme 2 page 50 et Th eor eme 3 page 114): Th eor eme 2 (Stabilit e du comportement (16) et (17)) Soit û0 la donn ee initiale construite au Th eor eme 1. Soit û(t) la solution de (15) avec donn ee initiale û0 qui explose en temps ni T̂ en un point â. Alors, pour tout > 0, il existe V voisinage de û0 dans L1\W 1;p+1(RN ) tel que pour 22 Introduction tout u0 2 V , la solution u(t) de (15) avec donn ee initiale u0 explose en temps ni T en un point unique a 2 RN tels que ja âj+ jT T̂ j : De plus, u(t) se comporte comme (16) et (17) avec (a; T ) rempla cant (â; T̂ ). Remarque: Par les techniques de localisation pr esent ees a la n de la th ese, on peut avoir le même r esultat dans l'espace d' energie H1. La preuve du Th eor eme 2 s'appuie fondamentalement sur la technique de r eduction en dimension nie du Th eor eme 1 ainsi que sur l'invariance de (6) sous l'action de la transformation g eom etrique (a; T )! wa;T (y; s) = (T t) 1 p 1u(x; t) o u y = x a pT t , s = log(T t), associ ee a une condition de non d eg en erescence lorsque u(x; t) est au voisinage du pro l (T̂ t) 1 p 1 f 0@ x â q(T̂ t)j log(T̂ t)j1A : b) Equation de la chaleur complexe La technique de r eduction a un probl eme de dimension nie s'applique en fait dans un cadre beaucoup plus g en eral que (15), celui des equations vectorielles avec une non-lin earit e ne d erivant pas n ecessairement d'un gradient (voir section 5 dans [59]). Un prototype d'une telle equation est le suivant: 8<: @u @t = u+ (1 + i )jujp 1u u(x; t) 2 C ; x 2 RN ; t 0 1 < p; et si N 3; p < N+2 N 2 : (18) Dans [59], une solution explosive stable de (18) est construite dans le cas o u est petit (voir Th eor eme 1 page 93 et Proposition 1 page 93): Th eor eme 3 (Existence) Il existe 0 > 0 et T0 > 0 tels que pour tous 2 [ 0; 0], 0 < T T0 et a 2 RN , il existe u0 2 L1 \ W 1;p+1(RN ; C ) telle que l' equation (18) avec donn ee initiale u0 admet une solution u(t) explosant en temps ni T uniquement au point a 2 RN et qui v eri e: i) sup x2RN (T t) 1+i p 1 u(x; t) f x a p(T t)j log(T t)j!1+i ! 0 quand t! T , o u f (z) = p 1 + (p 1)2 4(p 2) jzj2 1 p 1 ; ii) 8x 2 RN nfag, u(x; t)! u (x) quand t! T et u (x) 8(p 2)j log jx ajj (p 1)2jx aj2 1+i p 1 quand x! a: Existence et stabilit e d'une solution de (2) 23 Bien que ce Th eor eme soit d'apparence tr es similaire au Th eor eme 1, il en di ere sur deux points: 1) Le Th eor eme 3 pr esente un comportement compl etement complexe, au sens o u le pro l limite obtenu n'a pas de direction xe dans C . 2) La preuve du Th eor eme 3 qui s'appuie fondamentalement sur la technique de r eduction en dimension nie introduite dans le cas r eel, pr esente n eanmoins une di cult e de plus sous la forme d'une direction d eg en er ee suppl ementaire dans le probl eme. Cette di cult e est mâ tris ee grâce a la th eorie de la modulation (voir Filippas et Merle [12] pour un usage similaire de la th eorie de la modulation). Le Th eor eme 2 de [59] g en eralise ce r esultat au cas vectoriel (voir page 115). c) Un probl eme d'extinction en temps ni Comme autre application, le cas de l' equation (2) avec un terme d'amortissement de la non-lin earit e jruj2 u o u 2 (1; p) est trait e de fa con analogue dans [45] (voir Tayachi [51] o u un terme d'amortissement de la forme jrujq est consid er e). En e et, il est montr e dans [45] que l' equation 8>>><>>>: @u @t = u jruj2 u +N(u) u(x; t) 2 R; x 2 RN ; t > 0 N(u) up quand u! +1 jN(u)j C exp( 1 u ) si juj 1 1 < < p; (19) admet une solution explosive en temps ni stable avec des comportements analogues a (16) et (17) (voir Proposition 1 page 133). Remarque: Signalons que si 1 < p dans (19), alors des changements de fonctions evidents ram enent (19) au cas de (2) ou (3), deux cas o u l'on dispose d ej a de solutions explosives (voir la remarque apr es Proposition 1 page 133). Grâce a une transformation du type h(x; t) = 1 u(x; t) ; le r esultat pour l' equation (19) donne un r esultat d'extinction en temps ni pour le probl eme suivant: @h @t = h G(h) dans [0; T ) h(x; t) = 1 sur @ [0; T ) (20) o u est un ouvert born e, G(h) 1 h quand h! 0 et > 0. Si h est d e nie sur [0; T ) et lim t!T inf x2 h(x; t) = 0; alors on dit que h s' eteint en temps ni. L' equation (20) constitue un mod ele de reconnexion d'un vortex avec la paroi dans un semi-conducteur de type II si = 1 (voir Chapman, Hunton et 24 Introduction Ockendon [8]). Elle est egalement reli ee a l' equation de di usion g en er ee par des ph enom enes de polarisation dans des conducteurs ioniques (voir Kawarada [35]). Quelques crit eres d'extinction en temps ni pour (20) etaient d ej a connus en dimension 1 (voir Kawarada [35], Levine [39] (article de revue), [40]). Cependant, peu de choses etaient connues sur le comportement de la solution a l'extinction, sauf en ce qui concerne la localisation des points d'extinction (voir Guo [24], Deng et Levine [10]), ou le taux d'extinction (voir Guo [24], [25], [26]). Dans [45], une solution stable de (20) s' eteignant en temps ni en un seul point est construite. Son comportement au voisinage du point d'extinction (analogue du temps d'explosion) est d ecrit avec pr ecision (voir le Th eor eme de la page 130). Th eor eme 4 (Existence d'une solution de (20) s' eteignant en temps ni) Pour tout a 2 , l' equation (20) admet une solution h s' eteignant en temps ni T > 0. De plus, 8x 2 nfag, h(x; t)! h (x) quand t! T et h (x) ( + 1)2jx aj2 8 j log jx ajj 1 +1 quand x! a: 3 Estimations g en erales sur les solutions explosives positives de (2) On se propose maintenant d'a ner les estimations (7) et (8) de Giga et Kohn. Dans ce but, on s'int eresse d'abord au probl eme de classi cation des solutions de (6) globales en espace et en temps et uniform ement born ees. Dans [44], le r esultat suivant est etabli (Th eor eme 2 page 191): Th eor eme 5 (Th eor eme de Liouville pour (6)) Soit w une solution de (6) d e nie pour (y; s) 2 RN R telle que 8(y; s) 2 RN R, 0 w(y; s) C. Alors, on est n ecessairement dans l'un des cas suivants: i) w 0, ii) w o u = (p 1) 1 p 1 , iii) 9s0 2 R tel que w(y; s) = '(s s0) o u '(s) = (1 + es) 1 p 1 : Remarque: Remarquons que ' est une connexion dans L1 des deux points critiques de (6): 0 et . En e et, _ ' = ' p 1 + 'p; '( 1) = ; '(+1) = 0: Remarque: Il su t d'avoir une solution de (6) d e nie sur ( 1; s ) pour avoir un th eor eme de classi cation (voir Corollaire 2 page 191). A travers la transformation (5), ce Th eor eme a comme corollaire le r esultat suivant (voir Corollaire 3 page 191): Corollaire 1 Soit u une solution de (2) d e nie pour (x; t) 2 RN ( 1; T ) telle que 8(x; t) 2 RN ( 1; T ), 0 u(x; t) C(T t) 1 p 1 . Alors, soit u 0, soit 9T T tel que u(x; t) = (T t) 1 p 1 . Estimations g en erales sur les solutions explosives positives de (2) 25 La preuve du Th eor eme 5 s'appuie fondamentalement sur les points suivants: 1) une classi cation des comportements lin eaires de w(s) quand s ! 1 dans L2 (RN ) (L1loc(RN )) o u (y) = e jyj2 4 (4 )N=2 , 2) l'usage des transformations g eom etriques w(y; s)! wa;b(y; s) = w(y + ae s 2 ; s+ b) pour a 2 RN et b 2 R, 3) un crit ere d'explosion en temps ni au voisinage du point critique de la fonctionnelle d' energie associ ee a (6): Si w est solution de (6) d e nie pour tout temps s logT et que pour un certain s0 logT , R w(y; s0) (y)dy > R (y)dy, alors w(s) explose en temps ni. (Proposition 3.5 page 205). A l'aide d'un argument de compacit e, on obtient dans [44] les estimations uniformes suivantes sur les solutions positives de (2) (Th eor eme 1 page 189) Th eor eme 6 (Estimations optimales a l'ordre 0 sur u(t) a l'explosion) On suppose que est un domaine convexe born e de classe C2; dans RN ou que = RN . On consid ere u(t) une solution de l' equation (2) explosant en temps ni T . Si de plus, u(0) 0 et u(0) 2 H1( ), alors, (T t) 1 p 1 ku(t)kL1( ) ! et (T t) 1 p 1+1k u(t)kL1 + (T t) 1 p 1+ 12 kru(t)kL1 ! 0 quand t! T: De fa con equivalente, pour tout a 2 , kwa(s)kL1 ! et k wa(s)kL1 + krwa(s)kL1 ! 0 quand s! +1: Le Th eor eme 6 combin e avec des estimations a priori des solutions de (6) dans W 3;1(RN ) a permis dans [46] d'a ner les r esultats a l'ordre un dans le cas = RN (Th eor eme 1 page 231): Th eor eme 7 (Estimations uniformes optimales a l'ordre un sur les solutions positives de (2)) Il existe des constantes positives C1, C2 et C3 telles que pour toute solution positive de (2) explosant en temps ni v eri ant u(0) 2 H1(RN ) et pour tout > 0, il existe t0( ) < T tel que i) 8t 2 [t0( ); T ), ku(t)kL1 + (N 2p + ) 1 j log(T t)j (T t) 1 p 1 ; kriu(t)kL1 Ci (T t) ( 1 p 1+ i 2 ) j log(T t)ji=2 pour i = 1; 2; 3, ii) 8s log(T t0( )), 8a 2 RN , kwa(s)kL1 + (N 2p + )1s ; kriwa(s)kL1 Ci si=2 : 26 Introduction Remarque: Dans le cas N = 1, Herrero et Vel azquez [28] (Filippas et Kohn [11] aussi) ont prouv e des estimations reli ees au Th eor eme 7, grâce a une propri et e de Sturm utilis ee en premier par Chen et Matano [9] (le nombre d'oscillations en espace est une fonction d ecroissante du temps). Remarque: Il existe dans [44] et [46] des versions des Th eor emes 6 et 7 valables pour une suite de solutions de (2) et qui donnent de la compacit e (Th eor eme 1' page 190 et Th eor eme 1' page 232). Le Th eor eme 6 nous permet de comparer les tailles relatives des termes de di usion ( u) et de r eaction (up) dans (2) ponctuellement en espace-temps. En e et, on d emontre dans [44] le Th eor eme de localisation suivant (Th eor eme 3 page 192): Th eor eme 8 (Comparaison avec l' equation di erentielle ordinaire) On suppose que est un domaine convexe et born e de classe C2; ou = RN , et que u0 2 H1( ). Alors, 8 > 0, 9C > 0 tel que 8t 2 [T2 ; T ), 8x 2 , jut upj up + C : Ainsi, la solution de l' equation aux d eriv ees partielles (2) est comparable uniform ement et globalement en espace-temps a une solution de l' equation diff erentielle ordinaire (localis ee par d e nition) u0 = up: (21) Ce Th eor eme constitue ainsi une justi cation a posteriori du changement de variables (5) qui a permis toute l' etude de (2). En e et, le choix de (5) etait en quelque sorte motiv e par la recherche d'une comparaison de u(x; t) a (T t) 1 p 1 qui est justement la solution de (21) qui explose au temps T . Remarque: De multiples cons equences d ecoulent de ce th eor eme. Par exemple (Corollaire 1 de [44], page 188): Corollaire 2 On suppose que est un domaine convexe et born e de classe C2; ou = RN . Alors, pour toute solution positive u de (2) qui explose en temps ni T et qui v eri e u(0) 2 H1( ), pour tout 0 > 0, il existe t0( 0; u0) < T tel que pour tous a 2 et t 2 [t0; T ), si u(a; t) (1 0) (T t) 1 p 1 , alors a n'est pas point d'explosion de u. 4 Existence de pro l a l'explosion pour les solutions de (2) Grâce aux estimations de Th eor eme 7, on d emontre dans [46] un Th eor eme de classi cation des pro ls dans la variable y ps , qui s epare l'espace en partie singuli ere (l a o u il y a explosion) et partie r eguli ere dans le cas non d eg en er e (Th eor eme 2 page 234). Th eor eme 9 (Classi cation des pro ls a l'explosion) Il existe k 2 f0; 1; :::; Ng et une matrice N N orthogonale Q tels que w(Q(z)ps; s)! fk(z) uniform ement sur tout compact jzj C, o u fk(z) = (p 1+ (p 1)2 4p N k Xi=1 jzij2) 1 p 1 si k N 1 et fN (z) = = (p 1) 1 p 1 . Existence de pro l a l'explosion pour les solutions de (2) 27 Remarque: Ce r esultat a et e prouv e aussi par Vel azquez dans [55]. Cependant, grâce aux techniques uniformes de [46], on peut montrer que la vitesse de convergence est ind ependante du point d'explosion consid er e, alors qu'elle en d epend dans le r esultat de [55]. Un des probl emes int eressants qui en d ecoule est de relier toutes les notions de pro ls connues: pro l pour jyj born e, jzj = jyj ps born e ou x ' 0. On d emontre dans [46] que ces notions sont equivalentes dans le cas d'une solution qui explose en un point de fa con non d eg en er ee (cas g en erique), ce qui clari e de nombreux points evoqu es dans des travaux pr ec edents. On a nalement le Th eor eme suivant (Th eor eme 3 page 234). Th eor eme 10 ( Equivalence des comportements explosifs en un point) Soit u(t) une solution de (2) d e nie sur RN [0; T ), et a 2 RN . On a l' equivalence des trois comportements suivants de u(t) et de wa(s) (d e nie en (5)): i) 8R > 0, sup jyj R wa(y; s) + 2ps(N 1 2 jyj2) = o 1s quand s ! +1, ii) 8R > 0, sup jzj R wa(zps; s) f0(z) ! 0 quand s ! +1 avec f0(z) = (p 1 + (p 1)2 4p jzj2) 1 p 1 , iii) 9 0 > 0 tel que pour tout jx aj 0, u(x; t) ! u (x) quand t ! T et u (x) h 8pj log jx ajj (p 1)2jx aj2 i 1 p 1 quand x! a. Dans ce cas, a un point d'explosion isol e de u(t). Remarque: Grâce aux estimations uniformes utilis ees dans la preuve de ce th eor eme, on peut prouver que les vitesses de convergence dans chaque expression i), ii) ou iii) d epend de la vitesse de convergence dans les deux autres et d'une borne sur ku0kC2(RN) (et non sur u0). Les estimations de Vel azquez [55] permettent aussi d'avoir de r esultat d' equivalence, mais les convergences d ependent du point d'explosion consid er e. La th ese est organis ee en deux parties: Premi ere partie: Existence et stabilit e de solutions explosives pour des equations de type chaleur et description pr ecise de leur pro l a l'explosion. Organis ee en quatre articles [47], [48], [59] et [45] (dont trois en commun avec Frank Merle), elle reprend les r esultats de la section 2 de cette introduction. Deuxi eme partie: Estimations g en erales des solutions positives explosives de l' equation de la chaleur non lin eaire et notions de pro ls a l'explosion. Elle englobe les r esultats des sections 3 et 4 de l'introduction, sous la forme de trois articles ecrits en collaboration avec Frank Merle ([43], [44] et [46]). Bibliographie 29 Bibliographie [1] Ball, J., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford 28, 1977, pp. 473-486. [2] Bebernes, J., et Eberly, D., Mathematical problems from combustion theory. Applied Mathematical Sciences, 83. Springer-Verlag, New York, 1989. [3] Bebernes, J.W., et Kassoy, D.R., A mathematical analysis of blowup for thermal reactions|the spatially nonhomogeneous case, SIAM J. Appl. Math. 40, 1981, pp. 476-484. [4] Berger, M., et Kohn, R., A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math. 41, 1988, pp. 841-863. [5] Bricmont, J., et Kupiainen, A., \Renormalization group and nonlinear PDEs", in Quantum and non-commutative analysis (Kyoto, 1992), Math. Phys. Stud., 16, Kluwer, Dordrecht, 1993, pp. 113-118. [6] Bricmont, J., et Kupiainen, A., Universality in blow-up for nonlinear heat equations, Nonlinearity 7, 1994, pp. 539-575. [7] Bricmont, J., Kupiainen, A., et Lin, G., Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math. 47, 1994, pp. 893-922. [8] Chapman, S.J., Hunton, B.J., et Ockendon, J.R., Vortices and Boundaries, Quart. Appl. Math. a parâitre. [9] Chen, X., Y., et Matano, H., Convergence, asymptotic periodicity, and nite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations 78, 1989, pp. 160-190. [10] Deng, K., et Levine, H.A., On the blow-up of ut at quenching, Proc. Am. Math. Soc. 106, 1989, pp. 1049-1056 [11] Filippas, S., et Kohn, R., Re ned asymptotics for the blowup of ut u = up, Comm. Pure Appl. Math. 45, 1992, pp. 821-869. [12] Filippas, S., et Merle, F., Modulation theory for the blowup of vectorvalued nonlinear heat equations, J. Di erential Equations 116, 1995, pp. 119-148. 30 Introduction [13] Filippas, S., et Liu, W. X.,On the blowup of multidimensional semilinear heat equations, Ann. Inst. H. Poincar e Anal. Non Lin eaire 10, 1993, pp. 313-344. [14] Fisher, R.A., The advance of advantageous genes, Ann. of Eugenics 7, 1937, pp. 355-369. [15] Friedman, A. Partial di erential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. [16] Friedman, A., Remarks on nonlinear parabolic equations, Proc. of Sympos. Appl. Math., vol. 17, 1965, pp. 3-23, AMS. [17] Friedman, A., et McLeod, B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34, 1985, pp. 425-447. [18] Fujita, H., On the blowing-up of solutions of the Cauchy problem for ut = u+ u1+ , J. Fac. Sci. Univ. Tokyo Sect. I 13, 1966, pp. 109-124. [19] Galaktionov, V., A., Kurdyumov, S., P., et Samarskii, A., A., On approximate self-similar solutions for some class of quasilinear heat equations with sources, Math. USSR-Sb 52, 1985, pp. 155-180. [20] Galaktionov, V., A., et Vazquez, J., L., Regional blow-up in a semilinear heat equation with convergence to a Hamilton-Jacobi equation, SIAM J. Math. Anal., 24, 1993, pp. 1254-1276. [21] Giga, Y., et Kohn, R., Asymptotically self-similar blowup of semilinear heat equations, Comm. Pure Appl. Math. 38, 1985, pp. 297-319. [22] Giga, Y., et Kohn, R., Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36, 1987, pp. 1-40. [23] Giga, Y., et Kohn, R., Nondegeneracy of blow-up for semilinear heat equations, Comm. Pure Appl. Math. 42, 1989, pp. 845-884. [24] Guo, J., On the quenching behavior of the solution of a semilinear parabolic equation, J. Math. Anal. Applic. 151, 1990 pp 58-79. [25] Guo, J., On the quenching rate estimate, Quart. Appl. Math. 49, 1991, pp. 747-752. [26] Guo, J., On the semilinear elliptic equation w 1 2y:rw+ w w = 0 in Rn , Chinese J. Math. 19, 1991, pp. 355-377. [27] Henry, D., Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. [28] Herrero, M.A, et Vel azquez, J.J.L., Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincar eAnal. Non Lin eaire 10, 1993, pp. 131-189. [29] Herrero, M.A, et Vel azquez, J.J.L., Flat blow-up in one-dimensional semilinear heat equations, Di erential Integral Equations 5, 1992, pp. 973997. Bibliographie 31 [30] Herrero, M.A, et Vel azquez, J.J.L., Some results on blow up for semilinear parabolic problems, Degenerate di usions (Minneapolis, MN, 1991), IMA Vol. Math. Appl., 47, Springer, New York, 1993, pp. 105-125. [31] Kapila, A.K., Reactive-di usion system with Arrhenius kinetics: dynamics of ignition, SIAM J. Appl. Math. 39, 1980, pp. 21-36. [32] Kaplan, S.,On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math. 16, 1963, pp. 305-330. [33] Kassoy, D., et Poland, J. The thermal explosion con ned by a constant temperature boundary. I. The induction period solution, SIAM J. Appl. Math. 39, 1980, pp. 412-430. [34] Kassoy, D., et Poland, J. The thermal explosion con ned by a constant temperature boundary. II. The extremely rapid transient, SIAM J. Appl. Math. 41, 1981, pp. 231-246. [35] Kawarada, H., On solutions of initial boundary value problem for ut = uxx + 1 1 u , RIMS Kyoto Univ. 10, 1975, pp. 729-736. [36] Lacey, A.A., Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math. 43, 1983, pp. 1350-1366. [37] Levermore, C., D., et Oliver, M.,The complex Ginzburg-Landau equation as a model problem, Dynamical systems and probabilistic methods in partial di erential equations (Berkeley, 1994), Lectures in Appl. Math., 31, Amer. Math. Soc., Providence, RI, 1996, pp. 141-190. [38] Levine, H., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = Au+F (u), Arch. Rat. Mech. Anal. 51, 1973, p. 371-386. [39] Levine, H.A., Advances in quenching, Nonlinear di usion equations and their equilibrium states, 3 (Gregynog, 1989), pp. 319-346. [40] Levine H.A., Quenching, nonquenching, and beyond quenching for solutions of some parabolic equations, Annali Mat. Pura. Appl. 155, 1989, pp. 243-260. [41] McKean, H.P., Application of a Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math. 28, 1975, pp. 323-331. [42] Merle, F., Solution of a nonlinear heat equation with arbitrary given blow-up points, Comm. Pure Appl. Math. 45, 1992, pp. 263-300. [43] Merle, F., et Zaag, H., Estimations uniformes a l'explosion pour les equations de la chaleur non lin eaires et applications, S eminaire EDP, 19961997, Ecole Polytechnique, pp. XIX-1 XIX-8. [44] Merle, F., et Zaag, H., Optimal estimates for blow-up rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51, 1998, pp. 139-196. 32 Introduction [45] Merle, F., et Zaag, H., Reconnection of vortex with the boundary and nite time quenching, Nonlinearity 10, 1997, pp-1497-1550. [46] Merle, F., et Zaag, H., Re ned uniform estimates at blow-up and applications for nonlinear heat equations, Geom. Funct. Anal., a parâ tre. [47] Merle, F., et Zaag, H. Stabilit e du pro l a l'explosion pour les equations du type ut = u+ jujp 1u, C. R. Acad. Sci. Paris S er. I Math. 322, 1996, pp. 345-350. [48] Merle, F., et Zaag, H., Stability of the blow-up pro le for equations of the type ut = u+ jujp 1u, Duke Math. J. 86, 1997, pp. 143-195. [49] Nagasawa, M., A limit theorem of a pulse-like wave form for a Markov process, Proc. Japan Acad. 44, 1968 pp. 491-494 [50] Pazy, A. Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York-Berlin, 1983. [51] Tayachi, S., Solutions autosimilaires d' equations semi-lin eaires paraboliques avec terme de gradient non lin eaire, Th ese de doctorat, Universit e Paris-nord. [52] Vel azquez, J.J.L., Blow up for semilinear parabolic equations, Recent advances in partial di erential equations (El Escorial, 1992), RAM Res. Appl. Math., 30, Masson, Paris, 1994, pp. 131-145. [53] Vel azquez, J.J.L., Classi cation of singularities for blowing up solutions in higher dimensions, Trans. Amer. Math. Soc. 338, 1993, pp. 441-464. [54] Vel azquez, J.J.L., Estimates on the (n 1)-dimensional Hausdor measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J. 42, 1993, pp. 445-476. [55] Vel azquez, J.J.L., Higher-dimensional blow up for semilinear parabolic equations, Comm. Partial Di erential Equations 17, 1992, pp. 1567-1596. [56] Weissler, F.B., Local existence and nonexistence for semilinear parabolic equations in Lp, Indiana Univ. Math. J. 29, 1980, pp. 79{102. [57] Weissler, F.B., Single point blow-up for a semilinear initial value problem, J. Di erential Equations, 55, 1984, pp. 204-224. [58] Williams, F., Combustion theory, Addison-Wesley, Reading MA, 1983. [59] Zaag, H., Blow-up results for vector valued nonlinear heat equations with no gradient structure, Ann. Inst. H. Poincar e Anal. Non Lin eaire 15, 1998, a parâ tre. Premi ere partie Existence et stabilit e de solutions explosives pour des equations de type chaleur et description pr ecise de leur pro l a l'explosion Chapitre 1 Stabilit e du pro l a l'explosion pour les equations du type ut = u + jujp 1u 36 Stabilit e du pro l a l'explosion pour ut = u+ jujp 1u Equations aux d eriv ees partielles/Partial Di erential Equations Stabilit e du pro l a l'explosion pour les equations du type ut = u+ jujp 1u y Frank Merle et Hatem Zaag R esum e On consid ere dans cette note l' equation non lin eaire suivante: ut = u+ jujp 1u; u(:; 0) = u0; (et d'autres extensions de cette equation, o u le principe du maximum ne s'applique pas). On d ecrit d'abord le comportement au voisinage de l'explosion d'une solution explosant en temps ni. Ensuite, on montre que ce comportement est stable. Stability of blow-up pro le for equation of the type ut = u+ jujp 1u Abstract In this note, we consider the following nonlinear heat equation ut = u+ jujp 1u; u(:; 0) = u0; (and various extensions of this equation, where the maximum principle do not apply). We rst describe precisely the behavior of a blow-up solution near blow-up time and blow-up point. We then show a stability result on this behavior. Abridged English Version In this note, we consider the following nonlinear equation: (1) ut = u+ jujp 1u; u(:; 0) = u0 2 H; where u(t) : x 2 RN ! u(x; t) 2 R. We note H = W 1;p+1(RN ) \ L1(RN ). We assume in addition the exponent p subcritical: if N 3 then 1 < p < (N +2)=(N 2), otherwise, 1 < p < +1. Other types of equations will be also considered. We study the case where the solution u(t) of (1) blows-up in nite time in the sense that u exists on [0; T ) with T < +1, and ku(t)kH ! +1 when t ! T . (see Ball [1], Levine [13]). In this case, there is at least one blow-up point a (that is a 2 RN such that: ju(a; t)j ! +1 when t ! T ). We are interested in the structure of the solution near this point. We want to study the behavior of u(t) near blow-up time and point, and the stability of such behavior. This problem has been extensively studied in the last recent years (see [6], [8], [9], [10]). In [3], Bricmont and Kupiainen construct a blow-up solution u(t) for (1), which approaches a universal pro le with a boundary layer separating regions where u(x; t) is \large" and regions where u(x; t) is \small", and moving at the rate (2) p(T t)j log(T t)j: For that, they used ideas close to the renormalization theory, and some hard analysis. yNote parue dans C. R. Acad. Sci. Paris S er. I Math. 322, 1996, pp. 345-350. Abridged english version 37 In this note, we shall give an idea of a more elementary proof of their result, based on a more geometrical approach and on techniques of a priori estimates, which apply to other contexts. (In particular, we do not use maximum principle). Theorem 1 Existence of a blow-up solution with a boundary layer with the rate (2) There exists T0 > 0 such that for each T 2 (0; T0], 8g 2 H with kgkL1 (log T ) 2, one can nd d0 2 R and d1 2 RN such that for each a 2 RN , the solution u(t) of equation (1) with initial data u0(x) = T 1 p 1nf(z)(1 + d0 + d1z p 1 + (p 1)2 4p jzj2 ) + g(z)o; where: z = (x a)(j logT jT ) 1 2 , blows-up at T at only one blow-up point: a. Moreover, (3) lim t!T k(T t) 1 p 1 u(a+ ((T t)j log(T t)j) 12 z; t) f(z)kL1(RN) = 0; with f(z) = (p 1 + (p 1)2 4p jzj2) 1 p 1 : Remark: Such behavior is suspected to be generic. Remark: Related results using strongly dimension one and maximum principle were obtained in [11] for N = 1. In [19], second author shows analogous results for the following equations (where the maximum principle do not apply): ut = u+ jujp 1u+ ijujr 1u, Ut = U + jU jp 1U +F1(U) with U : RN ! RM , jF1(U)j CjU jr, where p < N+2 N 2 if N 3, and 1 r < p. Moreover, we suspect that the same analysis can be carried for other types of equations not satisfying maximum principle, for example: ut = 2u+ jujp 1u: As in the paper of Bricmont and Kupiainen [2], we won't use maximum principle in the proof. The technique used here will allow us using geometrical interpretation of quantities of the type of d0 and d1 to derive stability results concerning this type of behavior, with respect to perturbations of the initial data. Theorem 2 Stability of the blow-up behavior with respect to initial data Let û0 be an initial data constructed in Theorem 1. Let û(t) be the solution of equation (1) with initial data û0, T̂ its blow-up time and â its blow-up point. Then there exists a neighborhood V0 of û0 in H which has the following property: For each u0 in V0, u(t) blows-up in nite time T = T (u0) at only one blow-up point a = a(u0), where u(t) is the solution of equation (1) with initial data u0. Moreover, u(t) behaves near T (u0) and a(u0) in an analogous way as û(t): lim t!T k(T t) 1 p 1u(a+ ((T t)j log(T t)j) 12 z; t) f(z)kL1(RN) = 0: Remark: Theorem 2 yields the fact that the blow-up pro le f(z) is stable with respect to perturbations in initial data. Remark: From [14], we have T (u0)! T̂ , a(u0)! â, as u0 ! û0 in H . According to a result in [14], we have the following corollary: Corollary 1 (N 2) For arbitrary given set of k points x1,..., xk in RN , there exists initial data u0 such that the solution u of (1) with initial data u0 blows-up exactly at x1,..., xk. 38 Stabilit e du pro l a l'explosion pour ut = u+ jujp 1u I Introduction Dans cette note, on consid ere l' equation de la chaleur non lin eaire suivante:ut = u+ jujp 1u; u(:; 0) = u0 2 H; (1) avec u(t) : x 2 RN ! u(x; t) 2 R. On noteH =W 1;p+1(RN )\L1(RN ). De plus, on suppose l'exposant p sous-critique: si N 3 alors 1 < p < (N + 2)=(N 2), sinon, 1 < p < +1. D'autres types d' equations seront egalement consid er es. On etudie le cas o u la solution u(t) de (1) explose en temps ni, au sens o u u existe sur [0; T ) avec T < +1 et ku(t)kH ! +1 quand t ! T . (voir Ball [1], Levine [13]). Dans ce cas, il existe au moins un point d'explosion a (qui est un point a 2 RN satisfaisant: ju(a; t)j ! +1 quand t ! T ). On s'int eresse a la structure de la solution au voisinage de ce point. On etudie le comportement de u(t) au voisinage du temps et du point d'explosion, ainsi que la stabilit e d'un tel comportement. Ce probl eme a et e beaucoup etudi e ces derni eres ann ees (voir [6], [8], [9], [10]). Dans [3], Bricmont et Kupiainen ont construit une solution u(t) de (1), explosant en temps ni, et qui approche un pro l universel s eparant les r egions o u u(x; t) est \grande" des r egions o u u(x; t) est \petite", avec une interface se d epla cant selon p(T t)j log(T t)j: (2) Pour d emontrer ce r esultat, ils ont utilis e des id ees proches de la th eorie de la renormalisation, et des estimations assez di ciles. Dans cette note, on donne une id ee d'une preuve plus el ementaire de leur r esultat, s'appuyant sur une approche plus g eom etrique et sur des techniques d'estimations a priori, qui s'appliquent dans d'autres contextes. (En particulier, on n'utilise pas de principe du maximum). Th eor eme 1 Existence d'une solution explosant en temps ni avec une interface se d epla cant selon (2) Il existe T0 > 0 tel que pour tout T 2 (0; T0], 8g 2 H avec kgkL1 (logT ) 2, on peut trouver d0 2 R et d1 2 RN tels que pour tout a 2 RN , la solution u(t) de l' equation (1) avec donn ee initiale u0(x) = T 1 p 1nf(z)(1 + d0 + d1z p 1 + (p 1)2 4p jzj2 ) + g(z)o; avec: z = (x a)(j log T jT ) 12 , explose en temps ni T en un seul point d'explosion: a. De plus, lim t!T k(T t) 1 p 1u(a+ ((T t)j log(T t)j) 12 z; t) f(z)kL1(RN) = 0; (3) avec f(z) = (p 1 + (p 1)2 4p jzj2) 1 p 1 : Remarque: On soup conne ce comportement d'être g en erique. Remarque: Des r esultats similaires ont et e obtenus dans [11] pour N = 1 grâce au principe du maximum et a la dimension un. Dans [19], le deuxi eme auteur montre des r esultats analogues pour les equations suivantes (o u le principe du maximum ne s'applique pas): Id ees de la d emonstration du Th eor eme 1 39 ut = u+ jujp 1u+ ijujr 1u, Ut = U+ jU jp 1U+F1(U); avec U : RN ! RM , jF1(U)j CjU jr, p < N+2 N 2 si N 3, et 1 r < p. De plus, on pense qu'on peut obtenir par les mêmes m ethodes des r esultats d'explosion pour d'autres equations ne v eri ant pas le principe du maximum, par exemple: ut = 2u+ jujp 1u: Comme dans [2], on n'utilise pas de principe du maximum dans la preuve. Les techniques utilis ees ici permettront grâce a une interpretation g eom etrique de quantit es du type de d0 et d1 d'obtenir des r esultats de stabilit e concernant ce type de comportement par rapport aux donn ees initiales. Th eor eme 2 Stabilit e du comportement a l'explosion par rapport aux donn ees initiales Soit û0 une donn ee initiale construite au Th eor eme 1. Soit û(t) la solution de l' equation (1) avec donn ee initiale û0, T̂ son temps d'explosion, et â son point d'explosion. Alors il existe un voisinage V0 de û0 dans H avec les propri et es suivantes: Pour tout u0 dans V0, u(t) explose en temps ni T = T (u0) en un seul point d'explosion a = a(u0), o u u(t) est la solution de l' equation (1) avec donn ee initiale u0. De plus, le comportement de u(t) au voisinage de T (u0) et a(u0) est analogue au comportement de û(t) au voisinage de T̂ et â: lim t!T k(T t) 1 p 1u(a+ ((T t)j log(T t)j) 12 z; t) f(z)kL1(RN) = 0: Remarque: Le Th eor eme 2 implique que le pro l a l'explosion f(z) est stable par rapport a des perturbations dans les donn ees initiales. Remarque: Apr es [14], on a T (u0)! T̂ , a(u0)! â, quand u0 ! û0 dans H . D'apr es un r esultat dans [14], on a le corollaire suivant: Corollaire 1 (N 2) Pour tout ensemble de k points x1,..., xk dans RN , il existe une donn ee initiale u0 telle que la solution u de (1) avec donn ee initiale u0 explose exactement en x1,..., xk. II Id ees de la d emonstration du Th eor eme 1 Pour les preuves des autres r esultats et pour plus de d etails, voir [15]. Partie I : Transformation du probl eme Nous traitons le cas N = 1. On introduit les variables auto-similaires: y = x a pT t ; s = log(T t); w(y; s) = (T t) 1 p 1u(x; t); (4) o u a est le point d'explosion et T le temps d'explosion de la solution u(t) a construire. On introduit q(y; s) = w(y; s) '(y; s) = w(y; s) f 2ps + (p 1 + (p 1)2 4p y2 s ) 1 p 1 g: (5) q satisfait l' equation suivante: @q @s = (L+ V (y; s))q +E(q; y; s); (6) 40 Stabilit e du pro l a l'explosion pour ut = u+ jujp 1u avec L(q) = @2q @y2 12y @q @y + q, V (y; s) = p('p 1 1 p 1 ) et E(q; y; s) = fj'+qjp 1('+q) 'p p'p 1qg+f@2' @y2 1 2y @' @y 1 p 1'+'p @' @s g. On ecrit alors q(y; s) = q(y; s) 0( y K0ps ) + q(y; s)(1 0( y K0ps )) = qb(y; s) + qe(y; s), o u K0 > 0, 0 2 C1 0 (R), 0 1 sur [ 1; 1] et 0 0 sur Rn[ 2; 2]. On d ecompose ensuite qb suivant le spectre de L dans L2(R; d ) avec d (y) = ey2=4 p4 (spec(L) = f1 m2 jm 2 Ng). On obtient: q(y; s) = qb(y; s) + qe(y; s) = f 2 X m=0 qm(s)hm(y) + q (y; s)g+ qe(y; s); (7) o u hm est la fonction propre qui correspond a 1 m2 , q (y; s) est la projection de qb(y; s) sur l'espace des valeurs propres n egatives de L. On va construire u0 telle que u(t) v eri e une estimation plus forte que (3). D e nissons d'abord pour A > 0, s > 0: VA(s) = fr 2 L2(R; d ) j jrmj As 2;m = 0; 1; jr2j A2(log s)s 2; jr (y)j A(1+ jyj3)s 2; krekL1 A2s 1 2 ; avec r(y) =P2m=0 rmhm(y)+ r (y)+ re(y)g, V̂A(s) = [ A s2 ; A s2 ]2 R2 . On cherche A > 0 et S0 > 0 tels que pour tout s0 S0, g 2 H avec kgkL1 1 s20 , on peut trouver (d0; d1) 2 R2 tels que 8s s0, lim s!1 kwd0;d1(y; s) f( y ps )kL1 = lim s!1 kqd0;d1(y; s)kL1 = 0; o u qd0;d1 est la solution de l' equation (6) avec donn ee initiale a s = s0, qd0;d1(y; s0) = (p 1 + (p 1)2 4ps0 y2) p p 1 (d0 + d1y=ps0) 2ps0 + g(y=ps0): (8) On va en fait trouver (d0; d1) tels que qd0;d1(s) 2 VA(s) pour s s0, ce qui entrâ ne lim s!+1 kqd0;d1(s)kL1 = 0. Il est facile de v eri er alors que u(t) explose en temps ni T avec un seul point d'explosion: x = a, et v eri e (3). Partie II: R eduction a un probl eme de dimension nie C'est la partie cruciale de la preuve du Th eor eme 1. Ici, on montre a travers des estimations a priori que pour contrôler q(s) dans VA(s) (s s0), il su t de contrôler (q0; q1)(s) dans V̂A(s) (ainsi, on r eduit un probl eme de dimension in nie a un probl eme de dimension nie). Proposition 1 (Contrôle de q par (q0; q1)) Il existe A1 > 0 tel que pour tout A A1, il existe s1(A) > 0 tel que pour tout s0 s1(A), pour tout g 2 H avec kgkL1 1 s20 , on a la propri et e suivante: -si (d0; d1) est choisi tel que (q0(s0); q1(s0)) 2 V̂A(s0), et, -si pour s s0, on a 8s 2 [s0; s ], q(s) 2 VA(s), et q(s ) 2 @VA(s ), alors i) 8s 2 [s0; s ], jq2(s)j A2s 2 log s s 3, jq (y; s)j A2 (1 + jyj3)s 2, kqe(s)kL1 A2 2ps . ii) (q0(s ); q1(s )) 2 @V̂A(s ) et il existe 0 > 0 tel que 8 2 (0; 0), (q0; q1)(s + ) 62 V̂A(s + ). Partie III: Argument topologique en dimension nie On choisit A A1. On r esout maintenant le probl eme de dimension nie. On remarque par un calcul explicite: Id ees de la d emonstration du Th eor eme 1 41 Lemme 1 (Propri et e topologique pour s = s0) Il existe s2(A) > 0 tel que pour tout s0 s2(A), pour tout g 2 H avec kgkL1 1 s20 , il existe un ensemble Dg;s0 R2 topologiquement equivalent a un carr e, v eri ant la propri et e suivante: qd0;d1(s0) 2 VA(s0) si et seulement si (d0; d1) 2 Dg;s0 . On xe S0 > sup(s1(A); s2(A)) et consid ere s0 S0. On d emontre le Th eor eme 1 pour A, s0 et g 2 H avec kgkL1 1 s20 par un argument topologique. On proc ede par l'absurde, et on suppose que pour tout (d0; d1) 2 Dg;s0 , il existe s > s0 tel que qd0;d1(s) 62 VA(s). Soit s (d0; d1) l'in nimum de tous ces s. Grâce a la Partie II, on peut d e nir g : Dg;s0 ! @C (d0; d1) ! s (d0; d1)2 A (q0; q1)d0;d1(s (d0; d1)) o u C est le carr e unit e de R2 . On d emontre alors que g est continue et que sa restriction a @Dg;s0 est hom eomorphe a l'identit e. Ceci est une contradiction d'apr es la th eorie du degr e topologique, donc il existe (d0(g); d1(g)) tel que 8s s0, qd0;d1(s) 2 VA(s). Ceci termine la preuve. Bibliographie 43 Bibliographie [1] J. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford 28, 1977, p. 473-486. [2] M. Berger et R. Kohn, A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math. 41, 1988, p. 841-863. [3] J. Bricmont et A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity 7, 1994, p. 539-575. [4] J. Bricmont, A. Kupiainen et G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math. 47, 1994, p. 893-922. [5] J. Bricmont et A. Kupiainen, Renormalization group and nonlinear PDEs, Quantum and non-commutative analysis, past present and future perspectives, Kluwer (Boston), 1993. [6] S. Filippas et R. Kohn, Re ned asymptotics for the blowup of ut u = up, Comm. Pure Appl. Math. 45, 1992, p. 821-869. [7] S. Filippas et F. Merle, Modulation theory for the blowup of vectorvalued nonlinear heat equations J. Di . Equations 116, 1995, p. 119-148 [8] Y. Giga et R. Kohn, Nondegeneracy of blow-up for semilinear heat equations, Comm. Pure Appl. Math. 42, 1989, p. 845-884. [9] Y. Giga et R. Kohn, Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36, 1987, p. 1-40. [10] Y. Giga et R. Kohn, Asymptotically self-similar blowup of semilinear heat equations, Comm. Pure Appl. Math. 38, 1985, p. 297-319. [11] M. A. Herrero et J. J. L. Velazquez, Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincar e Anal. Non Lin eaire 10, 1993, p. 131-189. [12] M. A. Herrero et J. J. L. Velazquez, Flat blow-up in one-dimensional semilinear heat equations, Di erential Integral Equations 5, 1992, p. 973997. [13] H. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = Au+F (u), Arch. Rat. Mech. Anal. 51, 1973, p. 371-386. 44 Stabilit e du pro l a l'explosion pour ut = u+ jujp 1u [14] F. Merle, Solution of a nonlinear heat equation with arbitrary given blow-up points, Comm. Pure Appl. Math. 45, 1992, p. 263-300. [15] F. Merle et H. Zaag, Stability of the blow-up pro le for equations of the type ut = u+ jujp 1u, Duke Math. J. 86, 1997, p. 143-195. [16] Simon, B., Functional Integration and Quantum Physics, Academic press, New York, 1979. [17] J. J. L. Velazquez, Classi cation of singularities for blowing up solutions in higher dimensions Trans. Amer. Math. Soc. 338, 1993, p. 441-464. [18] F. Weissler, Single-point blowup for a semilinear initial value problem, J. Di erential Equations 55, 1984, p. 204-224. [19] H. Zaag, Blow-up results for vector-valued non-linear heat equations, Ann. Inst. H. Poincar e Anal. Non Lin eaire 15, 1998, a parâ tre. F. M. : D epartement de Math ematiques, Universit e de Cergy-Pontoise, 8 le Campus, 95 033 Cergy-Pontoise, France. H. Z. : D epartement de Math ematiques et Informatique, Ecole Normale Sup erieure, 45 rue d'Ulm, 75 230 Paris Cedex 05, France. Chapitre 2 Stability of the blow-up pro le for equations of the type ut = u + jujp 1u 46 Stability of the blow-up pro le for ut = u+ jujp 1u Stability of the blow-up pro le for equations of the type ut = u+ jujp 1u y Frank Merle Universit e de Cergy-Pontoise Hatem Zaag Universit e de Cergy-Pontoise, ENS, Paris VI Abstract In this paper, we consider the following nonlinear equation ut = u+ jujp 1u u(:; 0) = u0; (and various extensions of this equation, where the maximum principle do not apply). We rst describe precisely the behavior of a blow-up solution near blowup time and point. We then show a stability result on this behavior. Mathematics Subject Classi cation: 35K, 35B35, 35B40 Key words: Blow-up, Pro le, Stability 1 Introduction In this paper, we are concerned with the following nonlinear equation: ut = u+ jujp 1u u(:; 0) = u0 2 H; (1) where u(t) : x 2 RN ! u(x; t) 2 R, stands for the Laplacian in RN . We note H =W 1;p+1(RN )\L1(RN ). We assume in addition the exponent p subcritical: if N 3 then 1 < p < (N + 2)=(N 2), otherwise, 1 < p < +1. Other types of equations will be also considered. Local Cauchy problem for equation (1) can be solved in H . Moreover, one can show that either the solution u(t) exists on [0;+1), or on [0; T ) with T < +1. In this former case, u blows-up in nite time in the sense that ku(t)kH ! +1 when t! T . ( Actually, we have both ku(t)kL1(RN) ! +1 and ku(t)kW 1;p+1(RN) ! +1 when t! T ). Here, we are interested in blow-up phenomena (for such case, see for example Ball [1], Levine [14]). We now consider a blow-up solution u(t) and note T its blow-up time. One can show that there is at least one blow-up point a (that is a 2 RN such that: ju(a; t)j ! +1 when t ! T ). We will consider in this paper the case of a nite number of blow-up points (see [15]). More precisely, we will focus for simplicity on the case where there is only one blow-up point. We want to study the pro le of the solution near blow-up, and the stability of such behavior with respect to initial data. Standard tools such as center manifold theory have been proven non e cient in this situation (Cf [6] [3]). In order to treat this problem, we introduce yArticle paru dans Duke Math. J. 86, 1997, pp. 143-195. Introduction 47 similarity variables (as in [8]): y = x a pT t ; (2) s = log(T t); wT;a(y; s) = (T t) 1 p 1u(x; t); (3) where a is the blow-up point and T the blow-up time of u(t). The study of the pro le of u as t ! T is then equivalent to the study of the asymptotic behavior of wT;a (or w for simplicity), as s ! 1, and each result for u has an equivalent formulation in terms of w. The equation satis ed by w is the following: ws = w 1 2y:rw w p 1 + jwjp 1w: (4) Giga and Kohn showed rst in [8] that for each C > 0, lim s!+1 sup jyj C j w(y; s) j = 0; with = (p 1) 1 p 1 , which gives if stated for u: lim t!T sup jyj C j (T t)1=(p 1)u(a+ ypT t; t) j= 0: This result was speci ed by Filippas and Kohn [6] who established that in N dimension, if w doesn't approach exponentially fast, then for each C > 0 sup jyj C j w(y; s) [ + 2ps(N 12 jyj2)]j = o(1=s); which gives if stated for u: sup jyj C j (T t) 1 p 1u(a+ ypT t; t) [ + 2pj log(T t)j (N 12 jyj2)] j (5) = o(( log(T t)) 1): Velazquez obtained in [16] a related result, using maximum principle. Relaying on a numerical study, Berger and Kohn [2] conjectured that in the case of a non exponential decay, the solution u of (1) would approach an explicit universal pro le f(z) depending only on p and independent from initial data as follows: (T t) 1 p 1u(a+p(T t)j log(T t)jz; t) = f(z) +O(( log(T t)) 1) (6) in L1loc, with f(z) = (p 1 + (p 1)2 4p jzj2) 1 p 1 : (7) This behavior shows that in the case of one isolated blow-up point, there would be a free-boundary moving in (x; t) coordinates at the rate p(T t)j log(T t)j: 48 Stability of the blow-up pro le for ut = u+ jujp 1u This free-boundary roughly separates the space into two regions: 1) the singular one, at the interior of the free-boundary, where u can be neglected with respect to jujp 1u, so equation (1) behaves like an ordinary di erential equation, and blows-up. 2) the regular one, after the free-boundary, where u and jujp 1u are of the same order. Herrero and Velazquez in [12] and [13] showed in the case of dimension one (N = 1) using maximum principle that u behaves in three manners, one of them is the one suggested by Berger and Kohn, and they proved that estimate (6) is true uniformly on z belonging to compact subsets of R (without estimating the error). Going further in this direction, Bricmont and Kupiainen construct a solution for (1) satisfying (6) in a global sense. For that, they used on one hand ideas close to the renormalization theory, and on the other hand hard analysis on equation (4). In this paper, we shall give a more elementary proof of their result, based on a more geometrical approach and on techniques of a priori estimates: Theorem 1 Existence of a blow-up solution with a free-boundary behavior of the type (6) There exists T0 > 0 such that for each T 2 (0; T0], 8g 2 H with kgkL1 (logT ) 2, one can nd d0 2 R and d1 2 RN such that for each a 2 RN , the equation (1) with initial data u0(x) = T 1 p 1nf(z)(1 + d0 + d1z p 1 + (p 1)2 4p jzj2 ) + g(z)o; z = (x a)(j logT jT ) 12 ; has a unique classical solution u(x; t) on RN [0; T ) and i) u has one and only one blow-up point: a ii) a free-boundary analogous to (6) moves through u such that lim t!T (T t) 1 p 1u(a+ ((T t)j log(T t)j) 12 z; t) = f(z) (8) uniformly in z 2 RN , with f(z) = (p 1 + (p 1)2 4p jzj2) 1 p 1 : Remark: We took d0 and d1 respectively in the direction of h0(y) = 1 and h1(y) = y, the two rst eigenfunctions of L (Cf section 2), but we could have chosen other directions D0(y) and D1(y) (see Theorem 2). We can notice that we have a result in H = W 1;p+1(RN ) \ L1(RN ). We can also obtain blow-up results in H1(RN ) \ L1(RN ). If p < 1 + 4 N , then f(z) 2 H1, and we use the same arguments to solve the problem in H1(RN ) \ L1(RN ). If p 1 + 4 N , the result in H1 follows directly from the stability result (see Theorem 2 below). Remark: Such behavior is suspected to be generic. Remark 1.1 Introduction 49 One can ask the following questions: a) Why does the free-boundary move at such a speed? b) Why is the pro le precisely the function f? As in various physical situations, we suspect that the asymptotic behavior of w ! is described by self-similar solutions of equation (4). Since we are dealing with equation of the heat type (Cf (4)), the natural scaling is y ps . Let us hence try to nd a solution of the form v( y ps ), with v(0) = ; lim jzj!1 jv(z)j = 0: (9) A direct computation shows that v must satisfy the following equation, for each s > 0 and each z 2 RN : 1 2sz:rv(z) = 1s v(z) 12z:rv(z) 1 p 1v(z) + jv(z)jp 1v(z) (10) According to Giga and Kohn [10], the only solutions of (10) are the constant ones: 0; ; , which are ruled out by (9). We can then try to search formally regular solutions of (4) of the form V (y; s) = 1 Xj=0 1 sj vj( y ps ) and compare elements of order 1 sj ( in one dimension, in the positive case for simplicity). We obtain for j = 0: 0 = 1 2zv0 0(z) 1 p 1v0(z) + v0(z)p; and for j = 1 (z 6= 0) v0 1(z) + a(z)v1(z) = b(z) with a(z) = 2 z ( 1 p 1 pv0(z)p 1) and b(z) = v0 0(z) + 2 z v00 0 (z). The solution for v0 is given by v0(z) = (p 1 + c0z2) 1 p 1 for an integration constant c0 > 0. Using this to solve the equation on v1 yields v1(z) = v0(z)pz2[c1 + Z z 1 2v0( ) pb( )d ]; for another integration constant c1. Since we want V to be regular, it is natural to require that v1 is analytic at z = 0. v1 is regular if and only if the coe cient of in the Taylor expansion of v0( ) pb( ) near = 0 is zero which turns to be equivalent to c0 = (p 1)2 4p after simple calculation. Therefore, v0(z) = (p 1 + (p 1)2 4p z2) 1 p 1 . Hence, the rst term in the expansion of V is precisely the pro le function f . Carrying on calculus yields: v1(z) = p 1 2p f(z)p + (p 1)2 4p z2f(z)p log f(z) + c1z2f(z)p: (11) We note that v1(0) = 2p . Unfortunately, we are not able to calculate every vj . In conclusion, we take an 50 Stability of the blow-up pro le for ut = u+ jujp 1u other approach to obtain approximate self-similar solutions (see the proof of Theorem 1). As in the paper of Bricmont and Kupiainen [3], we won't use maximum principle in the proof. The technique used here will allow us using geometrical interpretation of quantities of the type of d0 and d1 to derive stability results concerning this type of behavior for the free-boundary, with respect to perturbations of initial data and the equation. Theorem 2 Stability with respect to initial data of the free boundary behavior Let û0 be initial data constructed in Theorem 1. Let û(t) be the solution of equation (1) with initial data û0, T̂ its blow-up time and â its blow-up point. Then there exists a neighborhood V0 of û0 in H which has the following property: For each u0 in V0, u(t) blows-up in nite time T = T (u0) at only one blow-up point a = a(u0), where u(t) is the solution of equation (1) with initial data u0. Moreover, u(t) behaves near T (u0) and a(u0) in an analogous way as û(t): lim t!T (T t) 1 p 1u(a+ ((T t)j log(T t)j) 12 z; t) = f(z) uniformly in z 2 RN . Remark: Theorem 2 yields the fact that the blow-up pro le f(z) is stable with respect to perturbations in initial data. Remark: From [15], we have T (u0)! T̂ , a(u0)! â, as u0 ! û0 in H . Remark: For this theorem, we strongly use a nite dimension reduction of the problem in R1+N , which is the space of liberty degrees of the stability Theorem: (T; a). Remark 1.2 Theorem 2 is true for a more general û0: It is enough that û(t) satis es the key estimate of the proof of Theorem 1. Remark: Since we do not use the maximum principle, we suspect that such analysis can be carried on for other type of equations, for example: ut = 2u+ juj2u; and ut = u+ jujp 1u+ ijujr 1u; (12) where 1 < r < p (p < N+2 N 2 if N 3). See also for other applications [18]. According to a result of Merle [15], we obtain the following corollary for Theorem 2: Corollary 1.1 Let D be a convex set in RN , or D = RN . For arbitrary given set of k points x1,..., xk in D, there exist initial data u0 such that the solution u of (1) with initial data u0 (with Dirichlet boundary conditions in the case D 6= RN ) blows-up exactly at x1,..., xk. Remark: The local behavior at each blow-up point xi (jx xij i) is also given by (8). Formulation of the problem 51 2 Formulation of the problem We omit the (T; a) or (d0; d1) dependence in what follows to simplify the notation. 2.1 Choice of variables As indicated before, we use similarity variables: y = x a pT t ; s = log(T t); w(y; s) = (T t) 1 p 1u(x; t): We want to prove for suitable initial data that: lim t!T k(T t) 1 p 1u a+ ((T t)j log(T t)j) 12 z; t f(z)kL1 = 0; or stated in terms of w: lim s!1 kw(y; s) f( y ps )kL1 = 0; where f(z) = (p 1 + (p 1)2 4p jzj2) 1 p 1 : We will not study as usually done, this limit di erence as s! +1 w(:; s) f( : ps ); but we introduce instead: q(y; s) = w(y; s) [N 2ps + (p 1 + (p 1)2 4ps y2) 1 p 1 ]: (13) The added term in (13) can be understood from Remark 1.1. There, we tried to obtain for w an expansion of the form P+1 j=0 1 sj vj( y ps ). We got v0 = f and for v1 the expression (11). Hence, it is natural to study the di erence w(y; s) (v0( y ps ) + 1sv1( y ps )). Since the expression of v1 is a bit complicated (see (11)), we study instead w(y; s) (v0( y ps ) + 1sv1(0)), which is (13) for N = 1. Now, if we introduce '(y; s) = N 2ps + f( y ps ) = N 2ps + (p 1 + (p 1)2 4ps jyj2) 1 p 1 ; (14) we have q(y; s) = w(y; s) '(y; s): Thus, the problem in Theorem 1 is to construct a function q satisfying lim s!+1 kq(:; s)kL1 = 0: 52 Stability of the blow-up pro le for ut = u+ jujp 1u From (4) and (13), the equation satis ed by q is the following: for s > 0, @q @s(y; s) = LV (q)(y; s) +B(q(y; s)) +R(y; s); (15) where { the linear term is LV (q) = L(q) + V (y; s)q (16) with L(q) = q 12y:rq + q and V (y; s) = p('p 1 1 p 1 ), { the nonlinear term (quadratic in q for p large) is B(q) = j'+ qjp 1(' + q) 'p p'p 1q; (17) { and the rest term involving ' is R(y; s) = ' 1 2y:r' 1 p 1'+ 'p @' @s : (18) It will be useful to write equation (15) in its integral form: for each s0 > 0, for each s1 s0, q(s1) = K(s1; s0)q(s0) + Z s1 s0 d K(s1; )B(q( )) + Z s1 s0 d K(s1; )R( ); (19) where K is the fundamental solution of the linear operator LV de ned for each s0 > 0 and for each s1 s0 by, @s1K(s1; s0) = LVK(s1; s0) (20) K(s0; s0) = Identity: 2.2 Decomposition of q Since LV will play an important role in our analysis, let us point some facts on it. i) The operator L is self-adjoint on D(L) L2(RN ; d ) with d (y) = e jyj2 4 dy (4 )N=2 : (21) Note here that there is a weight decaying at in nity. The spectrum of L is explicit. More precisely, spec(L) = f1 m2 jm 2 Ng; Formulation of the problem 53 and it consists of eigenvalues. The eigenfunctions of L are derived from Hermite polynomials: { N = 1: All the eigenvalues of L are simple. For 1 m2 corresponds the eigenfunction hm(y) = [m2 ] X n=0 m! n!(m 2n)! ( 1)nym 2n: (22) hm satis es Z hnhmd = 2nn! nm: (We will note also km = hm=khmk2L2 .) { N 2: We write the spectrum of L as spec(L) = f1 m1 + :::+mN 2 jm1; :::;mN 2 Ng: For (m1; :::;mN ) 2 N, the eigenfunction corresponding to 1 m1+:::+mN 2 is y ! hm1(y1):::hmN (yN ); where hm is de ned in (22). In particular, *1 is an eigenvalue of multiplicity 1, and the corresponding eigenfunction is H0(y) = 1; (23) * 1 2 is of multiplicity N , and its eigenspace is generated by the orthogonal basis fH1;i(y)ji = 1; :::; Ng, with H1;i(y) = h1(yi); we note H1(y) = (H1;1(y); :::; H1;N (y)); (24) *0 is of multiplicity N(N+1) 2 , and its eigenspace is generated by the orthogonal basis fH2;ij(y)ji; j = 1; :::; N; i jg, with H2;ii(y) = h2(yi), and for i < j, H2;ij(y) = h1(yi)h1(yj); we note H2(y) = (H2;ij(y); i j): (25) ii) The potential V (y; s) has two fundamental properties that will in uence strongly our analysis. a) We have V (:; s)! 0 in the L2(R; d ) when s! +1. In particular, the e ect of V on the bounded sets or in the \blow-up" region (jxj Cps) inside the free boundary will be a \perturbation" of the e ect of L. b) Outside the free boundary, we have the following property: 8 > 0, 9C > 0, 9s such that sup s s ; jyj ps C jV (y; s) ( p p 1)j 54 Stability of the blow-up pro le for ut = u+ jujp 1u with p p 1 < 1. Since 1 is the biggest eigenvalue of L, we can consider that outside the free boundary, the operator LV will behave as one with fully negative spectrum, which simpli es greatly the analysis in this region. Since the behavior of V inside and outside the free boundary is di erent, let us decompose q as the following: Let 0 2 C1 0 ([0;+1)), with supp( 0) [0; 2] and 0 1 on [0; 1]. We de ne then (y; s) = 0( j y j K0s 12 ); (26) where K0 > 0 is chosen large enough so that various technical estimates hold. We write q = qb + qe whereqb = q and qe = q(1 ). Let us remark that supp qb(s) B(0; 2K0ps) and supp qe(s) RnB(0;K0ps). Then we study qb using the structure of L. Since L has 1 + N expanding directions (corresponding to eigenvalues 1 and 1 2 ) and N(N+1) 2 neutral ones, we write qb with respect to the eigenspaces of L as follows: qb(y; s) = 2 X m=0 qm(s):Hm(y) + q (y; s) (27) where q0(s) is the projection of qb on H0, q1;i(s) is the projection of qb on H1;i, q1(s) = (q1;i(s); :::; q1;N (s)), H1(y) is given by (24), q2;ij(s) is the projection of qb on H2;ij , i j; q2(s) = (q2;ij(s); i j), H2(y) is given by (25), q (y; s) = P (qb) and P the projector on the negative subspace of L. In conclusion, we write q into 5 \components" as follows: q(y; s) = 2 X m=0 qm(s):Hm(y) + q (y; s) + qe(y; s): (28) (Note here that qm are coordinates of qb and not of q). In particular, if N = 1 and m = 0; 1; 2, qm(s) and Hm(y) are scalar functions, and Hm(y) = hm(y). We write in this case: q(y; s) = 2 X m=0 qm(s)hm(y) + q (y; s) + qe(y; s): (29) Let us now prove Theorem 1. 3 Existence of a blow-up solution with the given free-boundary pro le This section is devoted to the proof of Theorem 1. Existence of a blow-up solution 55 3.1 Transformation of the problem As in [3], we give the proof in one dimension (same proof holds in higher dimension). We also assume a to be zero, without loss of generality. Let us consider initial data: u0;d0;d1(x) = T 1 p 1nf(z)(1 + d0 + d1z p 1 + (p 1)2 4p z2 ) + g(z)o; where z = x(j log T jT ) 1 2 : We want to prove rst that there exists T0 > 0 such that for each T 2 (0; T0], for every g 2 H with kgkL1 (logT ) 2, we can nd (d0; d1) 2 R2 such that lim t!T(T t) 1 p 1ud0;d1(((T t)j log(T t)j) 12 z; t) = f(z) (30) uniformly in z 2 R, where ud0;d1 is the solution of (1) with initial data u0;d0;d1 , and f(z) = (p 1 + (p 1)2 4p z2) 1 p 1 : (31) This property will imply that ud0;d1 blows-up at time T at one single point: x = 0. Indeed, Proposition 3.1 Single blow-up point properties of solutions Let u(t) be a solution of equation (1). If u satis es the following property lim t!T k(T t) 1 p 1u(p(T t)j log(T t)jz; t) f(z)kL1 = 0 (32) then u(t) blows-up at time T at one single point: x = 0. Proof: For each b 2 R, we have from (32) lim t!T (T t) 1 p 1u(b; t) f( b p(T t)j log(T t)j ) = 0: Using (31), we obtain lim t!T(T t) 1 p 1u(0; t) = and for b 6= 0, lim t!T(T t) 1 p 1u(b; t) = 0: A result by Giga and Kohn in [8] shows that b is a blowup point if and only if lim t!T (T t) 1 p 1u(b; t) = . This concludes the proof of proposition 3.1. Therefore, it remains to nd (d0; d1) 2 R2 so that (30) holds to conclude the proof of Theorem 1. If we use the formulation of the problem in section 2, the problem reduces to nd S0 > 0 such that for each s0 S0, g 2 H with kgkL1 1 s20 , we can nd (d0; d1) 2 R2 so that the equation (15) @q @s (y; s) = LV (q)(y; s) +B(q(y; s)) +R(y; s); 56 Stability of the blow-up pro le for ut = u+ jujp 1u with initial data at s = s0, qd0;d1(y; s0) = (p 1 + (p 1)2 4ps0 y2) p p 1 (d0 + d1y=ps0) 2ps0 + g(y=ps0); (33) has a solution q(d0; d1) satisfying lim s!1 sup y2Rjqd0;d1(y; s)j = 0: (34) q will always depend on g, d0 and d1, but we will omit theses dependences in the notations (except when it is necessary). The convergence of q to zero in L1(R) follows directly if we construct q(s) solution of equation (15) satisfying a geometrical property, that is q belongs to a set VA C([s0;+1); L2(R; d )), such that VA shrinks to q 0 when s!1. More precisely we have the following de nitions: De nition 3.1 For each A > 0, for each s > 0, we de ne VA(s) as being the set of all functions r in L2(R; d ) such that jrm(s)j As 2;m = 0; 1; jr2(s)j A2(log s)s 2; jr (y; s)j A(1 + jyj3)s 2; kre(s)kL1 A2s 1 2 ; where r(y) =P2m=0 rm(s)hm(y) + r (y; s) + re(y; s) (Cf decomposition (29)). De nition 3.2 For each A > 0, we de ne VA as being the set of all functions q in C([s0;+1); L2(R; d )) satisfying q(s) 2 VA(s) for each s s0. Indeed, assume that 8s s0 q(s) 2 VA(s). Let us show that 8s s0, sup y2Rjq(y; s)j C(A) ps , which implies (34). We have from the de nitions of qb and qe q(y; s) = qb(y; s) + qe(y; s) = qb(y; s):1fjyj 2K0psg + qe(y; s) = 2 X m=0 qm(s)hm(y) + q (y; s) :1fjyj 2K0psg(y; s) + qe(y; s): Using the de nitions of hm (Cf (22)) and VA, the conclusion follows. 3.2 Proof of Theorem 1 Using these geometrical aspects, what we have to do is nally to nd A > 0 and S0 > 0 such that for each s0 S0, g 2 H with kgk1 1 s20 , we can nd (d0; d1) 2 R2 so that 8s s0, qd0;d1(s) 2 VA(s): (35) Let us explain brie y the general ideas of the proof. Existence of a blow-up solution 57 -In a rst part, we will reduce the problem of controlling all the components of q in VA to a problem of controlling (q0; q1)(s). That is, we reduce an in nite dimensional problem to a nite dimensional one. -In a second part, we solve the nite dimensional problem, that is to nd (d0; d1) 2 R2 such that (q0; q1)(s) satis es certain conditions. We will proceed by contradiction and use dynamics in dimension 2 of (q0; q1)(s) to reach a topological obstruction (using Index Theory). The constant C now denotes a universal one independent of variables, only depending upon constants of the problem such as p. Part I: Reduction to a nite dimensional problem In this section, we show that nding (d0; d1) 2 R2 such that 8s s0 q(s) 2 VA(s) is equivalent to nding (d0; d1) 2 R2 such that jqm(s)j A s2 8s s0, 8m 2 f0; 1g. For this purpose, we give the following de nition: De nition 3.3 For each A > 0, for each s > 0 we de ne V̂A(s) as being the set [ A s2 ; A s2 ]2 R2 . For each A > 0, we de ne V̂A as being the set of all (q0; q1) in C([s0;+1);R2 ) satisfying (q0; q1)(s) 2 V̂A(s) 8s s0. Step 1: Reduction for initial data Let us show that for a given A (to be chosen later), for s0 s1(A), the control of q(s0) in VA(s0) is equivalent to the control of (q0; q1)(s0) in V̂A(s0). Lemma 3.1 i) For each A > 0, there exists s1(A) > 0 such that for each s0 s1(A), g 2 H with kgkL1 1 s20 , if (d0; d1) is chosen so that (q0; q1)(s0) 2 V̂A(s0), then jq2(s0)j (log s0)s0 2; jq (y; s0)j C(1 + jyj3)s0 2; kqe(:; s0)kL1 s0 12 ii) There exists A1 > 0 such that for each A A1, there exists s1(A) > 0 such that for each s0 s1(A), g 2 H with kgkL1 1 s20 , we have the following equivalence: q(s0) 2 VA(s0) if and only if (q0; q1)(s0) 2 V̂A(s0). Proof: We rst note that part ii) of the lemma follows immediately from part i) and de nition 3.1. We prove then only part i). Let A > 0, s0 > 0 and g 2 H such that kgkL1 1 s20 . Let (d0; d1) 2 R2 . We write initial data (Cf (33)) as q(y; s0) = q0(y; s0) + q1(y; s0) + q2(y; s0) + q3(y; s0) where q0(y; s0) = d0F ( y ps0 ), q1(y; s0) = d1 y ps0F ( y ps0 ), q2(y; s0) = 2ps0 , q3(y; s0) = g( y ps0 ) and F ( y ps0 ) = (p 1 + (p 1)2 4ps0 y2) p p 1 . We decompose all the qi as suggested by (29). 58 Stability of the blow-up pro le for ut = u+ jujp 1u -From kgkL1 1 s20 we derive that jq3 0(s0)j+jq3 1(s0)j+jq3 2(s0)j+kq3 e(s0)kL1 C s20 , and then, jq3 (y; s0)j C s20 (1 + jyj3). -Using simple calculations we obtain jq2 0(s0)j C s0 , q2 1(s0) = 0, jq2 2(s0)j Ce s0 , jq2 (y; s0)j Cs 2 0 (1 + jyj3) and kq2 e(s0)kL1 Cs 1 0 . -For q0, we have q0 0(s0) = d0 R d (z) s0F ( z ps0 ) d0C(p) (s0 !1), q0 1(s0) = 0, q0 2(s0) = d0 R d (z) s0F ( z ps0 ) z2 2 8 d0C0(p) s0 (s0 !1), jq0 (y; s0)j d0 C s0 (1 + jyj3) and kq0 e(s0)kL1 Cd0. All theses last bounds are simple to obtain, perhaps except that for q0 . Indeed, we write q0 (y; s0) = d0 s0F ( y ps0 ) d0 R d (z) s0F ( z ps0 ) d0 R d (z) s0F ( z ps0 ) z2 2 8 (y2 2). The last term can be bounded by Cd0 s0 (1 + jyj3). We write the rst term as d0n s0(y)F ( y ps0 ) s0(0)F (0) R d (z)( s0F ( z ps0 ) s0(0)F (0))o. Using a Lipschitz property, we have j s0(y)F ( y ps0 ) s0(0)F (0)j Cy2 s0 , and the conclusion follows. -Similarly, we obtain for q1, q1 0(s0) = 0, q1 1(s0) = d1 ps0 R d (z) s0F ( z ps0 ) z2z d1C00(p) ps0 (s0 ! 1), q1 2(s0) = 0, jq1 (y; s0)j d1 C s3=2 0 (1 + jyj3) and kq1 e(s0)kL1 C d1 ps0 . Hence, by linearity, we write q0(s0) = d0a0(s0) + b0(g; s0) (36) q1(s0) = d1a1(s0) + b1(g; s0) with a0(s0) C(p), a1(s0) C00(p) ps0 , jb0(g; s0)j C s0 and jb1(g; s0)j C s20 . Therefore, we see that if (d0; d1) is chosen such that (q0; q1)(s0) 2 V̂A(s0) and if s0 s1(A), we obtain jdmj C s0 for m 2 f0; 1g. Using linearity and the above estimates, we obtain jq2(s0)j C s20 , jq (y; s0)j C s20 (1 + jyj3) and kqe(s0)k C s0 . Taking s1(A) larger we conclude the proof of lemma 3.1. Step 2: A priori estimates This step is the crucial one in the proof of Theorem 1. Here, we will show through a priori estimates that for s s0, the control of q in VA(s) reduces to the control of (q0; q1) in V̂A(s). Indeed, this result will imply that if for s s0, q(s ) 2 @VA(s ), then (q0(s ); q1(s )) 2 @V̂A(s ). (Compare with de nition 3.1). Remark 3.1 We shall note here that for each initial data q(s0), equation (15) has a unique solution on [s0; S] with either S = +1 or S < +1 and kq(s)kL1 ! +1, when s ! S . Therefore, in the case where S < +1, there exists s > s0 such that q(s ) 62 VA(s ) and the solution is in particular de ned up to s . Proposition 3.2 (Control of q by (q0; q1) in VA) There exists A2 > 0 such that for each A A2, there exists s2(A) > 0 such that for each s0 s2(A), for each g 2 H with kgkL1 1 s20 , we have the following property: -if (d0; d1) is chosen so that (q0(s0); q1(s0)) 2 V̂A(s0), and, Existence of a blow-up solution 59 -if for s1 s0, we have 8s 2 [s0; s1], q(s) 2 VA(s), then 8s 2 [s0; s1] , jq2(s)j A2s 2 log s s 3 jq (y; s)j A2 (1 + jyj3)s 2 kqe(s)kL1 A2 2ps : Proof: see Proof of Proposition 3.2 below. Step 3: Transversality Using now the fact that (q0; q1) controls the evolution of q in VA, we show a transversality condition of (q0; q1) on @V̂A(s ). Lemma 3.2 There exists A3 > 0 such that for each A A3, there exists s3(A) such that for each s0 s3(A), we have the following properties: i) Assume there exists s s0 such that q(s ) 2 VA(s ) and (q0; q1)(s ) 2 @V̂A(s ), then there exists 0 > 0 such that 8 2 (0; 0), (q0; q1)(s + ) 62 V̂A(s + ). ii) If q(s0) 2 VA(s0), q(s) 2 VA(s) 8s 2 [s0; s ] and q(s ) 2 @VA(s ) then there exists 0 > 0 such that 8 2 (0; 0), q(s + ) 62 VA(s + ). Proof: Part ii) follows from Step 2 and part i). To prove part i), we will show that for each m 2 f0; 1g, for each 2 f 1; 1g, if qm(s ) = A s2 , then dqm ds (s ) has the opposite sign of d ds ( A s2 )(s ) so that (q0; q1) actually leaves V̂A at s for s s0 where s0 will be large. Now, let us compute dq0 ds (s ) and dq1 ds (s ) for q(s ) 2 VA(s ) and (q0(s ); q1(s )) 2 @V̂A(s ). First, we note that in this case, kq(s )kL1 CA2 ps and jqb(y; s )j CA2 log s s2 (1 + jyj3) (Provided A 1). Below, the classical notation O(l) stands for a quantity whose absolute value is bounded precisely by l and not Cl. For m 2 f0; 1g, we derive from equation (15) and (22): R d (s )@q @skm = Z d (s )Lqkm + Z d (s )V qkm + Z d (s )B(q)km + Z d (s )R(s )km: We now estimate each term of this identity: a) j R d (s )@q @skm dqm ds j = j R d d ds qkmj j R d d ds qkmj R d jd ds jCA2 ps jkmj Ce s if s0 s3(A). b) Since L is self-adjoint on L2(R; d ), we write Z d (s )Lqkm = Z d L( (s )km)q: Using L( (s )km) = (1 m2 ) (s )km + @2 @s2 km + @ @y (2@km @y y2km), we obtain R d (s )Lqkm = (1 m2 )qm(s ) +O(CAe s ). c) We then have from (16): 8y; jV (y; s)j Cs (1 + jyj2). Therefore, j Z d (s )V qkmj Z d C s (1 + jyj5)CA2 log s s2 jkmj CA2 log s s3 60 Stability of the blow-up pro le for ut = u+ jujp 1u d) A standard Taylor expansion combined with the de nition of VA shows that j (y; s )B(q(y; s ))j Cjqj2 C(jqbj2 + jqej2) CA4(log s )2 s4 (1 + jyj3)2 + 1fjyj Kps g(y) A2 ps . Thus, j R d (s )B(q)kmj CA4(log s )2 s4 + Ce s . e) A direct calculus yields j R d (s )R(s )kmj C(p) s2 (Actually it is equal to 0 if m = 1). Indeed, in the case m = 0, we start from (18) and (14) and expand each term up to the second order when s!1. Since '(y; s) = f( y ps )+ 2ps , we derive: 1) R d (s)( ' p 1 ) = 1 p 1 ( 2ps + 2ps +O(Cs 2)) = p 1 +O(Cs 2), 2) R d (s)'p = R d fp+ 2ps R d pfp 1+O(Cs 2) = p 1 2(p 1)s+ 2ps p p 1+ O(Cs 2) = p 1 +O(Cs 2), 3) 's(y; s) = p 1 4ps2 y2fp 2ps2 and then R d (s)( 's) = O(Cs 2), 4) 'y(y; s) = p 1 2ps yfp and then R d (s)( 12y'y) = 2ps +O(Cs 2), 5) 'yy(y; s) = p 1 2ps fp + (p 1)2 4ps2 y2f2p 1, then R d (s)'yy = 2ps + O(Cs 2). Adding all these expansions, we obtain R d s R(s ) = O(C(p)s 2 ): Concluding steps a) to e), we obtain dqm ds (s ) = (1 m2 ) A s2 +O(C(p) s2 ) +O(CA4 log s s3 ) whenever qm(s ) = A s2 . Let us now x A 2C(p), and then we take s3(A) larger so that for s0 s3(A), 8s s0, C(p) s2 + O(CA4 log s s3 ) 3C(p) 2s2 . Hence, if = 1, dqm ds (s ) < 0, if = 1, dqm ds (s ) > 0. This concludes the proof of lemma 3.2.Now, let us x A sup(A2; A3). Part II: Topological argument Now, we reduce the problem to studying a two-dimensional one. Let us study now this problem. We give its initialization in the following lemma: Lemma 3.3 (Initialization of the nite dimensional problem) There exists s4(A) > 0 such that for each s0 s4(A), for each g 2 H with kgkL1 1 s20 , there exists a set Dg;s0 R2 topologically equivalent to a square with the following property:q(d0; d1; s0) 2 VA(s0) if and only if (d0; d1) 2 Dg;s0 . Proof: As stated by lemma 3.1 (ii), if we take s0 > s1(A) and g 2 H with kgkL1 1 s20 , then it is enough to prove that there exists a set Dg;s0 topologically equivalent to a square satisfying (q0; q1)(s0) 2 V̂A(s0) if and only if (d0; d1) 2 Dg;s0 . If we refer to the calculus of qm(s0) (Cf (36) and what follows), and take s4(A) s0(A) and s4(A) large enough, then this concludes the proof of lemma 3.3. Now, we x S0 > sup(s1(A); s2(A); s3(A); s4(A)) and take s0 S0. Then we start the proof of Theorem 1 for A and s0(A) and a given g 2 H with Existence of a blow-up solution 61 kgkL1 1 s20 . We argue by contradiction: According to lemma 3.3, for each (d0; d1) 2 Dg;s0 q(d0; d1; s0) 2 VA(s0). We suppose then that for each (d0; d1) 2 Dg;s0 , there exists s > s0 such that q(d0; d1; s) 62 VA(s). Let s (d0; d1) be the in mum of all these s. (Note here that s (d0; d1) exists because of remark 3.1). Applying proposition 3.2, we see that q(d0; d1; s (d0; d1)) can leave VA(s (d0; d1)) only by its rst two components, hence, (q0; q1)(d0; d1; s (d0; d1)) 2 @V̂A(s (d0; d1)): Therefore, we can de ne the following function: g : Dg;s0 ! @C (d0; d1) ! s (d0; d1)2 A (q0; q1)(d0; d1; s (d0; d1)) where C is the unit square of R2 . Now, we claim Proposition 3.3 i) g is a continuous mapping from Dg;s0 to @C. ii) The restriction of g to @Dg;s0 is homeomorphic to identity. From that, a contradiction follows (Index Theory). This means that there exists (d0(g); d1(g)) such that 8s s0, q(d0; d1; s) 2 VA(s), that is q 2 VA. In particular, kq(s)kL1 C(A) ps : Using Proposition 3.1, this concludes the proof of Theorem 1. Proof of Proposition 3.3: Step 1: i) We have (q0; q1)(s) is a continuous function of (w(s0); s) 2 H [s0;+1) where w(s0) is initial data for equation (4). Since w(s0) ( = q(y; s0) + '(y; s0), Cf (33) and (14) ) is continuous in (d0; d1) (it is linear), we have (q0; q1)(s) is continuous with respect to (d0; d1; s). Now, using the transversality property of (q0; q1) on @V̂A (lemma 3.2 ), we claim that s (d0; d1) is continuous. Therefore, g is continuous. Step 2: ii) If (d0; d1) 2 @Dg;s0 , then, according to the proof of lemma 3.3, (q0; q1)(s0) 2 @V̂A(s0). Therefore, using q(s0) 2 VA(s0) (lemma 3.1), we have q(s0) 2 @VA(s0). Applying ii) of lemma 3.2 with s0 and s = s0 yields 0 > 0 such that 8 2 (0; 0), q(s0 + ) 62 VA(s0 + ). Hence, s (d0; d1) = s0; and g(d0; d1) = s20 A (q0; q1)(s0). Formulas (36) show then that gj@Dg;s0 is homeomorphic to identity. This concludes the proof of Proposition 3.3. Let us now prove Proposition 3.2. 62 Stability of the blow-up pro le for ut = u+ jujp 1u 3.3 Proof of Proposition 3.2 For further purpose, we are going to prove a more general proposition which implies Proposition 3.2. Proposition 3.4 For each ~ A > 0 There exists ~ A2( ~ A) > 0 such that for each A ~ A2( ~ A), there exists ~ s2( ~ A;A) > 0 such that for each s0 ~ s2( ~ A;A), for each solution q of equation (15), we have the following property: -if jqm(s0)j As 2 0 ;m = 0; 1 (37) jq2(s0)j ~ As 2 0 log s0; jq (y; s0)j ~ As 2 0 (1 + jyj3); kqe(s)kL1 ~ As 1=2 0 ; -if for s1 s0, we have 8s 2 [s0; s1], q(s) 2 VA(s), then 8s 2 [s0; s1] , jq2(s)j A2s 2 log s s 3 jq (y; s)j A2 (1 + jyj3)s 2 kqe(s)kL1 A2 2ps : Proposition 3.4 implies Proposition 3.2. Indeed, referring to Lemma 3.1, we apply proposition 3.4 with ~ A = max(1; C). This gives ~ A2 > 0, and for each A ~ A2, ~ s2( ~ A;A). If we take s2(A) = max(~ s2(max(1; C); A); s1(A)) (Cf Lemma 3.1), then, applying proposition 3.4 and Lemma 3.1, one easily checks that Proposition 3.2 is valid for these values. Proof of Proposition 3.4 The proof is divided in two parts: In a rst part, we give a priori estimates on q(s) in VA(s): assume that for given A > 0 large, ~ A > 0, > 0 and initial time s0 s5(A; ~ A; ), we have q(s) 2 VA(s) for each s 2 [ ; + ], where s0. Using the equation satis ed by q, we then derive new bounds on q2, q and qe in [ ; + ] (involving A, ~ A and ). In a second part, we will use these new bounds to conclude the proof of Proposition 3.4. Step 1: A priori estimates of q. Let us recall the integral equation satis ed by q (Cf (19)): q(s) = K(s; )q( ) + Z s d K(s; )B(q( )) + Z s d K(s; )R( ); (38) where B(q) = j'+ qjp 1('+ q) 'p p'p 1q; R(y; s) = ' 1 2y:r' 1 p 1'+ 'p @' @s ; Existence of a blow-up solution 63 and K is the fundamental solution of LV (Cf (16)). We now assume that for each s 2 [ ; + ], q(s) 2 VA(s). Using (38), we derive new bounds on the three terms in the right hand side of (38), and then on q. In the case = s0, from initial data properties, it turns out that we obtain better estimates for s 2 [s0; s0 + ]. More precisely, we have the following lemma: Lemma 3.4 There exists A5 > 0 such that for each A A5, ~ A > 0, > 0, there exists s5(A; ~ A; ) > 0 with the following property: 8s0 s5(A; ~ A; ), 8 , assume 8s 2 [ ; + ], q(s) 2 VA(s) with s0. I)Case s0: we have 8s 2 [ ; + ]; i) (linear term) j 2(s)j A2 log s2 + (s )CAs 3; j (y; s)j C(e 1 2 (s )A+ e (s )2A2)(1 + jyj3)s 2; k e(s)kL1 C(A2e (s ) p +Ae(s ))s 1 2 ; where K(s; )q( ) = (y; s) = 2 X m=0 m(s)hm(y) + (y; s) + e(y; s): ii) (nonlinear term) j 2(s)j (s ) s3+1=2 ; j (y; s)j (s )(1 + jyj3)s 2 ; k e(s)kL1 (s )s 1 2 ; where = (p) > 0; andZ s d K(s; )B(q( )) = (y; s) = 2 X m=0 m(s)hm(y) + (y; s) + e(y; s): iii) (corrective term) j 2(s)j (s )Cs 3; j (y; s)j (s )C(1 + jyj3)s 2; k e(s)kL1 (s )s 3=4; whereZ s d K(s; )R(:; ) = (y; s) = 2 X m=0 m(s)hm(y) + (y; s) + e(y; s): 64 Stability of the blow-up pro le for ut = u+ jujp 1u II)Case = s0: Assume in addition that q(s0) satis es (37). Then, 8s 2 [s0; s0 + ], i) (linear term)j 2(s)j ~ A log s0 s2 + Cmax(A; ~ A)(s s0)s 3; j (y; s)j C ~ A(1 + jyj3)s 2; k e(s)kL1 C ~ A(1 + e(s s0))s 12 : We will give the proof of this lemma later. Step 2: Lemma 3.4 implies Proposition 3.4 Let ~ A be an arbitrary positive number. Let A > ~ A2( ~ A) where ~ A2( ~ A) will be de ned later. Let s0 > 0 to be chosen larger than ~ s2(A) (where ~ s2(A) will be de ned later). Let q be a solution of equation (15) satisfying (37), and s1 s0. Assume in addition that 8s 2 [s0; s1], q(s) 2 VA(s). We want to prove that 8s 2 [s0; s1] jq2(s)j A2 log s s2 1 s3 ; jq (y; s)j A 2s2 (1 + jyj3); kqe(s)kL1 A2 2ps : (39) Let 1 2 two positive numbers (to be xed in terms of A later). It is then enough to prove (39), on one hand for s s0 1, and on the other hand for s s0 2. In both cases, we use lemma 3.4. Hence, we suppose A A5, s0 max(s5(A; ~ A; 1); s5(A; ~ A; 2)). Case 1: s s0 1. Since we have 8 2 [s0; s], q( ) 2 VA( ), we apply lemma 3.4 (IIi), Iii), iii)) with A, = 1 and = s s0. From (38), we obtain: jq2(s)j ~ A log s0 s2 + C1(max(A; ~ A) + 1)(s s0)s 3 + (s s0)s 3 1=2 (40) jq (y; s)j (C1 ~ A+ C1(s s0))(1 + jyj3)s 2 + (s s0)(1 + jyj3)s 2 kqe(s)kL1 (C1 ~ A+ C1 ~ Aes s0)s 12 + (s s0)s 3=4 + (s s0)s 1 2 : To have (39), it is enough to satisfy ~ A log s0 s2 A2 2 log s s2 (41) C1 ~ As 2 + C1(s s0)s 2 A4 s 2 C1 ~ As 1=2 + C1 ~ Aes s0s 1=2 A2 4 s 12 ; on one hand, and C1(max(A; ~ A) + 1)(s s0)s 3 + (s s0)s 3 1=2 A2 2 log s s2 s 3 (42) (s s0)s 2 A4 s 2 (s s0)s 3=4 + (s s0)s 12 A2 4 s 12 Existence of a blow-up solution 65 on the other hand. If we restrict 1 to satisfy C1 1 A8 , C1 ~ Ae 1 A2 8 , (which is possible if we x 1 = 32 logA for A large), and A to satisfy ~ A A, ~ A A2 2 , C1 ~ A A8 and C1 ~ A A2 8 (that is A A6( ~ A)), then, since s s0 1, (41) is satis ed. With this value of 1, (42) will be satis ed if the following is true: C1(A+ 1)3 2 logAs 3 + 3 2 logAs 3 1=2 A2 2 log s s2 s 3 32 logAs 2 A4 s 2 32 logAs 3=4 + 3 2 logAs 1 2 A2 4 s 12 ; which is possible, if s0 s6(A). This concludes Case 1. Case 2: s s0 2. Since we have 8 2 [ ; s], q( ) 2 VA( ), we apply Part I) of lemma 3.4 with A, = = 2, = s 2. From (38), we derive: jq2(s)j A2 log(s 2) s2 + C2A 2s 3 + C2 2s 3 + 2s 3 1=2 (43) jq (y; s)j C2(e 1 2 2A+ e 22A2 + 2)(1 + jyj3)s 2 + 2(1 + jyj3)s 2 kqe(s)kL1 C2(A2e 2 p +Ae 2)s 12 + 2s 3=4 + 2s 12 ; To obtain (39), it is enough to have: fA; 2(s) 0 (44) C2(e 12 2A+ e 22A2 + 2) A4 C2(A2e 2 p +Ae 2) A2 4 ; with fA; 2(s) = A2 log s s2 s 3 [A2 log(s 2) s2 + C2(A+ 1) 2s 3 + 2s 3 1=2] on one hand, and 2s 2 A4 s 2 (45) 2s 3=4 + 2s 12 A2 4 s 12 ; on the other hand. Now, it is convenient to x the value of 2 such that C2Ae 2 = A2 8 , that is 2 = log A 8C2 . The conclusion follows from this choice, for A large. Indeed, for arbitrary A, we write jfA;log A 8C2 (s) s 3(A2 log A 8C2 1 C2(A+ 1) log A 8C2 )j CA2 s3+1=2 (log A 8C2 )2: 66 Stability of the blow-up pro le for ut = u+ jujp 1u Then, we take A A7 such that (A2 log A 8C2 1 C2(A+ 1) log A 8C2 ) 1 C2(( A 8C2 ) 1=2A+ e (log A 8C2 )2A2 + log A 8C2 ) A4 C2(A2( A 8C2 ) 1=p +A A 8C2 ) A2 4 : After, we introduce s7(A) > 0 such that for s s0 s7(A), we have s 3 1=2CA2(log A 8C2 )2 1 2s 3 and (45) satis ed. This way, (44) and (45) are satis ed, for A A7 and s0 s7(A), which concludes Case 2. We remark that for A A8, we have 1 = 3 2 logA 2 = log A 8C2 : If now we take A2 = sup(A5; A6( ~ A); A7; A8), and then s2 = max(s5(A; ~ A; 1(A)); s5(A; ~ A; 2(A)); s6(A); s7(A)), then this concludes the proof of Proposition 3.2. Proof of Lemma 3.4 Let A A5 with A5 > 0 to be xed later. Let ~ A > 0, > 0. We take and s0 s5(A; ~ A; ). We consider s0 such that 8s 2 [ ; + ], q(s) 2 VA(s). For each part Ii); ii); iii) and IIi), we want to nd s5(A; ~ A; 0) such that the concerned part holds for s0 s5(A; ~ A; ). The proof is given in two steps: -In a rst step, we give various estimates on di erent terms appearing in the equation (19). -In a second step, we use these estimates to conclude the proof. Step 1: Estimates for equation (38) i) Estimates on K: Lemma 3.5 (Bricmont-Kupiainen) . a) 8s 1 with s 2 , 8y; x 2 R, jK(s; ; y; x)j Ce(s )L(y; x), with e L(y; x) = e p4 (1 e ) exp[ (ye =2 x)2 4(1 e ) ]. b) For each A0 > 0, A00 > 0, A000 > 0, > 0, there exists s9(A0; A00; A000; ) with the following property: 8s0 s9, assume that for s0, jqm( )j A0 2;m = 0; 1; (46) jq2( )j A00(log ) 2; jq (y; )j A000(1 + jyj3) 2; kqe( )kL1 A00 1 2 ; then, 8s 2 [ ; + ] j 2(s)j A00 log s2 + (s )C max(A0; A000)s 3; Existence of a blow-up solution 67 j (y; s)j C(e 12 (s )A000 + e (s )2A00)(1 + jyj3)s 2; k e(s)kL1 C(A00e (s ) p +A000e(s ))s 12 ; where K(s; )q( ) = (y; s) = 2 X m=0 m(s)hm(y) + (y; s) + e(y; s): (47)c)8 > 0, 9s10( ) such that 8 s10( ), 8s 2 [ ; + ], j 2(s)j (s )Cs 3; j (y; s)j (s )C(1 + jyj3)s 2; whereZ s d K(s; )R( ) = (y; s) = 2 X m=0 m(s)hm(y) + (y; s) + e(y; s): Proof: see Appendix A. Using the above lemma and simple calculation, we derive the following: Corollary 3.1 8s 1 with s 2 , j R K(s; ; y; x)(1 + jxjm)dxj C Z e(s )L(y; x)(1 + jxjm)dx es (1 + jyjm): (48)ii) Estimates on B: Lemma 3.6 8A > 0, 9s11(A) such that 8 s11(A), q( ) 2 VA( ) implies j (y; )B(q(y; ))j Cjqj2 (49) and jB(q)j Cjqj p (50) with p = min(p; 2). Proof: Let A > 0. If q( ) 2 VA( ), then kq( )kL1 C(A) 1=2 1 2f(2K0), if s11(A) (Cf De nition 3.2, (7) for f and (26) for K0). (49) and (50) are equivalent to 1), 2) and 3), with 1) p 2 and jB(q)j Cjqj2, 2) p < 2 and j (y; )B(q(y; ))j Cjqj2, 3) p < 2 and jB(q)j Cjqjp. We prove 1), 2) and 3). For 1), we Taylor expand B(q), and use the boundedness of j'j and jqj. 2) holds if (y; ) = 0. Otherwise, we have jyj 2K0p . Again, we Taylor expand B(q): (y; )jB(q)j C (y; )jqj2 R 1 0 (1 )j'+ qjp 2d , and conclude writing (y; s)j'+ qjp 2 (y; s)(j'j jqj)p 2 (f(2K0) 1 2f(2K0))p 2 = C. For 3), we write B(q) jqjp = j1+ jp 1(1+ ) 1 p j jp by setting = q ' . We easily check that this expression is bounded for ! 0 and !1. 68 Stability of the blow-up pro le for ut = u+ jujp 1u iii) Estimate on R: Lemma 3.7 9s12 > 0 8 s12,jR(y; )j C : (51)Proof: From (18) and (14), we compute: 'yy = p 1 2p fp + (p 1)2 4p 2 y2f2p 1, 's = p 1 4p 2 y2fp + 2p 2 , and 'p ' p 1 12y'y = [f + 2p ]p 2p(p 1) f p 1 + p 1 4p y2fp = 2p(p 1) + [f + 2p ]p fp, using a Lipschitz property and simple calculations, the conclusion follows. iv) Estimates on q in VA: From De nition 3.2, we simply derive the following: Lemma 3.8 9s13 > 0 8A > 0, 8 s13, if q( ) 2 VA( ), then jq(y; )j CA2 2 log (1 + jyj3) (52) and jq(y; )j CA2 1=2: (53)Step 2: Conclusion of the Proof of Lemma 3.4 We choose s0 in all cases so that if s0 + and , we have 1 2s 1 and 1 2s 1. Ii) linear term in I) : We apply b) of lemma 3.5 with A0 = A, A00 = A2 and A000 = A. Take s5(A; ) = s9(A;A2; A; ). IIi) linear term in II) : We apply b) of lemma 3.5 with A0 = A, A00 = ~ A and A000 = ~ A. Iii) nonlinear term: 2(s): By de nition, 2(s) = R d (y)k2(y) (y; s) (y; s). = R d (y)k2(y) (y; s) R s d R K(s; ; y; x)B(q(x; ))dx = I + II , where I = R d (y)k2(y) (y; s) R s d R K(s; ; y; x) (x; )B(q(x; ))dx, and II = R d (y)k2(y) (y; s) R s d R K(s; ; y; x)(1 (x; ))B(q(x; ))dx. For I we write: jI j R d (y)jk2(y)j R s d R jK(s; ; y; x)j (x; )jB(q(x; ))jdx C R d (y)jk2(y)j R s d R jK(s; ; y; x)jjq(x; )j2dx (Cf (49)) C R d (y)jk2(y)j R s d R jK(s; ; y; x)jA4 4(log )2(1 + jxj6)dx (Cf (52)) CA4 R d (y)jk2(y)j R s d 4(log )2es (1 + jyj6) (Cf corollary 3.1) CA4 R d (y)jk2(y)j(1 + jyj6)(s ) 4(log s)2es CA4(s )es ( s 2 ) 4(log s)2 (we take s0 so that s + + s0 + = 2 ) For II , we use (50) and (53) to have: jII j C R e y2 4 dy (y; s)jk2(y)j R s d R dx(1 (x; )) Stability 69 es p4 (1 e (s )) exp[ (ye (s )=2 x)2 4(1 e (s )) ]A2 p p=2. Now, we have e 12 [ y2 4 (ye t=2 x)2 4(1 e t) ] e c(K0)s e Cs; for jyj 2K0ps and jxj K0p (if s0 ). Hence, we derive jII j C R e y2 8 dyjk2(y)j R s d R dx(1 (x; )) es p4 (1 e (s )) exp[ 1 2 (ye (s )=2 x)2 4(1 e (s )) ]e CsA2 p p=2. Using a variable change in x, and carrying all calculation, we bound jII j by (s )e Cs, for s s14(A; ). Adding the bounds for I and II, and taking s15(A; ), we obtain the estimate for 2(s). (y; s) : Using (50), (52), and (48), and computing as before yields j (y; s)j CA2 p(s )e(s )(1 + jyj3) p( log s s2 ) p. If we multiply this term by (s) and bound in it jyj3 p 3 by (ps)3 p 3, we obtain j b(y; s)j CA2 p(s )e(s )(1 + jyj3)(ps)3 p 3( log s s2 ) p, hence j b(y; s)j CA2 p(s )e(s )(1+jyj3) (log s) p s( p+3)=2 , which implies simply the estimate for (for s16( ) and some 1(p)). e(y; s): Using (50), (53), and (48), and computing as before yields j (y; s)j CA2 p(s )e(s )s 1 2 p. From this, we derive directly the estimate for e (for s17( ) and some 2(p)). Finally, we take max(s15; s16; s17)) = s5(A; ) and = min( 1; 2) to have the conclusion. iii) corrective term: For 2 and , we use c) of lemma 3.5. For e, we start from (51) and write e(y; s) = (1 (y; s)) (y; s) = (1 ) R s d R dxK(s; ; y; x)R(x; ), and then as in ii), j e(y; s)j C R s d R dxe(s )L(y; x)C = C R s d t es Cs (s )es (s )s 3 4 , if s10( ). 4 Stability In this section, we give the proof of Theorem 2. As in section 3, we consider N = 1 for simplicity, but the same proof holds in higher dimension. We will mention at the end of the section how to adapt the proof to the case N 2. 4.1 Case N = 1: Let us consider û0 an initial data in H , constructed in Theorem 1. Let û(t) be the solution of equation (1): ut = u+ jujp 1u; u(0) = û0: Let T̂ be its blow-up time and â be its blow-up point. We know from (35) that there exists  > 0, ŝ0 > log T̂ such that 8s ŝ0, q̂T̂ ;â(s) 2 VÂ(s), where q̂T̂ ;â is de ned in (13) by: q̂T̂ ;â(y; s) = e s p 1 û(â+ ye s 2 ; T̂ e s) [ 2ps + (p 1 + (p 1)2 4ps y2) 1 p 1 ]: 70 Stability of the blow-up pro le for ut = u+ jujp 1u Remark: Following Remark 1.2, we can consider a more general û0, that is û0 with the following property: 9(T̂ ; â), 9Â; ŝ0 such that 8s ŝ0, q̂T̂ ;â(s) 2 VÂ(s). From De nition 3.2, the de nition of q̂T̂ ;â(s), and Proposition 3.1, û(t) blows up at time T̂ at one single point â, and behaves as the conclusion of Theorem 1. We want to prove that there exists a neighborhood V0 of û0 in H with the following property: 8u0 2 V0, u(t) blows-up in nite time T at only one blow-up point a, where u(t) is the solution of equation (1) with initial data u(0) = u0. Moreover, u(t) satis es: lim t!T (T t) 1 p 1u(a+ ((T t)j log(T t)j) 12 z ; t) = f(z) (54) uniformly in z 2 R, with f(z) = (p 1 + (p 1)2 4p z2) 1 p 1 : The proof relays strongly on the same ideas as the proof of Theorem 1: use of nite dimensional parameters, reduction to a nite dimensional problem and continuity. For Theorem 2, we introduce a one-parameter group, de ned by: (T; a) ! qT;a; where qT;a is de ned by (13), for a given solution u(t) of equation (1) with initial data u0. This one-parameter group has an important property: 8(T; a), qT;a is a solution of equation (15). Therefore, our purpose is to ne-tune the parameter (T; a) in order to get (T (u0); a(u0)) such that qT (u0);a(u0)(s) 2 VA0(s), for s s0, A0 and s0 are to be xed later. Hence, through the reduction to a nite dimensional problem, we give a geometrical interpretation of our problem, since we deal with nite dimensional functions depending on nite dimensional parameters through a one-parameter group. As indicated in the formulation of the problem in section 2 and used in section 3 (De nitions 3.1 and 3.2), it is enough to prove the following: Proposition 4.1 (Reduction) There exist A0 > 0, s0 > 0, D0 neighborhood of (T̂ ; â) in R2 , and V0 neighborhood of û0 in H with the following property: 8u0 2 V0; 9(T; a) 2 D0 such that 8s s0, qT;a(s) 2 VA0(s), where qT;a is de ned by (13), and u(t) is the solution of equation (1) with initial data u(0) = u0. (We keep here the (T , a) dependence for clearness). Indeed, once this proposition is proved, (54) follows directly from (3), (13) and de nitions 3.1, 3.2. Proposition 3.1 applied to u(x a; t) then shows directly that u(t) blows-up at time T at one single point: x = a. The proof relays strongly on the same ideas as those developed in section 3, and geometrical interpretation of T and a. Let us explain brie y its main ideas: -In a rst part, as before, we reduce the control of all the components of q to a problem of control (q0; q1)(s), uniformly for u0 2 V1 and (T; a) 2 D1 (where V1 and D1 are respectively neighborhoods of û0 and (T̂ ; â)). Stability 71 -In a second part, we focus on the nite dimensional variable (q0; q1)(s), and try to control it. We study the behavior of q̂T;a under perturbations in (T; a) near (T̂ ; â) (and some topological structure related to these). We then extend the properties of q̂ to q, for u0 near û0. We conclude the proof proceeding by contradiction to reach a topological obstruction (using Index Theory). The constant C again denotes a universal one independent of variables, only depending upon constants of the problem such as p. For each initial data u0, u(t) denotes the solution of (1) satisfying u(0) = u0, and for each (T; a) 2 R2 , wT;a and qT;a denote the auxiliary functions derived from u by transformations (3) and (13). Part I: Initialization and reduction to a nite dimensional problem In this section, we rst use continuity arguments to show that for A, s0 large enough (to be xed later), for (u0; T; a) close to (û0; T̂ ; â), qT;a is de ned at s = s0, and satis es qT;a(s0) 2 VA(s0) (Step 1). After, we aim at nding (T; a) such that qT;a(s) in VA(s) for s s0. For this purpose, we reduce through a priori estimates the control of qT;a(s) in VA(s) to the control of (q0;T;a; q1;T;a)(s) in V̂A(s) for s s0 (Step 2). Step 1: Initialization We use here the fact that q̂T̂ ;â(s) 2 VÂ(s) for any s ŝ0, and the continuity of qT;a with respect to initial data u0 and (T; a), to insure that for xed s0 ŝ0, qT;a(s0) 2 V2 ~ A(s0), for (u0; T; a) close to (û0; T̂ ; â). Hence, if A is large enough, we have qT;a(s0) 2 VA(s0) and qT;a(s0) is \small" in a way. Lemma 4.1 (Initialization) For each s0 > ŝ0 there exist V1 neighborhood of û0 in H and D1(s0) neighborhood of (T̂ ; â) in R2 , such that for each u0 2 V1, (T; a) 2 D1(s0), q(T; a; s) is de ned (at least) for s 2 ( logT; s0], and qT;a(s0) 2 V2Â(s0). Proof of Lemma 4.1: 8T > 0, 8a 2 R, qT;a(s) is de ned on: ( logT;+1), if T T̂ , or ( logT; log(T T̂ )), if T > T̂ . Therefore, qT;a(s) is de ned on ( logT; s0] for T near T̂ . i) Reduction to the continuity of qT;a(s0) 2 L1(R) Let s0 > ŝ0. It is enough to prove that 8 > 0, there exist V and D such that 8u0 2 V , (T; a) 2 D, kqT;a(s0) q̂T̂ ;â(s0)kL1(R) : (55) Indeed, if it is the case, then, 8m 2 f0; 1; 2g; jqm;T;a(s0) q̂m;T̂ ;â(s0)j C ; (56) jq ;T;a(y; s0) q̂ ;T̂ ;â(y; s0)j C (1 + jyj2); (57) kqe;T;a(s0) q̂e;T̂ ;â(s0)kL1(R) C : (58) (56) and (58) follow directly from (55). For (57), write q (y; s) = (y; s)q(y; s) P2m=0 qm(s)hm(y), and use (55) and (56). Using q̂T̂ ;â(s0) 2 VÂ(s0) and taking > 0 small enough yields the conclusion of lemma 4.1. 72 Stability of the blow-up pro le for ut = u+ jujp 1u ii) Continuity of qT;a(s0) 2 L1(R) We have: qT;a(y; s0) q̂T̂ ;â(y; s0) = wT;a(y; s0) ŵT̂ ;â(y; s0) = e s0 p 1 fu(e s0=2y + a; T e s0) û(e s0=2y + â; T̂ e s0)g = e s0 p 1 fu(e s0=2y + a; T e s0) û(e s0=2y + a; T e s0)g +e s0 p 1 fû(e s0=2y + a; T e s0) û(e s0=2y + â; T e s0)g +e s0 p 1 fû(e s0=2y + â; T e s0) û(e s0=2y + â; T̂ e s0)g: Since u0 ! u(t) 2 C1([ T̂ e ŝ0 2 ; T̂ e s0 2 ]; C1(R)) is de ned and continuous (for u0 near û0), we have the conclusion. Step 2: Uniform nite dimensional reduction This step is similar to Step 2 of Part 1 in the proof of Theorem 1. Here we show that for A and s0 to be xed later, if qT;a(s0) is \small" in VA(s0), then, the control of qT;a(s) in VA(s) for s s0 reduces to the control of (q0;T;a; q1;T;a)(s) in V̂A(s). Lemma 4.2 (Control of q by (q0; q1) in VA) There exists A2 > 2 such that for each A A2, there exists s2(A) > 0 such that for each s0 s2(A), we have the following properties: i) For any q, solution of equation (15), satisfying q(s0) 2 V2Â(s0) and, for s1 s0, 8s 2 [s0; s1], q(s) 2 VA(s), we have: 8s 2 [s0; s1], jq2(s)j A2s 2 log s s 3 jq (y; s)j A2 (1 + jyj3)s 2 kqe(s)kL1 A2 2ps : Moreover, ii) For any q, solution of equation (15), satisfying q(s0) 2 V2Â(s0)( VA(s0)), For s > s0, q(s) 2 VA(s) 8s 2 [s0; s ], and -q(s ) 2 @VA(s ), we have (q0; q1)(s ) 2 @V̂A(s ), and there exists 0 > 0 such that 8 2 (0; 0), (q0; q1)(s + ) 62 V̂A(s + ), (hence, q(s + ) 62 VA(s + )). Proof: i) We apply Proposition 3.4 with ~ A = max(2Â; (2Â)2), and take A2 = max( ~ A2; 2Â), and s2(A) = max(ŝ0 + 1; ~ s2( ~ A;A)) to have the conclusion. ii) We apply i) with s1 = s , and use De nition 3.1. Then, we apply lemma 3.2. Part II: Topological argument Below, we use the notations qT;a(s) = q(T; a; s), qT;a(y; s) = q(T; a; y; s), qm;T;a(s) = qm(T; a; s). Stability 73 In Part 1, we have reduced the problem to a nite dimensional one: for each u0 close to û0, we have to nd a parameter (T; a) = (T (u0); a(u0)) near (T̂ ; â) such that (q0; q1)(T; a; s) 2 VA(s) for s s0. We rst study the behavior of q̂(T; a) for (T; a) close to (T̂ ; â). Then, we show a stability result on this behavior for u0 near û0. Therefore, for a given u0, we proceed by contradiction to prove Proposition 4.1, which implies Theorem 2. Step 1: Study of q̂(T; a) We study the behavior of q̂(T; a) for (T; a) close to (T̂ ; â) in R2 . Proposition 4.2 (Behavior of q̂(T; a) near (T̂ ; â)) There exists A4 > 0 such that for each A A4, there exists s4(A) > 0 with the following property: For each s0 s4(A), there exists D4(s0) neighborhood of (T̂ ; â) such that for each (T; a) 2 D4(s0)nf(T̂ ; â)g, i) q̂(T; a; s) is de ned for s 2 ( logT; s0] and q̂(T; a; s0) 2 VA(s0), ii) 9s (T; a) > s0 such that 8s 2 [s0; s (T; a)], q̂(T; a; s) 2 VA(s) and q̂(T; a; s (T; a)) 2 @VA(s (T;A)), and if we de ne û0 : D4(s0)nf(T̂ ; â)g ! R2 (59) (T; a) ! ŝ (T; a)2 A (q̂0; q̂1)(T; a; ŝ (T; a)) then Im( û0) @C, where C is the unit square of R2 . Moreover, iii) û0 is continuous, iv) 8 > 0, there exists a curve 2 D4(s0) such that d( ; û0 ; 0) = 1, and 8(T; a) 2 , j(T; a) (T̂ ; â)j . Proof: In order to prove i); ii), and iii), we take A A5 with A5 = max(2Â; A2; A3), s0 s5(A) = max(ŝ0 + 1; s2(A); s3(A)), D5(s0) = D1(s0) (with the notations of lemma 4.1). For such A and s0, we can apply lemma 4.1, and lemma 4.2. Proof of i): By lemma 4.1, 8(T; a) 2 D5(s0), q̂(T; a; s) is de ned (at least) for s 2 ( logT; s0] and q̂(T; a; s0) 2 V2Â(s0) VA(s0), which proves i). Proof of ii): We claim that 8(T; a) 2 D5(s0)nf(T̂ ; â)g, 9s(T; a) > s0such that q̂(T; a; s) 62 VA(s). Indeed: Case 1: T > T̂ : Since q̂(T; a; y; s) = e s p 1 û(a+ ye s 2 ; T e s) '(y; s), q̂(T; a; s) is de ned on [s0; log(T T̂ )) and not after. Suppose that q̂(T; a; s) does not leave VA(s) for s 2 [s0; log(T T̂ )), then, 8y 2 R, 8s 2 [s0; log(T T̂ )), jq̂(T; a; y; s)j C(A) ps (Cf De nition 3.2). Since û(x; t) = (T t) 1 p 1 (q̂(T; a; x a pT t ; log(T t))+'( x a pT t ; log(T t))), lim supt!T̂ kû(t)kL1(R) CT;T̂ ;A < +1. This contradicts the fact that û(t) blows up at time T̂ . Case 2: T T̂ and (T; a) 6= (T̂ ; â): q̂(T; a; s) is de ned on [s0;+1). Suppose that q̂(T; a; s) does not leave VA(s) for s 2 [s0;+1). Then, 8y 2 R, 8s 2 [s0;+1), jq̂(T; a; y; s)j C(A) ps (Cf De nition 3.2). Hence, by (13), 74 Stability of the blow-up pro le for ut = u+ jujp 1u limt!T k(T t) 1 p 1u(a +p(T t)j log(T t)jz; t) f(z)kL1 = 0, and from Proposition 3.1, u(t) blows up at time T at one single point; x = a. Since (T; a) 6= (T̂ ; â), we have a contradiction. Therefore, q̂(T; a; s) leaves VA(s) for s s0. In conclusion, we derive: 8(T; a) 2 Dnf(T̂ ; â)g, 9s (T; a) > s0 such that 8s 2 [s0; s (T; a)], q̂(T; a; s) 2 VA(s) and q̂(T; a; s (T; a)) 2 @VA(s (T;A)). ( ŝ (T; a) > s0 since q̂(T; a; s) is in V2Â(s0) which is strictly included in VA(s0)). If now we de ne û0 by (59), then we see from lemma 4.2 that Im( û0) @C. Proof of iii): Let (T; a) 2 D5(s0)n(T̂ ; â). We have explicitly for m = 0; 1: q̂m(T; a; s) = R d km(y) (y; s)q̂(T; a; y; s) = R d km(y) (y; s)e s p 1 û(a+ye s=2; T e s) R d km(y) (y; s)'(y; s). From the continuity of u(x; t) with respect to (x; t), and ii) of lemma 4.2, ŝ (T; a) and ŝ (T;a)2 A (q0; q1)(T; a; ŝ (T; a)) are continuous with respect to (T; a). Proof of iv): Let > 0. We now construct satisfying d( ; û0 ; 0) = 1 and 8(T; a) 2 A;s1 , j(T; a) (T̂ ; â)j : This will be implied by the following: Lemma 4.3 There exists A6 > 0 such that 8A A6, 9s6(A) > 0 satisfying the following property: 8s0 s6(A), 9D6(s0) neighborhood of (T̂ ; â) such that 8 > 0, 9s1(A; ; s0) > s0, 9 , a 1-manifold in D6(s0) satisfying: 8(T; a) 2 , j(T; a) (T̂ ; â)j 8s 2 [s0; s1], q̂(T; a; s) 2 VA(s), (q̂0; q̂1)(T; a; s1) 2 @V̂A(s1); d( ; (q̂0; q̂1)(:; :; s1); 0) = 1: (60)a) Proof of lemma 4.3: The proof is not di cult, but it is a bit technical. See Appendix B for more details. b) Lemma 4.3 implies iv): Let A4 = max(A5; A6), and A A4. Let s4(A) = max(s5(A); s6(A)), and s0 s4(A). Let D4(s0) = D5(s0) \D6(s0), and > 0. Then, according to the beginning of Proof of Proposition 4.2, i) ii) and iii) hold. We take now s1 = s1(A; ; s0) and . By lemma 4.3, we see that 8(T; a) 2 , s (T; a) = s1, and û0(T; a) = s21 A (q̂0; q̂1)(T; a; s1). From (60), we derive, d( ; û0 ; 0) = 1, which concludes the proof of Proposition 4.2. Step 2: Behavior of q(T; a) for u0 near û0. Now, we x A0 = 1 + sup(2Â; A2; A3; A4), and then s0 = s0(A0) = sup(ŝ0; s2(A0); s3(A0); s4(A0)). Applying lemma 4.1 gives us V1, and D1(s0). We then x D0 = D1(s0) \D4(s0). Applying proposition 4.2 with s0 and 0 > 0 small enough gives us the curve 0 = 0 , included in D0. We consider now 0 as xed. Our purpose is to show that for u0 near û0, the behavior of q(T; a) on the curve 0 = (û0) is the same as q̂(T; a). More precisely, we have: Stability 75 Proposition 4.3 (Stability result on the behavior on 0, for u0 near û0) 8 > 0, 9V V1, neighborhood of û0 such that 8u0 2 V , 8(T; a) 2 0, i) q(T; a; s) is de ned for s 2 ( logT; s0] and q(T; a; s0) 2 VA0(s0), ii) 9s (T; a) > s0 such that 8s 2 [s0; s (T; a)], q(T; a; s) 2 VA0(s), and (q0; q1)(T; a; s (T; a)) 2 @V̂A0(s (T; a)). Then we can de ne u0 : ! @C (61) (T; a) ! s (T; a)2 A0 (q0; q1)(T; a; s (T; a)) where C is the unit square of R2 . Moreover, iii) u0 is a continuous mapping from 0 to @C, iv) k u0 j 0 û0 j 0kL1( 0) Proof: We rst show a local result, then by compactness arguments we conclude the proof. We claim the following: Lemma 4.4 (Punctual stability on 0) 8 > 0, 8(T; a) 2 0, 9D ;T;a neighborhood of (T; a) in D0, 9V ;T;a neighborhood of û0 in V such that: 8(T 0; a0) 2 D ;T;a, 8u0 2 V ;T;a, i) q(T 0; a0; s) is de ned (at least) for s 2 ( logT; s0] and q(T 0; a0; s0) 2 VA0(s0), ii)9s (T 0; a0) > s0 such that 8s 2 [s0; s (T 0; a0)], q(T 0; a0; s) 2 VA0(s), and (q0; q1)(T 0; a0; s (T 0; a0)) 2 @V̂A0(s (T 0; a0)). Moreover, js (T 0; a0)2 A0 (q0; q1)(T 0; a0; s (T 0; a0)) s (T 0; a0)2 A0 (q̂0; q̂1)(T 0; a0; ŝ (T 0; a0))j : (62) We remark that Proposition 4.3 follows from lemma 4.4. Indeed, for > 0, from lemma we write: 0 [(T;a)2 0D ;T;a; and using the compactness of 0, we have the conclusion. Proof of Lemma 4.4 We have explicitly for u0 2 H , s 2 ( logT; log(T T̂ )) if T > T̂ , otherwise s 2 ( logT;+1), and m = 0; 1 qm;T;a(s) = R d km(y) (y; s)q(T; a; y; s) = R d km(y) (y; s)e s p 1u(a+ye s=2; T e s) R d km(y) (y; s)'(y; s). Therefore, using the continuity of u(x; t) with respect to (u0; x; t), (q0; q1)(T; a; s) is a continuous function of (u0; T; a; s). Using this fact and the transversality of (q̂0; q̂1)(T; a; ŝ (T; a)) on V̂A0(s (T; a)) (lemma 4.2 ii)), i) and ii) follow then easily. This concludes the proof of Proposition 4.3. Step 3: The conclusion of the proof From continuity properties of the topological degree, there exists 1 > 0 such that 8 2 C( 0;R2 ) satisfying k û0kL1( 0) 1, we have d( 0; ; 0) = 1. Applying Proposition 4.3, with = 1, we have 8u0 2 V 1 , d( 0; u0 ; 0) = 1. 76 Stability of the blow-up pro le for ut = u+ jujp 1u We claim that the conclusion of Proposition 4.1 follows with A0, s0, D0 and V0 = V 1 . Indeed, by contradiction as in section 3: suppose that for u0 2 V0, we have 8(T; a) 2 D0, there exists s s0, q(T; a; s) 62 VA0(s). Let s (T; a) be the in mum of all these s. We now remark that u0 is de ned on D0 (lemma 4.1 and lemma 4.2). u0 is continuous from D0 to @C (see proof of Proposition 4.2 iii), and d( 0; u0 ; 0) = 0, which is a contradiction. Hence Proposition 4.1 is proved, which concludes the proof of Theorem 2. 4.2 Case N 2: Let us consider û0 an initial data in H , constructed in Theorem 1. Let û(t) be the solution of equation (1): ut = u+ jujp 1u; u(0) = û0: Let T̂ be its blow-up time and â be its blow-up point. Although the proof of Theorem 1 was given in 1 dimension, we know that there exists  > 0, ŝ0 > log T̂ such that 8s ŝ0, q̂T̂ ;â(s) 2 VÂ(s), where: q̂T̂ ;â is de ned in (13) by: qT̂ ;â(y; s) = e s p 1 û(â+ ye s 2 ; T̂ e s) [N 2ps + (p 1 + (p 1)2 4ps jyj2) 1 p 1 ]; and -De nitions 3.1 and 3.2 are still good to de ne VÂ(s), if we understand qm(s) to be a vector valued function, as de ned in section 2 (see (27) and (28)), and jqm(s)j to be the supremum of of all coordinates of qm(s). (By the same way, the de nition of V̂A(s) given in 3.3 is good here). With these adaptations, our purpose is summarized in the following Proposition, analogous to Proposition 4.1: Proposition 4.4 (Reduction) There exist A0 > 0, s0 > 0, D0 neighborhood of (T̂ ; â) in R1+N , and V0 neighborhood of û0 in H with the following property: 8u0 2 V0; 9(T; a) 2 D0 such that 8s s0, qT;a(s) 2 VA0(s), where qT;a is de ned by (13), and u(t) is the solution of equation (1) with initial data u(0) = u0. Indeed, once this proposition is proved, from (3), (13) and de nitions 3.1, 3.2, we have: lim t!T (T t) 1 p 1u(a+ ((T t)j log(T t)j) 12 z ; t) = f(z) uniformly in z 2 RN , with f(z) = (p 1 + (p 1)2 4p jzj2) 1 p 1 : Proposition 3.1 (which is true in N dimensions) applied to u(x a; t) then shows directly that u(t) blows-up at time T at one single point: x = a. Formally, the proof in the case N 2 and in the case N = 1 have exactly the same steps with the same statements of Propositions and lemmas, under the following obvious changes: Proof of lemma 3.5 77 -(T̂ ; â), (T; a) and (T 0; a0) are in R1+N , and every neighborhood of such a point is a neighborhood in R1+N . -In Part 2, C denotes the unit (1+N)-cube of R1+N , (and , 0,...) is a Lipschitz N-submanifold of R1+N , forming the boundary of a bounded connected Lipschitz open set of R1+N , and all introduced topological degrees di erent from zero are equal to ( 1)N . Moreover, the proofs can be adapted without di culty to the case N 2, even: -the proof of Proposition 4.2, which relays on results of section 3 (subsection 3.3 and lemma 3.2) that are true in N dimensions (In particular, the lemma 3.5 of Bricmont and Kupiainen, with the adaptation R ! RN ). -the construction of given in Appendix B can be simply adapted to the case N 2. A Proof of lemma 3.5 In this appendix, we prove lemma 3.5. Equation (15) has been studied in [3], hence, our analysis will be very close to [3] (the proof is essentially the same as in [3]). Lemma 3.5 relays mainly on the understanding of the behavior of the kernelK(s; ; y; x) (see (20)). This behavior follows from a perturbation method around e(s )L(y; x). Step 1: Perturbation formula for K(s; ; y; x) Since L is conjugated to the harmonic oscillator e x2=8Lex2=8 = @2 x2 16 + 1 4 + 1, we use the de nition (20) of K and give a Feynman-Kac representation for K: K(s; ; y; x) = e(s )L(y; x) Z d s yx (!)eR s 0 V (!( ); + )d (63) where d s yx is the oscillator measure on the continuous paths ! : [0; s ] ! R with !(0) = x, !(s ) = y, i.e. the Gaussian probability measure with covariance kernel ( ; 0) = !0( )!0( 0) +2(e 1 2 j 0j e 12 j + 0j + e 1 2 j2(s ) 0+ j e 12 j2(s ) 0 j; (64) which yields R d s yx !( ) = !0( ) with !0( ) = (sinh s 2 ) 1(y sinh 2 + x sinh s 2 ). We have in addition e L(y; x) = e p4 (1 e ) exp[ (ye =2 x)2 4(1 e ) ]: Now, we derive from (63) a simpli ed expression for K(s; ; y; x) considered as a perturbation of e(s )L(y; x). In order to simplify the notation, we write from now on ( ; ') for R d (y) (y)'(y). 78 Stability of the blow-up pro le for ut = u+ jujp 1u Lemma A.1 (Bricmont-Kupiainen) 8s 1 with s 2 , the kernel K(s; ; y; x) satis es K(s; ; y; x) = e(s )L(y; x)(1 + 1sP1(s; ; y; x) + P2(s; ; y; x)) where P1 is a polynomial P1(s; ; y; x) = X m;n 0;m+n 2 pm;n(s; )ymxn with jpm;n(s; )j C(s ) and jP2(s; ; y; x)j C(s )(1 + s )s 2(1 + jyj+ jxj)4: Moreover, j(k2; (K(s; ) ( s 1)2)h2)j C(s )(1 + s )s 2. Proof: See lemma 5 in [3]. Step 2: Conclusion of the proof of lemma 3.5 Proof of a): From (16), it follows easily that V (y; s) Cs 1. Using this estimate and (63), we write: jK(s; ; y; x)j e(s )L(y; x) R d s yx (!)eR s 0 C( +t) 1dt e(s )L(y; x) R d s yx (!)(s 1)C Ce(s )L(y; x) since s 2 and d s yx is a probability. Proof of c): See lemma 2 in [3]. Proof of b): We consider A0 > 0, A00 > 0, A000 > 0 and > 0. Let s0 , s0 and q( ) satisfying (46). We want to estimate some components of (y; s) = K(s; )q( ) (see (47)) for each s 2 [ ; + ]. Since s0 , we have: 8 2 [ ; s], s 2 . Therefore, up to a multiplying constant, any power of any 2 [ ; s] will be bounded systematically by the same power of s during the proof. i) Estimate of 2(s): 2(s) = (k2; (:; s)K(s; )q( )) = 2s 2q2( ) + (k2; ( (:; s) (:; )) 2s 2q( )) +(k2; (:; s)(K(s; ) 2s 2)q( )). From (46), (21) and (26), we have j 2s 2q2( )j A00s 2 log and j(k2; ( (:; s) (:; )) 2s 2q( ))j Ce C 3=2(s ) 2s 2 max(A00;A000) p CA0(s )s 3 for s0 s1(A0; A00; A000; ). We write (k2; (:; s)(K(s; ) 2s 2)q( )) as P2r=0 br + b + be where br = (k2; (:; s)(K(s; ) 2s 2)hr)qr( ), b = (k2; (:; s)(K(s; ) 2s 2)q ( )) and be = (k2; (:; s)(K(s; ) 2s 2)qe( )). For r = 0 or 1, we use lemma A.1, corollary 3.1, (21), (46), the fact that e(s )Lhr = e(1 r=2)(s )hr and (k2; hr) = 0, and derive jbrj = j(k2; (:; s)(K(s; ) e(s )L)hr)qr( ) + (k2; (:; s)(e(s )L 2s 2)hr)qr( )j CA0(s )s 3 + Ce Cs(s ) CA0(s )s 3 CA0(s )s 3. Proof of lemma 3.5 79 We have by lemma A.1 and the same arguments jb2j = j(k2; (K(s; ) 2s 2)h2)q2( ) + (k2; ( 1 + (:; s))(K(s; ) 2s 2)h2)q2( )j C(s )(1 + s )s 2A00s 2 log s+ Ce Cs(s ) CA0(s )s 3 if s0 s2(A0; A00; ). b can be treated exactly as b0, it is bounded by C(s )A000s 3. Since K(s; ) 2s 2 = K(s; ) e(s )L + (e(s )L 1) + (1 2s 2), we write be = be;1 + be;2 + be;3 with be;1 = (k2; (:; s)(K(s; ) e(s )L)qe( )), be;2 = (k2; (:; s) R s 0 d Le Lqe( )), be;3 = (k2; (:; s)(1 2s 2)qe( )). From (46), we bound be;3 by C(s )s 1A00 1=2e C C(s )A0s 3 if s0 s3(A0; A00; ). Since L is self-adjoint, jbe;2j R e y2=4 p4 dyL(k2 (:; s))(y) R s 0 d R dx e(s ) p4 (1 e 1) exp[ (ye =2 x)2 4(1 e ) ]A00 1=2. Now, we have e 1 2 [ y2 4 (ye =2 x)2 4(1 e ) ] e C(K0)s e 2s, for jyj 2K0ps and jxj K0p (if K0 is big enough and s0 ). Hence, jbe;2j CA00s 1=2 R e y2=8dy R s 0 d R dx e s p4 (1 e 1) exp[ 12 (ye =2 x)2 4(1 e ) ] CA00s 1=2(s )e s CA0(s )s 3 if s0 s4(A0; A00; ). Using these techniques and lemma A.1 we bound be;1 in the same way. Adding all these bounds yields the bound for j 2(s)j. ii) Estimate of (y; s): By de nition, (y; s) = P ( (:; s)K(s; )q( )) = P ( (:; s)K(s; )q ( )) + 2 Xr=0 qr( )P ( (:; s)K(s; )hr) + P ( (:; s)K(s; )qe( )) (65) where P is the L2(R; d ) projector on the negative subspace of L (see subsection 2.2). In order to bound the rst term, we proceed as in [3] K(s; )q ( ) = Z dxex2=4K(s; ; :; x)f(x) (66) where f(x) = e x2=4q (x; ). From Step 1, we have ex2=4K(s; ; y; x) = N(y; x)E(y; x) with N(y; x) = [4 (1 e (s )] 1=2es ex2=4e (y e (s )=2x)2 4(1 e (s )) (67) and E(y; x) = R d s yx (!)eR s 0 V (!( ); + )d . Let f0 = f and for m 1, f ( m 1)(y) = R y 1 dxf ( m)(x). From (46) and the following lemma, we can bound f ( m): Lemma A.2 jf ( m)(y)j CA000s 2(1 + jyj3)3 me y2=4. Proof: See lemma 6 in [3]. By integrating by parts, we rewrite (66) as: (K(s; )q ( ))(y) = 2 Xr=0( 1)r+1 Z @r xN(y; x)@xE(y; x)f ( r 1)(x)dx Z @3 xN(y; x)E(y; x)f ( 3)(x)dx: (68) 80 Stability of the blow-up pro le for ut = u+ jujp 1u From (67), we get for s 1 and r 2 f0; 1; 2; 3g j@r xN(y; x)j Ce r(s ) 2 (1 + jyj+ jxj)rex2=4e(s )L(y; x). Using the integration by parts formula for Gaussian measures (see [11]), we have: @xE(y; x) = 12 Z s 0 Z s 0 d d 0@x ( ; 0) Z d s yx (!)V 0(!( ); + ) V 0(!( 0); + 0)eR s 0 d 00V (!( 00); + 00) (69)+ 12 Z s 0 d @x ( ; ) Z d s yx (!)V 00(!( ); + )eR s 0 d 00V (!( 00); + 00): By (16), we have V (y; s) Cs 1 and jdnV dyn j Cs n=2 for n = 0; 1; 2. Combining this with (64) and using s 2 we have R d s yx (!)eR s 0 d 00V (!( 00); + 00) C and j@xE(y; x)j Cs 1(s )(1 + s )(jyj+ jxj). Using (46), (68) and all these bounds, we get j(K(s; )q ( ))(y)j CA000s 2e (s )=2(1 + jyj3) if s0 s5( ) and s 1. This yields j(P (:; s)K(s; )q ( ))(y)j CA000s 2e (s )=2(1 + jyj3) if s 1. For s 1, we use directly lemma A.1, corollary 3.1, (46) and C e (s )=2 to get the same estimate. Now, we consider the second term in (65) (r = 0; 1; 2). From corollary 3.1, lemma A.1, and the fact that jyj 2K0s1=2, we obtain: jqr( )( (:; s)K(s; )hr)(y) qr( )e(s )(1 r=2)( (:; s)hr)(y)j Cmax(A0; A00)s 3+1=2 log s:(s )(1 + s )es (1 + jyj3) (70) Hence P fqr( )( (:; s)K(s; )hr)(y) qr( )e(s )(1 r=2)( (:; s)hr)(y)g satis es the same bound. Since P hr = 0 and j(1 (:; s))hrj Cs 1=2(1 + jyj3), we can bound qr( )e(s )(1 r=2)P ( (:; s)hr) by (70). Hence, the second term of (65) is bounded by CA000s 2e (s )=2(1 + jyj3) if s0 s6(A0; A00; A000; ). For the last term in (65), we use (46) and a) of lemma 3.5 to get k(1 + jyj3) 1 (:; s)K(s; )qe( )kL1 CA00es s 1=2 sup y;x (1 + jyj3) 1 : exp[ 1 2 (x ye (s )=2)2 4(1 e (s )) ] (y; + (s ))(1 (x; )) CA00s 2 s t0 e s s t0 for a suitable constant t0. This yields a bound on the last term in (65) which can be written as CA00e (s )2s 2(1 + jyj3) for s0 large enough. Hence, combining all bounds for terms in (65), we have j (y; s)j Cs 2(A000e (s )=2 + A00e (s )2)(1 + jyj3): Estimate of e(y; s): We write e(y; s) = (1 (y; s))K(s; )q( ) = (1 (y; s))K(s; )(qb( ) +qe( )). From (46) and corollary 3.1, we have jqb(y; s)j CA000 1=2 and k(1 Proof of lemma 4.3 81 (y; s))K(s; )qb( )kL1 A000es s 1=2 if s0 s7(A0; A00; A000). Using (46) and the following lemma from [3]: Lemma A.3 kK(s; )(1 ( ))kL1 Ce (s )=p we have k(1 (y; s))K(s; )qe( )kL1 A00e (s )=ps 1=2, which yields the conclusion. This concludes the proof of lemma 3.5. B Proof of lemma 4.3 Let us recall lemma 4.3: Lemma B.1 There exists A6 > 0 such that 8A A6, 9s6(A) > 0 satisfying the following property: 8s0 s6(A), 9D6(s0) neighborhood of (T̂ ; â) such that 8 > 0, 9s1(A; ; s0) > s0, 9 , a 1-manifold in D6(s0) satisfying: 8(T; a) 2 , j(T; a) (T̂ ; â)j 8s 2 [s0; s1], q̂(T; a; s) 2 VA(s), (q̂0; q̂1)(T; a; s1) 2 @V̂A(s1); (71) d( ; (q̂0; q̂1)(:; :; s1); 0) = 1: (72)In this lemma, we want to control the evolution of q̂(T; a; s) in VA(s), for (T; a) close to (T̂ ; â). Hence, in a rst step, we use q̂T̂ ;â(s) 2 VÂ(s) 8s ŝ0, to give estimates on di erent components of q̂T;a(s), for (T; a) near (T̂ ; â). From these estimates, we introduce a function (~ q0; ~ q1)(T; a; s) close to (q̂0; q̂1)(T; a; s), but much more simple, and show that (~ q0; ~ q1) satis es properties analogous to (71) and (72). Therefore, we extend this result to (q̂0; q̂1), by continuity, and then nish the proof of lemma 4.3. Step 1: Asymptotic development of q̂(T; a) for (T; a) near (T̂ ; â) Applying (13) and (3), one time to (T̂ ; â) and one time to (T; a), we write: q̂(T; a; y; s) = f(1 ) 1 p 1 q̂(T̂ ; â; y + p1 ; s log(1 ))g (73) + f(1 ) 1 p 1 (p 1 + (p 1)2(y + )2 4p(1 )(s log(1 )) ) 1 p 1 (p 1 + (p 1)2y2 4ps ) 1 p 1 g + f(1 ) 1 p 1 2p(s log(1 )) 2psg; with = (T T̂ )es, and = (a â)es=2. Now, we use q̂(T̂ ; â; s) 2 VÂ(s) for s ŝ0, to give a development of q̂T;a(y; s), when j j 12 , and j j 12 . 82 Stability of the blow-up pro le for ut = u+ jujp 1u Lemma B.2 (development of q̂(T; a) near (T̂ ; â)) There exists s7 > 0 such that 8s s7, 8(T; a) 2 R2 satisfying j(T T̂ )esj 1 2 and j(a â)e s 2 j 1 2 , we have: q̂0(T; a; s) = ~ q0(T; a; s) +O( log s s5=2 + ps + 2 + 2 1s ) (74) q̂1(T; a; s) = ~ q1(T; a; s) +O( log s s2 + 2 s + s + log s s3 ) @q̂0 @T (T; a; s) = @~ q0 @T (T; a; s) + es(O( + s 1=2)); (75) @q̂0 @a (T; a; s) = @~ q0 @a (T; a; s) + es=2O( log s s2 + j j s ); (76) @q̂1 @T (T; a; s) = @~ q1 @T (T; a; s) + esO( 1 ps ); (77) @q̂1 @a (T; a; s) = @~ q1 @a (T; a; s) + es=2O( j j s + 1 s2 + j j s ) (78) with ~ q0(T; a; s) = 5 8ps2 + p 1 (79) ~ q1(T; a; s) = s 2p; and = (T T̂ )es and = (a â)e s 2 . Moreover, jq̂2(T; a; s)j C log s s2 + C j j s + C 2 jq̂ (T; a; y; s)j C(1 + jyj3)( 1 s2 + j j+ j j s3=2 ) jq̂e(T; a; y; s)j C ps : Proof of lemma B.2: The idea is simple: for s ŝ0, , we try to express each component of q̂(T; a) in terms of the corresponding component of q̂(T̂ ; â), and bound the residual terms using q̂(T̂ ; â; s) 2 VÂ(s) and other estimates that follow from. Hence, we rst give various estimates following from q̂(T̂ ; â; s) 2 VÂ(s), and then , we prove only some of the estimates in lemma B.2, since the other estimates can be obtained in the same way. i) We write the estimates following from q̂(T̂ ; â; s) 2 VÂ(s). Lemma B.3 (Consequences of q̂(T̂ ; â; s) 2 VÂ(s)) 9s16 > 0, 8s s16, jq̂(T̂ ; â; y; s)j C ps ; (80) jq̂b(T̂ ; â; y; s)j C log s s2 (1 + jyj3); (81) q̂0(T̂ ; â; s) = 5 8ps2 + o( 1 s2 ); j@q̂0 @s (T̂ ; â; s)j C s2 ; (82) Proof of lemma 4.3 83 jq̂1(T̂ ; â; s)j C log s s3 ; (83) j@q̂ @s(T̂ ; â; y; s)j C 1 + jyj ps ; (84) Proof of lemma B.3: (80) and (81) follow directly from De nition 3.2. After some simple calculations, we show that R d (y; s)R(y; s) = 5 8ps2 + O(s 3). As in the proof of lemma 3.2, we write the equation satis ed by q0(s): dq̂0 ds (T̂ ; â; s) = q̂0(T̂ ; â; s) + 5 8ps2 +O( log s s3 ); which implies (82). By the same way, we write: dq̂1 ds (T̂ ; â; s) = 12 q̂1(T̂ ; â; s) +O( log s s3 ); which yields (83). From (80), we derive that r = @q @s satis es @r @s = @2r @y2 1 2y @r @y +A(y; s)r +D(y; s); with jA(y; s)j C and, if p 32 jD(y; s)j Cs , otherwise, jD(y; s)j C sp 12 . By parabolic regularity, (84) follows. ii) Proof of some estimates in lemma B.2: (74) and (75) (The other estimates follow from similar techniques). From (73), we have: q̂0(T; a; s) = I1 + I2 + I3, with I1 = (1 ) 1 p 1 R d (y) (y; s)q̂(T̂ ; â; y+ p1 ; s log(1 )), I2 = (1 ) 1 p 1 R d (y) (y; s)(p 1 + (p 1)2(y+ )2 4p(1 )(s log(1 ))) 1 p 1 R d (y) (y; s)(p 1 + (p 1)2y2 4ps ) 1 p 1 I3 = R d (y) (y; s)f(1 ) 1 p 1 2p(s log(1 )) 2psg. -I3: We have easily: jI3j Cj js 1. -I2: Since all quantities appearing in I2 are bounded, we can write: I2 = O(e s)+R d (y)f(p 1+ (p 1)2(y+ )2 4p(1 )(s log(1 ))) 1 p 1 (p 1+ (p 1)2y2 4ps ) 1 p 1 g + p 1 R d (y)(p 1 + (p 1)2(y+ )2 4p(1 )(s log(1 ))) 1 p 1 +O( 2); = O(e s) +O( 2) + R d (y)f (p 1)2(y+ )2 4p(1 )(s log(1 )) (p 1)2y2 4ps g 1 p 1 (p 1 + (p 1)2y2 4ps ) 1 1 p 1 +O(R d (y)f (p 1)2(y+ )2 4p(1 )(s log(1 )) (p 1)2y2 4ps g2) + p 1 + p 1fR d (y)(p 1 + (p 1)2(y+ )2 4p(1 )(s log(1 ))) 1 p 1 R d (y) g, hence, jI2 p 1 j Ce s+C 2 +Cj js 1 +C 2s 1+C 2s 2+C 2s 2+C 4s 2+ Cj js 1. Therefore, jI2 p 1 j Ce s + C 2 + Cj js 1 + C 2s 1: -I1: Using (80), we write: I1 = O( s 1=2) + R d (y) (y; s)q̂(T̂ ; â; y+ p1 ; s log(1 )). If we introduce a 84 Stability of the blow-up pro le for ut = u+ jujp 1u new integration variable: z = y+ p1 , we obtain: I1 = O( s 1=2) + L1 +L2 with L1 = R (z; s log(1 ))q̂(T̂ ; â; z; s log(1 )) exp( (zp1 )2 4 ) 4 dz, and L2 = R f (zp1 ; s) (z; s log(1 ))gq̂(T̂ ; â; z; s log(1 )) exp( (zp1 )2 4 ) 4 dz. L1 = R (z; s log(1 ))q̂(T̂ ; â; z; s log(1 )) exp( z2 4 ) 4 exp( z2 4 ) exp( 2 zp1 2 4 )dz = O( s 1=2) + R (z; s log(1 ))q̂(T̂ ; â; z; s log(1 )) exp( z2 4 ) 4 f1 + 2 zp1 2 4 + 12 ( 2 zp1 2 4 )2 R 1 0 exp( ( 2 zp1 2 4 ))d gdz: Using (81), we obtain: L1 = O( s 1=2) + q̂0(T̂ ; â; s log(1 )) + q̂1(T̂ ; â; s log(1 )) +O( 2s 2 log s): By (82) and (83), we have: L1 = 5 8ps2 +O( s 1=2) +O(s 3) +O( 2s 2 log s): jL2j C R j zp1 ps z ps log(1 ) j(jq̂b(T̂ ; â; z; s log(1 ))j+ jq̂e(T̂ ; â; z; s log(1 ))j) exp( Cz2)dz: Using (81) for qb, (80) for qe, and the fact that qe 0 for jzj K0ps yields: L2 Cfj js 1=2 + j js 1=2g(s 2 log s+ e s). In conclusion, I1 = 5 8ps2 +O( s 1=2)+O(s 5=2)+O( 2s 2 log s): Adding I1; I2 and I3 yields (74). We compute @q̂0 @ instead of @q̂0 @T , and then we use @q̂0 @T = es @q̂0 @ to conclude. With the previous notations, we write: @q̂0 @ (T; a; s) = @I1 @ + @I2 @ + @I3 @ . @I3 @ : @I3 @ = 1 p 1 (1 ) 1 1 p 1 2p(s log(1 ))(1 1 s log(1 ) ), and j@I3 @ j Cs 1. @I2 @ : @I2 @ = 1 p 1 (1 ) 1 1 p 1 R d (y) (y; s)(p 1+ (p 1)2(y+ )2 4p(1 )(s log(1 )) ) 1 p 1 +(1 ) 1 p 1 R d (y) (y; s) 1 p 1 (p 1)2(y+ )2(1 (s log(1 ))) 4p(1 )2(s log(1 ))2 (p 1+ (p 1)2(y+ )2 4p(1 )(s log(1 ))) 1 1 p 1 . Computing as for I2, we obtain: @I2 @ = O( ) + p 1 +O(s 1): @I1 @ : @I1 @ =M1 +M2 +M3 with M1 = 1 p 1 (1 ) 1 1 p 1 R d (y) (y; s)q̂(T̂ ; â; y+ p1 ; s log(1 )), M2 = 1 p 1 (1 ) 1 p 1 R d (y) (y; s) y+ 2(1 )3=2 @q̂ @y (T̂ ; â; y+ p1 ; s log(1 )), M3 = 1 p 1 (1 ) 1 p 1 R d (y) (y; s) 1 1 @q̂ @s (T̂ ; â; y+ p1 ; s log(1 )), From (80), (84), and integration by parts we derive: j@I1 @ j jM1j+jM2j+jM3j Cs 1=2. this concludes the proof of lemma B.2. Step 2: Behavior of (q̂0; q̂1) near blow-up We use the explicit asymptotic development given in lemma B.2 to construct a 1-manifold ~ that is mapped by (q̂0; q̂1) into @V̂A(s). Proof of lemma 4.3 85 Lemma B.4 (Behavior of (q̂0; q̂1)) 9C0 = C0(p), 9A9 > 0 8A A9, 9s9(A) > 0 8s s9(A), 9 A;s rectangle in DA;s = (T̂ ; â) + ( C0Ae ss 2; C0Ae ss 2) ( C0Ae s 2 s 1; C0Ae s2 s 1) such that 8(T; a) 2 A;s, (q̂0; q̂1)(T; a; s) 2 @V̂A(s), and d( A;s, (q̂0; q̂1)(:; :; s); 0) = 1. Proof: Since (~ q0; ~ q1) given in (79) is almost the linear part of (q̂0; q̂1) (see lemma B.2), we can rst show for (~ q0; ~ q1) an analogous version of lemma B.4, then use lemma B.2 to conclude. We use scaling arguments to get uniform estimates in s. Indeed, let us introduce: ~ Q = ( ~ Q0; ~ Q1) : ( C0A;C0A)2 ! R2 (85) (~ ; ~ ) ! 1 A ( 5 8p + ~ p 1 ; ~ 4p); and̂Qs = (Q̂0; Q̂1)s : ( C0A;C0A)2 ! R2 (~ ; ~ ) ! s2 A (q̂0; q̂1)(T̂ + ~ ess2 ; â+ ~ e s 2 s1 ; s); (86) where C0 = C0(p). Note that ~ Q is independent of s, and that (~ q0; ~ q1)(T; a; s) = A s2 ( ~ Q0; ~ Q1)((T T̂ )ess2; (a â)e s2 s): (q̂0; q̂1)(T; a; s) = A s2 (Q̂0; Q̂1)s((T T̂ )ess2; (a â)e s2 s): The conclusion of lemma B.4 follows if we show that there exists a 1-manifold ~ in ( C0A;C0A)2 such that 8(~ ; ~ ) 2 ~ , Q̂s(~ ; ~ ) 2 @C, and d(~ ; Q̂s; 0) = 1. From lemma B.2, we compute for s s17(A): k ~ Q Q̂skC1(( CA;CA)2) C log s Aps ! 0 when s! +1. It is easy to see that 8 2 [0; 1), 9~ rectangle such that 8(~ ; ~ ) 2 ~ , ~ Q(~ ; ~ ) 2 (1 + )@C, and d(~ ; ~ Q; 0) = 1. From the continuity of topological degree, we know that there exist 0 > 0, 0 > 0 such that for each curve ~ (indexed by @C) satisfying k~ ~ 0kL1(@C) 0p2 (~ 0 itself is indexed by @C), for each continuous function Q : ( C0A;C0A)2 ! R2 satisfying k ~ Q QkL1(( C0A;C0A)2) 0, we have: d(~ ; Q; 0) = 1. Since we have k ~ Q Q̂skL1(( C0A;C0A)2) C log s Aps , and from (85) Jac ~ Q = 2 4p(p 1)A2 < 0, we can take s large enough, (s s11(A; 0; 0)) so that: 8(~ ; ~ ) 2 ~ 0 ; Q̂s(~ ; ~ ) 2 ext(1 + 0 2 )C; (87) 8(~ ; ~ ) 2 ( C0A;C0A)2; JacQ̂s(~ ; ~ ) < 0; (88) -8! 2 ImQ̂s \ Im ~ Q, if ! = Q̂s( ) then j ~ Q 1(!)j 0; (89) k ~ Q Q̂skL1(( C0A;C0A)2) 0: (90) 86 Stability of the blow-up pro le for ut = u+ jujp 1u By (90) and (87), we have d(~ 0 ; Q̂s; 0) = 1. Therefore, by (87), 8! 2 (1 + 0 4 )C, d(~ 0 ; Q̂s; !) = 1 (the degree is the same in the same component of R2nQ̂s(~ 0)). Combining this with (88) and the de nition of topological degree for C1 functions yields 8! 2 (1 + 0 4 )C, there exists a unique(~ ; ~ ) 2 R2 such that Q̂s(~ ; ~ ) = !. Hence, Q̂s is a di eomorphism from (Q̂s) 1((1 + 0 4 )C) onto (1 + 0 4 )C. Thus there exists a piecewise C1 1-manifold ~ interior to ~ 0 , such that Q̂s maps ~ onto @C (~ is di eomorphic to @C). By (89), j~ ~ 0;Aj 0. Therefore, we derive: d(~ ; Q̂s; 0) = 1. This concludes the proof of lemma B.4. Step 3: Conclusion of the proof of lemma 4.3 We take A A9, s0 max(ŝ0 + 1; s7; s9(A)) and > 0. 8s1 > s0, we consider DA;s1 and A;s1 given by lemma B.4. If s1 s12(A; ; s0), then 8(T; a) 2 A;s1 , j(T; a) (T̂ ; â)j , and (T; a) 2 D1(s0) (with the notations of lemma 4.1). Therefore, for such s1, we have j(T T̂ )es1 j CA s21 and j(a â)e s1 2 j CA s1 . This implies 8s 2 [s0; s1], j(T T̂ )esj CA s2 and j(a â)e s2 j CA s . What we want to do now is to show that 8s 2 [s0; s1], q̂(T; a; s) 2 VA(s). By lemma B.2, we have: For s0 s13(A), 8(T; a) 2 A;s1 , 8s 2 [s0; s1]: jq̂0(T; a; s)j CA s2 (91) jq̂1(T; a; s)j CA s2 (92) jq̂2(T; a; s)j C log s s2 (93) jq̂ (T; a; y; s)j C(1 + jyj3) 1 s2 (94) jq̂e(T; a; y; s)j C ps : (95) Therefore, if A A14, jq̂2(T; a; s)j A2 log s s2 , jq̂ (T; a; y; s)j A(1 + jyj3) 1 s2 ; jq̂e(T; a; y; s)j A2 ps : (96) It remains for us to show that jq̂m(T; a; s)j A s2 , for m = 0; 1. Following the proof of lemma 3.2, we easily prove: Lemma B.5 (Transversality property) 9A15 > 0, 8A A15, 9s15(A) such that 8s0 s15(A), 8s1 > s0, for any solution q of (15), satisfying: -Properties (91) to (95), for s 2 [s0; s1], -9s 2 (s0; s1] such that (q0; q1)(s) 2 @V̂A(s), we have the following property: 9 > 0 such that 8s 2 (s ; s), (q0; q1)(s ) 2 int(V̂A(s )). If A A15 and s0 s15(A), then by lemma B.5, 8(T; a) 2 A;s1 8s 2 [s0; s1); (q̂0; q̂1)(T; a; s) 2 int(V̂A(s)): (97) Proof of lemma 4.3 87 Indeed, this follows if we apply lemma B.5 to s1 ((q̂0; q̂1)(s1) 2 @V̂A(s1) by lemma B.4) and to s 2 (s0; s1], and use I = fs 2 [s0; s1)j8s0 2 [s; s1); (q̂0; q̂1)(T; a; s0) 2 int(V̂A(s0))g. The conclusion of lemma 4.3 follows for A A6 = max(A9; A14; A15), s0 max(ŝ0 + 1; s7; s9(A); s13(A); s15(A)); D6(s0) = D1(s0), and for > 0, s1 = s12(A; ; s0) and = A;s1 . Bibliography 89 Bibliography [1] Ball, J., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford 28, 1977, pp. 473-486. [2] Berger, M., and Kohn, R., A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math. 41, 1988, pp. 841-863. [3] Bricmont, J., and Kupiainen, A., Universality in blow-up for nonlinear heat equations, Nonlinearity 7, 1994, pp. 539-575. [4] Bricmont, J., Kupiainen, A., and Lin, G., Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math. 47, 1994, pp. 893-922. [5] Bricmont, J., and Kupiainen, A., Renormalization group and nonlinear PDEs, Quantum and non-commutative analysis, past present and future perspectives, Kluwer (Boston), 1993. [6] Filippas, S., and Kohn, R., Re ned asymptotics for the blowup of ut u = up, Comm. Pure Appl. Math. 45, 1992, pp. 821-869. [7] Filippas, S., and Merle, F., Modulation theory for the blowup of vectorvalued nonlinear heat equations J. Di erential Equations 116, 1995, pp. 119-148 [8] Giga, Y., and Kohn, R., Nondegeneracy of blow-up for semilinear heat equations, Comm. Pure Appl. Math. 42, 1989, pp. 845-884. [9] Giga, Y., and Kohn, R., Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36, 1987, pp. 1-40. [10] Giga, Y., and Kohn, R., Asymptotically self-similar blowup of semilinear heat equations, Comm. Pure Appl. Math. 38, 1985, pp. 297-319. [11] Glimm, J., Quantum Physics, Springer, New York, 1981. [12] Herrero, M.A, and Velazquez, J.J.L., Blow-up behavior of one-dimensional semilinear parabolic equations, Ann. Inst. H. Poincar e Anal. Non Lin eaire 10, 1993, pp. 131-189. [13] Herrero, M.A, and Velazquez, J.J.L., Flat blow-up in one-dimensional semilinear heat equations, Di erential Integral Equations 5, 1992, pp. 973-997. 90 Stability of the blow-up pro le for ut = u+ jujp 1u [14] Levine, H., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = Au+F (u), Arch. Rat. Mech. Anal. 51, 1973, pp. 371-386. [15] Merle, F., Solution of a nonlinear heat equation with arbitrary given blow-up points, Comm. Pure Appl. Math. 45, 1992 pp. 263-300. [16] Velazquez, J.J.L., Classi cation of singularities for blowing up solutions in higher dimensions Trans. Amer. Math. Soc. 338, 1993, pp. 441-464. [17] Weissler, F., Single-point blowup for a semilinear initial value problem, J. Di erential Equations 55, 1984, pp. 204-224. [18] Zaag, H., Blow-up results for vector-valued nonlinear heat equations with no gradient structure, Ann. Inst. H, Poincar e Anal. Non Lin eaire 15, 1998, to appear. D epartement de Math ematiques, Universit e de Cergy-Pontoise, 8 le Campus, 95 033 Cergy-Pontoise, France. D epartement de Math ematiques et Informatique, Ecole Normale Sup erieure, 45 rue d'Ulm, 75 230 Paris Cedex 05, France. Chapitre 3 Blow-up results for vector-valued nonlinear heat equations with no gradient structure 92 Blow-up results for vector-valued nonlinear equations Blow-up results for vector-valued nonlinear heat equations with no gradient structure y Hatem Zaag Universit e de Cergy-Pontoise, Ecole Normale Sup erieure 1 Introduction We are interested in the following reaction-di usion equation: @u @t = u+ (1 + i )jujp 1u; u(0; x) = u0(x); (1) where, 2 R, p 2 (1;+1), p < (N + 2)=(N 2) if N 3, and u0 2 H = W 1;p+1(RN ; C ) \ L1(RN ; C ). (1) is a special case of the vector-valued equation: @u @t = u+ F (u); u(x; 0) = u0(x); (2) where u(t) : x 2 RN ! RM , F : RM ! RM is regular and F is not necessarily a gradient. For simplicity, we focus on the study of (1) (results for equation (2) will also be presented in section 5). Equation (1) appears in the study of various physical problems (plasma physics, nonlinear optics). See for example Levermore and Oliver [15] and the references inside. Blow-up results for vector-valued equations have been intensively studied in di erential geometry. See for example a review paper by Hamilton [12]. The Cauchy problem for equation (1) can be solved in H . u(t), solution of (1) would exist either on [0;+1) (global existence), or only on [0; T ), with 0 < T < +1. In this case, ju(t)jH ! +1 when t ! T , we say: u(t) blows-up in nite time T in H . In this paper, we are interested in the nite time blow-up for equation (1). If = 0 and u0(x) 2 R, then (1) can be considered as real-valued. Blowup in this real case has been studied by various authors. Relying on the use of monotony properties and maximum principle, Ball [1] and Levine [16] nd in this case obstructions to the global in time existence for (1). Other authors investigated the asymptotic behavior at blow-up of blow-up solutions of (1), = 0. See for example Weissler [20], see for a study in the scale of similarity variables Giga and Kohn [11], [10], [9], Filippas and Kohn [5], Filippas and Merle [6],... The notion of asymptotic pro le (that is a function from which, after a time dependent scaling, u(t) approaches as t ! T ) appears also in various papers: see for example Bricmont and Kupiainen [4], [3], Berger and Kohn [2] for a numerical study. In the scalar case and in one dimension, Herrero and Velazquez give a classi cation of possible blow-up pro les. They use the yArticle a parâ tre dans Ann. Inst. H. Poincar e Anal. Non Lin eaire 15, 1998. Introduction 93 maximum principle and the decay in time of the number of oscillations of the solution. Some of their results are generalized to N dimensions in [19]. Most of the techniques used for = 0 in the cited papers can not be applied in the case 6= 0, since (1) is complex-valued (no maximum principle applied), and the equation does not derive from a gradient. Another method has been introduced in [18] in the case = 0 (see also [4]): Once an asymptotic pro le is derived formally for (1), the existence of a solution u(t) which blows-up in nite time with the suggested pro le is proved rigorously, using a nonlinear analysis of equation (2) near the given pro le. This approach which does not use maximum principle allows us to nd blow-up solutions for vector-valued heat equations (even with no gradient structure). In this paper, we aim at adapting this method to show the existence of a blow-up solution for equation (1) with 6= 0. Let us remark that the scalar case provides us with a blow-up solution if = 0. Unfortunately, this result is a one dimensional result and it fails when we perturb slightly the nonlinearity. Indeed, let us mention the case of the following vectorial equation: @u @t = u+ jujp 1u+ ijujq 1u; uj@ = 0 (3) with 1 < q < (p + 1)=2, the method of Ball [1] yields a blow-up solution u(t) : ! C where is a bounded domain of RN , see appendix A for details. We show that there exists 0 > 0 such that for each 2 [ 0; 0], equation (1) has a blow-up solution. We give in addition a precise description of its blow-up behavior. Indeed, Theorem 1 (Existence of a blow-up solution for equation (1) for small ) There exists 0 > 0 such that for each 2 [ 0; 0], there exist initial data u0 such that equation (1) has a blow-up solution. This Theorem follows directly from the following proposition which speci es the behavior of u(t) near blow-up. Indeed, up to a time dependent scaling, u(t) approaches a universal pro le (p 1 + (p 1)2 4(p 2) jzj2) 1+i p 1 (4) when t! T . More precisely: Proposition 1 (Existence of a blow-up solution for equation (1) with the pro le (4) ) There exist 0 > 0, T0 > 0 such that for each 2 [ 0; 0], for each T 2 (0; T0], for each a 2 RN , i) there exist initial data u0 such that equation (1) has a blow-up solution u(x; t) on RN [0; T ) which blows-up in nite time T at only one blow-up point: a, ii) moreover, we have lim t!T k(T t) 1+i p 1 u(a+ ((T t)j log(T t)j) 12 z; t) f (z)1+i kL1(RN) = 0 (5) 94 Blow-up results for vector-valued nonlinear equations with f (z) = (p 1 + (p 1)2 4(p 2) jzj2) 1 p 1 : (6) iii) There exists u 2 C(RN nfag; C ) such that u(x; t) ! u (x) as t ! T uniformly on compact subsets of RN nfag, and u (x) 8(p 2)j log jx ajj (p 1)2jx aj2 1+i p 1 as x! a: (7) Remark: Estimate (5) is really uniform in z 2 RN . In previous papers dealing with the case = 0, only Bricmont and Kupiainen [4] and Merle and Zaag [18] give such a uniform convergence. In most papers, the same kind convergence is proved, but only uniformly on smaller subsets ( for jzj C=pj log(T t)j in [5],...). Remark: In fact, we show that property iii) is a consequence of ii). We want to point out that for the heat equation ( =0), iii) was known just in dimension one using the decay in time of the number of oscillations of the solution (Cf Herrero and Velazquez [13]). Remark: To prove Proposition 1, we linearize in a way equation (1) around f1+i , and give a nonlinear nite dimensional reduction of the problem. Then, we solve the nite dimensional problem using index theory. The proof is more di cult than in [18], because of the vectorial structure, the presence of a coupling between coordinates, and the presence of one more neutral direction. These techniques give then as in [18] a stability result with respect to the initial data of the behavior described in Proposition 1 (see section 5). Remark: Center manifold theory do not apply here. It fails to give a uniform estimate such as ii). One can point out that even if it works, a center manifold theory gives a convergence only uniform in the region fjzjpj log(T t)j Cg. For discussion in the case = 0, see Filippas and Kohn [5], page 834-835. Remark: We see from (6) that 0 < 0 < pp. Since equation (1) is rotation invariant, for each ! 2 S1, we can nd initial data u0 such that the corresponding solution has the pro le f1+i !. From this result, one can ask: what happens for > 0? Does equation (1) still have blow-up solutions? We conjecture the existence of ̂0 > 0 such that for j j < ̂0, equation (1) has blow-up solutions, while for j j > 0, no blow-up is possible for solutions of equation (1). That is, all solutions are globally de ned. Indeed, from the formal asymptotic analysis, one can remark that for j j > pp, f1+i is no longer bounded, and the analysis fails. Another question arises: what happens with the critical value = ̂0? Unfortunately, we are not able here to give a precise value of ̂0 and a rigorous proof of what is conjectured. As an extension of Theorem 1, one can mention that using the same techniques, we have the same result for the following vector-valued equation: du dt = u+ jujp 1u+G(u); u(x; 0) = u0(x) (8) where Formulation of the problem 95 1) u(t) : x 2 RN ! RM , p 2 (1;+1), p < (N + 2)=(N 2) if N 3, u0 2 H =W 1;p+1(RN ;RM ) \ L1(RN ;RM ), 2) G : RM ! RM is a perturbation of jujp 1u satisfying: G(u) = G1(juj2)u, jG(u)j Cjujr, jG( u1) G( u2)j C rju1 u2j for ju1j; ju2j 1, 1, r 2 [1; p), G1 : R+ ! R+ , Indeed, Theorem 2 (Existence of a blow-up solution for equation (8)) . There exist initial data u0 such that equation (8) has a blow-up solution. Let us mention brie y the organization of the paper. The proof of Proposition 1 relies strongly on a double-scale description of u(t), solution of (1). We rst give in section 2 an equivalent formulation of the problem in the scale of the well known similarity variables (see Giga and Kohn [11],..). Then, working in the original scale, we prove in section 3 the existence of a single-point blow-up solution for equation (1) such that (5) holds. In section 4, we return to the original scale u(x; t) and use the invariance of equation (1) under the transformation (t0; ) ! u (x; t) = 1+i p 1 u(p x; t0 + t) to show that estimate (5) yields the equivalent (7) for the pro le u in the original scale. We conclude in section 5 by giving some comments about the stability of the result of Proposition 1 and detailing the case of equation (8) (M 3). Without loss of generality, we can now assume that a = 0 and N = 1. The same proof holds in higher dimensions (see [18] for the analysis of the case N 2). We write each complex quantity (number or function) z as z = z1+ iz2 with z1; z2 2 R. The author wants to thank Professor F. Merle for his helpful suggestions and remarks. 2 Formulation of the problem As we mentioned just before, the proof of Proposition 1 will be completed in two steps. In the rst step (section 3), it is enough to construct u(t) a solution of equation (1) satisfying (5), since this implies directly that u(t) blows-up in nite time T at only one blow-up point: 0 (parts i) and ii) of Proposition 1). Indeed, it easily follows from (5) that limt!T ju(0; t)j = +1, which means that u(t) blows-up in time T at the point 0, and limt!T (T t) 1 p 1 ju(b; t)j = 0 for b 6= 0, which implies in turn that u(t) does not blow-up at b 6= 0, and therefore blows-up only at the point 0. This last result follows directly from a Theorem by Giga and Kohn (Theorem 2.1 in [11]). In a second step (section 4), we show how the behavior of the limiting pro le u (x) near the blow-up point (part iii) of Proposition 1) can be derived from the behavior of u(t) as t! T given by (5). Hence, our rst goal is to construct u(t) a solution of (1) satisfying (5). To have an idea about the blow-up growth of u, solution of equation (1), we compare this solution with a blow-up solution of the corresponding di erential equation du dt = (1 + i )jujp 1u: 96 Blow-up results for vector-valued nonlinear equations This solution is u(t) = ei ((p 1)(T t)) 1+i p 1 , with T > 0, 2 R. Now, we consider u, a solution of equation (1) which blows-up in nite time T > 0 at one blow-up point 0 2 R. We expect u to grow with a similar rate near blow-up. If we introduce convenient \similarity variables" y = x pT t s = log(T t) w(y; s) = (T t) 1+i p 1 u(x; t); (9) then, we can look for bounded non zero solutions of the following equation (which follows from (1) through (9)): @w @s = w 12yrw (1 + i ) w p 1 + (1 + i )jwjp 1w: (10) 2.1 Formal asymptotic analysis Since equation (10) is of heat type, one can ask whether it has self-similar solutions, or at least, approximate ones. We have the following lemma: Lemma 2.1 (Formal asymptotic behavior of w) . i) The only self-similar solutions w(y; s) = v0( y ps ) of (10) are the constant ones: v0 0, or v0 ei , with = (p 1) 1 p 1 and 2 R. ii) If equation (10) has a solution of the form w(y; s) = +1 Xj=0 1 sj vj( y ps ); (11) with vj regular and bounded, then, there exists 2 R such that v0(z) = ei (p 1 + (p 1)2 4(p 2)z2) 1+i p 1 = ei f (z)1+i ; (12) where f (z)1+i is the suggested pro le in (4). Proof: i) The equations satis ed by such a v0 are 0 = 1 2zv0 0(z) (1 + i ) v0 p 1 + (1 + i )jv0jp 1v0; (13) and 12zv0 0(z) = v00 0 (z). It is easy to see that the only solutions are the constant ones, and that v0 p 1 + jv0jp 1v0 = 0. This yields the conclusion. ii) If we substitute the form (11) in equation (10) and set z = y ps , we nd (if s ! +1) that v0 satis es (13). Searching a non constant solution v0(z) = (z)ei (z), with > 0, one nds that v0(z) = ei (p 1+ bz2) 1+i p 1 , with b > 0, 2 R. In fact, there is only one possible value of b. Indeed, if we substitute the expanded form (11) in equation (10) and compare elements of order 1s , we obtain F (z) = 0, where F (z) = 12zv0 0 + v00 0 1 2zv0 1 (1 + i ) v1 p 1 + (1 + i )f(p Formulation of the problem 97 1)jv0jp 3v0(v0;1v1;1+v0;2v1;2)+ jv0jp 1v1g, and vj = vj;1+ ivj;2, j = 1; 2. According to regularization properties of equation (10), it is natural to require that v1 is C3, which implies that F is C2. F 00(0) = 0 implies b = (p 1)2 4(p 2) . Remark: Looking for approximate solutions of (10) or for solutions of (10) in the expanded form (11) is a well known approach used in various problems such as nonlinear optics, and also nonlinear heat equations (see for instance Galaktionov, Kurdyumov and Samarskii [7] for approximate self-similar solutions in the case of global existence (in time), see also Galaktionov and Vazquez [8] where an approximate solution is shown to be an admissible blow-up pro le in the case of a heat equation with (1 + u) log2(1 + u) as a nonlinearity). Unfortunately, computation can not be carried out easily for the form (11) in the present case, and we are unable to show the existence of a solution for equation (10) with such a form. In fact, instead of using this linear approach, we use a nonlinear one in section 3 to show that (10) actually has a solution w(y; s) which approaches (in L1y ) f ( y ps )1+i as s ! +1. This approach (instead of the linear one) yields the stability of such a solution (see section 5). 2.2 Transformation of the problem Using similarity variables (see (9)), we see that proving (5) is equivalent to proving that (10) has a solution satisfying lim s!1 kw(y; s) f ( y ps )1+i kL1 = 0; (14) where f1+i is given by (4). In order to prove this, we will not linearize equation (10) around f1+i as it suggested by (14), because the linear operator of the linearized equation has two neutral modes which are di cult to control. We will instead use modulation theory and take advantage of the invariance of (10) under the action of S1 (T 0 : w ! ei 0w, for each 0 2 R): in fact, we introduce q(y; s) : [ logT;+1) ! C and (s) : [ logT;+1)! R such that w(y; s) = ('(y; s) + q(y; s))ei (s) 0 = R (y; s)(q2(y; s) q1(y; s))d (15) where '(y; s) = i (f ( y ps ) + 2(p 2)s )1+i ; = (p 1) 1 p 1 (16) (y; s) = 0( j y j K0s 1 2 ); (17) 0 2 C1 0 ([0;+1); [0; 1]), with 0 1 on [0; 1] and 0 0 on [2;+1], K0 is a constant large enough, and d (y) = e y2=4 p4 : (18)The introduced liberty degree (s) is xed by the second equation of (15). It will appear in the course of the proof that this second equation makes one of 98 Blow-up results for vector-valued nonlinear equations the neutral modes of the perturbation q to be zero, which simpli es greatly the control of q. One can remark that we don't linearize (10) around ei (s)f1+i , but around ei (s)'. Up to the natural action of S1 (multiplication by i ) which simpli es the study of the linear operator of the equation on q, these two expressions di er from each other by a term of order 1s , so that (at least) some components of q are smaller that 1s , which helps to have q(s)! 0 in L1y as s! +1. Now, we claim that proving parts i) and ii) of Proposition 1 reduces to proving the following proposition: Proposition 2.1 (Equivalent formulation of Proposition 1, i) and ii)) There exist 0 > 0, S0 > 0, such that 8 2 [ 0; 0], 8s0 S0, 9qs0 2 '(:; s0)+ H such that the system @q @s (y; s) = fL' id dsg(q)(y; s) +B(q)(y; s) +R( ; y; s) 0 = R (y; s)(q2(y; s) q1(y; s))d (y) (19) where >>>>><>>>>>: L'(q) = q 1 2y:rq (1 + i ) q p 1 +(1 + i )f(p 1)j'jp 3'('1q1 + '2q2) + j'jp 1qg; B(q) = (1 + i )fj'+ qjp 1('+ q) j'jp 1' (p 1)j'jp 3'('1q1 + '2q2) j'jp 1qg; R( ; y; s) = R (y; s) id ds'; R (y; s) = @' @s + ' 12y:r' (1 + i ) ' p 1 + (1 + i )j'jp 1'; (20) with initial data (q(y; s0); (s0)) = (qs0(y); 0) at s = s0, has a unique solution (q; ) for s s0, satisfying lim s!+1 kq(s)kL1 = 0, and 9 1 2 R such that (s)! 1 as s! +1. Indeed, due to (15), the rst equation in system (19) is equivalent to (10), hence, it is equivalent to (1) (use (9)). In addition, once proposition 2.1 is proved, we have: kw(y; s) ei( 1 log )f ( y ps )1+i kL1 kei (s)(q(y; s) + '(y; s)) ei( 1 log )f ( y ps )1+i kL1 (use (15)) kq(s)kL1 + k(ei (s) ei 1)'(y; s)kL1 + kei 1('(y; s) i f ( y ps )1+i )kL1 kq(s)kL1 + Cj (s) 1j+ Cs 1 ! 0 as s! +1 (see (16)). Therefore, w(y; s) approaches ei( 1 log )f ( y ps )1+i in L1(R) as s! +1. Since (10) is rotation invariant, we can replace w by e i( 1 log )w to obtain (14), which is equivalent to (5) through similarity variables (see (9)). Hence, we must study system (19) for (q; ) 2 L1(R) R to solve the problem. Its evolution is mostly in uenced by its linear part L'; (q) = (L' id ds )(q). Let us study more carefully this operator. L'; is a R-linear operator de ned on D(L'; ) L2(R; C ; d ). Since we are interested in the behavior of (q(s); (s)) in L1(R) R as s ! +1, let us consider the limit as s ! +1 of L'; (r) for a xed r 2 L1(R; C ) (note that L1(R; C ) L2(R; C ; d )). Since (s) will be shown to have a limit when s ! +1, we can think that the e ect of d ds appearing in the expression of L'; (see (20)) will be negligible. Therefore, L'; (r)! ~ L(r) = r 1 2y:rr+(1 + i )r1 as s! +1 (see (20) and (16)). The following lemma provides us with the spectral decomposition of ~ L: Formulation of the problem 99 Lemma 2.2 (Eigenvalues of ~ L) . i) ~ L is a R linear operator de ned on L2(R; C ; d ) and its eigenvalues are given by f1 m2 jm 2 Ng. Its eigenfunctions are given by f(1 + i )hm; ihmjm 2 Ng where hm(y) = [m2 ] X n=0 m! n!(m 2n)! ( 1)nym 2n: (21) We have: ~ L((1 + i )hm) = (1 m2 )(1 + i )hm and ~ L(ihm) = m2 ihm. ii) Each r 2 L2(R; C ; d ) can be uniquely written as r(y) = (1 + i )(P+1 m=0 r̂1;mhm(y)) + i(P+1 m=0 r̂2;mhm(y)), where r̂j;m 2 R. Proof: i) From [18], we know that fhmjm 2 Ng is a total family in L2(R;R; d ), and that ( 1 2y:r)hm = m2 hm. Hence, we decompose each r 2 L2(R; C ; d ) as r(y) =P+1 m=0(r1;m + ir2;m)hm(y). 2 R is an eigenvalue for ~ L () 9r 2 L2(R; C ; d ); r 6= 0; ~ Lr = r () 9r 6= 0 8m 2 N (1 m2 ) r1;m = 0 r1;m +( m2 )r2;m = 0 () 9m 2 N = 1 m2 The computation of eigenfunctions is easy and we shall skip it. ii) We write r = (1 + i )~ r1 + i~ r2, with ~ rj 2 L2(R;R; d ), and use the fact that fhmjm 2 Ng is a total family in L2(R;R; d ). Let us consider (q(s); (s)) a solution of system (19). We will use an integral formulation of its rst equation in terms of the fundamental solution of L'. We want kq(s)kL1 ! 0 as s ! +1. This L1 control will result from the L1 control of (1 (y; s))q(y; s) and (y; s)q(y; s) (see (17) for ): 1) in the \regular" region jyj K0ps, L' behaves in L2(R; C ; d ) like an operator with a fully negative spectrum. We will show from (20) that the fundamental solution of L' between s0 and s1 > s0 is a strict contraction from L1(jyj K0ps) to L1(R). Therefore, the control of (1 (y; s))q(y; s) in L1(R) will be done without di culties. 2) in the \singular" region jyj K0ps, L' behaves in L2(R; C ; d ) like ~ L. In order to control q(y; s), we expand it with respect to the spectrum of ~ L in L2(R; C ; d ), but we will control q in L1(R) and not only in L2(R; C ; d ) (see section 3 for the rigorous analysis). By lemma 2.2, ~ L has two expanding directions ((1 + i )h0; (1 + i )h1), two null ones ((1 + i )h2; ih0) and countably many negative ones. Here, the situation is a bit more complicated than in [18], because we have two null directions (instead of only one). Our strategy to control all the components of q so that k q(s)kL1 ! 0 as s! +1 is to control the part of q corresponding to the negative spectrum of ~ L and the one parallel to (1 + i )h2 (which corresponds to the null eigenvalue) as in [18]. The component parallel to ih0 (which corresponds also to the null eigenvalue) has been xed by the second equation of (19) to be zero (using modulation theory and the phase invariance of the equation). However, the analysis of system (19) is longer than the equivalent analysis in [18], because of terms with d ds , and the presence of strong coupling between 100 Blow-up results for vector-valued nonlinear equations the two scalar parts: ~ q1 and ~ q2 of q, satisfying: q = (1 + i )~ q1+ i~ q2. Fortunately, d ds will be controlled near the pro le ' (see 16), and, although the coupling will be of critical size, its e ect will be controlled by , which can be chosen small. 3 Existence of a blow-up solution for equation (2) In this section, we prove proposition 2.1, which implies parts i) and ii) of Proposition 1 and then Theorem 1. 3.1 Geometrical property for q As in [18], the convergence of kq(s)kL1 to zero as s! +1 will follow from a geometrical property: q(s) 2 VA(s), where VA(s) L1(R; C ) shrinks to q 0 as s! +1. The structure of VA(s) respects the free-boundary moving in q at the rate ps, and also the eigenfunctions of the operator ~ L (Cf lemma 2.2). In order to de ne VA(s), we introduce the following useful notations: For each g 2 L1(R;R) and s > 0, we de ne gb(y; s) = (y; s)g(y) and ge(y; s) = (1 (y; s))g(y). Since L1(R;R) L2(R;R; d ), we introduce for each m 2 N, gm(s) as the L2(R;R; d ) projection of gb(y; s) on hm, (Cf (21)). We also let g (y; s) = P (gb) and g?(y; s) = P?(gb), where P and P? are the L2(R;R; d ) projectors respectively on Vect fhmjm 3g and Vect fhmjm 1g. Thus, we write either g(y) = 2 X m=0 gm(s)hm(y) + g (y; s) + ge(y; s) (22) or g(y) = g0(s)h0(y) + g?(y; s) + ge(y; s): (23)For each z 2 C , we write in a unique way z = (1 + i )~ z1+ i~ z2, where ~ z1 and ~ z2 are real. Hence, if r 2 L1(R; C ), we write: r(y) = (1 + i )~ r1(y) + i~ r2(y) and expand ~ r1 and ~ r2 respectively as in (22) and (23). Thus, we write: r(y) = (1 + i )~ r1(y)+ i~ r2(y) = (1 + i )fP2m=0 ~ r1;m(s)hm(y) + ~ r1; (y; s) + ~ r1;e(y; s)g + if~ r2;0(s)h0(y) + ~ r2;?(y; s) + ~ r2;e(y; s)g: (24) De nition 3.1 For each A > 0, for each s > 0, let VA(s) be the set of all functions r in L1(R; C ) such that j~ r1;m(s)j As 2; for m = 0; 1; j~ r1;2(s)j A2(log s)s 2; j~ r2;0(s)j As 2; j~ r1; (y; s)j A(1 + jyj3)s 2; j~ r2;?(y; s)j A(1 + jyj3)s 2; k~ r1;e(s)kL1 A2s 1 2 ; k~ r2;e(s)kL1 A2s 12 ; where r is given by (24). Existence of a blow-up solution for equation (2) 101 Remark: We note that L1(R; C ) L2(R; C ; d ), which justi es the expansion with respect to the eigenvalues of ~ L in de nition 3.1. Remark: It is easy to see that if q(s) 2 VA(s), then 8y 2 R, jq(y; s)j C(A)s 1=2 (see [18] for details). Therefore, kq(s)kL1(R;C) ! 0 as s! +1, and we obtain a convergence in L1(R; C ) and not only in L2(R; C ; d ), as in other papers (see [5],..). We emphasize that a convergence in L2(R; C ; d ) or more generally in Hm(R; C ; d ) yields a convergence in L1([ R;R]; C ) for each R > 0, and never a uniform convergence on R. With this remark, we claim that proposition 2.1 follows from the following proposition: Proposition 3.1 Equivalent formulation of Proposition 1, i) and ii) There exists A > 0, 0 > 0, S0 > 0, such that 8 2 [ 0; 0], 8s0 S0, 9(d0; d1) 2 R2 such that system (19) with initial data at s = s0 8<: qd0;d1(y; s0) = (1 + i )f0( y ps0 )p(d0 + d1y=ps0) ( s0 )1+i + i s0 (sin[ log( s0 )] cos[ log( s0 )])f0( y ps0 )p (s0) (s0) = 0 (25) (where f0 is given by (6), = 2(p 2) ; (s0) = R f0( y ps0 )p (y; s0)d (y) R (y; s0)d (y) ) (26) has a unique solution (q; )d0;d1 for s s0, satisfying q(s) 2 VA(s), 8s s0. Indeed, once proposition 3.1 is proved, we take for qs0 the expression in (25). From q(s) 2 VA(s), 8s s0, we have kq(s)kL1 ! 0 as s! +1, and 9 1 such that (s)! 1 as s! +1. Indeed, we have the following lemma: Lemma 3.1 8A > 0, 9s3(A) > 0 such that 8 2 [ 1; 1], 8s s3(A), if q(s) 2 VA(s), then jd ds (s)j C s2 . This lemma implies R +1 s0 jd ds (s)jds < +1, which gives 1 such that (s)! 1 as s! +1. We give the proof of this lemma in the next subsection. In order to understand the dynamics of q and , we derive the equations satis ed by ~ q1 and ~ q2 (q(y; s) = (1 + i )~ q1(y; s) + i~ q2(y; s), Cf decomposition (24)) and : Lemma 3.2 (Equations satis ed by ~ q1, ~ q2 and ) If q satis es (19) for s s0, then: @~ q1 @s (y; s) = (L+ V1;1(y; s) + d ds (s))~ q1 + (V1;2(y; s) + d ds (s))~ q2 + ~ B1(q(y; s)) + ~ R1( ; y; s); (27)@~ q2 @s (y; s) = (V2;1 (1 + 2)d ds (s))~ q1 + (L 1 + V2;2(y; s) d ds (s))~ q2 + ~ B2(q(y; s)) + ~ R2( ; y; s); (28) 102 Blow-up results for vector-valued nonlinear equations d ds Z (y; s)((1 + 2) ~ '1 + ~ '2 + (1 + 2)~ q1 + ~ q2)d = Z (L 1)~ q2d + Z @ @s ~ q2d + Z (V2;1~ q1 + V2;2~ q2)d + Z ~ B2(q)d + Z (y; s) ~ R 2(y; s); (29) where L = 12y:r+ 1; (30) 8>><>>>: V1;1(y; s) = (1 2)(j'jp 1 1 p 1 ) + (p 1)j'jp 3('21 2'22) 1 V1;2(y; s) = (j'jp 1 1 p 1 ) + (p 1)j'jp 3('1 '2)'2 V2;1(y; s) = (1 + 2)f (j'jp 1 1 p 1 ) + (p 1)j'jp 3('1 + '2)'2g V2;2(y; s) = (1 + 2)f(j'jp 1 1 p 1 ) + (p 1)j'jp 3'22g; ' is given by (16), (1 + i ) ~ B1 + i ~ B2 = B, (1 + i ) ~ R1 + i ~ R2 = R, and B, R are given by (20). Proof: (27) and (28) follow directly from (19). For (29), we note that we derive form (19) d ds R (y; s)~ q2(y; s)d (y) = 0 (~ q2 = q2 q1). Therefore R (y; s)@~ q2 @s (y; s)d (y) = R @ @s (y; s)~ q2(y; s)d (y). Multiplying (28) by and integrating with respect to d yields (29). The proof of Proposition 3.1 follows the general ideas developed in [18]. Indeed, it is divided in two parts: -In a rst part, we reduce the problem of the control in VA(s) of all the components of q(s) to the problem of controlling (~ q1;0(s); ~ q1;1(s)), which are the components of q corresponding to expanding directions of ~ L (see (24) and lemma 2.2). That is, we reduce an in nite dimensional problem to a nite dimensional one. -The second part of the proof is devoted to the solving of the nite dimensional problem, using 2-dimensional dynamics of (~ q1;0; ~ q1;1)(s) and a topological argument (index theory) based on the variation of the 2-dimensional parameter (d0; d1) appearing in the expression (25) of initial data qd0;d1(y; s0). 3.2 Proof of the geometrical property on q(s) First, we prove lemma 3.1 which insures that proposition 3.1 implies proposition 2.1 and then Proposition 1 i) and ii). Proof of lemma 3.1: We control d ds thanks to equation (29). Let us estimate each term appearing in: If s0 s3(A), we have the following estimates. Since q 2 VA, the left-hand side of (29) is (in absolute value) greater than Cjd ds j where C > 0. Since L is self-adjoint in L2(R; d ), R (L 1)~ q2d = R (L 1) ~ q2d = R (@2 @y2 12y @ @y )~ q2e y2=8 e y2=8 p4 dy. From (17), j@2 @y2 12y @ @y j C, and @2 @y2 12y @ @y 0 for jyj K0ps. Hence, we can bound e y2=8 by e K2 0s=8, and use q(s) 2 VA(s) to obtain j R (L 1)~ q2d j Ce s (if K0 is large enough). Existence of a blow-up solution for equation (2) 103 The same argument yields j R @ @s ~ q2d j Ce s. We have jVi;j(y; s)j Cs 1(1 + jyj2) (see lemma B.1 in appendix B). Combining this with De nition 3.1, we get j R (V2;1~ q1+V2;2~ q2)d j Cs 3 log s. We have j (y; s)B(q(y; s))j Cjqj2 for q(s) 2 VA(s) (see lemma B.4). Therefore, j R ~ B2(q)d j R jqj2d Cs 3. From (20), j R (y; s) ~ R 2(y; s)j C s2 (see lemma B.5). Combining all the previous estimates gives: jd ds j C s2 . Now, we give the proof of proposition 3.1 following the plan announced in the previous subsection. Part I: Reduction to a nite dimensional problem Here, (q; ) stands for a solution of system (19) with initial data (25). We show through a priori estimates that nding (d0; d1) 2 R2 such that 8s s0 q(s) 2 VA(s) is equivalent to nding (d0; d1) 2 R2 such that 8s s0 (~ q1;0(s); ~ q1;1(s)) 2 V̂A(s), where De nition 3.2 For each A > 0, for each s > 0, we de ne V̂A(s) as being the set [ A s2 ; A s2 ]2 R2 . Proposition 3.2 (Control of q(s) by (~ q1;0(s); ~ q1;1(s)) in ~ VA(s)) There exists A1 > 0 such that for each A A1, there exists 1(A) > 0, s1(A) > 0 such that for each 2 [ 1; 1], s0 s1(A), we have the following properties: -if (d0; d1) is chosen so that (~ q1;0(s0); ~ q1;1(s0)) 2 V̂A(s0), and, -if for s1 s0, we have 8s 2 [s0; s1], q(s) 2 VA(s) and q(s1) 2 @VA(s1), theni) (~ q1;0(s1); ~ q1;1(s1)) 2 @V̂A(s1), ii) (transversality) there exists 0 > 0 such that 8 2 (0; 0), (~ q1;0(s1 + ); ~ q1;1(s1 + )) 62 V̂A(s1 + ) (hence, q(s1 + ) 62 VA(s1 + )). Proof: see Proof of Proposition 3.2 below. Now, we x A A1, and 0 = 1. We note q(d0; d1) = qd0;d1 (see proposition 3.1). Part II: Topological argument for the nite dimensional problem In the following proposition, we initialize the nite dimensional problem and study the Cauchy problem for system (19). Proposition 3.3 (Initialization and Cauchy problem for system (19)) There exists s2(A) > 0 such that for each 2 [ 0; 0], for each s0 s2(A), i) there exists a set Ds0 R2 topologically equivalent to a square with the following property: q(d0; d1; s0) 2 VA(s0) if and only if (d0; d1) 2 Ds0 . ii) For each (d0; d1) 2 Ds0 , 9S = S(d0; d1) > s0 (maximal) such that system (19) with initial data (25) at s = s0 has a unique solution (q; )(d0; d1) on [s0; S), with q and C2 and q(s) 2 VA+1(s), 8s 2 [s0; S). iii) (q; ) is continuous with respect to (d0; d1; s). 104 Blow-up results for vector-valued nonlinear equations Proof: i) From (25), we have ~ q1(d0; d1; y; s0) = f0( y ps0 )p(d0 + d1 y ps0 ) s0 cos[ log( s0 )] and ~ q2(d0; d1; y; s0) = s0 ( sin[ log( s0 )])(1 (s0)f0( y ps0 )p). The expression of ~ q1 is similar to the expression of initial data (33) for the similar equation (15) in [18]. ~ q2 is a sum of two terms appearing in the mentioned formula (33) in [18]. Hence, one can adapt without di culties lemmas 3.1 and 3.3 of [18] to conclude (note that ~ q2;0(d0; d1; s0) = 0). ii) As if to use (15) in a reverse way, we introduce w(y; s) = ei (s)(q(y; s) + '(y; s)): (31) Therefore, our problem is equivalent to the following system in (w; ): @w @s = w 12y:rw (1 + i ) w p 1 + (1 + i )jwjp 1w (32) F (( (s); s) = 0 where F ( ; s) = cos( )(w2;0(s) w1;0(s)) + sin( )( w1;0(s) w2;0(s)) ~ '2;0(s); (33) with initial data w(d0; d1; s0) = q(d0; d1; s0) + '(s0); (34) (s0) = 0: (35) By a simple calculation, we have w(d0; d1; s0) 2 H . Hence from classical theory, we have local existence and uniqueness of a C2 solution for (32) with initial data (34). In order to prove existence and uniqueness for (s), we apply the implicit function theorem to F near ( ; s) = (0; s0). First we compute @F @ ( ; s) = sin( )(w2;0(s) w1;0(s)) + cos( )( w1;0(s) w2;0(s)) and @F @ (0; s0) = '1;0(s0) '2;0(s0) (1 + 2)~ q1;0(s0) ~ q2;0(s0) (use (31)). By (16), '1;0(s0) '2;0(s0) ! as s0 ! +1. Hence, if s0 s2(A) and (d0; d1) 2 Ds0 , then q(s0) 2 VA(s0) VA+1(s0) and @F @ (0; s0) 6= 0. Since F (0; s0) = 0 (because ~ q2;0(d0; d1; s0) = 0), and F is C2, we have existence and uniqueness of C2 (s). We add that the solution (q; )(s) is well de ned if we require q(s) 2 VA+1(s). iii) Using again the equivalent formulation (31), we see that (q; )(d0; d1; s) is a continuous function of (q(d0; d1; s0); s). Since q(d0; d1; s0) is continuous in (d0; d1) (it is a ne, see (25)), we obtain iii). Now, we x S0 > max(s1(A); s2(A)), and take 2 [ 0; 0], s0 S0. Then we start the proof of Proposition 3.1 for A, and s0. We argue by contradiction: According to proposition 3.3, for each (d0; d1) 2 Ds0 , system (19) with initial data (25) has a unique solution on [s0; S(d0; d1)) and q(d0; d1; s0) 2 VA(s0). We suppose then that for each (d0; d1) 2 Ds0 , there exists Existence of a blow-up solution for equation (2) 105 s > s0 such that q(d0; d1; s) 62 VA(s). Let s (d0; d1) be the in mum of all these s. By proposition 3.2 (s1 = s ), we can de ne the following function: : Ds0 ! @C (d0; d1) ! s (d0; d1)2 A (~ q1;0; ~ q1;1)(d0; d1; s (d0; d1)) where C is the unit square of R2 . Now we claim Proposition 3.4 i) is a continuous mapping from Ds0 to @C. ii) There exists a non-trivial a ne function g : Ds0 ! C such that g 1 j@C = Idj@C . From that , a contradiction follows (Index Theory). Hence, there exists (d0; d1) such that 8s s0, q(d0; d1; s) 2 VA(s). This concludes the proof of proposition 3.1, and also the proof of parts i) and ii) of Proposition 1. Proof of proposition 3.4: i) Part iii) of proposition 3.3 implies that (~ q1;0(s); ~ q1;1(s)) is a continuous function of (d0; d1). Using the transversality property of (~ q1;0(s ); ~ q1;1(s )) on @V̂A(s ) ( ii) of proposition 3.2), we claim that s (d0; d1) is continuous. Therefore, is continuous. ii) If (d0; d1) 2 @Ds0 , then from i) of proposition 3.3 , q(d0; d1; s0) 2 VA(s0). According to the proof of lemma 3.3 in [18], (~ q1;0(s0); ~ q1;1(s0)) 2 @V̂A(s0). Applying ii) of proposition 3.2 with s0 and s1 = s0, we have s (d0; d1) = s0, and (d0; d1) = s20 A (~ q1;0(s0); ~ q1;1(s0)). Let T : (d0; d1) 2 Ds0 ! s20 A (~ q1;0(s0); ~ q1;1)(s0)) 2 C. From (25), T is a ne. Hence T 1 j@Ds0 = Idj@Ds0 . This concludes the proof of proposition 3.4. Now, we give the proof of proposition 3.2. 3.3 Proof of proposition 3.2 As we suggested in the formulation of the problem, the proof follows the general ideas of [18]. However, it is more complicated because of terms with d ds or because of strong interference between ~ q1 and ~ q2 (see (27), (28)). Therefore, we summarize arguments which are similar to those exposed in [18] by showing how to adapt them to the present context, and emphasize the arguments relative to these new terms. We divide the proof in three steps: In Step 1, we give a priori estimates on q(s) in VA(s): assume that for given A > 0 large, > 0 and an initial time s0 s4(A; ), we have q(s) 2 VA(s) for each s 2 [ ; + ], where s0. Using system (19) which is satis ed by q, we then derive new bounds on ~ q1;2, ~ q1; , ~ q1;e, ~ q2;? and ~ q2;e in [ ; + ] (involving A and ). -In Step 2, we show that these new bounds are better than those de ning VA(s) (see de nition 3.1) provided that (A). Since ~ q1;2(s) = 0 by hypothesis in (19), only ~ q1;0(s) and ~ q1;1(s) remain to be controlled: the problem 106 Blow-up results for vector-valued nonlinear equations is then reduced to the control of a two dimensional variable (~ q1;0(s); ~ q1;1(s)). Afterwards, we conclude the proof of part i) of proposition 3.2. -In Step 3, we use dynamics of (~ q1;0(s); ~ q1;1(s)) to prove its transversality on @VA(s) (part ii) of proposition 3.2). Step 1: A priori estimates of q. From equations (27) and (28) (which are equivalent to the rst equation of system (19)), we write the integral equations satis ed by ~ q1 and ~ q2: ~ q1(s) = K1(s; )~ q1( ) + Z s d K1(s; )V1;2( )~ q2( ) + Z s d K1(s; ) ~ B1(q)d + Z s d K1(s; ) ~ R 1( ) + Z s d K1(s; )d ds ( )f ~ '1( ) + ~ '2( ) + ~ q1( ) + ~ q2( )g (36)~ q2(s) = K2(s; )~ q2( ) + Z s d K2(s; )V2;1( )~ q1( ) + Z s d K2(s; ) ~ B2(q)d + Z s d K2(s; ) ~ R 2( ) Z s d K2(s; )d ds ( )f(1 + 2) ~ '1( ) + ~ '2( )) + (1 + 2)~ q1( ) + ~ q2( )g (37) whereK1 is the fundamental solution of L+V1;1, K2 is the fundamental solution of L 1 + V2;2, L is given by (30), B(q) = (1 + i ) ~ B1 + i ~ B2, R (y; s) = (1 + i ) ~ R 1 + i ~ R 2, B and R are given by (20). We now assume that for each s 2 [ ; + ], q(s) 2 VA(s). Using (36, 37), we derive new bounds on all terms in the right hand sides of (36, 37), and then on q. In the case = s0, from initial data properties, it turns out that we obtain better estimates for s 2 [s0; s0 + ]. More precisely, we have the following lemma: Lemma 3.3 There exists A4 > 0 such that for each A A4, > 0, there exists s4(A; ) > 0 with the following property: 8 2 [ 1=2; 1=2], 8s0 s4(A; ), 8 , assume 8s 2 [ ; + ], q(s) 2 VA(s) with s0. I)~ q1 estimates: We have 8s 2 [ ; + ]; i) (main linear term) j 1;2(s)j A2 log s2 + (s )CAs 3; j 1; (y; s)j C(e 1 2 (s )A+ e (s )2A2)(1 + jyj3)s 2; k 1;e(s)kL1 C(A2e (s ) 2p +Ae(s ))s 1 2 ; where, as in decomposition (22), K1(s; )~ q1( ) = 1(y; s) = 2 X m=0 1;m(s)hm(y) + 1; (y; s) + 1;e(y; s): Existence of a blow-up solution for equation (2) 107 If = s0, and q(s0) satis es (25), then j 1;2(s)j log s0 s2 + CA(s s0)s 3; j 1; (y; s)j C(1 + jyj3)s 2; k 1;e(s)kL1 C(1 + e(s s0))s 12 : ii)(interference term) j 1;2(s)j Cj jA(s )es s 3; j 1; (y; s)j Cj jA2(s )(1 + jyj3)s 2; k 1;e(s)kL1 Cj j(A2 + e(s )A)(s )s 1=2; where, as in decomposition (22), R s d K1(s; )V1;2( )~ q2( ) = 1(y; s) = 2 X m=0 1;m(s)hm(y) + 1; (y; s) + 1;e(y; s): iii) (nonlinear term) j 1;2(s)j (s ) s3+1=2 ; j 1; (y; s)j (s )(1 + jyj3)s 2 ; k 1;e(s)kL1 (s )s 1 2 ; where = (p) > 0; and as in (22), R s d K1(s; ) ~ B1(q( )) = 1(y; s) = 2 X m=0 1;m(s)hm(y) + 1; (y; s) + 1;e(y; s): iv) (main corrective term) j 1;2(s)j (s )Cs 3; j 1; (y; s)j (s )C(1 + jyj3)s 2; k 1;e(s)kL1 (s )s 3=4; where as in (22), Z s d K1(s; ) ~ R 1( ) = 1(y; s) = 2 X m=0 1;m(s)hm(y) + 1; (y; s) + 1;e(y; s): v) (small terms) j 1;2(s)j C(s )s 3; j 1; (y; s)j C(s )(1 + jyj3)s 3; k 1;e(s)kL1 C(s )s 3=2; where as in (22), R s d K1(s; )d ds ( )f ~ q1( ) + ~ q2( ) + ~ '1( ) + ~ '2( )g = 1(y; s) = 2 X m=0 1;m(s)hm(y) + 1; (y; s) + 1;e(y; s): 108 Blow-up results for vector-valued nonlinear equations II)~ q2 estimates: We have 8s 2 [ ; + ]; i) (main linear term) j 2;?(y; s)j C(e 12 (s )A+ e (s )2A2)(1 + jyj3)s 2; k 2;e(s)kL1 C(A2e (s ) p +A)s 1 2 ; where, as in decomposition (23), K2(s; )~ q2( ) = 2(y; s) = 2;0(s)h0(y) + 2;?(y; s) + 2;e(y; s): If = s0, and q(s0) satis es (25), then j 2;?(y; s)j C(1 + jyj3)s 2; k 2;e(s)kL1 Cs 1 2 : (38)ii) (interference term) j 2;?(y; s)j Cj jA(s )(1 + jyj3)s 2; k 2;e(s)kL1 Cj jA2(s )s 1=2; where as in (23), R s d K2(s; )V2;1( )~ q1( ) = 2(y; s) = 2;0(s)h0(y) + 2;?(y; s) + 2;e(y; s): iii) (nonlinear term) j 2;?(y; s)j (s )(1 + jyj3)s 2 ; k 2;e(s)kL1 (s )s 1 2 ; where = (p) > 0; and as in (23), Z s d K2(s; ) ~ B2(q( )) = 2(y; s) = 2;0(s)h0(y) + 2;?(y; s) + 2;e(y; s): iv) (main corrective term) j 2;?(y; s)j Cs 2(s )(1 + jyj3); k 2;e(s)kL1 (s )s 3=4; where as in (23), Z s d K2(s; ) ~ R 2( ) = 2(y; s) = 2;0(s)hm(y) + 2;?(y; s) + 2;e(y; s): v) (small terms) j 2;?(y; s)j C(s )(1 + jyj3)s 2; k 2;e(s)kL1 C(s )s 2; where R s d K2(s; )d ds ( )f ~ q2( ) (1 + 2)~ q1( ) ~ '2( ) (1 + 2) ~ '1( )g = 2(y; s) = 2;0(s)h0(y) + 2;?(y; s) + 2;e(y; s), as in (23). Proof: see appendix B . Step 2: Lemma 3.3 implies i) of proposition 3.2 Here, we derive i) of proposition 3.2 from lemma 3.3. We follow the method used in [18] to prove proposition 3.4 starting from lemma 3.12. Indeed, from integral equations (36, 37) and lemma 3.3, we derive new bounds on ~ q1;2(s), Existence of a blow-up solution for equation (2) 109 ~ q1; (y; s), ~ q1;e(y; s), ~ q2;?(s) and ~ q2;e(y; s), assuming that 8s 2 [ ; + ], q(s) 2 VA(s), for and s0 s4(A; ). The key estimate is to show that for s = + (or s 2 [ ; + ] if = s0), these new bounds are better than those de ning VA(s), provided that (A). Comparing lemma 3.3 here and lemma 3.4 in [18], we see that we have additional terms: -Interference terms Iii) and IIii), Small terms Iv) and IIv). If we try to adapt the proof of proposition 3.4 of [18] in order to prove a similar result, we see that the introduction of small terms does not change anything to the proof, since they are either of lower order, if compared for example with linear terms (speaking in terms of power of s): 1: , 1;e and 2;e, or of the same order, but with a \small" coe cient (compared with A): 1;2 and 2;?. This is not the case of interference terms Ii) and IIi), which have a critical growth in terms of power of s. But recalling that in the mentioned proof in [18], we have (s ) log A C , if we assume that: Cj jA log A C elog A C 1 (Cf 1;2), Cj jA2 log A C A4 (Cf 1; ), Cj j(A2 + elog A C A) log A C A2 4 (Cf 1;e), Cj jA log A C A4 (Cf 2;?) and Cj jA2 log A C A2 4 (Cf 2;e), which is possible if j j 5(A), with 5(A) > 0, then all these terms, while remaining with critical growth, have a reasonable coe cient (1, A4 or A2 4 ). Therefore, adapting the proof of Proposition 3.4 in [18] for j j 5(A), we prove a similar proposition: Proposition 3.5 There exists A5 > 0 such that for each A A5, there exists 5(A) > 0, s5(A) > 0 such that for each 2 [ 5; 5], s0 s5(A), we have the following property: -if (d0; d1) is chosen so that (~ q1;0(s0); ~ q1;1(s0)) 2 V̂A(s0), and, -if for s1 s0, we have 8s 2 [s0; s1], q(s) 2 VA(s), then 8s 2 [s0; s1] , j~ q1;2(s)j A2s 2 log s s 3, j~ q1; (y; s)j A2 (1 + jyj3)s 2, k~ q1;e(s)kL1 A2 2ps , j~ q2;?(y; s)j A2 (1 + jyj3)s 2, k~ q2;e(s)kL1 A2 2ps . By de nition of (q; ) (Cf system (19)), we have ~ q2;0(s) = 0. If in addition q(s1) 2 @VA(s1), we see from de nition 3.1 of VA(s) that the rst two components of q(s1), namely ~ q1;0(s1) and ~ q1;1(s1) are in @V̂A(s1). This concludes the proof of part i) of proposition 3.2. Step 3: Transversality property of (~ q1;0(s1); ~ q1;1(s1)) on @V̂A(s1) To prove part ii) of proposition 3.2, we show that for each m 2 f0; 1g, for each 2 f 1; 1g, if ~ q1;m(s1) = A s21 , then d~ q1;m ds (s1) has the opposite sign of d ds ( A s2 )(s1) so that (~ q1;0; ~ q1;1) actually leaves V̂A at s1 for s1 s0 where s0 will be large. Now, let us compute d~ q1;0 ds (s1) and d~ q1;1 ds (s1) for q(s1) 2 VA(s1) and (~ q1;0(s1); ~ q1;1(s1)) 2 @V̂A(s1). First, we note that in this case, kq(s1)kL1 CA2 ps1 110 Blow-up results for vector-valued nonlinear equations and jqb(y; s1)j CA2 log s1 s21 (1 + jyj3) (Provided A 1). Below, O(l) stands for a quantity whose absolute value is bounded precisely by l and not Cl. For m 2 f0; 1g, we derive from equation (27) and (21): R d (s1)@~ q1 @s km = Z d (s1)L~ q1km + Z d (s1)fV1;1~ q1 + V1;2~ q2gkm + Z d (s1) ~ B1(q)km + R d (s1) ~ R 1(s1)km + R d (s1)d ds (s1)f ~ q1 + ~ q2 + ~ '1 + ~ '2gkm, where km = hm=khmk2L2(R;R;d ) (see (21)). We now estimate each term of this identity: a) j R d (s1)@~ q1 @s km d~ q1;m ds j = j R d d ds ~ q1kmj R d jd ds jCA2 ps1 jkmj Ce s1 if s0 s3(A). b) Since L is self-adjoint on L2(R; d ), we write Z d (s1)L~ q1km = Z d L( (s1)km)~ q1: Using L( (s1)km) = (1 m2 ) (s1)km + @2 @s2 km + @ @y (2@km @y y2km), we obtain R d (s1)L~ q1km = (1 m2 )~ q1;m(s1) +O(CAe s1). c) We have 8y 2 R; jVi;j (y; s)j Cs (1 + jyj2). Therefore, j R d (s1)fV1;1~ q1 + V1;2~ q2gkmj R d Cs 1 1 (1 + jyj2)CA2s 2 1 log s1jkmj CA2s 3 1 log s1 d) A standard Taylor expansion combined with the de nition of VA shows that j (y; s1)B(q(y; s1))j Cjqj2 C(jqbj2 + jqej2) CA4(log s1)2 s41 (1 + jyj3)2 + 1fjyj Kps1g(y) A2 ps1 . Thus, j R d (s1) ~ B1(q)kmj CA4(log s1)2 s41 + Ce s1 . e) From lemma B.5 in appendix B, we have j R d (s1) ~ R 1(s1)kmj C(p) s21 (Actually it is equal to 0 if m = 1). f) From lemma 3.1, we have jd ds (s1)j Cs 2 1 . Hence, j R d (s1)d ds (s1)f ~ q1 + ~ q2 + ~ '1 + ~ '2)kmj Cs 2 1 . Putting together the estimates a) to f), we obtain d~ q1;m ds (s1) = (1 m2 ) A s21 +O(C(p) s21 ) +O(CA4 log s1 s31 ) whenever ~ q1;m(s1) = A s21 . Let us now x A 2C(p), and then we take s1(A) larger so that for s0 s1(A), 8s s0, C(p) s2 + O(CA4 log s s3 ) 3C(p) 2s2 . Hence, if = 1, d~ q1;m ds (s1) < 0, if = 1, d~ q1;m ds (s1) > 0. This concludes the proof of part ii) of proposition 3.2. It also concludes the proof of part ii) of Proposition 1, and then the proof of Theorem 1. 4 Blow-up pro le of u(t) solution of (2) near blow-up point We prove in this section part iii) of Proposition 1. We consider u(t) solution of (1) constructed in section 3, which blows-up in nite time T > 0 at only one blow-up point: 0. We know from section 3 that: sup z2R j(T t) 1+i p 1 u(zp(T t)j log(T t)j; t) f(z)j C pj log(T t)j (39) Blow-up pro le of u(t) solution of (2) 111 with f(z) = (p 1 + (p 1)2 4(p 2) jzj2) 1+i p 1 : (40)Adapting the techniques used by Merle in [17] to equation (1), we derive the existence of a pro le u 2 C(Rnf0g; C ) such that u(x; t) ! u (x) as t ! T uniformly on compact subsets of Rnf0g. We want to nd an equivalent function for u near the blow-up point: 0. For this purpose, we de ne for each t 2 [0; T ), a rescaled version of u(t): v(t; ; ) = (T t) 1+i p 1 u( pT t; t+ (T t) ) (41) where 2 R, 2 [ t T t ; 1) [0; 1). From equation (1), we see that v(t; ; ) satis es the same equation as u(t; x): 8 2 [ t T t ; 1); @v @ = v + (1 + i )jvjp 1v: (42) Stated in terms of v(t), (39) becomes: sup 2R j(1 ) 1+i p 1 v(t; ; ) f( p(1 )j logf(1 )(T t)gj )j (43) C pj logf(T t)(1 )gj . We proceed in two steps: rst, we consider r > 0 and estimate v(t; ; ) and its derivatives locally near (r; t) 2 R satisfying j (r; t)j = rpj log(T t)j. We show that v(t; ; ) is bounded, and that it does not vary much for j (r; t)j bounded and 2 [0; 1], then, we can identify v(t; ; 0) (approximated by (43)) and v(t; ; 1). For each x 2 Rnf0g, we write jxj as j (r; t)jp(T t) = rp(T t)j log (T t)j for some r > 0 and t < T and combine this identi cation with (41) to get the equivalent of u (x) for x! 0: u (x) 8(p 2)j log jxjj (p 1)2jxj2 1+i p 1 : (44)For simplicity, we omit t in the notation and write v( ; ) for v(t; ; ), (r) for (r; t). Part I: Estimate for v near rpj log(T t)j From (41), v blows-up at time = 1 at only one blow-up point: 0. Using (43) and a lower bound shown by Giga and Kohn in [11] on blow-up rate for v, we derive a local bound on v for 2 [0; 1), j (r)j bounded, independent from r and t. Using classical parabolic theory and the fact that v depends in a certain sense only on for j j small, we show that v actually does not depend much on 2 [0; 1) for j (r)j bounded. Proposition 4.1 (Estimate on @v @ ( (r); )) There exists r1 > 0 such that 8r r1, 9t1(r) < T such that 8t 2 [t1(r); T ), 8 2 [0; 1), j @v @ ( (r); )j Cjf(r)jp. 112 Blow-up results for vector-valued nonlinear equations Proof: Step 1: Local bounds on v near (r) for 2 [ 1=2; 1) We crucially use a lower bound on blow-up rate for v established by Giga and Kohn in [11] to show that jvj is bounded for near (r) and 2 [ 1=2; 1). Lemma 4.1 (Lower bound on blow-up rate for v) . i) (Giga-Kohn) There exists = (p; ;N) > 0 with the following property: If for j (r)j 3pj log(T t)j, 2 [ 1=2; 1) (1 ) 1 p 1 jv( ; )j ; then 8 2 R with j (r)j 2pj log(T t)j, 8 2 [ 1=2; 1), jv( ; )j C. ii) There exists r2 > 0 such that 8r r2, 9t2(r) < T such that 8t 2 [t2(r); T ), if j (r)j 2pj log(T t)j, 2 [ 1=2; 1) then jv( ; )j C: Proof: i) follows immediately from Theorem 2.1 in [11]. ii) is a direct consequence of i) and estimate (43). Indeed, if j (r)j 3pj log(T t)j and 2 [ 1=2; 1), then we have by (43) (1 ) 1 p 1 jv( ; )j Cjf(r)j + Cj log(T t)j 1=2. Step 2: Local bound on @v @ ( ; ) near (r) for 2 [0; 1) = 0: From a parabolic estimate and (43) considered for 0, we have for j (r)j pj log(T t)j: j@2v @ 2 ( ; 0) 1 j log(T t)j @2f @z2 ( pj log(T t)j )j C pj log(T t)j : Hence, from (42), we have for r r3, t t3(r), j (r)j pj log(T t)j: j @v @ ( ; 0)j Cjf(r)jp. 2 [0; 1): We use the equation satis ed by @v @ and standard tools of localization and local estimates with the semi-group e to conclude. Indeed, if z( ; ) = j @v @ j2, it follows from equation (42) and ii) of lemma 4.1 that 8 2 [0; 1), 8 2 R with j (r)j pj log(T t)j, @z @ z +Mz, where M = M(p; ;N). We can consider 2 C1 0 (R) satisfying ( ) = 0 if j (r)j pj log(T t)j, 0 1, ( ) = 1 if j (r)j pj log(T t)j=2, and jr j+ j j C. If w( ; ) = e M ( )z( ; ), then w satis es: @w @ w + e M ( z + 2rz:r ) and 8 2 R, jw( ; 0)j Cjf(r)j2p. If 2 [0; 1), then w( (r); ) (e w(0))( (r); ) + Z 0 d (4 ( ))1=2 Z dxe jx (r)j2 4( ) (zj j+ 2jrzjjr j)(x; ) Cjf(r)j2p + Z 0 d (4 ( ))1=2 Z dxe j log(T t)j=4 8( ) e jx (r)j2 8( ) C Blow-up pro le of u(t) solution of (2) 113 (lemma 4.1 ii) implies by parabolic regularity that for r r2, t t2(r), (zj j+ 2jrzjjr j)(x; ) C, for 2 [0; 1) and jx (r)j pj log(T t)j). Therefore, w( (r); ) Cjf(r)j2p + e j log(T t)j. If t t4(r), then w( ; ) Cjf(r)j2p, which implies 8 2 [0; 1), j @v @ ( (r); )j Cjf(r)jp. Taking r1 = max(r2; r3) and t1(r) = max(t2(r); t3(r); t4(r)) concludes the proof. Part II: Conclusion of the proof For each r r1 and each x 2 Rnf0g small enough, we de ne t(r; x) 2 [0; T ) by jxj = j (r)jpT t = rp(T t(r; x))j log(T t(r; x))j: (45) Applying proposition 4.1 to v(t(r; x)), we estimate the di erence between u (x) and u(x; t(r; x)) and then between u (x) and f(r). Then, by simple asymptotic calculation, we reach the equivalent (44). Lemma 4.2 (A rst estimate on the pro le u (x)) 8r r1, 9R2(r) > 0 such that 8x 2 R with 0 < jxj < R2 j(T t(r; x)) 1+i p 1 u (x) f(r)j Cjf(r)jp; where t(r; x) is uniquely determined by (45). Proof: Using proposition 4.1 and (43), we write for r r1, t t1(r): 8 2 [0; 1) jv( (r); ) f(r)j jv( (r); ) v( (r); 0)j + jv( (r); 0) f(r)j Cjf(r)jp + Cj log(T t)j 1=2. Stated in terms of u, this gives: 8 2 [0; 1) j(T t) 1+i p 1 u( (r)pT t; t+ (T t) ) f(r)j (46) Cjf(r)jp + Cj log(T t)j 1=2. From this estimate, we derive R2(r) > 0 such that 8x 2 R with 0 < jxj < R2, we have: 8 2 [0; 1) j(T t(r; x)) 1+i p 1 u(x; t(r; x) + (T t(r; x)) ) f(r)j Cjf(r)jp, where t(r; x) is given by (45). If we let go to 1, we have the conclusion of lemma 4.2. Now, we conclude the proof of estimate (44). For this purpose, we consider an arbitrary > 0 and look for R > 0 such that for 0 < jxj < R , j jxj2 log jxj 1+i p 1 u (x) 8(p 2) (p 1)2 1+i p 1 j : If we consider an arbitrary r r1, then by lemma 4.2, we have for 0 < jxj < R2 j jxj2 log jxj 1+i p 1 u (x) 8(p 2) (p 1)2 1+i p 1 j j jxj2 log jxj 1+i p 1 [2r2(T t(r; x))] 1+i p 1 j:ju (x)j (47) 114 Blow-up results for vector-valued nonlinear equations + [2r2] 1 p 1 j(T t(r; x)) 1+i p 1 u (x) f(r)j + j[2r2] 1+i p 1 f(r) 8(p 2) (p 1)2 1+i p 1 j We x r( ) r1 such that j[2r2] 1+i p 1 f(r) h 8(p 2) (p 1)2 i 1+i p 1 j and jf(r)jp 1 . From (45), we have jxj2 log jxj = 2r2(T t(r; x)) log(T t(r; x)) log(T t(r; x)) + log j log(T t(r; x))j + 2 log r : Let R > 0 su ciently small and smaller than R2(r( )) such that for 0 < jxj < R j jxj2 log jxj 1+i p 1 [2r2(T t(r; x))] 1+i p 1 j [2r2(T t(r; x))] 1 p 1 : Hence, for 0 < jxj < R , we have from (47): j[ jxj2 log jxj ] 1+i p 1 u (x) [ 8(p 2) (p 1)2 ] 1+i p 1 j [2r2(T t(r; x))] 1 p 1 ju (x)j+ C r 2 p 1 jf(r)j + C r 2 p 1 jf(r)j(1 + C ) + C (use lemma 4.2 and jf(r)jp 1 ) C . This concludes the proof of part iii) of Proposition 1. 5 Generalization and comments As a rst application of the techniques in previous sections, we have the following stability result concerning the behavior described in Proposition 1: Theorem 3 (Stability with respect to initial data of the pro le (4)) Let 2 ( 1; 1) where 1 > 0 and consider û0 initial data constructed in Proposition 1. Let û(t) be the solution of equation (1) with initial data û0, T̂ its blow-up time and â its blow-up point. Then there exists a neighborhood V of û0 in H with the following properties: For each u0 2 V, u(t) blows-up in nite time T = T (u0) at one single point a = a(u0), where u(t) is the solution of equation (1) with initial data u0. Moreover, u(t) approaches the pro les (6) and (7) near (T; a) similarly as û(t) does near (T̂ ; â). The proof of this theorem relies strongly on the techniques developed in sections 2, 3 and 4. We give just the key ideas of the proof. Consider initial data u0 in a neighborhood of û0 and u(t) the corresponding solution of (1). Then, for each (T; a) near (T̂ ; â), we introduce as in section 2 a two-parameter group acting on u(t): (T; a)! (q(T; a; y; s); (T; a; s)) where q(T; a; y; s) = w(T; a; y; s) '(y; s) ~ q2;0(s) = 0; Generalization and comments 115 w(T; a) is de ned similarly as in (9) by y = x a pT t s = log(T t) w(y; s) = (T t) 1+i p 1 u(x; t); and ' is given by (16). Therefore, our problem reduces to searching a parameter (T (u0); a(u0)) such that 8s s0; q(T; a; s) 2 VA(s) (48) for some s0 > 0 and A > 0 (see de nition 3.1). Indeed, T (u0) and a(u0) will be shown then to be respectively the blow-up time and point of u(t). Moreover, we derive directly form (48) an estimate analogous to (6) and then, by the techniques of section 4, an other estimate analogous to (7). By uniform a priori estimates analogous to proposition 3.2, we reduce this problem to a nite dimensional one. We solve it using a non-degeneration property of the two-parameter group acting on û(t) itself (see [18] for similar argument). Hence, we reach the conclusion of Theorem 3. The proof used for equation (1) applies in a more general case: consider the following vector-valued heat equation: du dt = u+ jujp 1u+G(u); u(x; 0) = u0(x) (49) where 1) u(t) : x 2 RN ! RM , p 2 (1;+1), p < (N + 2)=(N 2) if N 3, 2) G : RM ! RM is a perturbation of jujp 1u satisfying: G(u) = G1(juj2)u, jG(u)j Cjujr, jG( u1) G( u2)j C rju1 u2j for ju1j; ju2j 1, 1, r 2 [1; p), G1 : R+ ! R+ , G needs not be a gradient, 3) u0 2 H =W 1;p+1(RN ;RM ) \ L1(RN ;RM ). Using the same techniques as in the case M = 2 (equation (1) with = 0), we show the following blow-up result for equation (49): Theorem 2: (Existence of a blow-up solution for equation (49)) There exist initial data u0 such that equation (49) has a blow-up solution. This Theorem is a direct consequence of the following proposition which describes more precisely the behavior of u(t) near blow-up. Indeed, after a time dependent scaling, u(t) approaches a universal pro le (p 1 + (p 1)2 4p jzj2) 1 p 1!; (50) when t! T , where ! 2 SM 1. In fact, we have the more precise result: Proposition 2 (Existence of a blow-up solution for equation (49) with the pro le (50)) 116 Blow-up results for vector-valued nonlinear equations There exists T0 > 0 such that for each T 2 (0; T0], for each a 2 RN , for each ! 2 SM 1, there exist initial data u0 such that equation (49) has a blow-up solution u(x; t) on RN [0; T ) which blows-up in nite time T at only one blow-up point: a. Moreover, lim t!T(T t) 1 p 1u(a+ ((T t)j log(T t)j) 1 2 z; t) = f(z)! (51) uniformly in z 2 RN , with f(z) = (p 1 + (p 1)2 4p jzj2) 1 p 1 : (52) Remark: Structural stability: In [18], a particular version of this Proposition was shown in the case M = 1 and G = 0 (without perturbation): Single point blow-up and a blow-up pro le (52). There, this result was shown to be stable with respect to perturbations in initial data. With proposition 2, the blow-up solution constructed in [18] is shown to be structurally stable in a certain class of functions, since this solution behaves in the same way when we take a non zero G and consider a higher dimension (M 2): we still have single point blow-up with the same scalar pro le (52). A Appendix: A blow-up result for @u @t = u + jujp 1u +ijujq 1u on bounded domain for q small We consider the complex-valued heat equation (3): @u @t = u+ jujp 1u+ ijujq 1u (53) uj@ = 0; where u(t) : ! C , is a bounded domain of RN , p 2 (1;+1), p < (N + 2)=(N 2) if N 3, and q > 1. Proposition A.1 (Existence of blow-up solutions for equation (53)) Assume 1 < q < (p+ 1)=2. There exists A( ; p; q) > 0 such that for each u0 2 H1 0 ( ) with ku0kL2( ) A and E(u0) 0 where E(u0) = 1 2 Z jru0j2dx 1 p+ 1 Z ju0jp+1dx; (54) equation (53) with initial data u0 has a unique solution u 2 C([0; T ); H1 0( )) with 0 < T < +1, which blows-up in H1 0 ( ) as t! T . Proof: From classical theory, we know that if 1 < q p and u0 2 H1 0 ( ), then equation (53) with initial data u0 has a unique solution de ned on [0; T ) with T = Tu0 2 (0;+1] and u 2 C([0; T ); H1 0 ( )). Moreover, if T < +1, then u(t) blows-up in H1 0 ( ) as t! T . A blow-up result for ut = u+ jujp 1u+ ijujq 1u 117 Hence, proposition A.1 will be proved if we show that for 1 < q < (p+1)=2, ku0k2L2( ) A (to be chosen later) and E(u0) 0, we have Tu0 < +1. We proceed as follows: rst we give estimates on u(t) for t 2 [0; T ), then we use a blow-up result for an integral inequality to conclude. Lemma A.1 (Estimate for u(t), solution of (53)) If z(t) = (R ju(x; t)jp+1dx)2=(p+1), then 8t 2 [0; T ), z(t) c1A2 + c2 Z t 0 d z( )(p+1)=2 c3 Z t 0 d Z 0 dsz(s)q (55) where c1 = c1( ; p) > 0, c2 = c2( ; p) > 0 and c3 = c3( ; p; q) > 0. Proof: For simplicity, we omit x, and dx in following expressions of the type R ju(x; t)j2dx. From (54), d dtE(u(t)) = <( R ut(t) u(t) R ju(t)jp 1u(t) ut(t)). From (53), d dtE(u(t)) = <( R ut(t)ut(t) + i R ju(t)jq 1u(t) ut(t)) R jut(t)j2 + R ju(t)jq jut(t)j R jut(t)j2 + 12 (R jut(t)j2 + R ju(t)j2q) (Cauchy Schwartz), 1 2 R jut(t)j2 + c4( ; p; q)(R ju(t)jp+1)2q=(p+1) (H older). Integrating this inequality and using E(u0) 0 gives E(u(t)) c4( ; p; q) Z t 0 ds(Z ju(s)jp+1)2q=(p+1): (56) Now, if we multiply equation (53) by u(t) and take the real part, we obtain using expression (54) d dt Z ju(t)j2 = 4E(u(t)) + p 1 p+ 1 Z ju(t)jp+1: (57) Using (56), R ju(0)j2 A2 and (R ju(t)jp+1)2=(p+1) c1( ; p) R ju(t)j2 (Holder), we have the conclusion by integrating (57). Now, the conclusion follows directly from lemma A.1 and the following lemma: Lemma A.2 (Blow-up result for an integral inequality) Let z 2 C([0; T );R+) such that z(t) B + a Z t 0 dt0z(t0)(p+1)=2 b Z t 0 dt0 Z t0 0 dsz(t)q (58) where 1 < p, 1 < q < (p+ 1)=2, a > 0 and b > 0. There exists B0 > 0 such that if B B0, then T < +1. Proof: Let g(t) = a2 z(t)(p+1)=2 b R R t 0 dsz(s)q. Let us show that 8t 2 [0; T ), g(t) > 0. We proceed by a priori estimates. For B > 0, we can de ne T = supfT 0 2 [0; T )j8t 2 [0; T 0); R t 0 dt0g(t0) 0g > 0. Then we have 8t 2 [0; T ), g(t) > 0. 118 Blow-up results for vector-valued nonlinear equations Indeed, we have 8t 2 [0; T ) R t 0 dt0g(t0) 0. Therefore, (58) yields z(t) B + a2 R t 0 dsz(s)(p+1)=2 which gives z(t) B and z(t) a2 R t 0 dsz(s)(p+1)=2. Hence, g(t) = a2 z(t)(p+1)=2 b R R t 0 dsz(s)q a2B(p 1)=2z(t) b R R t 0 dsz(s)q > a2B(p 1)=2 a2 R t 0 dsz(s)(p+1)=2 b R R t 0 dsz(s)q a2 4 B(p 1)=2 R t 0 dsB(p+1)=2 qz(s)q b R R t 0 dsz(s)q = (a2 4 Bp q b) R t 0 dsz(s)q . Now, if B > (4ba 2)1=(p q), then 8t 2 [0; T ), g(t) > 0. This yields T = T and 8t 2 [0; T ), R t 0 dt0g(t0) 0. Therefore, (58) implies that 8t 2 [0; T ); z(t) B + a2 Z t 0 dsz(s)(p+1)=2: Hence, T 4B(1 p)=2 a(p 1) < +1 by classical arguments. B Appendix: Proof of lemma 3.3 Lemma 3.3 consists in a priori estimates on terms appearing in the integral equations satis ed by ~ q1 and ~ q2 (see (36), (37)). Let us recall them: ~ q1(s) = K1(s; )~ q1( ) + Z s d K1(s; )V1;2( )~ q2( ) + Z s d K1(s; ) ~ B1(q)d + Z s d K1(s; ) ~ R 1( ) + Z s d K1(s; )d ds ( )f ~ '1( ) + ~ '2( ) + ~ q1( ) + ~ q2( )g ~ q2(s) = K2(s; )~ q2( ) + Z s d K2(s; )V2;1( )~ q1( ) + Z s d K2(s; ) ~ B2(q)d + Z s d K2(s; ) ~ R 2( ) Z s d K2(s; )d ds ( )f(1 + 2) ~ '1( ) + ~ '2( )) + (1 + 2)~ q1( ) + ~ q2( )g whereK1 is the fundamental solution of L+V1;1, K2 is the fundamental solution of L 1 + V2;2, L is given by (30), B(q) = (1 + i ) ~ B1 + i ~ B2, R (y; s) = (1 + i ) ~ R 1 + i ~ R 2, B and R are given by (20). From these expressions, we obviously see that the main step in doing a priori estimates is the understanding of the behavior of the kernels K1 and K2. By de nition, K1 and K2 can be considered as perturbations of e L and e (L 1) respectively. Hence, we give the proof in two steps: -in Step 1, we give estimates on the integral operators K1 and K2, nonlinear term B(q) and corrective term R appearing in equations (36) and (37). -in Step 2, we use these estimates to prove lemma 3.3. Proof of lemma 3.3 119 Step 1: Estimates on linear, nonlinear and corrective terms of (36) and (37). In order to estimate K1 and K2, we follow the perturbation method used in [18] (and before in Bricmont and Kupiainen [4]). Since K1 and K2 correspond respectively to the operators L + V1;1 and L 1 + V2;2, we estimate rst the potentials Vi;j so we are able to adapt the cited method which comparesK1 and K2 to e L and e (L 1) respectively. Then, we show that B(q) can be considered in some sense as a quadratic term, and R is in fact small as s! +1. Lemma B.1 (Estimates on potentials Vi;j , j j 1=2) 8s 1, a)V1;1(y; s) Cs 1, jdnV1;1 dyn j Cs n=2, n = 0; 1; 2, jV1;1(y; s)j Cs 1(1 + jyj2), V1;1(y; s) = 1 4sh2(y) + ~ V1;1(y; s) with j ~ V1;1(y; s)j Cs 2(1 + jyj4), 8 > 0, 9C > 0, 9s such that sup s s ; jyj ps C jV1;1(y; s) ( p 2 p 1 )j with p 2 p 1 1 1=(2p). b)V2;2(y; s) Cs 1, jdnV2;2 dyn j Cs n=2, n = 0; 1; 2, jV2;2(y; s)j Cs 1(1 + jyj2), V2;2(y; s) = s 1Q (y) + ~ V2;2(y; s) with Q a polynomial of degree 2 with bounded coe cients and j ~ V2;2(y; s)j Cs 2(1 + jyj4), 8 > 0, 9C > 0, 9s such that sup s s ; jyj ps C j 1 + V2;2(y; s) ( 1 1 + 2 p 1 )j with 1 1+ 2 p 1 < 1 1=p. c) For V = V1;2 or V2;1, we have jV (y; s)j Cj j, and jV (y; s)j Cj js 1(1 + jyj2). Proof: The expressions of Vi;j are given in lemma 3.2. a) V1;1(y; s) (1 2)(j'(0; s)jp 1 1 p 1 ) + (p 1)j'(0; s)jp 3(j'(0; s)j2 0) 1 C( )s 1 Cs 1. We introduce W1;1(z; s) = V1;1(y; s) with z = y=ps. In order to prove the next estimate, it is enough to prove that jdnW2;2 dyn j C, n = 0; 1; 2. Since V1;1 is a sum of products of terms j'jp 1 and 'j=j'j, j = 1; 2, our problem reduces to proving that these terms have bounded rst and second derivatives with respect to z, which follows easily (see (16), the key estimates are @f @z = 2bz (p 1)(p 1+bz2)f and jf j j'j with b = (p 1)2 4(p 2) ). We introduce ~ W1;1(Z; s) = V1;1(y; s) with Z = jyj2=s. We can Taylor expand ~ W1;1 near Z = 0 to have ~ W1;1(Z; s) = ~ W1;1(0; s) + Z @ ~ W1;1 @Z (0; s) + O(Z2) with ~ W1;1(0; s) = 1=(2s) + O(s 2) and @ ~ W1;1 @Z (0; s) = 1=4 + O(s 1). Returning to V1;1, this yields the next two estimates. The last estimate is obvious from the expressions of V1;1 and '. 120 Blow-up results for vector-valued nonlinear equations b) For the rst term, we make a change of variables by setting Y = 1f (y=ps) + 1=(2(p 2)s) 2 (1=(2(p 2)s); 1=(2(p 2)s) + 1] and V̂2;2(Y; s) = V2;2(y; s). Then, it is easy to see that V̂2;2(:; s) is increasing. Therefore, V̂2;2(Y; s) V̂2;2(1=(2(p 2)s) + 1; s) C( )s 1 Cs 1. For next estimates, do exactly as for V1;1. c) Same proofs, one has to be careful with the parameter . Lemma B.2 (Estimates on K1, j j < 1=2) . a) 8s > 1 with s 2 , 8y; x 2 R, jK1(s; ; y; x)j Ce(s )L(y; x), with e L(y; x) = e p4 (1 e ) exp[ (ye =2 x)2 4(1 e ) ], kK1(s; )(1 ( ))kL1 Ce (s )=(2p). b) For each A0 > 0, A00 > 0, A000 > 0, > 0, there exists s9(A0; A00; A000; ) with the following property: 8s0 s9, assume that for s0, jqm( )j A0 2;m = 0; 1; jq2( )j A00(log ) 2; jq (y; )j A000(1 + jyj3) 2; kqe( )kL1 A00 1 2 ; then, 8s 2 [ ; + ] j 2(s)j A00 log s2 + (s )CA0s 3; j (y; s)j C(e 1 2 (s )A000 + e (s )2A00)(1 + jyj3)s 2; k e(s)kL1 C(A00e (s ) p +A000e(s ))s 1 2 ; where, as in decomposition (22), K1(s; )q( ) = (y; s) = 2 X m=0 m(s)hm(y) + (y; s) + e(y; s): (59)c) For each A0 > 0, A00 > 0, A000 > 0, > 0, there exists s10(A0; A00; A000; ) with the following property: 8s0 s10, assume that for s0, jqm( )j A0 2;m = 0; 1; jq2( )j A00 3; jq (y; )j A000(1 + jyj3) 3; kqe( )kL1 A0 2; then, 8s 2 [ ; + ] j 2(s)j A00s 3 + (s )CA0s 3; j (y; s)j CA000(1 + jyj3)s 3; where K1(s; )q( ) is expanded in (59). Proof of lemma 3.3 121 Proof: In [4] (proof of lemma 1), the authors prove the estimate for an integral operator K corresponding to L + V (see (30) for L), where V is a particular function. However, their result is in fact true for a larger class of operators satisfying estimates of the type a) in lemma B.1. Hence, lemma B.2 follows. Lemma B.3 (Estimates on K2, j j < 1=2) . a) 8s > 1 with s 2 , 8y; x 2 R, jK2(s; ; y; x)j Ce (s )e(s )L(y; x), with e L(y; x) = e p4 (1 e ) exp[ (ye =2 x)2 4(1 e ) ], kK2(s; )(1 ( ))kL1 Ce (s )=p. b) For each A0 > 0, A00 > 0, > 0, there exists s11(A0; A00; ) with the following property: 8s0 s11, assume that for s0, jq0( )j A0 2;m = 0; 1; jq?(y; )j A0(1 + jyj3) 2; kqe( )kL1 A00 1 2 ; then, 8s 2 [ ; + ] j ?(y; s)j C(e 12 (s )A0 + e (s )2A00)(1 + jyj3)s 2; k e(s)kL1 C(A00e (s ) p +A0)s 1 2 ; where, as in decomposition (23), K2(s; )q( ) = (y; s) = 0(s)h0(y) + ?(y; s) + e(y; s): Proof: Again, we can adapt the proof of lemma 1 in [4] with L replaced by L 1 and V replaced by V2;2, without di culties. Indeed, one checks easily that V2;2 satis es all useful estimates: b) of lemma B.1. Lemma B.4 (Estimates on B(q( )) for q( ) in VA( ), j j 1=2 ) . 8A > 0, 9s12(A) > 0 such that 8 s12(A), q( ) 2 VA( ) implies j (y; )B(q(y; ))j = j(1 + i ) ~ B1 + i ~ B2j Cjqj2, jB(q)j = j(1 + i ) ~ B1 + i ~ B2j Cjqj p with p = min(p; 2). Proof: Start with (20) and do the same as in the proof of lemma 3.6 in [18]. Lemma B.5 (Estimates on R (y; s), j j 1=2) 8s 1, if R is expanded as in (24), then: j ~ R 1;0(s)j Cs 2, ~ R 1;1(s) = 0, j ~ R 1;2(s)j Cs 3, j ~ R 1; (y; s)j Cs 2(1 + jyj3), k ~ R 1;e(s)kL1 Cs 1, and j ~ R 2;0(s)j Cs 2, j ~ R 2;?(y; s)j Cs 2(1 + jyj3), k ~ R 2;e(s)kL1 Cs 1. 122 Blow-up results for vector-valued nonlinear equations Proof: ~ R 1;1(s) = 0 since R is even. All the other estimates follow from the three following estimates: j (y; s)R (y; s)j Cs 2(1 + jyj2), jR (y; s)j Cs 1 and j ~ R 1;2(s)j Cs 3. Proof of j (y; s)R (y; s)j Cs 2(1 + jyj2): From (20), we have R (y; s) = @' @s + ' 12y:r' (1 + i ) ' p 1 + (1 + i )j'jp 1' = (1 + i ) i (f + 2(p 2)s )i ( 2(p 2)s2 + (p 1)y2 4(p 2)s2 fp ) + (1 + i ) i (f + 2(p 2)s )i (p 1)y2 4(p 2)sfp + (1 + i )i i (f + 2(p 2)s )i 1( (p 1)y 2(p 2)sfp )2 + (1 + i ) i (f + 2(p 2)s )i ( (p 1) 2(p 2)sfp + p(p 1)2y2 4(p 2)2s2 f2p 1 ) + (1 + i ) i ((f + 2(p 2)s )p+i 1 p 1(f + 2(p 2)s )1+i ): (60) Some of these terms are easily seen to be bounded by Cs 2(1 + jyj2), whereas others need some calculation: we divide the others by (1 + i )(f + 2(p 2)s )i i and obtain Q(y; s) = (p 1)y2 4(p 2)sfp (p 1) 2(p 2)sfp 1 p 1 (f + 2(p 2)s ) +(f + 2(p 2)s )p. It remains to prove that j (y; s)Q(y; s)j Cs 2(1+jyj2). We write Q(y; s) = (f + 2(p 2)s )p fp (p 1) 2(p 2)sfp 2(p 2)(p 1)s . Setting z = jyj2 s 0 and Q̂(z; s) = Q(y; s), we have jQ̂(0; s)j Cs 2 and j@Q̂ @z (z; s)j = pj@f @z f(f + 2(p 2)s )p 1 fp 1 (p 1) 2(p 2)sfp 1 gj Cs 1 if z 2K0, (Taylor expansion). Therefore, if z 2K0, jQ̂(z; s)j Cs 2+O(jzjs 1). Returning to Q, this gives the result. Proof of jR (y; s)j Cs 1: Thinking of R as a function of jyj2s 1 and s (see (60)), this estimate is obvious for all terms except (1 + i ) i (f + 2(p 2)s )i ( (p 1)jyj2 4(p 2)s fp + (f + 2(p 2)s )p 1 p 1f ) = (1 + i ) i (f + 2(p 2)s )i ((f + 2(p 2)s )p fp ). We conclude using a Taylor expansion. Proof of j ~ R 1;2(s)j Cs 3: From (60), we have ~ R 1(y; s) = @'1 @s + '1 1 2y:r'1 + (j'jp 1 1 p 1)('1 '2): Starting from ~ R 1;2(s) = R d (y) (y; s) ~ R 1(y; s)h2(y) 8 , one carries out easy but long asymptotic calculation to get the result. Step 2: Conclusion of the proof of lemma 3.3 We now prove lemma 3.3. Proof of lemma 3.3 123 Ii) Case s0: Apply b) of lemma B.2 with A0 = A, A00 = A2 and A000 = A. Case = s0: From (25), ~ q1(y; s0) = f0( y ps0 )p(d0+d1y=ps0) <(( 2(p 2)s0 )1+i ). Since (d0; d1) is chosen so that (~ q1;0(s0); ~ q1;1(s0)) 2 V̂A(s0), we have from lemma 3.1 in [18], j~ q1;m(s0)j As 2 0 , m = 0; 1, j~ q1;2(s0)j (log s0)s 2 0 , j~ q1; (y; s0)j Cs 2 0 (1 + jyj3) and k~ q1;e(s0)kL1 s 1=2 0 . We apply b) of lemma B.2 with A0 = A, A00 = C, A000 = 1 to conclude Iii): We have from lemma B.1 jV1;2(y; s)j Cj js 1(1 + jyj2). Since q( ) 2 VA( ), jV1;2(y; )~ q2(y; )j CAj j 3(1 + jyj4). Hence, j 1;2(s)j = jC R d h2(y) R s d K1(s; )V1;2( )~ q2( )j C R d (1 + jyj2) R s d e(s )LCAj j 3(1 + jxj4) CAj j 3 R d (1 + jyj6)(s )es CAj js 3(s )es , if s0 . If we set Q(y; ) = V1;2(y; )~ q2(y; ), we have by lemma B.1 jV1;2(y; )j Cj j and then jQm( )j Cj jA 2,m = 0; 1; 2, jQ (y; )j Cj jA(1+jyj3) 2, jQe(y; )j Cj jA2 1=2. Applying lemma B.2 and integrating between and s yields good estimates for 1; and 1;e. Iiii): Using lemma B.4 and a) of lemma B.2, we do the same as for the nonlinear term in Proof of lemma 3.4 in [18]. Iiv): From lemma B.5, we have j ~ R 1;0( )j C 2, ~ R 1;1( ) = 0, j ~ R 1;2( )j C 3, j ~ R 1; (y; )j C 2(1 + jyj3), j ~ R 1;e(y; )j C 1. Applying lemma B.2 b) and integrating between and s gives the results for 1;2 and 1; . For 1;e,we use the following estimate: jR (y; )j C 1, and compute: j 1;ej = j R s d K1(s; )R 1( )j R s d e(s )LC 1 (use lemma B.2 a)) C 1(s )es Cs 3=4(s ) if s0 s5( ). Iv): We set Q(y; ) = d ds ( )f ~ q1 + ~ q2 + ~ '1 + ~ '2g. By lemma 3.1, we have jd ds ( )j C 2. Using q( ) 2 VA( ), ' bounded and a simple calculation, we have: jQm( )j CA 2, m = 0; 1, jQ2( )j Cj j 3, jQ (y; )j CA(1 + jyj3) 3, jQe(y; )j C 2. Using lemma B.2 c), we obtain estimates for 1;2 and 1; . For 1;e, use jQ(y; )j C 2 and do as for 1;e. IIi): For s0, use lemma B.3. For = s0, we have from (25) ~ q2(y; s0) = s0 ( cos[ log( s0 )] sin[ log( s0 )])(1 (s0)f0( y ps0 )) where and (s0) are given by (26). It follows easily that ~ q2;0(s0) = 0, j~ q2;?(y; s0)j Cs 2 0 (1 + jyj3) and j~ q2;e(y; s0)j Cs 1 0 s1=2 0 . Apply b) of lemma B.3 to conclude. 124Blow-up results for vector-valued nonlinear equationsIIii): we have by lemma B.1 jV2;1(y; )j Cj j and jV2;1(y; )j Cj j 1(1+jyj2). If Q(y; ) = V2;1(y; )~q1(y; ), thenjQ0( )j Cj jA2s 3 log s, jQ?(y; )j Cj jAs 2 andjQe(y; )j Cj jA2s 1=2.Using lemma B.3 b) yields the conclusion.IIiii): Using lemmas B.4 and lemma B.3 a), we do the same as for Iiii).IIiv): Same estimates as Iiv).IIv): By lemma 3.1, we have jdds ( )j C 2. Using lemma B.3 a) andintegrating over [ ; s] yields the conclusion. Bibliography125Bibliography[1] Ball, J., Remarks on blow-up and nonexistence theorems for nonlinearevolution equations, Quart. J. Math. Oxford 28, 1977, pp. 473-486.[2] Berger, M., and Kohn, R., A rescaling algorithm for the numerical cal-culation of blowing-up solutions, Comm. Pure Appl. Math. 41, 1988, pp.841-863.[3] Bricmont, J., and Kupiainen, A., Renormalization group and nonlinearPDEs, Quantum and non-commutative analysis, past present and futureperspectives, Kluwer (Boston), 1993.[4] Bricmont, J., and Kupiainen, A., Universality in blow-up for nonlinearheat equations, Nonlinearity 7, 1994, pp. 539-575.[5] Filippas, S., and Kohn, R., Re ned asymptotics for the blowup of utu = up, Comm. Pure Appl. Math. 45, 1992, pp. 821-869.[6] Filippas, S., and Merle, F., Modulation theory for the blowup of vector-valued nonlinear heat equations J. Di erential Equations 116, 1995, pp.119-148[7] Galaktionov, V., A., Kurdyumov, S., P., and Samarskii, A., A., On ap-proximate self-similar solutions for some class of quasilinear heat equa-tions with sources, Math. USSR-Sb 52, 1985, pp. 155-180.[8] Galaktionov, V., A., and Vazquez, J., L., Regional blow-up in a semi-linear heat equation with convergence to a Hamilton-Jacobi equation,SIAM J. Math. Anal., 24, 1993, pp. 1254-1276.[9] Giga, Y., and Kohn, R., Asymptotically self-similar blowup of semilinearheat equations, Comm. Pure Appl. Math. 38, 1985, pp. 297-319.[10] Giga, Y., and Kohn, R., Characterizing blowup using similarity variables,Indiana Univ. Math. J. 36, 1987, pp. 1-40.[11] Giga, Y., and Kohn, R., Nondegeneracy of blow-up for semilinear heatequations, Comm. Pure Appl. Math. 42, 1989, pp. 845-884.[12] Hamilton, R., S., The formation of singularities in the Ricci ow, Sur-veys in di erential geometry, Vol. II, Internat. Press, Cambridge, 1995,pp. 7-136. 126Blow-up results for vector-valued nonlinear equations[13] Herrero, M.A, and Velazquez, J.J.L., Blow-up behavior of one-dimen-sional semilinear parabolic equations, Ann. Inst. H. Poincar e Anal. NonLin eaire 10, 1993, pp. 131-189.[14] Herrero, M.A, and Velazquez, J.J.L., Flat blow-up in one-dimensionalsemilinear heat equations, Di erential Integral Equations 5, 1992, pp.973-997.[15] Levermore, C., D., and Oliver, M.,The complex Ginzburg-Landau equa-tion as a model problem, Dynamical systems and probabilistic me-thods in partial di erential equations (Berkeley, 1994), Lectures in Appl.Math., 31, Amer. Math. Soc., Providence, RI, 1996, pp. 141-190.[16] Levine, H., Some nonexistence and instability theorems for solutions offormally parabolic equations of the form Put = Au+F (u), Arch. Rat.Mech. Anal. 51, 1973, pp. 371-386.[17] Merle, F., Solution of a nonlinear heat equation with arbitrary givenblow-up points, Comm. Pure Appl. Math. 45, 1992, pp. 263-300.[18] Merle, F., and Zaag, H., Stability of the blow-up pro le for equations ofthe type ut = u+ jujp 1u, Duke Math. J. 86, 1997, pp. 143-195.[19] Velazquez, J.J.L., Classi cation of singularities for blowing up solutionsin higher dimensions Trans. Amer. Math. Soc. 338, 1993, pp. 441-464.[20] Weissler, F., Single-point blowup for a semilinear initial value problem,J. Di erential Equations 55, 1984, pp. 204-224.The author is supported by a grant from the M.E.S.R (Laboratoire d'analysenum erique, Paris VI).AddressD epartement de Math ematiques, Universit e de Cergy-Pontoise, 2 avenue Adol-phe Chauvin, Pontoise, 95302 Cergy-Pontoise cedex, France.D epartement de Math ematiques et Informatique, Ecole Normale Sup erieure, 45rue d'Ulm, 75 230 Paris cedex 05, France. Chapitre 4Reconnection of vortexwith the boundary andnite time Quenching 128Reconnection of vortex with the boundary and quenchingReconnection of vortex with the boundaryand nite time Quenching yFrank MerleInstitute for Advanced Study and Universite de Cergy-PontoiseHatem ZaagEcole Normale Superieure and Universite de Cergy-PontoiseAbstract: We construct a stable solution of the problem of vortex reconnectionwith the boundary in a superconductor under the planar approximation. That is asolution of@h@t = h+ ehH0 1hsuch that h(0; t) ! 0 as t ! T . We give a precise description of the vortex near thereconnection point and time.We generalize the result to other quenching problems.Mathematics subject Classi cation: 35K, 35B40, 35B45Key words: quenching, blow-up, pro le1 Introduction1.1 The physical motivation and resultsWe consider a Type II superconductor located in the region z > 0 of thephysical space R3 . Under some conditions, the magnetic eld develops a particu-lar type of line singularity called vortex (see Chapman, Hunton and Ockendon[5] for more details and discussion). In general, a vortex is not situated in aplane, but under some reasonable physical conditions, the planar approxima-tion is relevant. In this case, a vortex line at time t 0 can be viewed asL(t) = f(x; y; z) = (x; 0; h(x; t))jx 2 g where = ( 1; 1) or = R, andh > 0 is a regular function. The physical derivation gives that h(x; t) satis esthe following equation: ht = hxx + ehH0 F0(h)(I)where H0 is the applied magnetic eld assumed to be constant, F0 is a regularfunction satisfying F0(k) 1k and F 00(k)1k2 as k ! 0:(1)We assume :i) In the case where = R8>><>>: F0(k)Ce 2kas k ! +1jF 00(k)jCe 2kas k ! +1h(x; t)a1x+ b1 as x! +1h(x; t)a2x+ b2 as x! 1(2)where a1 > 0 and a2 > 0. For simplicity, we take b1 = b2 = 0 and a1 = a2.yArticle paru dans Nonlinearity 10, 1997, pp. 1497-1550. Introduction129ii) In the case where = ( 1; 1),h(1; t) = h( 1; t) = 1:(3)One can remark that boundary conditions of the type i) are closer to the physicalcontext. Nevertheless, boundary conditions of the type ii) are mostly consideredin the literature in order to simplify the mathematical approach of the problem.Similar results can be shown with other types of boundary conditions (mixedboundary conditions on bounded domains). Indeed, our analysis will be localand therefore will not depend on boundary conditions.Classical theory gives for any initial vortex line L(0) = f(x; 0; h0(x))jx 2 gwhere h0 is positive, regular and satis es boundary conditions, the existence anduniqueness of a solution to (I)-(2) and (I)-(3) locally in time. Therefore, thereexists a unique solution to (I) on [0; T ) and either T = +1 or T < +1 and inthis case limt!T infx2 h(x; t) = 0, i.e. h extinguishes in nite time, and if x0 2 issuch that there exists (xn; tn) ! (x0; T ) as n ! +1 satisfying h(xn; tn) ! 0as n! +1, then x0 is an extinction point of h.This phenomenon is called a vortex reconnection with the boundary (theplane z = 0). Two questions arise:Question 1: Are there any initial data such that T < +1?Question 2: What does the vortex look like at the reconnection time?Equation (I) with a more general exponent can also appear in various phy-sical contexts (combustion for example), and the problem of reconnection isknown as the quenching problem.Indeed, we consider ht = h F (h);h 0(II)where(H1) F 2 C1(R + ); F (k) 1k and F 0(k) k +1 as k ! 0with > 0 and h is de ned on a bounded domainRN with boundarycondition h 1 on @ . The case = RN can also be considered with hypothesis(H1) and (H2) where(H2) jF (k)j+ jF 0(k)jCe k as k ! +1h(x; t)a1jxj as jxj ! +1Few results are known on equation (II). For > 0, some criteria of quenchingare known for solutions de ned on ( 1; 1) with Dirichlet boundary conditions(or mixed boundary conditions) in dimension one (see Deng and Levine [6], Guo[12], Levine [18]). Even in that case, few informations are known on the solutionat quenching except on the quenching rate (See also Keller and Lowengrub [17]for formal asymptotic behavior). In particular, there is no answer to questions1 and 2 for problem (I).To answer questions 1 and 2, we will not use the classical approach whichconsists in nding a general quenching criterion for initial data and in studyingthe quenching behavior of the solution. As in [22] and [25], the techniques weuse here are the reverse: we study the quenching behavior of a solution a priori, 130Reconnection of vortex with the boundary and quenchingand using this information, we prove by a priori estimates the existence of asolution which has all the properties we expect. Using this type of approach, weprove then that this behavior is stable. Let us rst introduce:̂(z) = ( + 1 + ( + 1)24 jzj2)1=( +1);(4)and H x0(x) de ned by:i) In the case = RN : H x0(x) = H (x x0) where H is de ned by:H (x) = h ( +1)2jxj28 log jxj i 1+1 for jxj C(a1; )H (x) = a1jxjfor jxj 1H (x) > 0; jrH (x)j > 0 for x 6= 0 and H 2 C1(RN ):(5)ii) In the case where is bounded:Hx0(x) = h ( +1)2jx x0j28 log jx x0j i 1+1 for jx x0j min C( );14d(x0; @ )H x0(x) = 1for jx x0j 12d(x0; @ )H x0(x) > 0; jrH (x)j > 0for x 6= x0 and H x0 2 C1( nfx0g):We also introduce H , the set to initial data:H = fk 2 +H1 \W 2;1(RN ) j 1=k 2 L1(RN )g if = RN(6)where 2 C1(RN ),0 for jxj 1, (x) = a1jxj for jxj 2 and a1 isde ned in (H2),H = fk 2 H1 \W 2;1( ) j 1=k 2 L1( )g if is bounded:(7) We claim the following:Theorem (Existence and stability of a vortex reconnection with theboundary or quenching for equation (II) with > 0)Assume that = RN and F is satisfying (H1) and (H2), or is bounded andF is satisfying (H1).1) (Existence)For all x0 2 , there exists a positive h0 2 H such that for aT0 > 0, equation (II) with initial data h0 has a unique solution h(x; t) on [0; T0)satisfying limt!T0 h(x0; t) = 0.Furthermore,i)limt!T0 k(T0 t)1= +1h(x0 + zp (T0 t) log(T0 t); t) 1̂(z)kL1 = 0;ii) h (x) = limt!T0 h(x; t) exists for all x 2 and h (x) H x0(x) as x! x0.2)(Stability) For every > 0, there exists a neighborhood V0 of h0 in Hwith the following property:for each ~h0 2 V0, there exist ~T0 > 0 and ~x0 satisfyingjT0 ~T0j+ jx0 ~x0j Introduction131such that equation (II) with initial data ~h0 has a unique solution ~h(x; t) on[0; ~T0) satisfying limt! ~T0 ~h(t; ~x0) = 0. In addition,limt! ~T0 k( ~T0 t)1= +1~h(~x0 + zq ( ~T0 t) log( ~T0 t); t)1̂(z)kL1 = 0;~h (x) = limt! ~T0 ~h(x; t) exists for all x 2 and ~h (x) H ~x0(x) as x! ~x0.Remark: In the case = 1 (equation (I)), this Theorem implies that thevortex connects with the boundary in nite time. Let us note that the pro lewe obtain is C1 (which is not true for > 1). Using the precise estimate of thebehavior of h at extinction, it will be interesting to check the validity of theplanar approximation in the physical problem near the reconnection time for abehavior like the one described in the theorem.Remark: We can also consider a larger class of equations:@h@t = r:(A(x)rh(x)) b(x)F (h)where F satis es (H1) and (H2) with > 0, A(x) is a uniformly elliptic N Nmatrix with bounded coe cients, b(x) is bounded, and b(x0) > 0.Using the stability result and techniques similar to [21], we can construct forarbitrary given k points in a quenching solution h of equation (II) whichquenches at time T exactly at the given points. The local quenching behaviorof h near each of these points is the same as the one given in the Theorem.Remark: We have two types of informations on the singularity:Part i): it describes the singularity in some re ned scale variable at x0 wherewe can observe the quenching dynamics. We point out that the estimate weobtain is global (convergence takes place in L1).Part ii): it describes the singularity in the original variables and shows itsin uence on the regular part of the solution.We see in the estimates that these two descriptions are related.In order to see why such a pro le is selected, see [22] and [25] for similar discus-sions.Remark: Part ii) is valid only for some extinction solutions. We suspect thiskind of extinction behavior to be generic (see [15] for a related problem). Indeed,we suspect ourselves to be able to show existence of extinction solutions of (I)-(2)such that:h(x; t)! h k(x)where h k(x) Cjxjk , k 2 N and k 2. Unfortunately, this kind of behavior issuspected to be unstable.1.2 Mathematical setting and strategy of the proofThe case = RN is di erent from the case is a bounded domain in theway how to treat the Cauchy problem outside the singularity.Let us consider the problem of the existence of a solution such that i) andii) of the Theorem hold. We rst note that once the existence result is proved, 132Reconnection of vortex with the boundary and quenchingthe stability result can be proved in the same way as in [22]. In order to provethe Theorem, we use the following transformation:u(x; t) =+1h(x; t)(8)where h is the extinction solution of (II) to be constructed, and > 0. On itsexistence interval [0; T ), u(t) satis es@u@t = u a jruj2u + f(u)(III)where a = a( ; ) = 1 + 1,f(u) = +1u1+ 1F ( 1+1u 1) = up + f1(u)(9)with p = p( ; ) = 1+ +,(H3) f1 2C1(R+ ); f1(v) = o(vp) and f 01(v) = o(vp 1) as v ! +11 < a < p;and in the case = RN ,(H4) ( jf(v)j+ jf 0(v)j Cv1+ 1exp(1+1 v 1) as v ! 0;u(x; t)1a1jxj as jxj ! +1Now, with the transformation ( ; ) ! (a( ; ); p( ; )), the problem of ndinga solution h of (II) such that limt!T infx2Rh(x; t) = 0 is equivalent to the problem ofnding a solution u of (III) such thatlimt!T ku(t)kL1 = +1;(that is a solution of (III) which blows-up in nite time).Problem (III) can be viewed as a gradient perturbation of the nonlinear heatequation (a = 0)@u@t = u+ jujp 1u(IV)where u(x; t) is de ned for x 2 RN , t 0, p > 1 and p < (N + 2)=(N 2) ifN 3.For this equation, Ball [1], Kavian [16] and Levine [20] obtained obstructionsto global existence in time, using monotony properties and the maximum prin-ciple. Another method has been followed by Merle and Zaag in [22] (see alsoGiga and Kohn [10], [9] and [8], Bricmont and Kupiainen [4], Zaag [25]). Oncean asymptotic pro le (that is a function from which, after a time dependentscaling, u(t) approaches as t ! T ) is derived formally, the existence of a solu-tion u(t) which blows-up in nite time with the suggested pro le is then provedrigorously, using analysis of equation (IV) near the given pro le and reductionof the problem to a nite dimensional one. Introduction133In the case a = 0, the existence and stability of a blow-up solution u(t) of(IV) such that at the blow-up point x0:limt!T k(T t) 1p 1u(x0 +p(T t) log(T t)z; t)0(z)kL1 = 0where0(z) = (p 1 + (p 1)24p z2) 1=(p 1)is proved in [22]. Bricmont and Kupiainen obtained the existence result usingrenormalization group theory (see [4]).In these new variables, and with the introduction of(z) = (p 1 + (p 1)24(p a) jzj2) 1p 1 :(10)andUx0(x) =+1Hx0(x) ;(11)= h 8(p a)j log jxjj(p 1)2jxj2 i 1p 1 if = RN , x0 = 0 and jxj C(a1; ), the Theorem isequivalent to the following Proposition:Proposition 1 (Existence of blow-up solutions for equation (III))Assume that = RN and f is satisfying (H3) and (H4), or is bounded andf is satisfying (H3).For each a 2 (1; p), for each x0 2 , there exist regular initial data u0 such thatequation (III) has a unique solution u(x; t) which blows-up at a time T0 > 0only at the point x0.Moreover,i) limt!T0 u(x; t) = u (x) exists for all x 2 nfx0g and u (x) Ux0(x) as x! x0.ii) limt!T0 (T0 t) 1p 1u(x0 + ((T0 t)j log(T0 t)j) 12 z; t) (z) L1 = 0:Remark: This proposition provides us with a blow-up solution of (III) in thecase a 2 (1; p). Let us remark that we already know that blow-up occurs in thecase a 1:If a < 1 and v = (1 a) 1 ap 1u1 a, then v satis es:@v@t = v + vp0 with p0 = p a1 a > 1:(12)If a = 1 and v = (p 1) logu, then v satis es@v@t = v + ev:(13)It is well-known that equations (12) and (13) (and then (III)) have blow-upsolutions.We introduce similarity variables (see [10], [8] and [9])):y = x x0pT t ; s = log(T t); wT;x0 (y; s) = (T t) 1p 1u(x; t);(14) 134Reconnection of vortex with the boundary and quenchingwhere x0 is the blow-up point and T the blow-up time of u(t), a blow-up solutionof (III) to be constructed (we will focus on the study of solutions that blow-upat one single point). We now assume x0 = 0.The study of the pro le of u as t ! T is then equivalent to the study of theasymptotic behavior of wT;x0 (noted w) as s!1, and each result for u has anequivalent formulation in terms of w. From equation (III), the equation satis edby w is the following: 8y 2 RN , 8slogT :@w@s = w 12y:rw wp 1 a jrwj2w + wp + e psp 1 f1(e sp 1w)(15)where f1(v) = f(v) vp and f satis es (H3) and (H4).The problem is then to nd w a solution of (15) such thatkw(y; s) ( yps )kL1 ! 0 as s! +1:We introduce'(y; s) = ( yps) + (p 1) 1p 12(p a)s and q(y; s) = w(y; s) '(y; s)(16)where is introduced in (10) (the introduction of the term (p 1) 1p 12(p a)s is notnecessary but it simpli es the calculations).Then q satis es: 8y 2 RN , 8slogT :@q@s = (L+ V (y; s))q +B(q) + T (q) +R(y; s) + e psp 1 f1(e sp 1 ('+ q))(17)with L =12y:r+ 1, V (y; s) = p'(y; s)p 1pp 1 ,B(q) = ('+ q)p 'p p'p 1q,T (q) = a jr'+rqj2'+q +a jr'j2' , R(y; s) = @'@s + ' 12y:r' 'p 1+'p a jr'j2' .Therefore, the question is to nd w a solution of (15) or q a solution of (17)such thatlims!1 kq(s)kL1 = 0:(18)The equation satis ed by q is almost the same as in [22], except the termT (q). As in [22], we introduce estimates on q in the blow-up region jzj K0 orjyjK0ps, and in the regular region jzj K0 or jyjK0ps where z = yps isthe self-similar variable for q. The estimates of T (q) in the region jyjK0psfollow from regularizing e ect of the heat ow. One can remark that the Cauchyproblem for an equation of the type @u@t = u jruj2 + up is suspected not tobe solved in H1 or W 1;p+1.In the analysis of [22], the estimates in the region jyjK0ps imply smallnessof q only, and do not allow any control of T (q) in this region. In other words,the analysis based on the method of [22], that is to estimate the solution in thez variable is not su cient and must be improved. For this, we add estimatesin three regions in a di erent variable scale (centered in the original x variablenot necessarily at the considered blow-up point) using techniques similar tothose used in [25] to derive the exact pro le in x variable: u(x; t) ! u (x) as Existence of a blow-up solution for equation (16)135t ! T where u (x) U (x) as x ! 0 (see (11) for U ). This part makes theoriginality of the paper. We expect that such techniques can be useful in varioussupercritical problems.We rst de ne for K0 > 0, 0 > 0 and t 2 [0; T ) given, three regions coveringRN : P1(t) = fx j jxjK0p (T t) log(T t)g= fx j jyjK0psg = fx j jzj K0g;P2(t) = fx j K04 p (T t) log(T t) jxj0g= fx j K04 ps jyj0e s2 g = fx j K04 jzj e s2psg;P3(t) = fx j jxj0=4g = fx j jyj04 e s2 g = fx j jzj e s2psg;for i = 1; 2; 3; Pi = f(x; t) 2 RN [0; T )jx 2 Pi(t)g;where s = log(T t), y = xpT t , z = yps =xp(T t)j log(T t)j .In P1, the \extinction region" of h (which is also the blow-up region of u),we make the change of variables (14) and (16) to do an asymptotic analysisaround the pro le (y=ps).Outside the singularity in region P2, we control h using classical parabolic esti-mates on k, a rescaled function of h de ned for x 6= 0 byk(x; ; ) = (T t(x)) 1+1h(x+pT t(x) ; (T t(x)) + t(x))where K04 p(T t(x))j log(T t(x))j = jxj . From equation (II), we see that ksatis es almost the same equation as h: 8 2 RN , 8 2 [ t(x)T t(x) ; 1):@k@ = k (T t(x))+1F ((T t(x)) 1+1 k)where (T t(x)) +1F ((T t(x)) 1+1 k)1k as (T t(x)) 1+1 k ! 0.We will in fact prove that h behaves for j j0pj log(T t(x))j and 2[ t0 t(x)T t(x) ; 1) for some t0 < T , like the solution of@k̂@ = 1̂k :In P3, the regular region, we estimate directly h. This will give the desiredestimate.The proof of the existence result of the Theorem will be presented in section2. Assuming some a priori estimates in P1, P2 and P3, we show in section 2that h(t) can be controlled near the pro le by a nite dimensional variable.Adjusting the nite dimensional parameters, we then conclude the proof. Wepresent a priori estimates in P1 in section 3, and in P2 and P3 in section 4.The authors thank R. Kohn who pointed out various references on this pro-blem. Part of this work was done while the second author was visiting theInstitute for Advanced Study. 136Reconnection of vortex with the boundary and quenching2 Existence of a blow-up solution for equation(16)In this section, we give the proof of the existence result of the Theorem. Theproof will be given in the case = RN (we will mention the di erences withthe case is bounded, when it is necessary, see section 4). We assume N = 1in order to simplify the notations. The same calculations and proof hold in ahigher dimension (see [22] and [25]). We assume x0 = 0 since (II) is translationinvariant. For simplicity in notations, we simplify hypothesis (H1) and assumethat8v 2 (0; 1]; F (v) = 1v :(19)Same calculations holds without this simpli cation.Let us rst remark on the following about the Cauchy problem for equation(II).Lemma 2.1 (Local Cauchy Problem for equation (II)) The local in timeCauchy problem for equation (II) is well-posed in H where H is de ned by (7)if is bounded, and by (6) if = R.Moreover, in both cases, either the solution h exists for all time t > 0 oronly on [0; T ) with T < +1, and in this case limt!T infx2 h(x; t) = 0.Proof: The case is bounded follows from classical arguments.For the case = R, we de ne ~h(x; t) by h(x; t) = (x) + ~h(x; t). This way,(II) is equivalent to ~ht =~hxx F ( (x) + ~h) + xx:(20)Using (H1) and (H2), we see by classical arguments that this equation canbe solved in H .Let us consider > 0 and T > 0, all xed. The problem is to nd t0 < Tand h0 such that the solution of equation (II) with data at t0 h(x; t0) = h0extinguishes in nite time T > 0 at only one extinction point x = 0 and:limt!T k(T t)1= +1h(zp (T t) log(T t); t) 1̂(z)kL1(R) = 0(21)h (x) = limt!T h(x; t) exists for all x 2 R andh (x) > 0 for x 6= 0; h (x) H (x) as x! 0(22)where ̂ and H are introduced in (4) and (5).As explained in the introduction, (21) and (22) follow from the control ofh(x; t) for t 2 [t0; T ) in three di erent scales, depending on the three regionsP1, P2, and P3.a) In P1, the extinction region, we rescale h by means of (8), (14) and (16)in order to de ne for t 2 [t0; T ), q(s) where s = log(T t) and8>>>><>>>>: 8y 2 R;q(y; s) = (T t) 1p1u(ypT t; t) '(y; s);8x 2 R;u(x; t) =+1h(x; t) and > 0;'(y; s) = ( yps ) + (p 1) 1p 12(p a)s ;p = + +1; a = +1;and is given in (10):(23) Existence of a blow-up solution for equation (16)137Remark: To prove the Theorem, we can take = 1. Nevertheless, we needto keep > 0 general, if we want to deduce directly Proposition 1 from theTheorem.The equation satis ed by q is (17): 8y 2 R, 8slog(T t0):@q@s = (L+ V (y; s))q +B(q) + T (q) +R(y; s) + e psp 1 f1(e sp 1 ('+ q))(24)with L =12y:r+ 1, V (y; s) = p'(y; s)p 1pp 1 ,B(q) = ('+ q)p 'p p'p 1q,T (q) = a jr'+rqj2'+q +a jr'j2' , R(y; s) = @'@s + '12y:r' 'p 1+'p a jr'j2' ,f1(u) =+1u1+ 1F ( 1+1u 1) up.We note that L is self-adjoint on D(L) L2(R; d ) withd (y) = e jyj24p4(25)and that its eigenvalues are f1m2 jm 2 Ng.In one dimension, hm(y) = [m2 ]Xn=0 m!n!(m 2n)! ( 1)nym 2n is the eigenfunctioncorresponding to 1m2 . We introduce also km =hm=khmk2L2(R;d ) and notethat Vect fhm j m 2 Ng is dense in L2(R; d ).We are interested in obtaining L1(R) estimates for q. Since L1(R)L2(R; d ), we will expand q (actually, a cut-o of q) with respect to the ei-genvalues of L. Nevertheless, the estimates we will obtain will be L1 and notL2(R; d ).The control of h(t) for t 2 [t0; T ) in this region P1 is equivalent to the controlof q(s) for s 2 [ log(T t0);+1) in a set VK0;A(s) so that lims!1 kq(s)kL1 = 0.The de nition of VK0;A(s) requires the introduction of a cut-o function(y; s) = 0( jyjK0ps)(26)where0 2 C1(R+ ; [0; 1]); 0 1 on [0; 1]; 0 0 on [2;+1):(27)b) In P2, we control a rescaled function of h de ned for x 6= 0 by 8 2 R,8 2 [ t0 t(x)T t(x) ; 1):k(x; ; ) = (T t(x)) 1+1h(x+pT t(x) ; (T t(x)) + t(x));(28)where t(x) is de ned byjxj = K04 p(T t(x))j log(T t(x))j = K04 p (x)j log (x)j(29)with(x) = T t(x):Let us note that (x) is related to the asymptotic pro le H (x). 138Reconnection of vortex with the boundary and quenchingLemma 2.2 For xed K0, we have:i) H (x) k̂(1) (x) 1+1 as x! 0,ii) jrH (x)j8( +1)K0 k̂(1)pj log (x)j (x) 1+1 12 as x! 0 wherek̂( ) = (( + 1)(1 ) + ( + 1)24 K2016 ) 1+1 :(30)Proof: From (29), we write:log jxj = log K04 + 12 log (x) +12 log j log (x)j andjxj2log jxj = 2K2016 (x)log (x)log (x)+log j log (x)j+2 log K04 . Therefore,log (x) 2 log jxj and (x) 8K20 jxj2j log jxjj as x! 0:(31)Since H (x) = k̂(1) h 8jxj2K20 j log jxjji 1+1 andjrH (x)j4p2( +1)K0 k̂(1)pj log jxjj h 8jxj2K20 j log jxjji 1+1 12 when x is small (see (5)), weget the conclusion.k satis es almost the same equation as h: 8 2 [ t0 t(x)(x) ; 1), 8 2 R,@k@ = k (x)+1F ( (x) 1+1 k):(32)We will see that the estimates on k allow us to write (x)+1F ( (x) 1+1 k) = 1kfor suitable . If we show that k( ) behaves like k̂ (see (30)) which is a solutionof the ODEdk̂d = 1̂kde ned for 2 [0; T̂ ) with T̂ = 1+ ( +1)K2064 > 1, and that jr k( )jC(K0;A)pj log (x)j ,then according to lemma 2.2, this yields that h(x; t) behaves in P2 like H (x)and jrh(x; t)j C(K0; A)jrH (x)j if x and T t0 are small, which is almostthe estimate ii) of the Theorem.c) In P3, we estimate directly h using the local in time well posedness of theCauchy problem for equation (III).More formally, we de ne for each t 2 [t0; T ) a set S (t) depending on someparameters so that h(t) 2 S (t) means that h is controlled in the three regionsas described before. We show then that if 8t 2 [t0; T ), h(t) 2 S (t), then (21)and (22) hold and the Theorem follows.Let us de ne S (t):De nition of S (t) and SI) For all t0 < T , K0 > 0, 0 > 0, 0 > 0, A > 0, 0 > 0, C 00 > 0, C0 > 0and 0 > 0, for all t 2 [t0; T ), we de ne S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t) asbeing the set of all functions h 2 H satisfying:i) Estimates in P1: q(s) 2 VK0;A(s) where s = log(T t), q(s) is de nedin (23) and VK0;A(s) is the set of all functions r in W 1;1(R) such that8<: jrm(s)jAs 2 (m = 0; 1); jr2(s)jA2s 2 log s;jr (y; s)jAs 2(1 + jyj3); jre(y; s)jA2s 1=2j(@r@s )?(y; s)jAs 2(1 + jyj3);(33) Existence of a blow-up solution for equation (16)139wherere(y; s) = (1 (y; s))r(y);r (s) = P ( (s)r);for m 2 N; rm(s) = R d km(y) (y; s)r(y); r?(s) = P?( (s)r);(34)is de ned in (26), P and P? are the L2(R; d ) projectors respectively onVect fhmjm 3g and Vect fhmjm 2g, d , hm and km are introduced in (25).ii) Estimates in P2: For all jxj 2 [K04 p(T t)j log(T t)j; 0],= (x; t) = t t(x)(x) , and j j0pj log (x)j,jk(x; ; ) k̂( )j0, jr k(x; ; )jC00pj log (x)j , and jr2 k(x; ; )j C0where k, k̂, t(x) and (x) are de ned in (28), (30) and (29).iii) Estimates in P3: For all jxj04 , jh(x; t) h(x; t0)j0 and jrh(x; t)rh(x; t0)j0.II) For all t0 < T we de ne S (t0;K0; 0; 0; A; 0; C 00; C0; 0) =fk 2 C([t0; T ); H) j 8t 2 [t0; T ); k(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t)g.Remark: Note that according to (25) and (34), we have for all r 2 L1(R),r(y) = 2Xm=0 rm(s)hm(y) + r (y; s) + re(y; s);(35)r(y) = 1Xm=0 rm(s)hm(y) + r?(y; s) + re(y; s):(36)Therefore, i) yields an estimate onkq(s)kL1 and k @q@y ? (s)kL1 .Remark: The estimates on h are in W 1;1(R). In particular, they are global.The estimates on @q@y in P1, r k in P2 and on rh in P3 allow us to controlthe term T (q) appearing in the equation satis ed by q (see (24)). We remarkthat the estimate q(s) 2 VK0;A(s) describes h mainly in P1. The estimate on qeinvolved in de nition (33) is useful only in the frontier between P1 and P2.Now we show that if we nd suitable parameters and initial data such thath 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0), then the Theorem holds.Proposition 2.1 (Reduction of the proof) For given t0 < T , K0, 0, 0,A, 0, C 00, C0 and 0 such that 0 12 k̂(1) and 012 infjxj0=4h(x; t0), assumethat h 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0). Then h(t) extinguishes in nite timeT only at the point x0 = 0, that is limt!T h(0; t) = 0 and 8x 6= 0, there exists(x) > 0 such thatlim inft!T minjx0 xj (x)h(x0; t) > 0:(37)Moreover, with ̂ and H de ned by (4) and (5),limt!T k(T t) 1+1h(zp (T t) log(T t); t)1̂(z)kL1(R) = 0;(38) 140Reconnection of vortex with the boundary and quenchingh (x) = limt!T h(x; t) exists for all x 2 R andh (x) > 0 for x 6= 0 and h (x) H (x) as x! 0:(39)Proof: We assume that h 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0). One can remarkthat once (38), (37) and (39) are proved, it follows thati) limt!T h(0; t) = 0: h(t) extinguishes at time T at the point x = 0,ii) x = 0 is the only extinction point of h.It remains then to prove (37), (38) and (39).Proof of (37):From iii) of De nition of S (t), we know that if jxj04 , then 8t 2 [t0; T ),h(x; t) h(x; t0)0 inf jxj 04 h(x; t0)012 inf jxj 04 h(x; t0)) > 0. Thisyields (37) for jxj0.From ii) of De nition of S (t), we have 8jxj 2 (0; 0], for t close enough toT , jk(x; 0; (x; t)) ĥ( (x; t))j0 where (x; t) = t t(x)(x) . Therefore,k(x; 0; (x; t)) k̂( (x; t)) 0 k̂(1) 0 12 k̂(1) (from (30) and 0 12 k̂(1)).From (28), it follows: h(x; t) 12 k̂(1) (x) 1+1 > 0. This yields (37) for 0 < jxj <0. Proof of (38):We consider q(s), the function introduced in (23). Let us show thatkq(s)kL1(R)! 0 as s! +1:(40)From i) of the de nition of S (t) and (35), we have 8s 2 [ log(T t0);+1),q(s) 2 VK0;A(s) andjq(y; s)j = j1fjyj2K0psg 2Xm=0 qm(s)hm(y) + q (y; s)!+ qe(y; s)j1fjyj2K0psg(As 2(1+jyj)+A2s 2 log s(jyj2+2)+As 2(1+jyj3))+A2s 1=2C(K0; A)s 1=2 and (40) follows.Let z 2 R and g(z) = j(T t)1= +1=h(zp (T t) log(T t); t)1̂(z) j. Wehaveg(z)Cj(T t) +1+1h(zp (T t) log(T t); t)+1 ̂(z) j 1where  = max( ; 1).Using (4) and (23), we have = 1=(p a) and = (p a)=(a 1), therefore+1 = 1p 1 ,+1 ̂(z) = ( + 1+ ( + 1)24 jzj2) 1p 1 = '(zps; s) (p 1) 1=(p 1)2(p a)s ;and (T t) +1+1h(zp (T t) log(T t); t)= (T t) 1p1u(zp (T t) log(T t); t) with s = log(T t).Combining this with (23) again, we getg(z) C( ; ) jq(zp log(T t); log(T t))j+ 1=j log(T t)j 1Ckq(s)kL1(R)+ 1=j log(T t)j 1! 0 as t! T by (40). This yields (38). Existence of a blow-up solution for equation (16)141Proof of (39): From the proof of (37) and classical theory (see Merle [21]for a similar problem), there exists a pro le function h (x) such that 8x 6= 0,limt!T h(x; t) = h (x) > 0. To show that h (x) H (x) as x ! 0, we give thefollowing localization estimate:Proposition 2.2 (Localization in P2) Assume that k is a solution ofequationk = k 1k(41)for 2 [0; 0) with 0 1(< T̂ ). Assume in addition: 8 2 [0; 0],i) For j j 2 0, jk( ; 0) k̂(0)j and jrk( ; 0)j ,ii) For j j 7 04 , k( ; ) 12 k̂( ).iii) For j j 7 04 , jr2k( ; )j C0,where k̂ is introduced in (30). Then there exists = ( ; 0) such that 8 2 [0; 0],for j j0,jk( ; ) k̂( )j and jrk( ; )j , where ! 0 as ! 0 and 0 ! +1.Proof: We prove in section 4 a more accurate version of this Proposition (Propo-sition 4.1). One can adapt without di culties the proof to the present context.Let us apply this Proposition to k(x; ; ) when x is near zero with 0 = 1and 0 = j log (x)j1=4. We rst check all the hypothesizes of the Proposition:Lemma 2.3 If x is small enough, then k(x; ; ) satis es (41) forj j j log (x)j1=4 and 2 [0; 1). Moreover,i)supj j j log (x)j1=4 jk(x; ; 0) k̂(0)j+ jr k(x; ; 0)j (x)! 0 as x! 0;(42)ii) for j j j log (x)j 14 , 8 2 [0; 1), k(x; ; )12 k̂( ),iii) for j j j log (x)j 14 , 8 2 [0; 1), jr2 k(x; ; )j C0.Combining this lemma and Proposition 2.2, we get 8 2 [0; 1), jk(x; ; )k̂( )j (x)! 0 as x! 0. Using (28), (30) and letting ! 1, we obtain(x) 1+1h (x) k̂(1) = ( + 1)2K20641+1 :(43)By lemma 2.2, we obtain h (x) H (x) as x ! 0, which concludes the proofof Proposition 2.1.Proof of lemma 2.3:i) and iii): Since (29) implies that (x)! 0 as x! 0, we have by combining(38) and (28):supj j j log (x)j1=4 j1=k(x; ; 0) 1=̂( x+ p (x)p (x)j log (x)j )j ! 0 as x ! 0. Hence, from(4), the rst part of (42) follows.From ii) of the De nition of S (t), we have jr k(x; ; 0)jC00pj log (x)j andjr2 k(x; ; 0)j C0 for j j j log (x)j1=4, if x is small. This yields the secondpart of i) and iii). 142Reconnection of vortex with the boundary and quenchingii): From ii) of the De nition of S (t), it follows that for x small en-ough, we have jk(x; ; ) k̂( )j0 for j j j log (x)j1=4 and 2 [0; 1).Hence, ii) follows from (30) since 012 k̂(1). By the way, this implies thatj (x) 1+1 k(x; ; )j 1 for j j j log (x)j1=4 and 2 [0; 1). Therefore, it followsfrom (32) and (19) that k satis es (41).From this Proposition, the proof of the Theorem reduces to nd suitableparameters t0 < T , K0, 0, 0, A, 0, C 00, C0, 0 and h0 2 H so that thesolution h of equation (II) with data h(t0) = h0 belongs toS (t0;K0; 0; 0; A; 0; C 00; C0; 0).Unfortunately, the spectrum of L which greatly determines the dynamicof q (and then the dynamic of h too) contains two expanding eigenvalues: 1and 1=2. Therefore, we expect that for most choices of initial data h0, thecorresponding q0(s) and q1(s) with s = log(T t) will force h(t) to exitS (t0;K0; 0; 0; A; 0; C 00; C0; 0; t).As a matter of fact, we will show through a priori estimates that for sui-tably chosen t0 < T , K0, 0, 0, A, 0, C 00, C0 and 0, the control of h(t)in S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t) for t 2 [t0; T ) reduces to the control of(q0(s); q1(s)) inV̂A(s) [ As 2; As 2]2(44)for slog(T t0) (q0(s) and q1(s) correspond to expanding eigenvaluesin the q variable). Hence, we will consider initial data h0 depending on twoparameters (d0; d1) 2 R2 , and then, we will x (d0; d0) using a topological ar-gument so that (q0(s); q1(s)) 2 V̂A(s) for all slog(T t), which yieldsh(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t), thanks to the nite dimensional re-duction.Let us de neh0(d0; d1; x) = (T t0) 1+11+1 (z) + (d0 + d1z) 0( jzjK0=16)11(x; t0)+H (x)(11(x; t0))(45)where z = x=p(T t0)j log(T t0)j,1(x; t0) = 0x(T t0) 12 j log(T t0)j p2 ;(46), 0 and H are de ned in (10), (27) and (5). The problem now reduces tond (d0; d1) in some D R2 such thath(d0; d1) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0).The proof is divided in two parts:i) Finite dimensional reduction:From the technique of a priori estimates, we nd suitable parameters t0 < T ,K0, 0, 0, A, 0, C 00, C0 and 0 so that the following property is true: Assumethat for t 2 [t0; T ), we have 8t 2 [t0; t ],h(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t) andh(t ) 2 @S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t ), then Existence of a blow-up solution for equation (16)143(q0(s ); q1(s )) 2 @V̂A(s ) where s = log(T t ), q0 and q1 follow from q by(34), q and V̂A(s) are de ned in (23) and (44).ii) Solution of the nite dimensional problem:We use a topological argument to nd a parameter (d0; d1) 2 R2 such that(q0(s); q1(s)) 2 V̂A(s) for all slog(T t0), and therefore,h 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0). This yields the Theorem.Part I: A priori estimates of h(t), solution of equation (II) andnite dimensional reductionStep 0: Initialization of the problemWe claim the following lemma:Lemma 2.4 (Initialization of the problem) There exists K01 > 0 suchthat for each K0 K01 and 1 > 0, 9 1(K0; 1) > 0 and C (K0) > 0 such that8 01(K0; 1), 9 1(K0; 1; 0) > 0, such that 8 01(K0; 1; 0), 8C1 > 0,8A 1, 9t1(K0; 1; 0; A; C1) < T such that 8t0 2 [t1; T ), there exists a rectangleD(t0;K0; A) R2 with the following properties:If h(x; t0) is de ned by (45), then:i) 8(d0; d1) 2 D(t0;K0; A), h(t0) 2 H de ned in (6), (q0(s0); q1(s0)) 2V̂A(s0) de ned in (44) and h(t0) 2 S (t0;K0; 0; 0; A; 1; C (K0); C1; 0; t0),with s0 = log(T t0). More precisely:jq0(s0)jAs 20jq1(s0)jAs 20jq2(s0)js 20 log s0 jq (y; s0)jCs 20 (1 + jyj3)jqe(y; s0)j s 1=20j @q@y ? (y; s0)js 20 (1 + jyj3);j @q@y (y; s0)j s 120for jyjK0ps0;for all jxj 2 [K04 p(T t)j log(T t)j; 0], 0 = t0 t(x)(x) , andj j 20pj log (x)j, jk(x; ; 0) k̂( 0)j1, jr k(x; ; 0)jC (K0)pj log (x)j andjr2 k(x; ; 0)j C1 where k, k̂, t(x) and (x) are de ned in (28), (30) and(29). ii)(d0; d1) 2 D(t0;K0; A), (q0(s0); q1(s0)) 2 V̂A(s0);(d0; d1) 2 @D(t0;K0; A), (q0(s0); q1(s0)) 2 @V̂A(s0);(q0(s0); q1(s0)) is an a ne function of (d0; d1) when (d0; d1) 2 @D(t0;K0; A).Proof: See Appendix A.Step 1: A priori estimatesWe now claim the following estimates:Proposition 2.3 (A priori estimates in P1) There exists K02 > 0 such thatfor each K0 K02, there exists A2(K0) > 0 such that for each A A2(K0),0 > 0 and C 00 A3, there exist 2( 0) > 0 and t2(K0; 0; A; C 00) < T such thatfor each t0 2 [t2(K0; 0; A; C 00); T ), 012 k̂(1), 0 > 0, C0 > 0 and 02( 0),we have the following property:if h(x; t0) is given by (45) and if (d0; d1) is chosen so that(q0(s0); q1(s0)) 2 V̂A(s0) de ned in (44) with s0 = log(T t0), 144Reconnection of vortex with the boundary and quenchingif for some t 2 [t0; T ), we have8t 2 [t0; t ], h(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t) thenjq2(s )jA2s 2log s s 3; jq (y; s )jA2 s 2(1 + jyj3)jqe(y; s )jA22 s 1=2;j( @q@y )?(y; s )jA2 s 2(1 + jyj3);where s = log(T t ), q is de ned in (23) and the notation is given in (34).Proof: See section 3.Proposition 2.4 (A priori estimates in P2) There exists K03 > 0 such thatfor all K0 K03, 1 1, 0 1, C 0 > 0, C 00 > 0 and C 000 > 0 we have thefollowing property:Assume that k is a solution of equation@k@ = k 1k(47)for 2 [ 1; 2) with 012 1 (< T̂ ).Assume in addition: 8 2 [ 1; 2],i) 8 2 [ 2 0; 2 0], jk( ; 1) k̂( 1)j1 and jrk( ; 1)j C0000 ,ii) 8 2 [ 7 04 ; 7 04 ], jrk( ; )j C000 and jr2k( ; )j C 0 ,iii) 8 2 [ 7 04 ; 7 04 ], k( ; ) 12 k̂( ),where k̂ is given by (30). Then, for 003(C 00 ; C 0 ; C 000 ) there exists =(K0; C 00 ; 1; 0) such that 8 2 [ 0; 0], 8 2 [ 1; 2],jk( ; ) k̂( )j and jrk( ; )j 2C0000 , where ! 0 as ( 1; 0)! (0;+1).Proof: See section 4.Proposition 2.5 (A priori estimates in P3) For all > 0, 0 > 0, 0 > 0,and 1 > 0, there exists t4( ; 0; 0; 1) < T such that 8t 2 [t4; T ), if h is asolution of (II) on [t0; t ] for some t 2 [t0; T ) satisfyingi) for jxj 2 [ 06 ; 04 ], 8t 2 [t0; t ],0 h(x; t)1; jrh(x; t)j1 and jr2h(x; t)j1;(48)ii) h(x; t0) = H (x) for jxj06 where H is de ned by (5),then for jxj 2 [ 04 ;+1), 8t 2 [t0; t ],jh(x; t) h(x; t0)j+ jrh(x; t) rh(x; t0)j :Proof: See section 4.Step 2: Finite dimensional reductionFrom Propositions 2.3, 2.4 and 2.5, we have the following:Proposition 2.6 (Finite dimensional reduction) We can choose parame-ters t0 < T , K0, 0, 0, A, 0, C 00 and C0 and 0 such that the following pro-perties hold: Assume that h(x; t0) is given by (45) and (d0; d1) 2 D(t0;K0; A).Then,i) h(t0) 2 H \ S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t0). Existence of a blow-up solution for equation (16)145Assume in addition that for some t 2 [t0; T ), we have 8t 2 [t0; t ],h(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t) andh(t ) 2 @S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t ) thenii) (q0(s ); q1(s )) 2 @V̂A(s ) where q is de ned in (23) and s = log(T t ).iii) (Transversality) there exists 0 > 0 such that 8 2 (0; 0),(q0(s + ); q1(s + )) 62 V̂A(s + ) (henceh(t + ) 62 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t + )).Proof: We proceed in two steps: we rst show that we can x K0, 0 and C0independently from A, take A A7 and choose 0, 0, C 00, 0 and t0 in terms ofA, so that i) and ii) hold. In the second step, we x A and t0 so that iii) holdstoo.Proof of i) and ii)It follows from the following lemma:Lemma 2.5 There exist constants K0, 0, C0, and A7 > 0 such that for allA A7, there exist 0(A) > 0, 0(A), C 00(A), 7(A) and t7(A) < T such thatfor all 07 and t0 2 [t7; T ), and under the hypotheses of Proposition 2.6, i)and ii) hold.ProofLet us rst choose suitably the constants, and then show that i) and ii) ofProposition 2.6 follow for this choice.All the constants we are referring to below appear either in lemma 2.4 orPropositions 2.3, 2.4 or 2.5.We proceed in ten steps:i) Fix K0 = 4max(K01;K02;K03).ii) Fix 0 = 14 min(k̂(1); 1) (note that k̂(1) depends only on K0). Fix C0 = 1. LetA7(K0) be large enough so that A7 max(1; A2(K0)) and for all A A7(K0),A3 C 00(A) where we introduceC 00(A) = 4maxC3A2K30 +kr̂kL1(B(0;2K0)); 20k̂(1)( +1)K0 ; C (K0) withC (K0) de ned in lemma 2.4 and C3 a constant which is independent of all theparameters and appears in lemma 2.6.Consider A any number larger than A7(K0), and consider C 00(A).iii) Applying Proposition 2.4 with K0, C 0 = 2, C 00 (A) = 2C 00(A) and C 000 (A) =14C 00(A), we get 0(A) 1 and 1(A) 1 such that for all 00 and 11 ,the conclusion of the Proposition holds with = 02 .iv) Let 1(A) = min(12 1(A); 0) and C1 = 12 .v) We claim the following lemma:Lemma 2.6 8A A7, there exist 5(K0; 1(A)) > 0 such that for all 05,there exists 5( 0; A) > 0 such that for all 05( 0; A), there are t5( 0; A) < Tand 5( 0; A) > 0 such that for all 05( 0; A) and t0 2 [t5( 0; A); T ),if for all t 2 [t0; t ], h(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t) for some t 2[t0; T ), then we have for jxj 2 [K04 p(T t )j log(T t )j; 0]:i) For j j74 0pj log (x)j and for all 2 [max(0; t0 t(x)(x) ); t t(x)(x) ]:k(x; :; :) satis es (47) and, jr k(x; ; )j2C00(A)pj log (x)j , jr2 k(x; ; )j 2C0 andk(x; ; ) 12 k̂( ). 146Reconnection of vortex with the boundary and quenchingii) For j j 20pj log (x)j and = max( t0 t(x)(x) ; 0): jk(x; ; ) k̂( )j1and jr k(x; ; )jC00(A)4pj log (x)j .Proof: We focus on the proof of the fact that for jxj 2 (0; 0],for j j74 0pj log (x)j, for t 2 [max(0; t(x)); T ), we havejr k(x; ; )j2C 00(A)pj log (x)j(49)where = t t(x)(x) , and : for jxj 2 (0; 0], for j j 20pj log (x)j,jk(x; ; 0(x)) k̂( 0(x))j1(50)and jr k(x; ; 0(x))j14C 00(A)pj log (x)j(51)where 0(x) = max( t0 t(x)(x) ; 0).The other estimates follow by similar techniques.Let > 0 to be xed later. If 07(K0; ) for some 7(K0; ) > 0, thenwe have from (29): for j j 20pj log (x)j,(1 )jxj jx+ p (x)j (1 + )jxj:(52)Proof of (49):From (28), we haver k(x; ; ) = (x) 1+1+ 12rh(x+ p (x); t):(53)Let us denote x+ p (x) by X and distinguish three cases:Case where jX j K04 p(T t)j log(T t)j:From (8), we write rh(X; t) = C ruu1+ 1(X; t).From i) of the De nition of S (t), we getj(T t) 1p 1u(X; t) (Xp(T t)j log(T t)j )j= jq( XpT t ; log(T t)) + 2(p a)j log(T t)j jCA2K30pj log(T t)j by lemma B.1. Mo-reover,jru(X; t) (T t) 1p 1 12 j log(T t)j12r (Xp(T t)j log(T t)j )j =(T t) 1p 1 12 jrq( XpT t ; log(T t))j(T t) 1p 1 12 j log(T t)j12CA2K30 (see the proof of lemma B.1)Hence, by (9), we obtain:j(T t) 1+1+12rh(X; t) j log(T t)j 12r̂(Xp(T t)j log(T t)j)jC3A2K30pj log(T t)j andjrh(X; t)jC3A2K30 +kr̂kL1(B(0;K0)) (T t) 11 12 j log(T t)j 12 .This gives by (53):jr k(x; ; )jT t(x) 1+1 12 j log(T t)j 12C 00(A). Existence of a blow-up solution for equation (16)147Since (1 )jxj jX j (see (52)) and jX jK0p(T t)j log(T t)j, we havejxjK04(1 )p(T t)j log(T t)j.From (29), we have jxj ! (x) is an increasing function. Therefore,(x) ( K04(1 )p(T t)j log(T t)j)8K20 K20 (T t)j log(T t)j16(1 )2 12 j log(T t)j = (T t)(1 )2 by (31).Moreover, we have t t(x), therefore, T t (x). Hence,jr k(x; ; )j 2C 00(A)j log (x)j 12 if is small enough.Case where jX j 2 [K04 p(T t)j log(T t); 0]:We write rh(X; t) = (X) 1+1 12r k(X; 0; t t(X)(X) ). This gives by (53):r k(x; ; t) = (X)(x)1+1 12 r k(X; 0; t t(X)(X) ).From ii) of the De nition of S (t), we obtain:jr k(x; ; )j C 00(A)j log (x)j 12(X) 1+1 12 j log (X)j 12(x) 1+1 12 j log (x)j 12 .Using (52) and taking small enough, this yieldsjr k(x; ; )j 2C 00(A)j log (x)j 12 .Case jX j0: If 0minjx0j 0 jrh(x0; t0)j, then we have from iii) of theDe nition of S (t):jrh(X; t)j (1 + )jrh(X; t0)j (1 + )jrh( x; t0)j where = 1 if > 1and = 1 + if1 (see (52)).From lemma 2.2, we get:jrh(X; t)j (1 + ) 10k̂(1)( +1)K0 ( x) 1+1 12 j log ( x)j 12 .Arguing as before, we obtain from (53):jr k(x; ; )j20k̂(1)( +1)K0 j log (x)j 12 2C 00(A)j log (x)j 12 if is smallenough. This concludes the proof of (49).Proof of (50):If jxjK04 p(T t0)j log(T t0)j, then (29) yields t(x) t0 and 0(x) =t0 t(x)(x) . Hence, (50) follows from lemma 2.4.If jxj K04 p(T t0)j log(T t0)j, then t(x) t0 and 0(x) = 0. From (28) and(30), we let X = x+ p (x) and write:jk(x; ; 0) k̂(0)j = j (x) 1+1h(X; t(x)) ( + 1) + ( +1)24 K20161+1 j I + IIwhere I = j (x) 1+1h(X; t(x)) ( + 1) + ( +1)24jXj2(X)j log (x)j 1+1 jand II = j ( + 1) + ( +1)24jXj2(X)j log (x)j 1+1 ( + 1) + ( +1)24 K20161+1 j.From i) of the De nition of S (t), (23) and the fact thatjX j (1 + )jxj (1+ )K04 p (x)j log (x)jK0p (x)j log (x)j, we getI CA2K30 j log (x)j 12CA2K30 j log(T t0)j 12 , sincejxj K04 p(T t0)j log(T t0)j. Now, if T t0 is small enough, then I12 .From (52) and (29), we have (1 )2K2016jXj2(X)j log (X)j (1 + )2K2016 . Hence,if is small enough, we obtain II12 .This concludes the proof of (50).Proof of (51):If jxjK04 p(T t0)j log(T t0)j, then (29) yields t(x) t0 and 0(x) = 148Reconnection of vortex with the boundary and quenchingt0 t(x)(x) . Hence, lemma 2.4 yields: for j j 20pj log (x)j,jr k(x; ; 0(x))j C (K0)j log (x)j 12 14C 00(A).If jxj K04 p(T t0)j log(T t0)j, then t(x) t0 and 0(x) = 0. With X =x + p (x), we write: r k(x; ; 0) = (x) 1+1+ 12rh(X; t(x)). Arguing as forthe rst case in the proof of (49), we get:jr k(x; ; 0)jhC3A2K30 +kr̂kL1(B(0;K0))i j log (x)j 1214C 00(A)j log (x)j 12 .This concludes the proof of (51) and the proof of lemma 2.6.vi) We now x 0(A) = min(12 1(K0; 1(A)); 5(K0; 1(A)); 1). We also x0(A) min( 1(K0; 1(A); 0(A)); 5( 0(A); A)) such that0(A)pj log ( 0)j0(A).vii) Then, we take 7(A) =12 min( 2( 0(A)); 5( 0(A); A)) and consider 07.viii) By direct parabolic estimates, it is easy to see that there exists t6(A) < Tsuch that for all t0 2 [t6; T ), ifh(t0) 2 S (t0;K0; 0; 0; A; 0; C 00; C1; 0; t0) and 8t 2 [t0; t0],h(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t), thenh(t0) 2 S (t0;K0; 0; 0; A; 0; C 00; 34 ; 0; t0).ix) Let 0(A) =12 k̂(1) ( 06 ) 1+1 and 1(A) = max(32 k̂(0) ( 04 ) 1+1 ;C 00 ( 06 ) 1+1 12pj log ( 06 )j ; C 00 ( 04 ) 1+1 12pj log ( 04 )j ; C0 ( 06 ) 1+11).x) Let t7(A) = max(t1(K0; 1(A); 0(A); A; C1); t2(K0; 0(A); A; C 00(A));t4( 02 ; 0; 0; 1); t5( 0(A); A); t6(A)), and consider t0 an arbitrary number in[t7(A); T ).Now, we show that i) and ii) of Proposition 2.6 hold for this choice. Let usassume that h(t0) is given by (45) and (d0; d1) 2 D(t0;K0; A). Then, lemma2.4 applies and h(t0) 2 H \ S (t0:K0; 0: 0; A; 1; C (K0); 0; t0). Since 10,C (K0) C 00 and 0 < 0, i) follows.We now assume that in addition, we have 8t 2 [t0; t ],h(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t) andh(t ) 2 @S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t ) for some t 2 [t0; T ). Accordingto the De nition of S (t), three cases may occur:Case 1: q(s ) 2 @VK0;A(s ). From ii) of lemma 2.4, Proposition 2.3 andi) of the De nition of S (t), we have (q0(s ); q1(s )) 2 @V̂A(s ) which is i) ofProposition 2.6.Case 2: There exist x and such thatjxj 2 [K04 p(T t )j log(T t )j; 0] and j j0pj log (x)j, and eitherjk(x; ; 1) k̂( 1)j = 0 or jr k(x; ; 1)j =C00pj log (x)j or jr2 k(x; ; 1)j = C0 =1, where 1 = t t(x)(x) < 1.According to viii) and lemma 2.4, we have jr2 k(x; ; )j34 . Let 0 =max( t0 t(x)(x) ; 0) and 0 =0pj log (x)j. Note that 00pj log ( 0)j0 .Since 0 1, it follows from lemma 2.6:For j j 20pj log (x)j, jk(x; ; 0) k̂( 0)j1 and jr k(x; ; 0)jC00(A)4pj log (x)j C00(A)4 0 . Existence of a blow-up solution for equation (16)149For j j74 0pj log (x)j and for all 2 [ 0; 1]: k(x; :; :) satis es (87) andjr k(x; ; )j 2C00(A)0 , jr2 k(x; ; )j 2C0 and k(x; ; ) 12 k̂( ).Applying Proposition 2.4 yields:For j j0pj log (x)j, jk(x; ; 1) k̂( 1)j02 and jr k(x; ; 1)j2 14C00(A)pj log (x)j< C00(A)pj log (x)j , which contradicts the hypotheses of Case 2.Case 3: There exists x 2 R such that jxj04 and jh(x; t ) h(x; t0)j = 0or jrh(x; t ) rh(x; t0)j = 0. From ii) of the De nition of S(t), we have:8t 2 [t0; t ], for jxj 2 [ 06 ; 04 ]: jk(x; 0; ) k̂( )j0, jr k(x; 0; )jC00pj log (x)jand jr2 k(x; 0; )j C0, where = t t(x)(x) . Using (28) and the fact that 012 k̂(1)12 k̂(0), we obtain:12 k̂(1) (x) 1+1 h(x; t)32 k̂(0) (x) 1+1 , jrh(x; t)j C 00 (x) 1+1 12pj log (x)j andjr2h(x; t)j C0 (x) 1+11. Therefore, 0(A) h(x; t)1(A),jrh(x; t)j1 andjr2h(x; t)j1. From (45), we have h(x; t0) = H (x)for jxj06 . Hence, Proposition 2.5 applies and we get: jh(x; t) h(x; t0)j +jrh(x; t) rh(x; t0)j02 < 0, which contradicts the hypotheses of Case 3.This concludes the proof of i) and ii) of Proposition 2.6.Proof of iii):Let us recall that K0, 0 and C0 are xed independently of A, where A istaken larger than some A7 > 0, 0, 0 and C 00 are xed in terms of A, andt0 2 [t7(A); T ), 07(A), for some t7(A) < T . and 7(A) > 0. Let us provethis lemma:Lemma 2.7 There exists A8 > 0 such that for all A A8, there exist t8(A) 0.From (24) and (34), we have: R (s )@q@s (s )kmd = R (s )Lq(s )kmd +R (s ) hV (s )q(s ) +B(q) + T (q) +R(s ) + e psp 1 f1(e sp 1 ('+ q))i kmd .If we take t0 2 [t11(K0; 0(A); A; 0; C 00); T ) and 011( 0(A)), then we getfrom lemma 3.2 (see section 3):jdqmds (s ) (1m2 )qm(s )j C6s2for some C6 independent from all the other constants. Since qm(s ) = As 2,we have dqmds (s ) > 0 for A 4C6.Conclusion of the proof: If we take A = max(A7; A8) and0 = min( 7(A); 8(A); 12 minjxj 04 h(x; t0)) ( minjxj 04 h(x; t0) > 0 according to (45)and (5)), and t0 = max(t7(A); t8(A)), then both i) and ii) of Proposition 2.6hold. This concludes the proof of Proposition 2.6. Let us note that with thischoice, the reduction of the proof of Proposition 2.1 holds. 150Reconnection of vortex with the boundary and quenchingPart II: Topological argumentFrom Proposition 2.6, we claim that there exist (d0; d1) 2 D(t0;K0; A) suchthat h(d0; d1) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0). The proof is similar to theanalogous one in [22], let us give its main ideas.We proceed by contradiction: From i) of Proposition 2.6, we have8(d0; d1) 2 D(t0;K0; A),h(d0; d1; t0) 2 H \ S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t0). Therefore, we de nefor each (d0; d1) 2 D(t0;K0; A) a time t (d0; d1) as being the in nitum of allt 2 [t0; T ) such thath(d0; d1; t) 62 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t). By ii) of Proposition 2.6, wehave(q0; q1)(d0; d1; s (d0; d1)) 2 @V̂A(s (d0; d1)) where s = log(T t ).Hence, we can de ne from (44) the following function:: D(t0;K0; A) ! @C(d0; d1)! s(d0;d1)2A (q0; q1)(d0; d1; s (d0; d1))where C is the unit square of R2 .Now we claimProposition 2.7 i) is a continuous mapping from D(t0;K0; A) to @C.ii) There exists a non trivial a ne function g : D(t0;K0; A) ! C such thatg 1j@C = [email protected]: The proof is very similar to the proof of Proposition 3.3 in [22], that isthe reason why we give only the important arguments.i) follows from the continuity in H of the solution h(t) at a xed time t withrespect to initial data, and the transversality property iii) of Proposition 2.6.From ii) of lemma 2.4, we have 8(d0; d1) 2 @D(t0;K0; A), s (d0; d1) = s0and ii) follows.From Proposition 2.7, a contradiction follows (Index Theory). Therefore,there exist (d0; d1) 2 D(t0;K0; A) such thath(d0; d1) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0). By Proposition 2.1 and the Con-clusion of the proof of Proposition 2.6, the main Theorem follows.3 A priori estimates of u(t) in the blow-up zoneThis section is devoted to the proof of Proposition 2.3. Let us consider t0 0, there exists s2(A) > 0 such that for each s0s2(A) and K0 > 20, if h(x; t0) is given by (45) and (d0; d1) is chosen so that(q0(s0); q1(s0)) 2 V̂A(s0), thenjq2(s0)j s 20 log s0; jq (y; s0)j Cs 20 (1 + jyj3);jqe(y; s0)j s 1=20 ; jr?(y; s0)j s 20 (1 + jyj3); 152Reconnection of vortex with the boundary and quenchingand jr(y; s0)j s 1=20 for jyjK0ps0.Proof: The proof is included in the proof of lemma 2.4: See the end of its Step2. Now we consider s0 and 2 [0; ]. We suppose that8t 2 [T e ; T e ( + )] h(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t). Thenwe give bounds on terms in right hand sides of equations (54), (55) and (56),expanded as in (34).Remark: In fact, we give in lemma 3.2 estimates on equation (54) projectedon hm with m = 0; 1 or 2. Only m = 2 is useful for the proof of Proposition 2.3.The estimates for m = 0 or 1 are in a large part the same, they are useful forthe proof of Proposition 2.6.Lemma 3.2 There exists K11 > 0 and A11 > 0 such that for each K0 K11,0 > 0, A A11, > 0, C 00 > 0, there exists t11(K0; 0; A; ; C 00) with thefollowing property:8t0 2 [t11(K0; 0; A; ; C 00); T ), 8 2 [0; ], for all 012 k̂(1), 0 > 0,C0 > 0 and 011( 0) for some 11( 0) > 0, assume thath(x; t0) is given by (45) and (d0; d1) is chosen so that (q0(s0); q1(s0)) 2V̂A(s0)for somelog(T t0), we have 8t 2 [T e ; T e ( + )] h(t) 2S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t). Then, 8s 2 [ ; + ],I) Equation (54): If m = 0; 1 or 2,j Z (y;s)km(y)@q@s (y; s)d q0m(s)je s(57) j Z (y; s)km(y)Lq(y; s)d (1m2 )qm(s)je s(58)j Z (y; s)km(y)V (y; s)q(y; s)d js 5=2(59)j Z (y; s)km(y)B(q)(y; s)d jCs 3(60)j Z (y; s)km(y)T (q)(y; s)d js 2 1=4(61)j Z (y; s)km(y)R(y; s)d jCs 2(62)j Z (y; s)km(y)e psp 1 f1(e sp 1 ('+ q))d je s:(63)If m = 2, then we have more precisely:j Z (y; s)k2(y)V (y; s)q(y; s)d + 2ps(p a)q2(s)jCAs 3(64)j Z (y; s)k2(y)T (q)(y; s)d2as(p a)q2(s)jCAs 3(65)j Z (y; s)k2(y)R(y; s)d jCs 3(66)II) Equation (55): A priori estimates of u(t) in the blow-up zone153Case s0:j (y; s)jC(Ae (s )=2 +A2e (s)2)s 2(1 + jyj3)(67)j e(y; s)jC(A2e (s )=p +AK30es )s 1=2(68)where (s) = K(s; )q( ) is expanded as in (35),j (y; s)jC(s )s 2(1 + jyj3)(69)j e(y; s)j(s )s 1=2(70)where (s) = R sd K(s; ) (B(q( )) + T (q( ))),j (y; s)jC(s )s 2(1 + jyj3)(71)j e(y; s)jCK30 (s )es s 1=2(72)(73)where (s) = R sd K(s; )R( ) is expanded as in (35),j (y; s)jC(s )s 2(1 + jyj3)(74)j e(y; s)jC(s )s 1=2(75)where (s) = R sd K(s; )e psp 1 f1(e p 1 ('+ q)) is expanded as in (35).Case = s0: More precisely,j (y; s)jCs 2(1 + jyj3)(76)j e(y; s)jCK30es s 1=2:(77)III) Equation (56):Case s0:jP?( (s)K1(s; )r( ))j C(Ae (s )2 + C(K0)C 00e (s)2)1 + jyj3s2(78)jP?( (s) Z sd K1(s; ) @@y (B(q) + T (q))( ))j C(s )1=2 1 + jyj3s2(79)jP?( (s) Z sd K1(s; )R1( ))j C(s )1 + jyj3s2(80)jP?( Z sd K1(s; )e (@'@y + r)f 01(e p 1 ('+ q))j(81)C(s ) 1+jyj3s2 where P? is de ned in (34).Case = s0: More precisely,jP?( (s)K1(s; )r( ))j Cs 2(1 + jyj3):(82)Proof: See Appendix B.Step 2: Lemma 3.2 implies Proposition 2.3Let K0 K02 > 0, 0 > 0, A A2(K0) > 0 where A2(K0) will be xedlater, and C 00 A3. Let t0 > 0 to be xed in [t2(K0; 0; A; C 00); T ) (wheret2(K0; 0; A; C 00) will be de ned later). Consider 0 12 k̂(1), 0 > 0, C0 > 0 and 154Reconnection of vortex with the boundary and quenching02( 0). Let h(d0; d1) be a solution of equation (II) with initial data (45)de ned on [t0; t ] with t 2 [t0; T ), such that(d0; d1) is chosen so that (q0(s0); q1(s0)) 2 V̂A(s0) (s0 = log(T t0) and q isde ned by (23)),8t 2 [t0; t ], h(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t) andq(s ) 2 @VK0;A(s ).We want to show thatjq2(s )jA2s 2log s s 3; jq (y; s )jA2 s 2(1 + jyj3)jqe(y; s )jA22 s 1=2;jr?(y; s )jA2 s 2(1 + jyj3)(83)wherer(y; s) = @q@y (y; s):We consider 1(K0; A)2(K0; A) two positive numbers (which will be xedlater in terms of K0 and A). The conclusion follows if we treat Case 1 wheres s01 and then Case 2 where s s02. The proof relies stronglyon estimates of lemma 3.2. Therefore, we suppose K0 K11, A A11, C 00A3, t0 max(t11(K0; 0; A; 1; C 00); t11(K0; 0; A; 2; C 00)), s0 = log(T t0)max( 1; 2), 0 > 0, 0 12 k̂(1), C0 > 0 and 011( 0).Case 1: s s01(K0; A)We apply lemma 3.2 with A, = 1, = s s0 and = s0.From equation (54) with m = 2, we obtain: 8s 2 [s0; s ],jq02(s) + 2s1q2(s)j CAs 3 + 2e s CAs 3. Therefore, 8s 2 [s0; s ],j dds (s2q2(s))j CAs 1, and then, using s2s0 (indeed, s = s0 +s0 + 1 2s0), we obtain jq2(s )j s 2s20jq2(s0)j + 2A(s s0)s 3. Usingjq2(s0)j s 20 log s0 which follows from lemma 3.1, we get jq2(s )j s 2log s +CA(s s0)s 3. Together with estimates concerning equations (55) and (56) inlemma 3.2, we obtain:jq2(s )js 2log s + 2C1As 2jq (y; s )jC1(1 + s s0)s 2(1 + jyj3)jqe(y; s )jC1K30es s0(1 + s s0)s 1=2jr?(y; s )jC1(1 + (ss0)1=2 + (s s0))s 2(1 + jyj3)2C1(1 + s s0)s 2(1 + jyj3):To have (83), it is enough to have1 A22 ; 2C1(1 + s s0)A2 ; andC1K30es s0(1 + s s0) A22(84)on one hand and2C1As 2A22 log ss2 s 3(85)on the other hand.If we restrict 1 to satisfy 2C1(1 + 1) A=2 andC1K30e 1(1 + 1) A2=2(which is possible with 1 = 3=2 logA for A A6(K0) large enough), then (84)is satis ed, since s s01. Now if s0 s6(A), then (85) is satis ed. Thus(83) is satis ed also. This concludes Case 1. A priori estimates of u(t) in the blow-up zone155Case 2: s s02(K0; A)We apply lemma 3.2 with A, = = 2 and = s2. From equation(54) with m=2, we obtain 8s 2 [ ; s ], jq02(s) + 2s1q2(s)j CAs 3. Using thesame argument as Case 1 and jq2( )j A2 2 log , and then estimates onequation (55) and (56), we obtain:jq2(s )jA2s 2log(s2) + 2C2A 2s 3jq (y; s )jC2(Ae 2=2 +A2e 22 + 2)s 2(1 + jyj3)jqe(y; s )jC2(A2e 2=p +AK30e 2+K30 2e 2)s 1=2jr?(y; s )jC2(Ae 2=2 + C(K0)C 00e 22 +21=2 + 2)s 2(1 + jyj3):Since C 00 A3, in order to obtain (83), it is enough to havefA;2(s )0C2(Ae 2=2 +A2e 22 + 2)A2C2(A2e 2=p +AK30e 2+K30 2e 2)A22C2(Ae 2=2 +C(K0)A3e 22 +21=2 + 2)A2(86)with fA; 2(s ) = A2s 2log s s 3A2s 2log(s2) + 2C2A 2s 3.We now x 2 so thatC2K30Ae 2 = A2=8, i.e. 2 = logA=(8C2K30 ) . Then,the conclusion follows if A is large enough. Indeed, for all A > 1, we writejfA;log A8C2K30 (s ) s 3A2 log A8C2K30 2C2A log A8C2K30 1 jA2(log A8C2K30 )2s2(s log A8C2K30 )2 :Then we take A A7(K0; C 00) for some A7(K0) such thatA2 log A8C2K30 2C2A log A8C2K30 11C2(A( A8C2K30 ) 1=2 +A2e (log A8C2K30 )2 + log A8C2K30 )A2C2(A2( A8C2K30 ) 1=p +AK30 A8C2K30 +K30 log A8C2K30 A8C2K30 )A22C2(A( A8C2K30 ) 1=2 +C(K0)A3e (log A8C2K30 )2 + (log A8C2K30 )1=2+ log A8C2K30 )A2 .Afterwards, we take s0 s7(K0; A) so that 8s s0,A2(log A8C2K30 )2s 2(s log A8C2K30 ) 2 s 3=2.This way, (86) is satis ed for A A7(K0) and s0 s7(K0; A).This concludes Case 2.We remark that for A A8(K0), we have 1 = 32 logA log A8C2K30 = 2.If we take now K02 = K11, A2(K0) = max(A11; A6(K0); A7(K0); A8(K0))and t2 = max(t11(K0; 0; A; 1(A); C 00); T e 1(A);t11(K0; 0; A; 2(K0; A); C 00); T e 2(K0;A); T e s6(A); T e s7(K0;A)),2( 0) = 11( 0), then we conclude the proof of Proposition 2.3. 156Reconnection of vortex with the boundary and quenching4 A priori estimates in P2 and P3In this section, we estimate directly the solutions of equation (II).4.1 Estimates in P2Let us recall that k̂( ) = ( + 1)(1 ) + ( +1)24 K20161+1 and that it isde ned for 2 [0; T̂ ] with T̂ > 1.Proposition 4.1 There exists K03 > 0 such that for all K0 K03, 1 1,0 1 and C 0 > 0, C 00 > 0, C 000 > 0 we have the following property:Assume that k is a solution of equation@k@ = k 1k(87)for 2 [ 1; 2) with 012 1 (< T̂ ). Assume in addition: 8 2 [ 1; 2],i) 8 2 [ 2 0; 2 0], jk( ; 1) k̂( 1)j1 and jrk( ; 1)j C0000 ,ii) 8 2 [ 7 04 ; 7 04 ], jrk( ; )j C000 and jr2k( ; )j C 0 ,iii) 8 2 [ 7 04 ; 7 04 ], k( ; )12 k̂( ). Then, for 003(C 00 ; C 0 ; C 000 ) thereexists = (K0; C 00 ; 1; 0) such that 8 2 [ 0; 0], 8 2 [ 1; 2],jk( ; ) k̂( )j and jrk( ; )j 2C0000 , where ! 0 as ( 1; 0)! (0;+1).Proof: We can assume 1 = 0 and 2 = 0 1.Step 1: Gradient estimateLemma 4.1 Under the assumptions of Proposition 4.1, we have8 2 [ 5 04 ; 5 04 ], 8 2 [0; 0] jrk( ; )j 2C0000 , if 003(C 00 ; C 0 ; C 000 ).Proof: We have 8 2 [ 2 0; 2 0], 8 2 [0; 0],@@ rk = (rk) + rkk +1 :From iii), we have for j j7 04 , j 1k +1 j 1 for K0 large. If = jrhj2, then,by a direct calculation, 2@k@@k@and+ C for jxj 7 04 . Letus consider 1 2 C1(RN ) such that 1(x) = 1 for jxj3 02 , 1(x) = 0 forjxj 7 04 , 01 1, jr 1j C0 and j 1j C20 . Then, 1 = 1 satis es thefollowing inequality:11 2r 1:r1 + C 11 + C(C 00 ; C 0 ) 201f 3 02 jxj 2 0g + C 1. With 2 = e C 1, we have22 + C(C 00 ; C 0 ) 201f 3 02 jxj 2 0g and 02(0) C 000 220 :Therefore, by the maximum principle, 8 2 [ 5 04 ; 5 04 ],8 2 [0; 0], ( ; )C000 220 + C(C 00 ; C 0 )2 20 e C0 20 . Hence, for j j5 04 , 8 2 [0; 1], jrk( ; )jC0000 +C(C00 ;C 0 )0 e C0 20 2C0000 , if 003(C 00 ; C 0 ; C 000 ), which yields the conclu-sion. A priori estimates in P2 and P3157Step 2: Estimates on kWe are now able to conclude the proof of Proposition 4.1.Lemma 4.2 For j j0, 8 2 [0; 0], we have jk( ; ) k̂( )j , where ! 0as 0 ! +1 and 1 ! 0.Proof: Let us consider k1 a solution of equation (87) such that 8 2 [ 2; 2],8 2 [0; 0], jk1( ; 0) k̂(0)j1, jrk1( ; )j . Let us show that for j j 2,8 2 [0; 0], jk1(0; ) k̂( )j C(K0) + 1 where C(K0) is independent from .We have 8 2 [0; 0], k1(0; ) = 1jB2(0)j Rj j 2 k1( ; )dx + k2( ), and1k1(0; ) = 1jB2(0)j Rj j 2 1k1( ; ) d + k3( ), where jB2(0)j is the volume of thesphere of radius 2 in RN ,kk2kL1 2 andkk3kL1 C .In the distribution sense, for small enough, considering~k( ) = 1jB2(0)j Rj j 2 k1( ; )d , we have1~k C d~kd1~k + Cand j~k(0) k̂(0)j C + 1.Together with (87), we obtain by classical a priori estimates that 8 2 [0; 0],j~k( ) k̂( )j C(K0) + 1 (since C1 jk̂( )j C 01(K0)) and therefore 8j j 2,8 2 [0; 0], jh1(0; ) ĥ( )j C(K0) + 1. Applying this result to h1( ; ) =h( ; x0) for x0 2 [ 0+2; 0 2], from the assumption and step 1 we obtainlemma 4.2.Lemmas 4.1 and 4.2 yield Proposition 4.1.4.2 Estimates in P3We claim the followingProposition 4.2 For all > 0, 0 > 0, 0 > 0, and 1 > 0, there existst4( ; 0; 0; 1) < T such that 8t 2 [t4; T ), if h is a solution of (II) on [t0; t ] forsome t 2 [t0; T ) satisfyingi) for jxj 2 [ 06 ; 04 ], 8t 2 [t0; t ],0 h(x; t)1; jrh(x; t)j1 and jr2h(x; t)j1;(88)ii) h(x; t0) = H (x) for jxj06 where H is de ned by (5), then for jxj 2[ 04 ;+1), 8t 2 [t0; t ],jh(x; t) h(x; t0)j+ jrh(x; t) rh(x; t0)j :Proof:Let us obtain the estimates on h for jxj04 . The estimates on rh can beobtained similarly. We argue by contradiction. Let us consider t 2 (t0; t ) suchthat8t 2 [t0; t ); kh(x; t) h(x; t0)kL1(jxj 04 )(89)and kh(x; t ) h(x; t0)kL1(jxj 04 ) = .We can assume14 minjxj 06 H (x). We can remark that (5) implies that 158Reconnection of vortex with the boundary and quenchingjh(x; t0)j = H (x) > C0( 0) > 0 for jxj06 , therefore, we havejF (h(x; t))j C( 0) for jxj06 and t 2 [t0; t ).From assumption i), we have in fact 8t 2 [t0; t ], for 06 jxj04 , h(x; t)0 > 0 and jF (h(x; t))j C( 0). We then consider h1(x; t) = 1(x)h(x; t) where1 2 C1(RN ; [0; 1]), 1 1 for jxj05 , 1 0 for jxj06 , jr 1j C0 andj 1j C20 . We then have:@h1@t = h1 2r 1:rh1h1F (h):Since 8t 2 [t0; t ], j2r 1:rhj+ j 1hj C( 0; 1)1f 06 jxj 05 g(x), we write@h1@t = h1 + ~f1(x; t)1F (h)with j ~f1(x; t)j C( 0; 1)1f 06 jxj 05 g(x).Let us now consider the case of a bounded domain and the case = RN ,since there is a small di erence in the proof.i) is a bounded domain:In this case,8t 2 [t0; t ), h1(t) S(t t0)h1(t0) = R tt0 dsS(t s)[ ~f1(x; t)1F (h)] whereS(:) is the linear heat ow. Hence,jh1(t) h1(t0)jL1 jh1(t) S(tt0)h1(t0)jL1 + jS(t t0)h1(t0) h1(t0)jL1R tt0 ds[jS(t s) ~f1(s)jL1 + jS(t s)C( 0; 0) 1F (h)jL1 ]+jS(t t0)h1(t0) h1(t0)jL1R tt0 ds[ dspt s j ~f1(s)jLN + jS(t s)C( 0; 0)1f gjL1 ]+jS(t t0)h1(t0) h1(t0)jL1C( 0; 0; 1)pt t0 + jS(t t0) 1H1H jL1 .Now, if t0 2 [t5( ; 0; 0; 1); T ), then we have jh1(t )h1(t0)jL1 2 , which isa contradiction with (89).Therefore, 8t 2 [t0; t ] jh(x; t) h(x; t0)jL1(jxj 04 ) .ii) Case = RN : we de ne h2(x; t) = (x)+h1(x; t) where (x) is introducedin the introduction (such that 2 C1(RN ),0 on [ 1; 1], (x) = a1jxj forjxj 2). From the fact that @h2@t = h2 + F (h2(x) + (x)) + and that forjvj 1, jF (v)j+ jF 0(v)j Ce v , we obtain using similar techniques:8t 2 [t0; t ), jh2(x; t) h2(x; t0)jL1 or equivalently: 8t 2 [t0; t ),jh1(x; t) h1(x; t0)jL1 . This concludes the proof of Proposition 4.2.A Proof of lemma 2.4We must show that for suitable (d0; d1) 2 R2 , the estimates of the De nitionof S (t) hold for t = t0. Since estimate iii) holds obviously, we show in arst step that h(t0) 2 H and estimate ii) holds, for all choices of (d0; d1),provided that t0 is near T . Then, in step 2, we nd D(t0;K0; A) such that8(d0; d1) 2 D(t0;K0; A), q(s0) 2 VK0;A(s0), where q is the function introducedin (23).Step 1: Estimate ii) of the De nition of S (t) Proof of lemma 2.4159Let us rst remark that from (45), (5) and (6), we have h(t0) 2 + H1 \W 2;1(R). Moreover, one can see from (45), (10), (27) and (5) that 8x 2 R,h(x; t0) C(t0; d0; d1; 0) > 0. Therefore, h(t0) 2 H .Let us consider t0 < T , K0, 0, 0, 1, and C1, and show that if theseconstants are suitably chosen, then for jxj 2 [K04 p(T t0)j log(T t0); 0] andj j 20pj log (x)j, we havejk(x; ; t0 t(x)(x) ) k̂ t0 t(x)(x) j1; j@k@ j C (K0)pj log (x)j ;(90)and jr2 kj C1 where k, k̂, t(x) and (x) are de ned in (28), (30) and (29).Let us rst introduce some useful notations:0 = T t0; r(t0) = K04 p 0j log 0j and R(t0) = 120 j log 0j p2 ;(91)and remark that thanks to (31), we have for xed K0:(r(t0))0; (R(t0)) 16K20 0j log0jp 1; (2R(t0)) 64K20 0j log0jp 1;log (r(t0)) log (R(t0)) log (2R(t0)) log 0 as t0 ! T:(92)If 0K016 and 023C(a1; ), then it follows from (29) that for jxj 2[r(t0); 0] and j j 20pj log (x)j, we have j p (x)j jxj2 andr(t0)2 jxj2 = jxj jxj2 jx+ p (x)j 32 jxj32 0 C(a1; ):(93)Therefore, we get from (28), (45), and (27): for jxj 2 [r(t0); 0] and j j20pj log (x)j, k(x; ; t0 t(x)(x) ) =(I) 1(x + p (x); t0) + (II)(11(x+ p (x); t0))(94)with (I) =0(x) 1+1 ̂( x+ p (x)p 0j log 0j ) and (II) = (x) 1+1H (x+ p (x)).Estimate on k:By linearity and (46), it is enough to prove that for jxj 2 [r(t0); 2R(t0)] andj j 20pj log (x)j,(I) k̂ t0 t(x)(x)12(95)and for jxj 2 [R(t0); 0] and j j 20pj log (x)j,(II) k̂ t0 t(x)(x)12 :(96)We begin with (95). From (4) and (30), we have:(I) k̂ t0 t(x)(x) = j ( + 1)( 0(x)) + ( +1)24 jx+ p (x)j2(x)j log 0j1+1( + 1)( 0(x)) + ( +1)24 K20161+1 j Cj jx+ p (x)j2(x)j log 0j K2016 j 1+1 160Reconnection of vortex with the boundary and quenchingCK 2+10 jq log (x)log 0 +4K0pj log 0j j2 1 1+1 .Since jxj 2 [r(t0); R(t0)] and j j 20pj log (x)j, we haveq log( (R(t0)))log 0 (1 4 0K0 ) 2 1 q log (x)log 0 +4K0plog 0 2 1s log( (r(t0)))log 0 (1 + 4 0K0 )!2 1:(97)From (97) and (92), we nd 5(K0; 1) and t5(K0; 1) < T such that 8 05,8t0 2 [t5; T ),j(I) k̂ t0 t(x)(x) j CK 2+10 jjq log (x)log(T t0) +4K0pj log(T t0)j j2 1j 1+112 .Now, we treat (96). Let jxj 2 [R(t0); 0] and j j 20pj log (x)j. We havefrom (94), (5), (29) and (30),(II) = ( +1)2jx+ p (x)j28 (x)j log jx+ p (x)jj 1+1 = ( +1)2jK04 pj log (x)j+ j28 j log jx+ p (x)jj1+1 and(II) k̂ t0 t(x)(x)= ( +1)2jK04 pj log (x)j+ j28 j log jx+ p (x)jj1+1h( + 1)0(x) + ( +1)2K2064 i 1+1( +1)28K04 pj log (x)j+ 2j log jx+ p (x)jjK208 ! ( + 1)0(x)1+1C [(I1) + (I2)]with (I1) = K04 pj log (x)j+ 2j log jx+ p (x)jjK2081+1 and (I2) =0(x) 1+1 .Let us bound (I1). Since j j 20pj log (x)j, we have from (29),j(I1)jK04 pj log (x)j+0pj log (x)j 2j log jx+0p (x)j log (x)jjjK2081+1=log (x)log x+log(1+ 4 0K0 ) K04 + 0 2K2081+1 .Since jxj0 and log (x) 2 log jxj as x ! 0 (see (31)), we nd 6(K0; 1)such that for each 06(K0; 1), there is 6(K0; 1; 0) such that for all06(K0; 1; 0), for jxj 2 [R(t0); 0] and j j 20pj log (x)j, we havej(I1)j12 :(98)Let us bound (I2). Since jxj R(t0), we have from (92), j(I2)j0(R(t0)) 1+1C(K0)j log 0j (p 1)+1 . Therefore, if t0 t6(K0; 1), thenj(I2)j12 :(99) Proof of lemma 2.4161Combining (98) and (99), we get: If 06(K0; 1), 06(K0; 1; 0) andt0 t6(K0; 1), then for jxj 2 [R(t0); 0] and j j 20pj log (x)j, (96) holds.Estimate on @k@ :From (94), we have for jxj 2 [r(t0); 0] and j j 20pj log (x)j,@k@ (x; ; t0 t(x)(x) ) = E1 +E2 +E3 where E1 =0(x)1+1 p (x)p 0j log0jr̂ x+ p (x)p 0j log0j! 1(x + p (x); t0);(100) E2 = (x) 121+1rH (x+ p (x))(11(x+ p (x); t0));(101)E3 = E4 (x) 12 1+1 @ 1@x (x + p (x); t0) with(102)E4 =1+10 ̂ x+ p (x)p 0j log0j! H (x+ p (x)):In order to get the estimate on @k@ , it is enough to show that forjxj 2 [r(t0); 2R(t0)] and j j 20pj log (x)j; jE1jC(K0)pj log (x)j ;(103) jxj 2 [R(t0); 0] and j j 20pj log (x)j; jE2jC(K0)pj log (x)j ;(104) jxj 2 [R(t0); 2R(t0)] and j j 20pj log (x)j; jE3jC(K0)pj log (x)j :(105)We begin with E1. Let jxj 2 [r(t0); 2R(t0)] and j j 20pj log (x)j. From (4),it follows that jr̂(z)j Cjzj 1+1 . Therefore, by (100),jE1j0(x) 1+1 p (x)p 0j log 0j jx+ p (x)j 1+1( 0j log 0j) 12( +1)j log 0j 1+1 (x) 121+1C( )jxj 1+1 (by (93))C(K0)j log (x)j 12 j log 0j 1+1 j log (x)j 1+1 (by (29))C(K0)j log (x)j 12 j log 0j 1+1 j log (r(t0)) j 1+1 (since jxj r(t0))C(K0)j log (x)j 12 for t0 t7(K0) (use (92)), which implies (103).Now we treat E2. Let jxj 2 [R(t0); 0] and j j 20pj log (x)j. From (101),we have jE2j (x) 121+1 jrH (x+ p (x))j(x) 121+1 jrH ( x)j with =32 is1 and = 12 if > 1 (use (93) and(5)). According to lemma 2.2,jrH ( x)j C(K0) ( x) 1+1 12pj log ( x)j C0(K0) (x) 1+1 12pj log (x)j as x ! 0. This implies(104) for7(K0).Now we show the bound on E3. We consider jxj 2 [R(t0); 2R(t0)] and j j20pj log (x)j, and nd a bound on E4. From (102), 162Reconnection of vortex with the boundary and quenchingE4 = ( + 1) 0 + ( +1)24 jx+ p (x)j2j log 0j1+1( +1)28jx+ p (x)j2j log jx+ p (x)jj 1+1 . From(93) and (91), we have(t0) ( + 1) 0 + ( + 1)24 jx+ p (x)j2j log 0jC (t0)and (t0) ( + 1)28jx+ p (x)j2j log jx+ p (x)jj C (t0)with (t0) C 0j log0jp 1. Therefore, jE4j C 0j log0jp 1 +1( + 1) 0 + ( +1)28 jx+ p (x)j22j log 0j1j log jx+ p (x)jjC 0j log0jp 1 +1 0 +jx+ p (x)j2j log 0jj log jx+ p (x)jj log jx+ p (x)j20C 0j log0jp 1 +1 0 + 0j log0jp 2 log log 0 (use (93), (91) andjxj 2 [R(t0); 2R(t0)]). HencejE4j C 1+10 j log 0j (p 1)+1 1 + j log0jp 2 log log 0 :(106)Using (46) and (27), we have@ 1@x C 120 j log 0j p2 :(107)From (92) and the fact that jxj 2 [R(t0); 2R(t0)], we have:(x) 121+1( R(t0)) 121+1 C(K0) 0j log0jp 1121+1 if t0 t8(K0),with = 2 if1 and = 1 if < 1.Combining this with (102), (106) and (107), we getjE3j C(K0)j log 0j p+ 12 1 + j log0jp 2 log log 0 j log 0j 12 ift0 t8(K0).Since log 0 log (R(t0)) as t0 ! T (see (92)) and R(t0) jxj, this yields(105) for t0 t9(K0).The expected bound (90) on @k@ follows from (103), (104) and (105).Estimate on k:In the same way, we show that if t0 t10(K0; 0; C1), thenfor jxj 2 [r(t0); 0] and j j 20pj log (x)j, we have jr2 k(x; ; t t(x)(x) j C1.Step 2: Estimate i) of the De nition of S (t)From (23) and (45), we have (y; s0)q(y; s0) =(d0 + d1 yps0 ) 0( jyjps0K0=16) 2(p a)s0 0( jyjps0K0 ):(108)Using (34), (26), (25) and simple calculations, and taking K0 20, we have: ift0 t11, thenq0(s0) = d0 R 0( jyjps0K0=16 )d2(p a)s0 R 0( jyjps0K0 )d ;q1(s0) = d1ps0 R y22 0( jyjps0K0=16 )d ;(109) Proof of lemma 2.4163andq0(s0) = d0(1 +O(e s0 )) 2(p a)s0 +O(e s0)(110)q1(s0) = d1ps0 (1 +O(e s0 ))(111)q2(s0) = d0O(e s0) +O(e s0 );(112)jq (y; s0)j Ces0(1 + jd0j)(1 + jyj2) + Cjd1js 120 e s0 jyj+ 2(p a)s0 (10( jyjK0ps0 )) + (jd0j+ jd1 yps0 j)(10( jyjps0K0=16 )):Since 8n 2 N, j 0(z) 1jCnjzjn, and K0 20, we getjq (y; s0)j (jd0j+ jd1j+ Cs0 ) (1 + jyj3)s3=20 :(113)Let us show thatjqe(y; s0)j Cs0 :(114)From (23), we have qe(y; s0) = Q1 + Q2 where Q2 = 2(p a)s0 (1 (y; s))Cs 10 and Q1 = (1 (y; s)) e s0p 1 +1h(x;t0)( yps0 ) with x = ye s0=2 andt0 = T e s0 .If jxj R(t0) (see (91) for R(t0)), then we have from (45), (46) and (27) Q1 = 0.If jxj R(t0), then we have from (10), (45), (91), (9) and easy calculations:( yps0 ) ( e s02 R(t0)ps0 ) Cs 10 andh(x; t0)1(x; t0)(T t0) 1+1C ( e s02 R(t0)ps0 )1+ (11(x; t0))H (R(t0))C(T t0) 1+1 s 10 .Therefore, by (9), jQ1j Cs 10 , which yields (114).By analogous calculations, one can easily obtain:j @q@y ? (y; s0)j C (jd0j+ jd1j+ 1=s0)ps0(1 + jyj3)s3=20(115)and j @q@y (y; s0)j s 10 for jyjK0ps0.From (109), one sees that g : (d0; d1)! (q0(s0); q1(s0)) is an a ne function.Let us introduce D(t0;K0; A) = g 1 [ As20 ; As20 ]2 . D(t0;K0; A) is obviously arectangle.If (d0; d1) 2 D(t0;K0; A), or equivalently jqm(s0)j As20 for m = 0; 1, then,from (110) and (111), we obtain jd0j Cs 10 and jd1j CAs 3=20 . Combiningthis with (112), (113), (114) and (115), we obtain 8A > 0, there exists t12(A) 0, there exists t1(K0; 0) such that8t0 2 [t1; T ), for all A 1, 0 > 0, C0 > 0, C 00 > 0, 012 k̂(1) and01( 0) for some 1( 0) > 0, we have the following property: Assumethat h(x; t0) is given by (45) and that for some t 2 [t0; T ), we have h(t) 2S (t0;K0; 0; 0; A; 0; C 00; C0; t), then:i) jq(y; s)j CA2K30s 1=2 and jq(y; s)j CA2s 2 log s(1 + jyj3),ii) jrq(y; s)j C(K0; C 00)A2s 1=2, jrq(y; s)j C(K0; C 00)A2s 2 log s(1 + jyj3),j(1 (y; s))rq(y; s)j C(K0)C 00s 12 , where s = log(T t) and q is de nedin (23).Proof:i): From i) of the de nition of S (t), we have q(s) 2 VK0;A(s). Therefore, theproof of lemma 3.8 in [22] holds.ii): Arguing similarly as for i), we obtain from i) of the de nition of S (t) and(26):j (y; s)rq(y; s)j CA2 log ss2 (1 + jyj3) and j (y; s)rq(y; s)j CA2K30ps :Since jr'(y; s)j Cs 1=2 and s 1=2 s 2jyj3 for jyjK0ps and K0 1,we have to prove that j(1 (y; s))r(q +')(y; s)j C(K0)C 00s 1=2 in order toconclude the proof.From (23), this reduces to show that 8t t0, for jxj r(t),jruj(x; t) = C( ) jrhjh +1 (x; t) C(K0; C 00) (T t) ( 1p 1+ 12 )pj log(T t)j(116)wherer(t) =K0p(T t)j log(T t)j:(117)Let us consider two cases: Proof of lemma 3.2165Case 1: jxj 2 [r(t); 0]. We use the information contained in ii) of the de ni-tion of S (t). From (28), we haveh(x; t) = (x) 1+1 k(x; 0; (x; t))(118) andrxh(x; t) = (x) 1+1 12r k(x; 0; (x; t))(119)with (x; t) = t t(x)(x) . Therefore, since p = + +1,jrhjh +1 (x; t) = (x) ( 1p 1+ 12 ) jr kjk +1 (x; 0; (x; t)):(120)Using the de nition of S (t), we have for jxj 2 [r(t); 0]jk(x; 0; (x; t)) k̂( )j0 and jr k(x; 0; (x; t))jC 00pj log (x)j :(121)Since 012 k̂(1), (120) and (29) yield for jxj 2 [r(t); 0]:jrhjh +1 (x; t) C(K0)C 00 (x) ( 1p 1+ 12 )pj log (x)j C(K0)C 00 (r(t)) ( 1p 1+ 12 )pj log (r(t)) j(122)with C(K0) = Ck̂(0) +1 . Since r(t)! 0 as t! T (see (117)), we have from (31)(r(t)) 2K20 r(t)2j log r(t)j and log (r(t)) log r(t) as t! T:(123)Using (117), we get( (r(t))) ( 1p 1+ 12 )pj log( (r(t)))j C4 (T t) ( 1p 1+ 12 )pj log(T t)j as t! Tfor some constant C4. Therefore, if t0 2 [t2(K0); T ) for some t2(K0) < T , thenwe have for t t0 ( (r(t))) ( 1p 1+ 12 )pj log( (r(t)))j 2C4 (T t) ( 1p 1+ 12 )pj log(T t)j :(124)Using (122) and (124), we nd (116) for jxj 2 [r(t); 0], provided that t0 2[t2(K0); T ).Case 2: jxj0. We use here the information contained in iii) of the de ni-tion of S (t), which asserts thatjh(x; t) h(x; t0)j0 and jrh(x; t) rh(x; t0)j0for jxj0. Let 1( 0) = 12 minf minjxj 0 jh(x; t0)j; minjxj 0 jrh(x; t0)jg. According to(45) and (5), we have 1( 0) > 0. If 01( 0), we get for jxj0:jrhjh +1 (x; t) C jrhjh +1 (x; t0) = C jrH jH ( +1) (x) 166Reconnection of vortex with the boundary and quenchingfrom (45). Therefore, proving (116) for all t t0 reduces to prove it for t = t0.From (5), one easily remarks that jrH jH ( +1) (x) C( 0) for jxj0. Therefore, ift0 2 [t4( 0); T ) for some t4( 0) < T , then we get (116) for t = t0.This concludes the proof of (116) for t = t0 and jxj0, hence for t t0and jxj0. Thus, with t1(K0; 0) = max(t2(K0); t4( 0)), this concludes theproof of (116) and the proof of lemma B.1.ii) Estimates on K and K1:As we remarked before, K1(s; ) = e (s )=2K(s; ). Hence, any estimateon K holds for K1 with the adequate changes.SinceK1 is the fundamental solution of L 1=2+V and L 1=2 is conjugatedto the harmonic oscillator e x2=8(L 1=2)ex2=8 = @2 x2=16 + 1=4 + 1=2, wegive a Feynman-Kac representation for K1:K1(s; ; y; x) = e(s )(L 1=2)(y; x)E(s; ; y; x)(125)whereE(s; ; y; x) = Z d syx (!)eR s0 V (!( ); + )d(126)and d syx is the oscillator measure on the continuous paths ! : [0; s ] !R with !(0) = x, !(s ) = y, i.e. the Gaussian probability measure withcovariance kernel ( ; 0) = !0( )!0( 0)+2(e 12 j 0j e 12 j + 0j + e 12 j2(s ) 0+ j e 12 j2(s ) 0 j;(127)which yields R d syx !( ) = !0( ) with!0( ) = (sinh s 2 ) 1(y sinh 2 + x sinh s 2 ).We have in additione (L 1=2)(y; x) =e =2p4 (1 e ) exp[ (ye =2 x)24(1 e ) ]:(128)Using this formulation for K1, we give estimates on the dynamics of K andK1 in the following lemma:Lemma B.2 i) 8s1 with s 2 , R jK(s; ; y; x)j(1+jxjm)dx es (1+jyjm).ii) There exists K2 > 0 such that for each K0 K2, A0 > 0, A00 > 0,A000 > 0, > 0, there existss2(K0; A0; A00; A000; ) with the following property: 8s0 s2, assume that fors0, q( ) is expanded as in (35) and satis esjqm( )jA0 2;m = 0; 1; jq2( )jA00(log ) 2;jq (y; )jA000(1 + jyj3) 2; jqe(y; )jA0012 ;then, 8s 2 [ ; + ]j (y; s)jC(e 12 (s )A000 +A00e (s)2)(1 + jyj3)s 2;j e(y; s)jC(A00e (s )p+A000K30e(s ))s 12 ;where (y; s) = K(s; )q( ) is expanded as in (35). Proof of lemma 3.2167iii) There exists K3 > 0 such that for each K0 K3, A0 > 0, A00 > 0,A000 > 0, A0000 > 0, > 0, there existss3(K0; A0; A00; A000; A0000; ) with the following property: 8s0 s3, assume thatfor s0, r( ) is expanded as in (36) and satis esjr0( )jA0 2;jr1( )jA00(log ) 2;jr (y; )jA000(1 + jyj3) 2; jre(y; )jA000012 ;then, 8s 2 [ ; + ]jP?( (s)K1(s; )r( ))j C(e 12 (s )A000 +A0000e (s)2)(1 + jyj3)s 2:Proof: See corollary 3.1 in [22] for i). See Lemma 3.5 in [22] for ii).Since K1(s; ) = e (s )=2K(s; ), and ii) and iii) have similar structure,one can adapt without di culty the proof of ii) (given in [22]) to get iii).iii) Estimates on B(q):Lemma B.3 8K0 1, 8A 1, 9s5(K0; A) such that 8s s5(A;K0), q(s) 2VK0;A(s) implies j (y; s)B(q(y; s))jC(K0)jqj2 and jB(q)j Cjqj p with p =min(p; 2).Proof: See Lemma 3.6 in [22].iv) Estimates on T (q):Lemma B.4 For all K0 1, A 1 and 0 > 0, there exists t6(K0; 0; A) < Tand 6( 0) such that for each t0 2 [t6(K0; 0; A); T ), 0 > 0, C 00 > 0, 0 12 k̂(1),C0 > 0 and 06( 0):if h(x; t0) is given by (45) and h(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; 0; t) forsome t 2 [t0; T ), thenj (y; s)(T (q) + 2ar'' :rq)j C(K0; A)( jyj2s2 jqj+ jqj2s + jrqj2);(129)j (y; s)T (q)jC(K0; A) (y; s) s 1jqj+ s 1=2jrqj(130) j(1 (y; s))T (q)jC(K0; C 00)min(s 1; s 5=2jyj3)(131)where s = log(T t) and q is de ned in (23).Proof:Proof of (129) and (130): They both follow from the Taylor expansion ofF ( ) = jr'+ rqj2'+ q + jr'j2' for 2 [0; 1]. Let us computeF 0( ) = q jr'+ rqj2('+ q)2 2rq:(r'+ rq)'+ q ,F 00( ) = 2q2 jr'+ rqj2('+ q)3 + 4qrq:(r'+ rq)('+ q)22 jrqj2'+ q .From F (1) = F (0) + F 0(0) + R 10 (1 )F 0( )d , we write(y; s)T (q) = a (y; s)(q jr'j2'2 2rq:r'' ) + a Z 10 (1 ) (y; s)F 00( )d : 168Reconnection of vortex with the boundary and quenchingUsing (23), lemma B.1 and (26), we claim that for s0 s7(A;K0), 8s s0,8 2 [0; 1], jr'j Cs 12 , jr'j2'2 C jyj2s2 andj (y; s)F 00( )j C(K0; A) (y; s)(s 1jqj2 + jrqj2)C(K0; A)(s 1jqj+ s 12 jrqj). Therefore, (129) and (130) follow.Proof of (131): From (23), we have jr'j2' (y; s) Cs 1. Therefore, if K0 1,(26) implies that (1 (y; s)) jr'j2' (y; s) min(Cs 1; Cs 5=2jyj3). In order toprove (131), it then remains to prove that(1 (y; s)) jr'+rqj2'+q (y; s) min(Cs 1; Cs 5=2jyj3), or simply, for jyjK0ps,jr'+rqj2'+q (y; s) Cs 1, since Cs 1 Cs 5=2jyj3 for jyjK0ps, if K0 1.From (23), this reduces to show that 8t t0, for jxj r(t),jruj2u (x; t) = C( ) jrhj2h +2 (x; t) C(K0; C 00) (T t) pp 1j log(T t)j(132)where r(t) is introduced in (117). The proof of (132) is in all its steps completelyanalogous to the proof of (116) given during the course of the proof of lemmaB.1, that is the reason why we escape it here.v) Estimates on R(y; ):Lemma B.5 8y 2 R, 8s 1,jR(y; s)j Cs 1,jR(y; s) C1(p; a)s 2j Cs 3(1 + jyj4) for some C1(p; a) 2 R, andj@R@y (y; s)j Cs 1 p(jyj+ jyj3) where p = min(p; 2).Proof: From (54), we haveR(y; s) = @'@s + '12y:r' 'p 1 + 'p a jr'j2' where'(y; s) = p + s ; = (p 1 + bz2) 1p 1 ; b = (p 1)24(p a) ; z = yps ;(133)= 2(p a) and = (p 1) 1p 1 . Therefore,R(y; s) =bz2(p 1)s p + s22b(p 1)s p + 4pb2z2(p 1)2s 2p 1p (p 1)s + 'p 4ab2z2(p 1)2s 2p' :(134)Proof of jR(y; s)j Cs 1: It follows form (134), and the fact that jzj2 p 1+C, ' 11 and j p 'pj C s 1.Proof of jR(y; s) C1(p; a)s 2j Cs 3(1+ jyj4): If jzj 1, then 1 s 1jyj2and jR(y; s) C1(p; a)s 2j Cs 1 s 1jyj2 2 Cs 3(1 + jyj4).Let us focus on the case jzj 1. The method we use consists in expandingeach term of (134) in terms of powers of s 1 and z2. From (133), one can easilyobtain the following bounds: for jzj 1, 8s 1,j p p + pb(p 1)3 z2j Cz4, j 2p 12p 1j Cz2,j'pp ps p 1 p(p 1) 22s2p 2j Cs 2, j p 11p 1 + b(p 1)2 z2j Cz4, Proof of lemma 3.2169j p 2 p 2j Cz2 and j 2p'2p 1j Cs 1.Combining all these bounds with (134) and (133), and using jzj 1, we get theresult.Proof of j@R@y (y; s)j Cs 1 p(jyj+ jyj3): The proof is completely similar tothe above estimates. We just give its main steps. First, use (134) to compute@R@y . Then, show that 8y 2 R, 8s 1, j@R@y (y; s)j Cs 12 p, in the same wayas for jR(y; s)j Cs 1. Therefore, if jzj 1, this gives the expected bound. Ifjzj 1, expand all the terms with respect to s and z2 to conclude.vi) Estimates on f1:Lemma B.6 8u 0, jf1(u)j+ jf 01(u)j C.Proof: According to (24), (H2) and (19), we have:f1(u) = +1u1+ 1F ( 1+1u 1) up, f 01(u) = F 0( 1+1u 1) pup 1,8v 2 (0; 1], F (v) = v 8v 1, jF (v)j Ce v C. Therefore,if1+1u 11, then f1(u) = f 01(u) = 0,if1+1u 11, then u+1 and jf1(u)j+ jf 01(u)j C( ).Step 2: Conclusion of the proofHere, we use the lemmas of Step 1 in order to conclude the proof. Therefore,we assume that K0 max(1;K2;K3), 0 > 0, A 1, t0 max (t1(K0; 0) ,T exp( s2(K0; A;A2; A; )); T exp( s2(K0; C; C; C; ));T exp( s2(K0; A; 1; C; )); T exp( s3(K0; CA;CA2; CA;C(K0)C 00; ));T exp( s3(K0; CA;C; 1; 1; )); T exp( s5(K0; A)); t6(K0; 0; A)),0 12 k̂(1), 0 > 0, C0 > 0, C 00 0, 0 min ( 1( 0); 6( 0)).We considerlog(T t0) and, and suppose that 8t 2 [Te ; T e ( + )], h(t) 2 S (t0;K0; 0; 0; A; 0; C 00; C0; t). Using the de nitionof S (t), and the lemmas of Step 1, we start the proof of the estimates of lemma3.2.Below, O(f) stands for a function bounded by f and not by Cf . We use thenotations introduced in (34).I) Equation (54)Since q0m(s) = dds R (y; s)km(y)q(y; s)d = R @@s ( q)kmd , we obtain:j R (y;s)km(y)@q@s (y; s)d q0m(s)j = j R @@s (y; s)km(y)q(y; s)d jC A2K30s1=2 e 2s by lemma B.1, (25) and (26). If s0 s12(K0; A), then (57) follows.Since L is self adjoint and Lkm = (1m2 )km, there exist two polynomials Pmand Qm such that j R (y; s)km(y)Lq(y; s)d (1m2 )qm(s)j = j R [L( (s)km)(s)km]q(s)d j = j R (@@yPm(y) + @2@y2Qm(y))q(s)d jCA2K30s 1=2e 2s by lemma B.1, (25) and (26). Therefore,j R (y; s)km(y)Lq(y; s)d j e s if s0 s13(K0; A), which yields (58).From (54), jV (y; s)j Cs 1(1 + jyj2). Therefore,j R (y; s)km(y)V (y; s)d j CA2s 3 log s s 5=2 for s s34(A), by lemmaB.1 and (25). This yields (59).From lemmas B.3 and B.1, and (25), we havej R (y; s)km(y)B(q)(y; s)d jC(K0)A4s 4(log s)2.Now, if s0 s15(K0; A), then (60) follows. 170Reconnection of vortex with the boundary and quenchingBy lemmas B.4 and B.1, and (25), we write:j R (y; s)k2(y)T (q)(y; s)d j s 2 1=4 for s0 s36(K0; A), which is (61).From (54), jV (y; s) + 2p=(s(p a))k2j Cs 2(1 + jyj4). Since jq0(s)j +jq4(s)j CAs 2 follows from q(s) 2 VK0;A(s), and since R (s)k22q(s)d =q2(s) + c0q0(s) + c4q4(s), we get (64) for s0 s7(A).From lemma B.5, we have jR(y; s)j C(s 2 + s 3jyj4). Using (25), we get(62).From lemma B.6, we have je psp 1 f1(e sp 1 (' + q))j Ce psp 1 . Therefore, asbefore, j R (y; s)km(y)e psp 1 f1(e sp 1 ('+q))d Ce psp 1 e s for s large and(63) follows.From (54), jV (y; s) + 2p=(s(p a))k2j Cs 2(1 + jyj4). Since jq0(s)j +jq4(s)j CAs 2 follows from q(s) 2 VK0;A(s), and since R (s)k22q(s)d =q2(s) + c0q0(s) + c4q4(s), we get (64) for s0 s7(A).By lemmas B.4 and B.1, and (25), we write:j R (y; s)k2(y)T (q)(y; s)d +Ej s 3 for s0 s16(K0; A; C 00), whereE = a=4 R rq(y; s)( (y; s)r'' (y2 2)e jyj2=4=p4 )dya=4 R q(y; s)r:( (y; s)r'='(y2 2)e jyj2=4=p4 )dy= O(e s) a=4 R q(y; s) (y; s)r:(r'='(y2 2)e jyj2=4=p4 )dy.By simple calculation,jr:(r'='(y2 2)ejyj2=4=p4 ) (h2(y) + h4(y)=4)=(s(p a)):e jyj2=4=p4 jP (jyj)e jyj2=4=s2 where P is a polynomial. Hence E = O(CA2s 4 log s)a=(4s(p a))(8q2(s)+c4q4(s)) = O(CAs 3) 2a=(s(p a))q2(s) and (65) holds.(66) follows from lemma B.5, (26) and (25).II) Equation (55)(67) and (68) follow from lemma B.2 ii) applied with A0 = A000 = A andA00 = A2.Lemmas B.3 and B.1 yieldjB(q(x; ))j Cjq(x; )j p CA2p 2p(log ) p(1 + jxj3) p.Lemmas B.4 and B.1 yieldjT (q(x; ))j j (x; )T (q(x; ))j + j(1 (x; ))T (q(x; ))jC(K0; A) 5=2 log (1 + jxj3) + C(K0; C 00) 5=2jxj3.Therefore, jB(q( )) + T (q( ))jC(K0; A; C 00) (log ) p2p (1 + jxj3p) + log5=2 (1 + jxj3) :(135)This way, j (y; s)j = j R sd K(s; ) (B(q( )) + T (q( ))) jR sd R dxjK(s; ; y; x)j jB(q(x; )) + T (q( ))jC(K0; A; C 00) R sd2p(log ) p R dxjK(s; ; y; x)j(1 + jxj3p)+ 5=2 log R dxjK(s; ; y; x)j(1 + jxj3)C(K0; A; C 00)(s )es s 2p(log s) p(1 + jyj3p) + s 5=2 log s(1 + jyj3)if s0(Indeed, s+++ s0 2 2 , and lemma B.2applies). Hence,j (y; s) (y; s)j C(K0; A; C 00)(s )es s 2p(log s) p(1 + jyj3jyj3p 3) Proof of lemma 3.2171+s 5=2 log s(1 + jyj3)C(K0; A; C 00)(s )es s 2p(log s) p(1 +jyj3(K0ps)3p 3)+s 5=2 log s(1 + jyj3) (s )s 2(1 + jyj3), if s0 s17(K0; A; ; C 00) (usep > 1). This yields j m(s)j C(s )s 2 for m = 0; 1; 2 and then (69).Lemmas B.3 and B.1 yield jB(q(x; ))j Cjq(x; )j pCK3p0 A2p p=2.Lemmas B.4 and B.1 yieldjT (q(x; ))j C(K0; A) 1 + C(K0; C 00) 1.Therefore, jB(q( )) + T (q( ))j C(K0; A; C 00) p=2.This way, j R sd K(s; )(B(q( )) + T (q( )))jR sd R dxjK(s; ; y; x)jjB(q(x; )) + T (q(x: ))jC(K0; A; C 00) R sd p=2 R dxjK(s; ; y; x)jC(K0; A; C 00)s p=2(s )es if s0(Indeed, s 2 and lemma B.2applies). Hence j e(y; s)j C(K0; A; C 00)s p=2(s )e (s )s 1=2 if s0s18(A; ; C 00) (use p > 1). This yields (70).Lemma B.5 implies that 8 > 1, 8x 2 R, jRm( )j C 2, m = 0; 1,jR2( )j C 2 log , jR (x; )j C 2(1 + jxj3) and jRe(x; )j C 1=2.Applying lemma B.2 ii) with A0 = A00 = A000 = C and then integrating withrespect to 2 [ ; s] yields (71) and (72).From lemma B.6, we have je pp 1 f1(e p 1 (' + q))j Ce pp 1 . Therefore,j (y; s)j = jK(s; )e pp 1 f1(e p 1 (' + q))j Ces e pp 1 according to i) oflemma B.2. Hence,j R sK(s; )e pp 1 f1(e p 1 ('+ q))j C(s )es e pp 1C(s )e e pp 1 s2 if s0,(s )s 2 if s s19(A; ). As before, this implies (74) and (75).From lemma 3.1 we have jqm(s0)j As 20 , m = 0; 1,jq2(s0)j s 20 log s0, jq (y; s0)j Cs 20 (1 + jyj3) and jqe(y; s0)j s 1=20 . If weapply lemma B.2 ii) with A0 = A, A00 = 1, A000 = C, then (76) and (77) follow.III) Equation (56)From de nition 34, we have for m = 0; 1,rm( ) = R rq(y; ) (y; )km(y)d= R q(y; )r( (y; )kme y2=4=p4 )dy= O(e ) R q(y; ) (y; )r(kme y2=4=p4 )dy= O(e ) + (m + 1) R q(y; ) (y; s)km+1(y)d = O(e ) + (m + 1)qm+1( ).Hence, if s0 s21, then jr0( )j CA 2 and jr1( )j CA2 2 log . Wehave jr?(y; )j A 2(1 + jyj3) since q( ) 2 VK0;A( ) (see the de nition ofS (t)), and jre(y; )j C(K0)C 00 1=2 by lemma B.1. Now, we apply lemmaB.2 iii) with A0 = A000 = CA, A00 = CA2 and A0000 = C(K0)C 00 to conclude theproof of (78)Estimate (79) is harder than estimate (78) because it involves a parabolicestimate on the kernel K1.Setting I(x; ) = B(q(x; )) + T (q(x; )), we writeK1(s; ) @@y (B(q) + T (q))( ) = R dxe(s )(L 1=2)(y; x)E(s; ; y; x) @I@x(x; )= (I) + (II) with (I) = R dx@xe(s )(L 1=2)(y; x)E(s; ; y; x)I(x; ) and(II) = R dxe(s )(L 1=2)(y; x)@xE(s; ; y; x)I(x; ). Let us rst bound (I).From (128), (I) = 172Reconnection of vortex with the boundary and quenchingR dx e(s )=2p4 (1 e (s )) 2(x ye (s )=2)4 (1 e (s )) exp (ye (s )=2 x)24 (1 e (s )) E(s; ; y; x)I(x; ).If s0, then 0 E(s; ; y; x) C (use for this V (x; ) C 1 which is aconsequence of (54), (126), d syx is a probability and s+++ s0 2 2 ). Using (135), we getj(I)j C(K0; A; C 00) e(s )=2p4 (1 e (s )) Rdxp4 (1 e (s )) 2jye (s )=2 xjp4 (1 e (s ))exp (ye (s )=2 x)24 (1 e (s ))2p(log ) p(1 + jxj3p) + 5=2 log (1 + jxj3)where p = min(p; 2) > 1. With the change of variables = x ye (s )=2p4 (1 e (s )) ,j(I)j C(K0; A; C 00) e(s )=2p4 (1 e (s ))2p(log ) pR d j je 2(1 + j p4 (1 e (s )) ye (s )=2j3p) + 5=2 logR d j je 2(1 + j p4 (1 e (s )) ye (s)=2j3)o, hence j(I)jC(K0; A; C 00)e(s )=2p4 (1 e (s )) (log ) p2p (1 + jyj3p) + log5=2 (1 + jyj3) :(136)Let us bound (II) now. Using the integration by parts formula for Gaussianmeasures (see [11]), we have @xE(s; ; y; x):=12 Z s0 Z s0 d 1d 2@x ( 1; 2) Z d syx (!)V 0(!( 1); + 1)V 0(!( 2); +2)eR s0 d 3V (!( 3); + 3)(137)+12 Z s0 d 1@x ( 1; 1) Z d syx (!)V 00(!( 1); +1)eR s0 d 3V (!( 3); + 3):By (54), we have j@nV@yn j Cs n=2 for n = 0; 1; 2. Combining this with (127)and (137), we get (for s0)j@xE(s; ; y; x)j Cs 1(s )(1 + s )(jyj+ jxj).Using this, (128) and (135), we obtainj(II)j e(s )=2 Rdxp4 (1 e (s )) exp (y (s )=2 x)24 (1 e (s )) (jyj+ jxj)Cs 1(s )(1 + s )C(K0; A; C 00)2p(log ) p(1 + jxj3p)+ 5=2 log (1 + jxj3) .Arguing as for (I), we get:j(II)j C(K0; A; C 00)e(s )=2(s )(1 + s )s 1(1 + jyj)(log ) p2p (1 + jyj3p) + log5=2 (1 + jyj3) :(138)Combining (136) and (138), we obtainj R sd K1(s; ) @I@y ( )j C(K0; A; C 00) s 2p(log s) p(1 + jyj3p)+s 5=2 log s(1 + jyj3)R se(s )=2p4 (1 e (s )=2) + e(s )=2(s )(1 + s )s 1(1 + jyj) dC(K0; A; C 00) s 2p(log s) p(1 + jyj3p) + s 5=2 log s(esps + e(s )=2((s )2+(s )3)s 1(1+ jyj)) (s0, which implies Proof of lemma 3.21732 s). Multiplying this by (y; s) and replacing some jyj by2K0ps, we get:8s 2 [ ; + ],j (y; s) R sd K1(s; ) @I@y ( )j C(K0; A; C 00) s (p+3)=2 + s 5=2 (1 + jyj3)ps (e + e =2( 3=2 + 5=2)s 1=2). If s s0 s22(A; ), thenj (y; s) R sd K1(s; ) @I@y ( )j Cs 2ps (1 +jyj3) (use p > 1). Therefore,jP?( (y; s) R sd K1(s; ) @I@y ( ))j Cs 2ps (1 +jyj3).This concludes the proof of (79).By de nition, R1(x; ) = @R@y (x; ) + @V@y q(x; ). From (54), we havej@V@y (x; )j = 2pb'(x; )p 2(p 1 + bx2= ) p=(p 1)x 1 withb = (p 1)2=(4(p a)). Setting z = x 1=2, we easily see thatj@V@y (x; )j C 1=2. Using lemmas B.1 and B.5, we getjR1(x; )j C (1+p)(jxj + jxj3) + CA2 5=2 log (1 + jxj3)C (2+ 2(p))(1 + jxj3) with 2(p) > 0 if s0 s33(A). Therefore,jK1(s; )R1( )j = j R K1(s; ; y; x)R1(x; )dxjC (2+ 2(p)) R jK1(s; ; y; x)(1 + jxj3)dxC (2+ 2(p))e(s )=2(1 + jyj3) by lemma B.2 i). Hence,j R s d K1(s; )R1( )j C(1 + jyj3) R sd(2+ 2(p))e(s )=2C(s )e(s )=2s (2+ 2(p))(1 + jyj3) if s0.Now, if s0 s23( ), thenj R s d K1(s; )R1( )j C(s )e =2s (2+ 2(p))(1 + jyj3)(s )s 2(1 + jyj3). By classical arguments, this yields (80).From lemmas B.2 and B.6, and the fact that @'@y C 1=2, we have:je (@'@y + r)f 01(e p 1 (' + q)j C(K0; C 00)A2 1=2e . Therefore, i) of lemmaB.2 yields:K1(s; )e (@'@y + r)f 01(e p 1 ('+ q))j C(K0; C 00)A2e s 21=2e . Hence,j R sd K1(s; )e (@'@y + r)f 01(e p 1 ('+ q))j C(K0; C 00)A2(s )e s 2 epC(K0; C 00)A2(s )e s 1=2e s2 if s0(s )s 2 if s s24(K0; A; ). Thus, by classical arguments, (81) follows.Since rm(s0) = O(es0)+(m+1)qm+1(s0), we have from lemma 3.1 jr0(s0)jCAs 20 , jr1(s0)j Cs 20 log s0, jr?(y; s0)j s 20 (1+jyj3) and jre(y; s0)j s 1=20 .Applying lemma iii) of B.2 with A0 = CA, A00 = C, A000 = A0000 = 1 yields (82). Bibliography175Bibliography[1] Ball, J., Remarks on blow-up and nonexistence theorems for nonlinearevolution equations, Quart. J. Math. Oxford 28, 1977, pp. 473-486.[2] Berger, M., and Kohn, R., A rescaling algorithm for the numerical cal-culation of blowing-up solutions, Comm. Pure Appl. Math. 41, 1988, pp.841-863.[3] Bricmont, J., and Kupiainen, A., Renormalization group and nonlinearPDEs, Quantum and non-commutative analysis, past present and futureperspectives, Kluwer (Boston), 1993.[4] Bricmont, J., and Kupiainen, A., Universality in blow-up for nonlinearheat equations, Nonlinearity 7, 1994, pp. 539-575.[5] Chapman, S.J., Hunton, B.J., and Ockendon, J.R., Vortices and Boun-daries, Quart. Appl. Math. to appear.[6] Deng, K., Levine, H.A., On the blow-up of ut at quenching, Proc. Am.Math. Soc. 106, 1989, pp. 1049-1056[7] Filippas, S., and Kohn, R., Re ned asymptotics for the blowup of utu = up, Comm. Pure Appl. Math. 45, 1992, pp. 821-869.[8] Giga, Y., and Kohn, R., Asymptotically self-similar blowup of semilinearheat equations, Comm. Pure Appl. Math. 38, 1985, pp. 297-319.[9] Giga, Y., and Kohn, R., Characterizing blowup using similarity variables,Indiana Univ. Math. J. 36, 1987, pp. 1-40.[10] Giga, Y., and Kohn, R., Nondegeneracy of blow-up for semilinear heatequations, Comm. Pure Appl. Math. 42, 1989, pp. 845-884.[11] Glimm, J., Quantum Physics, Springer, New York, 1981.[12] Guo, J., On the quenching behavior of the solution of a semilinear para-bolic equation, J. Math. Analysis Applic. 151, 1990, pp 58-79.[13] Guo, J., On the quenching rate estimate, Quart. Appl. Math. 49, 1991,pp. 747-752.[14] Guo, J., On the semilinear elliptic equation w 12y:rw+ w w = 0in Rn , Chinese J. Math. 19, no 4, 1991, pp. 355-377. 176Reconnection of vortex with the boundary and quenching[15] Herrero, M.A, and Velazquez, J.J.L., Blow-up pro les in one-dimensionalsemilinear parabolic problems, Comm. Partial Di erential Equations 17,1992 pp. 205-219.[16] Kavian, O., Remarks on the large time behavior of a nonlinear di usionequation, Ann. Inst. H. Poincar e Anal. Non Lin eaire 4, 1987, pp. 423-452.[17] Keller, J.B., Lowengrub, J.S., Straight forward asymptotics and numercifor blowing-up solutions to semilinear heat equations, Proc. from NATOSing. Conf. 1992[18] Levine, H.A., Advances in quenching, Nonlinear di usion equations andtheir equilibrium states, 3 (Gregynog, 1989), pp. 319-346.[19] Levine H.A., Quenching, nonquenching, and beyond quenching for solu-tions of some parabolic equations, Annali Mat. Pura. Appl. 155, 1989,pp. 243-260.[20] Levine, H., Some nonexistence and instability theorems for solutions offormally parabolic equations of the form Put = Au+F (u), Arch. Rat.Mech. Anal. 51, 1973, pp. 371-386.[21] Merle, F., Solution of a nonlinear heat equation with arbitrary givenblow-up points, Comm. Pure Appl. Math. 45, 1992, pp. 263-300.[22] Merle, F., and Zaag, H., Stability of the blow-up pro le for equations ofthe type ut = u+ jujp 1u, Duke Math. J. 86, 1997, pp. 143-195.[23] Souplet, P., Tayachi, S., and Weissler, F., B., Exact self-similar blow-upsolutions of a semilinear parabolic equation with a nonlinear gradientterm, Indiana Univ. Math. J. 45, 1996, pp. 655-682.[24] Velazquez, J.J.L., Classi cation of singularities for blowing up solutionsin higher dimensions Trans. Amer. Math. Soc. 338, 1993, pp. 441-464.[25] Zaag, H., Blow-up results for vector-valued nonlinear heat equations withno gradient structure, Ann. Inst. H. Poincar e Anal. Non Lin eaire 15,1998, to appear.AddressD epartement de math ematiques, Universit e de Cergy-Pontoise, 2 avenue Adol-phe Chauvin, Pontoise, 95 302 Cergy-Pontoise cedex, France.D epartement de math ematiques et d'informatique, Ecole Normale Sup erieure,45 rue d'Ulm, 75 230 Paris cedex 05, France. Deuxieme partieEstimations g enerales dessolutions positivesexplosives de l' equation dela chaleur non lin eaire etnotions de pro l al'explosion Chapitre 1Estimations uniformes al'explosion pour lesequations de la chaleur nonlin eaires et applications 180Estimations uniformes a l'explosion et applicationsEstimations uniformes a l'explosion pour lesequations de la chaleur non lin eaires et applications yFrank MerleUniversite de Cergy-PontoiseHatem ZaagEcole Normale Superieure et Universite de Cergy-PontoiseOn s'int eresse a l' equation de la chaleur non lin eaireut = u+ upu(0) = u0 0;(1)ou u est d e nie pour (x; t) 2 RN [0; T ), 1 < p et (N 2)p < N +2. Di erentesg en eralisations de cette equation peuvent être consid er ees (voir [12] pour plusde d etails):ut = r:(a(x)ru) + b(x)upu(0) = u0 0;(2)ou u est d e nie pour (x; t) 2 [0; T ), 1 < p et (N 2)p < N +2, = RN ouest un ouvert convexe born e et r egulier, a(x) est une matrice sym etrique etuniform ement elliptique, a(x) et b(x) sont C2 et born ees.Plus pr ecis ement, on s'int eresse au ph enom ene d'explosion en temps ni. Unelitt erature importante est consid er ee a ce sujet. On pourra citer les travaux deBall [1], Bricmont et Kupiainen et Lin [3] [2], Chen et Matano [4], Galaktionovet Vazquez [6], Giga et Kohn [7] [8] [9], Herrero et Velazquez [10] [11] (voir [12]et [13] pour les r ef erences). Dans la suite, on note T le temps d'explosion deu(t), une solution explosive de (1).Le probl eme qui nous int eresse est celui d'obtenir des estimations uniformesoptimales et de donner des applications de telles estimations.Pour de telles estimations, on est amen e a consid erer l' equation (1) dans saforme auto-similaire: pour tout a 2 RN , on posey = x apT ts = log(T t)wa(y; s) = (T t) 1p 1u(x; t):(3)On a alors que wa = w satisfait 8slogT , 8y 2 RN :@w@s = w12y:rw wp 1 + wp:(4) Le probl eme est d'estimer wa(s) quand s! +1, que a soit un point r egulierou un point d'explosion (a est dit point d'explosion lorsqu'il existe (an; tn) !(a; T ) tel que u(an; tn)! +1) de faCcon uniforme.yNote parue dans les actes du seminaire EDP 1996-1997, Ecole Polytechnique, pp. XIX-1XIX-8. Un theoreme de Liouville pour l'equation (4)181Giga et Kohn ont d emontr e qu'en fait les variables auto-similaires sont lesbonnes variables pour mesurer les solutions explosives dans les sens suivant: ilexiste 0 > 0 tel que 8s s 0, 0 jw(s)jL1 10 :On se propose dans un premier temps d'a ner ce r esultat pour obtenir dela compacit e dans le probl eme.1 Un th eor eme de Liouville pour l' equation (4)Pour ceci, on s'int eresse a un probl eme de classi cation de solutions globales.On a le r esultat suivant:Theoreme 1 (Theoreme de Liouville pour (4)) Soit w une solutionde (4) d e nie pour (y; s) 2 RN R telle que 8(y; s) 2 RN R,0 w(y; s) C. Alors, on est n ecessairement dans l'un des cas suivants:i) w 0,ii) wou = (p 1) 1p 1 ,iii) 9s0 2 R tel que w(y; s) = '(s s0) ou'(s) = (1 + es) 1p 1 :Remarque: Remarquons que ' est une connexion dans L1 des deux pointscritiques de (4): 0 et . En e et,_' = 'p 1 + 'p; '( 1) = ; '(+1) = 0:Remarque: Il su t d'avoir une solution de (4) d e nie sur ( 1; s ) pour avoirun th eor eme de classi cation (voir [12]).On peut obtenir comme corollaireCorollaire 1 Soit u une solution de (1) d e nie pour (x; t) 2 RN ( 1; 0)telle que 8(x; t) 2 RN ( 1; 0), 0 u(x; t) C(T t) 1p 1 . Alors,soit u 0,soit 9T 0 tel que u(x; t) = (T t) 1p 1 .Pour les d emonstrations, voir [12]. Les outils clefs de la d emonstration sont:i) une classi cation des comportements lin eaires de w(s) quand s ! 1dans L2 (RN )(L1loc(RN )) ou (y) = e jyj24(4 )N=2 ,ii) les transformations g eom etriquesw(y; s)! wa;b(y; s) = w(y + ae s2 ; s+ b)pour a 2 RN et b 2 R,iii) un crit ere d'explosion en temps ni dans les variables auto-similaires: sipour un certain s0 2 R, R w(y; s0) (y)dy > R (y)dy, alors w(s) explose entemps ni. 182Estimations uniformes a l'explosion et applications2 Estimations optimales a l'explosionPar un argument de compacit e, on obtient les estimations uniformes sui-vantes sur la solution w(s) de (4):Theoreme 2 (Estimations optimales a l'ordre zero sur w(s))Si w(s0) 2 H1(RN ), alorskw(s)kL1(RN) ! et krw(s)kL1(RN) + k w(s)kL1(RN) ! 0 quand s! +1.Remarque: Cette estimation est aussi valable pour un ensemble de solutions(voir [12]).Cette estimation est tr es importante car elle donne pour une solution laconvergence de wa(s) vers un ensemble limite dansL1loc uniform ement par rap-port a a 2 RN . Ceci nous permet ensuite par lin earisation autour de cet ensemblede d emontrer leTheoreme 3 (Estimation optimale a l'ordre un sur w(s)) Sous les hypo-th eses du Th eor eme 2, 8 0 > 0, il existe s( 0) tel que 8s s( 0), 9C1; C2 > 0tels quekw(s)kL1+ (N2p +0)1skrw(s)kL1C1pskr2w(s)kL1C2s :Remarque: Dans le casN = 1, en utilisant une propri et e de Sturme D evelopp eepar Chen et Matano (qui a rme que le nombre d'oscillations en espace dela solution est une fonction d ecroissante du temps), Herrero et Velazquez (etFilippas et Kohn) ont montr e des estimations de ce type.Remarque: La constante N2p est optimale (voir Herrero et Velazquez, Bricmontet Kupiainen, Merle et Zaag).3 Localisation a l'explosionLe Th eor eme 2 implique que dans la zone singuli ere du type fy j w(y; s)2 g, w est petit devant wp (ou de faCcon equivalente, u est petit devantup). Un ph enom ene de localisation sous critique introduit par Zaag [15] (sous leseuil de la constante) nous permet de propager ces estimations dans les zonessinguli eres : \u(x; t) grand". Il en d ecoule le th eor eme suivant:Theoreme 4 (Comparaison avec l'equation di erentielle ordinaire) Siu0 2 H1(RN ), alors 8 > 0, 9C > 0 tel que 8t 2[T2 ; T ), 8x 2 RN ,jut upj up + C :Remarque: Ainsi, on d emontre que la solution de l' equation aux d eriv ees par-tielles est comparable uniform ement et globalement en espace-temps a une equa-tion di erentielle ordinaire (localis ee par d e nition). Notion de Pro l au voisinage d'un point d'explosion183On peut noter que le r esultat reste vrai pour une suite de solutions souscertaines conditions.Remarque: De multiples corollaires d ecoulent de ce th eor eme. Par exemple,8 0 > 0, il existe t0( 0) < T tel que pour tout a 2 RN , t 2 [t0; T ), si u(a; t)(1 0) (T t) 1p 1 , alors, a n'est pas point d'explosion. (Ceci pr ecise un r esultatde Giga et Kohn ou t0 = t0( 0; a).4 Notion de Pro l au voisinage d'un point d'ex-plosionOn consid ere maintenant a 2 RN un point d'explosion de u(t) solution de(1). Par invariance par translation, on se ram ene a a = 0. La question est desavoir si u(t) (ou w0(s) d e nie en (3)) a un comportement universel ou pasquand t! T (ou s! +1).Filippas, Kohn, Liu, Herrero et Velazquez ont d emontr e que w evoluait sui-vant l'une des deux possibilit es suivantes:8R > 0, supjyj R w(y; s)+ 2ps trAk 12yTAky = O 1s1+quand s! +1 pour un certain > 0 avecAk = Q IN k 00 0 Q 1;k 2 f0; 1; :::; N 1g, Q une matrice N N orthogonale et IN k l'identit e desmatrices (N k) (N k).8R > 0, supjyj R jw(y; s) j C(R)e 0s pour un certain 0 > 0.Dans un certain sens, ces r esultats d emarquent mal d'un point de vue phy-sique la transition entre les zones singuli ere (wou > 0) et r eguli ere(w ' 0). En utilisant la th eorie de la renormalisation, Bricmont et Kupiainenont d emontr e dans [3] l'existence d'une solution de (4) telle que8s s0; 8y 2 RN ; jw(y; s) f0( yps )j Cpsou f0(z) = (p 1 + (p 1)24p jzj2) 1p 1 . Merle et Zaag ont d emontr e dans [14]le même r esultat grâce a des techniques de r eduction en dimension nie. Ils yd emontrent aussi la stabilit e par rapport aux donn ees initiales de telles compor-tements.Dans [15], Zaag montre que dans ce cas, u(x; t) ! u (x) quand t ! Tuniform ement sur RN nf0g et que u (x) h 8pj log jxj(p 1)2jxj2 i 1p 1 quand x! 0.Dans un premier temps, on est en mesure de d emontrer grâce aux estimationsdu Th eor eme 4, un th eor eme de classi cation des pro ls dans la variable yps (quis epare partie singuli ere et r eguli ere dans le cas non d eg en er e).Theoreme 5 (Classi cation des pro ls a l'explosion)Il existe k 2 f0; 1; :::; N 1g et une matrice N N orthogonale Q tels que 184Estimations uniformes a l'explosion et applicationsw(Q(z)ps; s)! fk(z) uniform ement sur tout compact jzj C, oufk(z) = (p 1+ (p 1)24p N kXi=1 jzij2) 1p 1 si k N 1 et fN (z) = = (p 1) 1p 1 .Un des probl emes int eressants qui en d ecoule est de relier toutes les notionsde pro ls connues: pro l pour jyj born e, jyjps born e ou x ' 0. On d emontreque ces notions sont equivalentes dans le cas d'une solution qui explose en unpoint de faCcon non d eg en er ee (cas g en erique), ce qui r epond a de nombreusesquestions pos ees dans des travaux pr ec edents.Theoreme 6 ( Equivalence des comportements explosifs en unpoint)Soit a un point d'explosion isol e de u(t) solution de (1). On a l' equivalencedes trois comportements suivants de u(t) et de wa(s) (d e nie en (3)):i) 8R > 0, supjyj R w(y; s)+ 2ps (N 12 jyj2) = o1s quand s! +1,ii) 8R > 0, supjzj R w(zps; s) f0(z) ! 0 quand s ! +1 avec f0(z) =(p 1 + (p 1)24p jzj2) 1p 1 ,iii) 9 0 > 0 tel que pour tout jx aj0, u(x; t) ! u (x) quand t ! T etu (x) h 8pj log jx ajj(p 1)2jx aj2 i 1p 1 quand x! a.Remarque: . Dans le cas N = 1, certaines implications etaient d eja d emon-tr ees. Bibliographie185Bibliographie[1] Ball, J., Remarks on blow-up and nonexistence theorems for nonlinearevolution equations, Quart. J. Math. Oxford 28, 1977, pp. 473-486.[2] Bricmont, J., Kupiainen, A., et Lin, G., Renormalization group andasymptotics of solutions of nonlinear parabolic equations, Comm. PureAppl. Math. 47, 1994, pp. 893-922.[3] Bricmont, J., et Kupiainen, A., Universality in blow-up for nonlinearheat equations, Nonlinearity 7, 1994, pp. 539-575.[4] Chen, X., Y., et Matano, H., Convergence, asymptotic periodicity, andnite-point blow-up in one-dimensional semilinear heat equations, J. Dif-ferential Equations. 78, 1989, pp. 160-190.[5] Filippas, S., et Kohn, R., Re ned asymptotics for the blowup of ut u =up, Comm. Pure Appl. Math. 45, 1992, pp. 821-869.[6] Galaktionov, V., A., et Vazquez, J., L., Geometrical properties of thesolutions of one-dimensional nonlinear parabolic equations, Math. Ann.303, 1995, pp. 741-769.[7] Giga, Y., et Kohn, R., Nondegeneracy of blow-up for semilinear heatequations, Comm. Pure Appl. Math. 42, 1989, pp. 845-884.[8] Giga, Y., et Kohn, R., Characterizing blowup using similarity variables,Indiana Univ. Math. J. 36, 1987, pp. 1-40.[9] Giga, Y., et Kohn, R., Asymptotically self-similar blowup of semilinearheat equations, Comm. Pure Appl. Math. 38, 1985, pp. 297-319.[10] Herrero, M.A, et Velazquez, J.J.L., Blow-up behavior of one-dimensionalsemilinear parabolic equations, Ann. Inst. H. Poincar eAnal. Non Lin eaire10, 1993, pp. 131-189.[11] Herrero, M.A, et Velazquez, J.J.L., Flat blow-up in one-dimensional se-milinear heat equations, Di erential Integral Equations 5, 1992, pp. 973-997.[12] Merle, F.,et Zaag, H., Optimal estimates for blow-up rate and behaviorfor nonlinear heat equations, Comm. Pure Appl. Math. 51, 1998, pp.139-196. 186Estimations uniformes a l'explosion et applications[13] Merle, F., et Zaag, H., Re ned uniform estimates at blow-up and appli-cations for nonlinear heat equations, Geom. Funct. Anal., a parâ tre.[14] Merle, F., et Zaag, H., Stability of the blow-up pro le for equations ofthe type ut = u+ jujp 1u, Duke Math. J. 86, 1997, pp. 143-195.[15] Zaag, H., Blow-up results for vector valued nonlinear heat equations withno gradient structure, Ann. Inst. H. Poincar e Anal. Non Lin eaire 15,1998, a parâ tre.Adresses:D epartement de math ematiques, Universit e de Cergy-Pontoise, 2 avenue Adol-phe Chauvin, Pontoise, 95 302 Cergy-Pontoise cedex, France.D epartement de math ematiques et informatique, Ecole Normale Sup erieure, 45rue d'Ulm, 75 230 Paris cedex 05, France.e-mail: [email protected], [email protected] Chapitre 2Optimal estimates forblow-up rate and behaviorfor nonlinear heat equations 188Optimal estimates for blow-up rate and behaviorOptimal estimates for blow-up rate and behaviorfor nonlinear heat equations yFrank MerleInstitute for Advanced Study and Universite de Cergy-PontoiseHatem ZaagEcole Normale Superieure and Universite de Cergy-PontoiseAbstract: We rst describe all positive bounded solutions of@w@s = w12y:rw wp 1 + wp:where (y; s) 2 RN R, 1 < p and (N 2)p N+2. We then obtain for blow-upsolutions u(t) of@u@t = u+ upuniform estimates at the blow-up time and uniform space-time comparison withsolutions of u0 = up.1 IntroductionWe consider the following nonlinear heat equation:@u@t = u+ jujp 1u in[0; T )u = 0on @ [0; T )(1)where u(t) 2 H1( ) and = RN (or is a convex domain).We assume in addition that1 < p; (N 2)p < N + 2 and u(0) 0:In this paper, we are interested in blow-up solutions u(t) of equation (1): u(t)blows-up in nite time T if u exists for t 2 [0; T ) and limt!T ku(t)kH1 = +1. Inthis case, one can show that u has at least one blow-up point, that is a 2 suchthat there exists (an; tn)n2N satisfying (an; tn)! (a; T ) and ju(an; tn)j ! +1.We aim in this work at studying the blow-up behavior of u(t). In particular, weare interested in obtaining uniform estimates on u(t) at or near the singularity,that is estimates \basically" independent of initial data.We will give two types of uniform estimates: the rst one holds especiallyat the singular set (Theorem 1) and the other one consists in surprising globalestimates in space and time (Theorem 3). It will be deduced from the formerby some strong control of the interaction between regular and singular parts ofthe solution. Various applications of this type of estimates will be given in [12].For the rst type of estimates, we introduce for each a 2 (a may be ablow-up point of u or not) the following similarity variables:y = x apT ts = log(T t)wa(y; s) = (T t) 1p 1u(x; t):(2) yArticle paru dans Comm. Pure Appl. Math. 51, 1998, pp. 139-196. Introduction189wa (= w) satis es 8slogT , 8y 2 Da;s:@w@s = w 12y:rw wp 1 + jwjp 1w(3)whereDa;s = fy 2 RN j a+ ye s=2 2 g:(4)We introduce also the following Lyapunov functional:E(w) = 12 Z jrwj2 dy + 12(p 1) Z jwj2 dy1p+ 1 Z jwjp+1 dy(5)where (y) = e jyj2=4(4 )N=2(6)and the integration is done over the de nition set of w.The study of u(t) near (a; T ) where a is a blow-up point is equivalent tothe study of the long time behavior of wa. Note that Da;s 6= RN in the case6= RN . This in fact is not a problem since we know from [8] that a 62 @ inthe case is C2; , and therefore, for a given a 2 , Da;s ! RN as s ! +1.Let a 2 be a blow-up point of u.If is a bounded convex domain in RN or = RN , then Giga and Kohnprove in [7] that:8slogT; kwa(y;s)kL1(Da;s)C or equivalently8t 2 [0; T ); ku(x;t)kL1( )C(T t) 1p 1 :(7)They also prove in [7] and [8] (see also [6]) that for a given blow-up point a 2 ,lims!+1wa(y; s) = limt!T(T t) 1p 1u(a+ ypT t; t) =where = (p 1) 1p 1 , uniformly on compact subsets of RN . The result ispointwise in a. Besides, for a.e y, lims!+1rwa(y; s) = 0.Let us denoteL1(Da;s) by L1.In this paper, we rst obtain uniform (on a and in some sense on u(0))sharp estimates on wa, and we nd a precise long time behavior forkwa(s)kL1 ,krwa(s)kL1 and kwa(s)kL1 (global estimates).Theorem 1 (Optimal bound on u(t) at blow-up time) Assume thatis a convex bounded C2; domain in RN or = RN . Consider u(t) a blow-upsolution of equation (1) which blows-up at time T . Assume in addition u(0) 0and u(0) 2 H1( ). Then(T t) 1p 1 ku(t)kL1( ) ! = (p 1) 1p 1 as t! Tand(T t) 1p1+1k u(t)kL1( ) + (T t) 1p 1+ 12 kru(t)kL1( ) ! 0 as t! T;or equivalently for any a 2 ,kwa(s)kL1 ! as s! +1andkwa(s)kL1 +krwa(s)kL1 ! 0 as s! +1: 190Optimal estimates for blow-up rate and behaviorRemark: We can point out that we do not consider local norm in w variablesuch as L2(d ) with d = e jyj2=4dy as a center manifold theory for equation(3) would suggest. Instead, we use L1 norm which yields results uniform withrespect to a 2 . Indeed, we have from (2) that 8a; b 2 , 8(y; s) 2 Db;s,wb(y; s) = wa(y + (b a)e s2 ; s);which yieldskwakL1 =kwbkL1 ,krwakL1 =krwbkL1 and kwakL1 =kwbkL1).One interest of Theorem 1 is that in fact, its proof yields the followingcompactness result:Theorem 1' (Compactness of blow-up solutions of (1)) Assume thatis a convex bounded C2; domain in RN or = RN . Consider (un)n2N asequence of nonnegative solutions of equation (1) such that for some T > 0 andfor all n 2 N, un is de ned on [0; T ) and blows-up at time T . Assume also thatkun(0)kH2( ) is bounded uniformly in n. Thensupn2N(T t) 1p 1 kun(t)kL1( ) ! as t! Tand supn2N (T t) 1p1+1k un(t)kL1( ) + (T t) 1p 1+ 12 krun(t)kL1( ) ! 0as t! T .Remark: The same results can be proved for the following heat equation:@u@t = r:(a(x)ru) + b(x)f(u); u(0) 0where f(u) up as u ! +1, (a(x)) is a symmetric, bounded and uniformlyelliptic matrix, b(x) is bounded, and a(x) and b(x) are C1.Let us point out that this result is optimal. One way to see it is by the fol-lowing Corollary which improves the local lower bound on the blow-up solutiongiven in [8] by Giga and Kohn.Corollary 1 (Lower bound on the blow-up behavior for equation (1))Assume that is a convex bounded C2; domain in RN or = RN . Then forall nonnegative solution u(t) of (1) such that u(0) 2 H1( ) and u(t) blows-upat time T , and for all 0 2 (0; 1), there exists t0 = t0( 0; u0) < T such that iffor some a 2 and some t 2 [t0; T ) we have0 u(a; t) (1 0) (T t) 1p 1 ;(8)then a is not a blow-up point of u(t).Remark: The result is still true for a sequence of nonnegative solutions unblowing-up at T > 0 and satisfying the assumptions of Theorem 1', with a t0independent of n.Remark: is the optimal constant giving such a result. The result of [8] wasthe same except that (1 0) was replaced by 0 small and it was required that Introduction191(8) is true for all (x; t) 2 B(a; r) [T r2; T ) for some r > 0 (no sign conditionwas required there).The proof of Theorem 1 relies strongly on the characterization of all connec-tions between two critical points of equation (3) inL1loc. Due to [6], the onlybounded global nonnegative solutions of the stationary problem associated to(3) in RN are 0 and , provided that (N 2)p N +2. Here we classify the so-lutions w(y; s) of (3) de ned on RN R and connecting two of the cited criticalpoints between them, and we obtain the surprising result:Theorem 2 (Classi cation of connections between critical points of(3)) Assume that 1 < p and (N 2)p < N+2 and that w is a global nonnegativesolution of (3) de ned for (y; s) 2 RN R bounded in L1. Then necessarily oneof the following cases occurs:i) w 0 or w ,or ii) there exists s0 2 R such that 8(y; s) 2 RN R, w(y; s) = '(s s0) where'(s) = (1 + es) 1p 1 :(9)Note that ' is the unique global solution (up to a translation) of's = 'p 1 + 'psatisfying '! as s! 1 and '! 0 as s! +1:Remark: This result is in the same spirit as the result of Berestycki and Ni-renberg [1], and Gidas, Ni and Nirenberg [5]. Here, the moving plane techniqueis not used, even though the proof uses some elementary geometrical transfor-mations. It is unclear whether the result holds without a sign condition or not.The assumption w is bounded in L1 and is de ned for s up to +1 is not reallyneeded, in the following sense:Corollary 2 Assume that 1 < p and (N 2)p < N+2 and that w a nonnegativesolution of (3) de ned for (y; s) 2 RN ( 1; s ) where s is nite or s = +1.Assume in addition that there is a constant C0 such that 8a 2 RN , 8s s ,Ea(w(s)) C0, where Ea(w(s)) = E(w(: + ae s2 ; s))(10)and E is de ned in (5). Then, one of the following cases occurs:i) w 0 or w ,or ii) 9s0 2 R such that 8(y; s) 2 RN ( 1; s ), w(y; s) = '(s s0) where'(s) = (1 + es) 1p 1 ;or iii) 9s0 s such that 8(y; s) 2 RN ( 1; s ), w(y; s) = (s s0) where(s) = (1 es) 1p 1 :Theorem 2 has an equivalent formulation for solutions of (1):Corollary 3 (A Liouville theorem for equation (1)) Assume that 1 < pand (N 2)p < N +2 and that u is a nonnegative solution in L1 of (1) de nedfor (x; t) 2 RN ( 1; T ). Assume in addition that 0 u(x; t) C(T t) 1p 1 .Then u 0 or there exist T0 T such that 8(x; t) 2 RN ( 1; T ), u(x; t) =(T0 t) 1p 1 . 192Optimal estimates for blow-up rate and behaviorRemark: u 0 or u blows-up in nite time T0 T .The third main result of the paper shows that near blow-up time, the solu-tions of equation (1) behave globally in space like the solutions of the associatedODE:Theorem 3 Assume that is a convex bounded C2; domain in RN or =RN . Consider u(t) a nonnegative solution of equation (1) which blows-up attime T > 0. Assume in addition that u(0) 2 H1(RN ) if = RN . Then 8 > 0,9C > 0 such that 8t 2[T2 ; T ), 8x 2 ,j@u@t jujp 1uj jujp + C :(11)Remark: (11) is true until the blow-up time. Let us point out that the result isglobal in time and in space. The same result holds for a sequence un as before(Theorem 1'). For clear reasons, the result is optimal.Remark: Let us note that the result is still true for equation@u@t = r:(a(x)ru) + b(x)f(u)where f(u) up as u ! +1, (a(x)) is a symmetric, bounded and uniformlyelliptic matrix, b(x) is bounded, and a(x) and b(x) are C1.The conclusion in this case isj@u@t b(x)f(u)j jf(u)j+ C :It is unclear whether Theorems 1, 2 and 3 hold without a sign condition.Remark: u0 = up is a reversible equation. Therefore the non reversible equationbehaves like a reversible equation near and at the blow-up time. Theorem 3localizes the equation. In particular, it shows that the interactions between twosingularities or one singularity and the \regular" region are bounded up to theblow-up time.Note that Theorem 3 has obvious corollaries. For example:If x0 is a blow-up point, thenu(x; t)! +1 as (x; t)! (x0; T ) (In other words, u is a continuous functionin R of (x; t) 2(0; T )).9 0 > 0 such that for all x 2 B(x0; 0) and t 2 (T0; T ), we have@u@t (x; t) > 0.Let us notice that theorems 1 and 3 have interesting applications in theunderstanding of the asymptotic behavior of blow-up solutions u(t) of (1) near agiven blow-up point x0. Various points of view has been adopted in the literature([8], [2], [9], [14]) to describe this behavior. In [12], we sharpen these estimatesand put them in a relation.In the second section, we see how Theorems 1 and 3 are proved using Theo-rem 2. The third section is devoted to the proof of Theorem 2. Optimal blow-up estimates for equation (1)1932 Optimal blow-up estimates for equation (1)In this section, we assume that Theorem 2 holds and prove Theorems 1 and1', Corollary 1 and Theorem 3. The rst three are mainly a consequence ofcompactness procedure and Theorem 2. Theorem 3 follows from Theorem 1 andscaling properties of equation (1) used in a suitable way.2.1 L1 estimates for the solution of (1)We prove Theorems 1 and 1' and Corollary 1 in this subsection.Proof of Theorem 1: Let u(t) be a nonnegative solution of equation (1) de -ned on [0; T ), which blows-up at time T and satis es u(0) 2 H1( ). It is clearthat the estimates on wa for all a 2 follow from the estimates on u by (2).In addition, the estimates on u follow from the estimates on wa for a particulara 2 still by (2). Hence, we consider a 2 a blow-up point of u and prove theestimates on this particular wa de ned bywa(y; s) = e sp 1u(a+ ye s2 ; T e s):Note that we have 8a; b 2 , 8(y; s) 2 Db;s,wb(y; s) = wa(y + (b a)e s2 ; s):We proceed in three steps: in a rst step, we show that wa, rwa and wa areuniformly bounded (without any precision on the bounds). Then, we show inStep 2 that blow-up for equation (1) must occur inside a compact set Kand that u, ru and u are bounded in nK. We nally nd the optimal boundson wa through a contradiction argument.Let us recall the expression of the energy E(w) introduced in (5), since itwill be useful for further estimates:E(wa) = 12 Z jrwaj2 dy + 12(p 1) Z jwaj2 dy1p+ 1 Z jwajp+1 dy(12)where is de ned in (6) and integration is done over the de nition set of w. Bymeans of the transformation (2), (12) yields a local energy for equation (1):Ea;t(u) = t 2p 1 N2 +1 Z 12 jru(x)j21p+ 1 ju(x)jp+1 (x apt )dx+12(p 1) t 2p 1 N2 Z ju(x)j2 (x apt )dx:(13)Without loss of generality, we can suppose a = 0. We recall that the notationL1 stands forL1(D0;s).Step 1: Preliminary estimates on wLemma 2.1 (Giga-Kohn, Uniform estimates on w) There exists apositive constant M such that 8slogT + 1, 8y 2 D0;s,jw0(y; s)j+ jrw0(y; s)j+ j w0(y; s)j+ jr w0(t; s)j Mand j@w@s (y; s)j M(1 + jyj): 194Optimal estimates for blow-up rate and behaviorLet us recall the main steps of the proof:Since u(0) 0, we know from Giga and Kohn [8] that there exists B > 0 suchthat8t 2 [0; T ); 8x 2 ; ju(x; t)j B(T t) 1p 1 :(14)In order to prove this, they argue by contradiction and construct by scalingproperties of equation (3) a solution of8<: 0 = v + vp in RNv0v(0)12which does not exist if (N 2)p < N + 2 and p > 1.The estimate on w0 is equivalent to (14).For s0logT +1 and y0 2 D0;s0 , consider W (y0; s0) =w0(y0 + y0e s2 ; s0 +s0). Then W (0; 0) = w0(y0; s0) and W satis es also (3). If y0e s02 (which is in) is not near the boundary, then we have jW (y0; s0)j M for all (y0; s0) 2B(0; 1) [ 1; 1]. By parabolic regularity (see lemma 3.3 in [7] for a statement),we obtain jrW (0; 0)j + j W (0; 0)j + jr W (0; 0)j M 0 = M 0(M). If y0e s02is near the boundary, then lemma 3.4 in [7] allows to get the same conclusion.Since this is true for all (y0; s0), we have the bound for rw0, w0 and r w0.The estimate on @w0@s follows then by equation (3).Step 2: No blow-up for u outside a compactProposition 2.1 (Uniform boundedness of u(x; t) outside a compact)Assume that = RN and u(0) 2 H1(RN ), or that is a convex bounded C2;domain. Then there exist C > 0, t1 < T and K a compact set of such that8t 2 [t1; T ), 8x 2 nK, ju(x; t)j+ jru(x; t)j+ j u(x; t)j C.Proof: Case = RN and u(0) 2 H1(RN ): Giga and Kohn prove in [8] thatuniform estimates on Ea;t (13) give uniform estimates inL1loc on the solution of(1). More precisely,Proposition 2.2 (Giga-Kohn) Let u be a solution of equation (1).i) If for all x 2 B(x0; ), Ex;T t0(u(t0)), then 8x 2 B(x0; 2 ), 8t 2( t0+T2 ; T ), ju(t; x)j ( )(T t) 1p 1 where ( ) c , > 0, and c anddepend only on p.ii) Assume in addition that 8x 2 B(x0; ), ju( t0+T2 ; x)j M . There exists0 = 0(p) > 0 such that if0, then 8x 2 B(x0; 4 ), 8t 2 ( t0+T2 ; T ),ju(t; x)j M where M depends only on M , , T and t0.Proof: see Proposition 3.5 and Theorem 2.1 in [8].Now, since u(0) 2 H1(RN ), we have u(t) 2 H1(RN ) for all t 2 [0; T ).Therefore, for xed t0 and0, (13), (6) and the dominated convergencetheorem yield the existence of a compact K0 RN such that 8x 2 RN nK0,Ex;T t0(u(t0)) .Hence, ii) of Proposition 2.2 applied to u(:+ x1; :) for x1 2 K0 and with = 1,asserts the existence of a compact K1 RN such that 8x 2 RN nK1, 8t 2( t0+T2 ; T ), ju(x; t)j M .Parabolic regularity (see lemma 3.3 in [7] for a statement) implies the esti-mates on ru and u on nK with a compact K containing K1. Optimal blow-up estimates for equation (1)195Case is a bounded convex C2; domain: The main feature in the proofof the estimate on ju(x; t)j is the result of Giga and Kohn which asserts thatblow-up can not occur at the boundary (Theorem 5.3 in [8]). The bounds onru and u follow from a similar argument as before (see lemma 3.4 in [7]).Step 3: Conclusion of the proofThe result has been proved pointwise. Therefore, the question is in somesense to prove it uniformly.We want to prove thatkw0(s)kL1 ! as s! +1.From [7] and [8], we know that jwb(0; s)j ! as s ! +1 if b is a blow-uppoint. Sincekw0(s)kL1 jw0(ae s2 ; s)j = jwa(0; s)j, this implies thatlim infs!+1 kw0(s)kL1and lim infs!+1 kw0(s)kL1 +krw0(s)kL1 + kw0(s)kL1 :(15)The conclusion will follow if we show thatlim sups!+1 kw0(s)kL1 +krw0(s)kL1 + kw0(s)kL1 :(16)Let us argue by contradiction and suppose that there exists a sequence(sn)n2N such that sn ! +1 as n! +1 andlimn!+1 kw0(sn)kL1 +krw0(sn)kL1 + kw0(sn)kL1 = + 3 0 where 0 > 0:We claim that (up to extracting a subsequence), we haveeither limn!+1 kw0(sn)kL1 = + 0orlimn!+1 krw0(sn)kL1 = 0orlimn!+1 kw0(sn)kL1 = 0:(17)From Proposition 2.1 and the scaling (2), we deduce for n large enough theexistence of y(0)n , y(1)n and y(2)n in D0;sn such thatkw0(sn)kL1 =jw0(y(0)n ; sn)j;orkrw0(sn)kL1 =jrw0(y(1)n ; sn)j;or kw0(sn)kL1 = jw0(y(2)n ; sn)j:(18)Let yn = y(i)n where i is the number of the case which occurs. Since yn 2 D0;sn ,(4) implies that yne sn=2 2 . Therefore, we can use (2) and de ne for eachn 2 N vn(y; s) = wyne sn=2(y; s+ sn)= e s+snp 1 u(ye s+sn2 + yne sn=2; T e (s+sn))= w0(y +ynes=2; s+ sn)(19)We claim that (vn) is a sequence of solutions of (3) which is compact inC3loc(RN R). More precisely, 196Optimal estimates for blow-up rate and behaviorLemma 2.2 (vn)n2N is a sequence of solutions of (3) with the following pro-perties:i) limn!+1 jvn(0; 0)j = + 0 or limn!+1 jrvn(0; 0)j = 0or limn!+1 j vn(0; 0)j = 0.ii) 8R > 0, 9n0 2 N such that 8n n0,vn(y; s) is de ned for (y; s) 2 B(0; R) [ R;R],vn 0 and kvnkL1(B(0;R) [ R;R]) B where B is de ned in (14).9m(R) > 0 such that kvnkC3(B(0;R) [ R;R]) m(R).Proof: i) vn satis es (3) since wyne sn=2 does the same. From (19), (17) and (18),we obtain i): limn!+1 jvn(0; 0)j = + 0 or limn!+1 jrvn(0; 0)j = 0or limn!+1 j vn(0; 0)j = 0.ii) Let R > 0.If = RN , then it is obvious form (19) that vn is de ned for (y; s) 2B(0; R) [ R;R] for large n.If is bounded, then we can suppose that up to extracting a subsequence,yne sn=2 converges to y1 2 as n ! +1. In fact y1 2 . Indeed, sinceu(y(0)n e sn=2; T e sn) = e snp 1 vn(0; 0)! +1 as n! +1(or jru(y(1)n e sn=2; T e sn)j = esn( 1p 1+ 12 )jrvn(0; 0)j ! +1, orj u(y(2)n e sn=2; T e sn)j = esn( 1p1+1)j vn(0; 0)j ! +1), in all cases, y1 is ablow-up point of u. Therefore, Step 2 implies that y1 2 K and that B(y1; 0)for some 0 > 0. Together with (19), this implies that vn is de ned for(y; s) 2 B(0; R) [ R;R] for large n.From (19), (14) and the fact that u 0, it directly follows that vn(y; s) 0and kvnkL1(B(0;R) [ R;R]) B.From lemma 2.1 and (19), it directly follows that 8(y; s) 2 B(0; R) [ R;R],jvn(y; s)j+jrvn(y; s)j+j vn(y; s)j+jr vn(y; s)j M and j@vn@s j M (1+R).Since w 0, parabolic estimates and strong maximum principle imply thatkvnkC3(B(o;R) [ R;R]) m(R) for some m(R) > 0. Just take m(R) =M (1 +R).Now, using the compactness property of (vn) shown in lemma 2.2, we ndv 2 C2(RN R) such that up to extracting a subsequence, vn ! v as n! +1in C2loc(RN R). From lemma 2.2, it directly follows thati) v satis es equation (3) for (y; s) 2 RN Rii) v 0 and kvkL1(RN R) Biii) jv(0; 0)j = + 0 or jrv(0; 0)j = 0 or j v(0; 0)j = 0 with 0 > 0.By Theorem 2, i) and ii) imply v 0 or vor v = '(s s0) where'(s) = (1 + es) 1p 1 . In all cases, this contradicts iii). Thus, Theorem 1 isproved.Proof of Theorem 1': The proof of Theorem 1' is similar to the proof ofTheorem 1. Let us sketch the main di erences.Step 1: One can remark that a uniform estimate on E(wn;a(s0)) wheres0 = logT is needed. Sinceku0kH2( ) is uniformly bounded, we have theconclusion.Step 2: One can use a uniform version of Giga and Kohn's estimates, asthey are stated (for example) in [11]. Optimal blow-up estimates for equation (1)197Step 3: Same proof.Proof of Corollary 1: Let us prove Corollary 1 now.We argue by contradictionand assume that for some 0 > 0, there is tn ! T and (an)n a sequence of blow-up points of u in such that8n 2 N; 0 u(an; tn) (1 0) (T tn) 1p 1 :Let us give two di erent proofs:Proof 1: Consider the following solution of equation (3):vn(y; s) = wan(y; s log(T tn)):From Proposition 2.1, an 2 K, since it is a blow-up point of u. As before, wecan use a compactness procedure on vn to get a nonnegative bounded solutionv of (3) de ned for (y; s) 2 RN R such that jv(0; 0)j (1 0) and vn ! v inC2loc. Therefore, Theorem 2 implies that v 0 or v = '(s s0) for some s0 2 R.In particular, E(v(0)) < E( ). Since E(vn(0))! E(v(0)) as n! +1, we havefor n large E(wan( log(T tn))) = E(vn(0)) < E( ), and in particular an cannot be a blow-up point of u (we have from [6], for any blow-up point a of u,E(wa(s)) E( ) for all slogT ). From this fact, a contradiction follows.Proof 2: It is a more elementary proof based on Theorem 3. Since an is ablow-up point and that the blow-up set is closed and bounded (see Proposition2.1), we can assume that an ! a1 where a1 is a blow-up point.We know from Theorem 3 that for some C 02 , we have 8x 2 , 8t 2[T2 ; T ),@u@t (x; t) up(x; t)202 ju(x; t)jp + C202 :(20)In particular, u(x; t)! +1 as (x; t)! (a1; T ) (see next subsection for a proofof Theorem 3 and this fact (22)-(23)). Let > 0 such that8(x; t) 2 B(0; ) (T ; T ); C202 < 202 up(x; t):(21)For large n, an 2 B(a1; ) and tn 2 [T ; T ). Therefore (20) and (21) yield8t 2 [tn; T ); @u@t (an; t) (1 +20)up(an; t):Since 0 < u(an; tn)(10)(T tn) 1p 1 , we get by direct integration:8t 2 [tn;min(T; T ( 0))),0 u(an; t)T tn(10)p 1 (1 +20)(t tn)1p 1with T ( 0) = tn +T tn(1+ 0)(1 0)p 1 > T if 0 < 1(p) for some positive 1(p).Thus, an is not a blow-up point and a contradiction follows. 198Optimal estimates for blow-up rate and behavior2.2 Global approximated behavior like an ODEWe prove Theorem 3 in this subsection. It follows from Theorem 1 andpropagation of atness (through scaling arguments) observed in [14].Let us rst show how to derive the consequences of Theorem 3 announcedin the introduction:If x0 is a blow-up point of u(t), thenu(x; t)! +1 as (x; t)! (x0; T )(22) and 9 0 > 0 such that 8(x; t) 2 B(x0; 0) (T0; T ); @u@t (x; t) > 0:(23)Proof of (22) and (23):From Theorem 3 applied with > 0, there exists C such that 8(x; t) 2[T2 ; T )@u@t (x; t) (1 )up(x; t) C :(24)Let A be an arbitrary large positive number satisfying(1 )Ap C > 0:(25)From the continuity of u(x; t), there exist 1 > 0 and 2 > 0 such that 8x 2B(x0; 1),u(x; T2) > A:(26)From (24) and (25), we have 8x 2 B(x0; 1), @u@t (x; T2) > 0. Now we claimthat 8(x; t) 2 B(x0; 1) (T 2; T ), u(x; t) > A (which yields (22) and (23) also,by (24) and (25)). Indeed, if not, then there exists (x1; t1) 2 B(x0; 1) (T 2; T )such that u(x1; t1) A. From the continuity of u, we get t2 2 (T2; t1] suchthat 8t 2 (T2; t2), u(x1; t) > A and u(x1; t2) = A. From (24) and (25), wehave 8t 2 (T2; t2), @u@t (x1; t) > 0, therefore, u(x1; t2) > u(x1; T2) > A by(26). Thus, a contradiction follows, and (22) and (23) are proved.We now prove Theorem 3.Proof of Theorem 3: Let us argue by contradiction and suppose that for some0 > 0, there exist (xn; tn)n2N a sequence of elements of[T2 ; T ) such that8n 2 N,j u(xn; tn)j0ju(xn; tn)jp + n:(27)Since ku(t)kL1( ) is bounded on compact sets of[T2 ; T ), we have that tn ! Tas n ! +1. We can also assume the existence of x1 2 such that xn ! x1as n ! +1. Indeed, if not, then either d(xn; @ ) ! 0 (if is bounded) orjxnj ! +1 (if = RN ) as n ! +1, and in both cases, (27) is no longersatis ed for large n, thanks to Proposition 2.1.We claim that x1 is a blow-up point of u. Indeed, if not, then parabolicregularity implies the existence of a positive such thatku(:; t)kW2;1(B(x1; )) C for some positive C, which is a contradiction by (27).Theorem 1 implies that u(xn; tn)(T tn) 1p 1 is uniformly bounded, therefore,we can assume that it converges as n! +1.Let us consider two cases:Case 1: u(xn; tn)(T tn) 1p 1 ! 0 > 0 ((xn; tn) is in some sense in the singularregion \near" (x1; T )). From (27), it follows that ku(tn)kL1 j u(xn; tn)j Optimal blow-up estimates for equation (1)1990 02 p (T tn) pp 1 with tn ! T , which contradicts Theorem 1.Case 2: u(xn; tn)(T tn) 1p 1 ! 0 ((xn; tn) is in the transitory region betweenthe singular and the regular sets).Let us rst de ne (t(xn))n such that t(xn) tn, t(xn)! T andu(xn; t(xn))(T t(xn)) 1p 1 = 0(28)where 0 2 (0; ) satis es 8t > 0, 8a 2 , Ea;t( 0t 1p 1 )202(p 1)p+10p+102and 0 is de ned in Proposition 2.2.Step 1: Existence of t(xn)Since x1 is a blow-up point of u, limt!T u(x1; t)(T t) 1p 1 = . It followsthat for any > 0 small enough, there exists a ball B(x1; 0) such that 8x 2B(x1; 0),1p 1u(x; T ) 3 + 04 . Since xn ! x1 as n ! +1, this impliesthat8n n1;1p 1u(xn; T )+ 02(29)for some n1 = n1( ) 2 N. Since u(xn; tn)(T tn) 1p 1 ! 0, we have the existenceof t (xn) 2 [T ; tn] [T ; T ) such that u(xn; t (xn))(T t (xn)) 1p 1 = 0,for all n n2( ), where n2( ) 2 N. Since was arbitrarily small, it follows froma diagonal extraction argument that there exists a subsequence t(xn) ! T asn! +1 such that t(xn) tn andu(xn; t(xn))(T t(xn)) 1p 1 = 0:Now, we claim that a contradiction follows if we prove the following Propo-sition:Proposition 2.3 Letvn( ; ) = (T t(xn)) 1p 1u(xn + pT t(xn); t(xn) + (T t(xn))):(30)Then, vn is a solution of (1) for 2 [0; 1), and there exists n0 2 N such that8n n0,8 2 [0; 1); j vn(0; )j02 jvn(0; )jp:(31)Indeed, from (31) and (30), we obtain: 8n n0, 8t 2 [t(xn); T ),j u(xn; t)j = (T t(xn)) ( 1p1+1)j vn(0; (t; n))j02 (T t(xn)) pp 1 jvn(0; (t; n))jp = 02 ju(xn; t)jp with (t; n) = t t(xn)T t(xn) ,which contradicts (27), since tn t(xn). Thus, Theorem 3 is proved.Step 2: Flatness of vnIn this Step we prove Proposition 2.3.We claim that the following lemma concludes the proof of Proposition 2.3:Lemma 2.3 i) 8 0 > 0, 8A > 0, 9n3( 0; A) 2 N such that 8n n3( 0; A),for all j j A and 2 [0; 34 ], jvn( ; 0)0j0, jr vn( ; )j0 andj vn( ; )j0.ii) 8 > 0, 8A > 0, 9n4( ; A) 2 N such that 8n n4, 8 2 [0; 1), forj jA4 , jvn( ; ) v̂( )j , jrvn( ; )j and j vn( ; )j where v̂( ) =0 p 11p 1 is a solution of dv̂d = v̂p with v̂(0) = 0. 200Optimal estimates for blow-up rate and behaviorIndeed, if is small enough and n is large enough, then 8 2 [0; 1), vn(0; )12 v̂(0) = 02 and j vn(0; )j02 p 0202 jvn(0; )jp.Proof of lemma 2.3: i) Let 0 > 0 and A > 0. From (28) and (30), we have:for all j j A and 2 [0; 34 ]:vn(0; 0) = 0,jvn( ; 0) vn(0; 0)j (T t(xn)) 1p 1+12Akru(t(xn))kL1( ),rvn( ; ) = (T t(xn)) 1p 1+12ru xn +pT t(xn); t(xn) + (T t(xn))= 111p 1+ 12 (T (t(xn) + (T t(xn)))) 1p 1+ 12ru xn + pT t(xn); t(xn) + (T t(xn)) andvn( ; ) = (T t(xn)) 1p 1+1 u xn + pT t(xn); t(xn) + (T t(xn)) =111p 1+1 (T (t(xn) + (T t(xn)))) 1p 1+1u xn + pT t(xn); t(xn) + (T t(xn)) .Since34 , t(xn)! T as n! +1, and (T t) 1p 1+ 12 kru(t)kL1( ) +(T t) 1p1+1k u(t)kL1( ) ! 0 as t! T (Theorem 1), i) is proved.ii) From i) and continuity arguments, it follows that for all j j A,E ;1 (vn(0)) 2E ;1( 0)0 for n large enough, by de nition of 0. Therefore,from Proposition 2.2 (applied with = 1 and using translation invariance), wehave 8 2 [ 12 ; 1), 8j jA2 , jvn( ; )j M(p).By classical parabolic arguments, we get8 2 [ 34 ; 1); 8j jA3 ; jvnj+ jrvnj+ j vnj M(p):(32)Now, using i), (32) and classical estimates for the heat ow, we get for all > 0:8j jA4 , 8 2 [0; 1), jrvn( ; )j and j vn( ; )j if n n5( ; A).Since vn is a solution of equation (1), combining this with i) and ODEestimates yields for all > 0: 8j jA4 , 8 2 [0; 1), jvn( ; ) v̂( )jifn n6( ; A). This concludes the proof of ii).3 Classi cation of connections between criticalpoints of equation (3) inL1locWe prove Theorem 2 and Corollaries 2 and 3 in this section.We rst prove Theorem 2, and then we show how Corollaries 2 and 3 can bededuced from Theorem 2.Proof of Theorem 2: We assume that 1 < p and (N 2)p < N + 2, andconsider w(y; s) a nonnegative global bounded solution of (3) de ned for (y; s) 2RN R. Our goal is to show that w depends only on time s.We proceed in 5 steps.In Step 1, we show that w has a limit w 1 as s ! 1, where w 1 is acritical point of (3), that is w 1 0 or w 1 . We focus then on the nontrivial case, that is w 1and w+1 0.In Step 2, we investigate the linear problem around , as s! 1, and showthat w would behave at most in three ways. Connections between critical points of equation (3)201In Step 3, we show that among these three ways we have the situationw(y; s) = '(s s0) with '(s) = (1 + es) 1p 1 . We then show (respectively inStep 4 and in Step 5) that the two other ways actually can not occur, we nd infact a contradiction through a blow-up argument for w(s) using the geometricaltransformation: a! wa de ned by wa(y; s) = w(y + ae s2 ; s)(33)(wa is also a solution of (3)) and a blow-up criterion for equation (3).Step 1: Behavior of w as s! 1This step can be found in Giga and Kohn [6]. The results are mainly conse-quences of parabolic estimates and the gradient structure of equation (3). Letus recall them brie y. We rst restate lemma 2.1 of section 2:Lemma 3.1 (Parabolic estimates) There is a positive constantM such that8(y; s) 2 RN R,jw(y; s)j + jrw(y; s)j+ j w(y; s)j M and @w@s (y; s) M(1 + jyj):Lemma 3.2 (Stationary solutions) Assume p (N+2)=(N 2) or N 2.Then the only nonnegative bounded global solutions in RN of0 = w 12y:rw wp 1 + jwjp 1w(34)are the trivial ones: w 0 and w .Proof: The following Pohozaev identity can be derived for each bounded solu-tions of equation (3) in RN (see Proposition 2 in [6]):(N + 2 p(N 2)) Z jrwj2 dy + p 12 Z jyj2jrwj2 dy = 0:Hence, for (N 2)p N + 2, w is constant. Thus, w 0 or w .Lemma 3.3 (Gradient structure) Assume p < (N + 2)=(N 2) or N 2.We de ne for each w solution of (3)E(w) = 12ZRN jrwj2 dy + 12(p 1)ZRN jwj2 dy1p+ 1ZRN jwjp+1 dy(35)where (y) = e jyj2=4(4 )N=2 :(36)Then, 8s1; s2 2 R,Z s2s1 ZRN @w@s 2 dyds = E(w(s1)) E(w(s2))(37) 202Optimal estimates for blow-up rate and behaviorOutline of the proof: (see Proposition 3 in [6] for more details).One may multiply equation (3) by @w@s and integrate over the ball B(0; R)with R > 0. Then, using lemma 3.1 and the dominated convergence theoremyields the result.Proposition 3.1 (Limit of w as s! 1) Assume p < (N + 2)=(N 2) orN 2. Let w be a bounded nonnegative global solution of (3) in RN+1 . Thenw+1(y) = lims!+1w(y; s) exists and equals 0 or . The convergence is uniformon every compact subset of RN . The corresponding statements hold also for thelimit w 1(y) = lims! 1w(y; s).Outline of the proof: (see Propositions 4 and 5 in [6] for more details).Let (sj) be a sequence tending to +1, and let wj(y; s) = w(y; s+ sj). Fromlemma 3.1, (wj) converges uniformly on compact sets to some w+1(y; s) andrwj ! rw+1 a.e. Assuming that sj+1 sj ! +1, one can use lemma 3.3 toshow that wj does not depend on s. Therefore, w+1 0 or w+1 = by lemma3.2. The continuity of w then asserts that w+1 does not depend on the choiceof the subsequence (sj). The analysis in 1 is completely parallel.According to (37) (with s1 ! 1 and s2 ! +1), there are only two cases:E(w 1) E(w+1) = 0: hence, @w@s 0. Therefore, w is a bounded globalsolution of (34). Thus, w 0 or waccording to lemma 3.2. This case hasbeen treated by Giga and Kohn in [6].E(w 1) E(w+1) > 0: since E( ) = (121p+1 ) p+1 R dy > 0 = E(0), wehave (w 1; w+1) = ( ; 0). It remains to treat this case in order to nish theproof of Theorem 2.In the following steps, we consider the case(w 1; w+1) = ( ; 0):Step 2: Classi cation of the behavior of w as s! 1:Since w is globally bounded in L1 and w ! as s ! 1, uniformly oncompact subsets of RN , we have lims! 1 kwkL2 = 0 where L2 is the L2-spaceassociated to the Gaussian measure (y)dy and is de ned in (36).In this part, we classify the L2 behavior of w as s! 1. Let us introducev = w . From (3), v satis es the following equation: 8(y; s) 2 RN+1 ,@v@s = Lv + f(v)(38)whereLv = v 12y:rv + v and f(v) = jv + jp 1(v + ) p p p 1v:(39)Since w is bounded in L1, we can assume jv(y; s)j M , and then jf(v)j Cv2with C = C(M).L is self-adjoint on D(L) L2 . Its spectrum isspec(L) = f1m2 jm 2 Ng; Connections between critical points of equation (3)203and it consists of eigenvalues. The eigenfunctions of L are derived from Hermitepolynomials:{ N = 1:All the eigenvalues of L are simple. For 1m2 corresponds the eigenfunctionhm(y) = [m2 ]Xn=0 m!n!(m 2n)! ( 1)nym 2n:(40)hm satis es R hnhm dy = 2nn! nm. Let us introducekm =hm=khmk2L2 :(41){ N 2:We write the spectrum of L asspec(L) = f1 m1 + :::+mN2jm1; :::;mN 2 Ng:For (m1; :::;mN ) 2 NN , the eigenfunction corresponding to1 m1+:::+mN2isy ! hm1(y1):::hmN (yN );where hm is de ned in (40). In particular,*1 is an eigenvalue of multiplicity 1, and the corresponding eigenfunctionisH0(y) = 1;(42)* 12 is of multiplicity N , and its eigenspace is generated by the orthogonalbasis fH1;i(y)ji = 1; :::; Ng, with H1;i(y) = h1(yi); we noteH1(y) = (H1;1(y); :::; H1;N (y));(43)*0 is of multiplicity N(N+1)2 , and its eigenspace is generated by the ortho-gonal basis fH2;ij(y)ji; j = 1; :::; N; i jg, with H2;ii(y) = h2(yi), and fori < j, H2;ij(y) = h1(yi)h1(yj); we noteH2(y) = (H2;ij(y); i j):(44)Since the eigenfunctions of L constitute a total orthonormal family of L2 ,we expand v as follows:v(y; s) = 2Xm=0 vm(s):Hm(y) + v (y; s)(45)wherev0(s) is the projection of v on H0,v1;i(s) is the projection of v on H1;i, v1(s) = (v1;i(s); :::; v1;N (s)), H1(y) is givenby (43),v2;ij(s) is the projection of v on H2;ij , i j; v2(s) = (v2;ij(s); i j), H2(y) is 204Optimal estimates for blow-up rate and behaviorgiven by (44),v (y; s) = P (v) and P the projector on the negative subspace of L.With respect to the positive, null and negative subspaces of L, we writev(y; s) = v+(y; s) + vnull(y; s) + v (y; s)(46)where v+(y; s) = P+(v)=P1m=0 vm(s):Hm(y),vnull(y; s) = Pnull(v) = v2(s):H2(y), P+ and Pnull are the L2 projectors respec-tively on the positive subspace and the null subspace of L.Now, we show that as s ! 1, either v0(s), v1(s) or v2(s) is predominantwith respect to the expansion (45) of v in L2 . At this level, we are not able touse a center manifold theory to get the result (see [3] page 834-835 for moredetails). In some sense, we are not able to say that the nonlinear terms in thefunction of space are small enough. However, using similar techniques as in [3],we are able to prove the result. We have the following:Proposition 3.2 (Classi cation of the behavior of v(y; s) as s ! 1 )As s! 1, one of the following situations occurs:i) jv1(s)j+ kvnull(y;s)kL2 + kv (y;s)kL2 = o(v0(s)),ii) jv0(s)j+ kvnull(y;s)kL2 + kv (y;s)kL2 = o(jv1(s)j),iii) kv+(y;s)kL2 + kv (y;s)kL2 = o(kvnull(y;s)kL2 ).Proof: See Appendix A.Now we handle successively the three cases suggested by proposition 3.2 toshow that only case i) occurs.In case i), we end up to show that w(y; s) = '(s s0) for some s0 2 R, where' is de ned in (9). In cases ii) and iii), we show that the solutions satisfy throughan elementary geometrical transformation a blow-up condition for equation (3)considered for increasing s, which contradicts their boundedness, and concludesthe proof of Theorem 2.Step 3: Case i) of Proposition 3.2: 9s0 2 R such that w(y; s) = '(s s0)Proposition 3.3 Suppose that jv1(s)j+ kvnull(y;s)kL2 + kv (y;s)kL2= o(v0(s)) as s! 1, then there exists s0 2 R such that:i) 8 > 0, v0(s) = p 1es s0 +O(e(2 )s) as s! 1,ii) 8(y; s) 2 RN+1 w(y; s) = '(s s0) where '(s) = (1 + es) 1p 1 .Remark: This proposition asserts that if a solution of (38) behaves like aconstant independent of y (that is like v0(s)), then it is exactly a constant.Proof: i) See Step 3 of Appendix A and take s0 = log( (p 1)C0).We remark that we already know a solution of equation (38) which behaveslike i). Indeed, '(s s0)= ('(s s0) )h0 is a solution of (38) whichsatis es '(s s0) = p 1es s0 +O(e(2 )s) as s! 1:From a dimension argument, we expect that for each parameter, there is at mostone solution such that:v0(s) p 1es s0 as s! 1: Connections between critical points of equation (3)205(if for example, center manifold analysis applies). We propose to prove this fact.In other words, our goal is to show that8(y; s) 2 RN+1 ; v(y; s) = '(s s0) :Since (38) is invariant under translations in time, we can assume s0 = 0 withoutloss of generality.For this purpose, we introduceV (y; s) = v(y; s) ('(s) ) = w(y; s) '(s):(47)From (3), V satis es the following equation:@V@s = (L+ l(s))V + F (V )where L =12y:r+ 1, l(s) =pes(p 1)(1+es) andF (V ) = j'+ V jp 1('+ V ) 'p p'p 1V . Note that 8s 0, jF (V )j CjV j2where C = C(M) and MkvkL1 .We know from Step 3 in Appendix A thatjV0(s)j+ jV1(s)j = O(e(2 )s); kVnull(s)kL2 = o(es) as s! 1:The following Proposition asserts that V 0, which concludes the proof ofProposition 3.3:Proposition 3.4 Let V be an L1 solution of@V@s = (L+ l(s))V + F (V )de ned for (y; s) 2 RN R such that V ! 0 as s! 1 uniformly on compactsets of RN ,jV0(s)j+ jV1(s)j = O(e(2 )s) andkVnull(s)kL2 = o(es) as s! 1:Then V 0.Proof: see Appendix B.Step 4: Irrelevance of the case where v1(s) is preponderantIn this case ii) of Proposition 3.3, we use the main term in the expansion ofv(s) as s! 1 to nd a0 and s0 such thatZ wa0(y; s0) (y)dy >(48)where wa0 is de ned in (33). Since w 0, we nd that (48) implies that wa0(which is also a solution of (3)) blows-up in nite time S > s0 (and so does w),which contradicts the fact that w is globally bounded. It is in fact mainly theonly place where the hypothesis w 0is used. More precisely, let us state the following Proposition:Proposition 3.5 (A blow-up criterion for equation (3)) Consider W 0a solution of (3) and suppose that for some s0 2 R,R W (y; s0) (y)dy > R dy = . Then W blows-up in nite time S > s0. 206Optimal estimates for blow-up rate and behaviorProof: We argue by contradiction and suppose that W is de ned for all s 2[s0;+1). If V =W , then V satis es equation (38). Let us de nez0(s) = Z V (y; s) (y)dy:Integrating (38) with respect to dy, we obtainz00(s) = z0(s) + Z f(V (y; s)) dywhere f(x) = ( + x)p p p p 1x for + x 0:It is obvious that f is nonnegative and convex on [ ;+1). SinceW = +V0, 0 and R dy = 1, we have the following Jensen's inequality:Z f(V (y; s)) dy f(Z V (y; s) dy) = f(z0(s)):Therefore,z00(s) z0(s) + f(z0(s)):(49)Since f(x) > 0 for x > 0 (f is strictly convex and f(0) = f 0(0) = 0) andz0(s0) > 0 by the hypothesis, by classical arguments, we have 8s s0, z00(s) 0,therefore, 8s s0, z0(s) > 0. By direct integration, we have 8s s0,s s0 Z z0(s)z0(s0) dxf(x) Z +1z0(s0) dxf(x) :Since 1f(x)1jxjp as s ! +1, a contradiction follows and Proposition 3.5 isproved.Proposition 3.6 (Case where v1(s) is preponderant) Suppose thatjv0(s)j+ kvnull(y;s)kL2 + kv (y;s)kL2 = o(jv1(s)j), then:i) 9C1 2 RN nf0g such that v0(s) pjC1j2ses and v1(s) C1es=2 as s !1.ii) 9a0 2 RN , 9s0 2 R such that R wa0(y; s0) (y)dy > where wa0 introducedin (33) is a solution of equation (3) de ned for (y; s) 2 RN R satisfyingkwa0kL1(RN R) B.From Proposition 3.5, ii) is a contradiction.Remark: wa has a geometrical interpretation in terms of w(y; s). Indeed, fromw(y; s), we introduce u(x; t) (as in (2)) de ned for (x; t) 2 RN ( 1; 0) by:x = yp t ; s = log( t); u(x; t) = ( t) 1p 1w(y; s):Now, if we de ne ŵa(y; s) from u(x; t) by (2) asx = y ap t ; s = log( t); ŵa(y; s) = ( t) 1p 1u(x; t);then, ŵa wa. Connections between critical points of equation (3)207Proof of Proposition 3.6: i) follows from Step 3 in Appendix A.Therefore, we prove ii). It is easy to check that wa satis es (3). Moreover,from (33) we get kwakL1(RN R) = kwkL1(RN R) B. We want to show thatthere exist a 2 RN and s0 2 R such that R wa(y; s0) (y)dy > . From (33), wehave:R wa(y; s) dy = R w(y + aes=2; s) dy.Let us note = aes=2. The conclusion follows if we show that there exist s0 2 Rand (s0) 2 RN such that R w(y + (s0); s0) dy > .For this purpose, we search an expansion for R w(y + ; s) dy as s ! 1and ! 0.R w(y + ; s) dy = R w(y; s) (y )dy = + R v(y; s) e jy j24(4 )N=2 dy= + e j j24 R v(y; s) (y)e :y2 dy= + e j j24 R v(y; s) (y) 1 + :y2 + ( :y)28 R 10 (1 )e :y2 d dy= + (1 +O(j j2)) (v0(s) + :v1(s) + (I))where (I) = e j j24 R dyv(y; s) (y) ( :y)28 R 10 d (1 )e :y2 .Using Schwartz's inequality, we havej(I)j R v(y; s) (y)dy 1=2 R dy ( :y)464 (y) R 10 d (1 )e :y2 2 1=2kv(s)kL2 j j28 R dyjyj4 (y) R 10 d (1 )e jyj2 2 1=2kv(s)kL2 j j216 R dyjyj4 (y)ejyj 1=2 = Cjj2kv(s)kL2 .Therefore, using the fact thatkv(s)kL2 2jv1(s)j = O(es=2) and i), we get:R w(y + ; s) dy = + v0(s) + :v1(s) +O(j j2es=2)= + pjC1j2ses + o(ses) +:C1es=2 + o(j jes=2).Now, if we make = (s) =1s C1jC1j and take s large enough, then R w(y+(s); s)12 (s):C1es=2 = es=22s jC1j > 0, and the existence of a0 and s0 isproved.This concludes the proof of Proposition 3.6.Step 5: Irrelevance of the case where v2(s) is preponderantAs in the previous part, we use the information given by the linear theoryat 1 to nd a contradiction in the case where iii) holds in Proposition 3.2.Proposition 3.7 (Case where v2(s) is preponderant) Assume thatkv+(y;s)kL2 + kv (y;s)kL2 = o(kvnull(y;s)kL2 ), then:i) there exists0, k 2 f0; 1:::; N 1g and Q an orthonormal N Nmatrix such thatvnull(y; s) = yTA(s)y 2trA(s)where A(s) = 4psA0 +O( 1s1+ ) as s! 1;A0 = Q IN k 00 0 Q 1and IN k is the (N k) (N k) identity matrix. Moreover,kv(s)kL2 =psrN k2 +O 1jsj1+ ; v0(s) = O( 1s2 ) and v1(s) = O( 1s2 ): 208Optimal estimates for blow-up rate and behaviorii) 9a0 2 RN , 9s0 2 R such that R wa0(y; s0) (y)dy > where wa0 de ned in(33) is a solution of equation (3) satisfyingkwa0kL1(RN R) B.From ii) and Proposition 3.5, a contradiction follows.Proof of i) of Proposition 3.7:The rst part of the proof follows as before the ideas of Filippas and Kohnin [3]. Then, we carry on the proof similarly as Filippas and Liu did in [4] forthe same equation when the null mode dominates as s ! +1. Since the usedtechniques are the same than in [3] and [4], we leave the proof in Appendix C.Proof of ii) of Proposition 3.7:We proceed exactly in the same way as for the proof of ii) of Proposition3.6. wa satis es equation (3), and the L1 bound on wa follows as before.By setting = aes=2, the proof reduces then to nd s0 and = (s0) suchthat R w(y + (s0); s0) dy > .For this purpose, we search an expansion for R w(y + ; s) dy as s ! 1and ! 0.R w(y + ; s) dy = R w(y; s) (y )dy = + R v(y; s) e jy j24(4 )N=2 dy= + e j j24 R v(y; s) (y)e :y2 dy= + e j j24 R v(y; s) (y) 1 + :y2 + ( :y)28 + ( :y)316 R 10 (1 )2e :y2 d dy.We writeZ w(y + ; s) dy = + (I) + (II);(50)where(I) = e j j24 (v0(s) + :v1(s)) + e j j24 R dyv(y; s) (y) ( :y)316 R 10 d (1 )2e :y2and (II) =18e j j24 R v(y; s)( :y)2 (y)dy.From i) of Proposition 3.7 and Schwartz's inequality, we havej(I)j Cs2 + C j j3jsj :(51)Since v = v + vnull + v+ = v + vnull + v1:y + v0, we have from theorthogonality of v and vnull + v+:(II) = e j j248 R v(y; s)( :y)2 dy= e j j248 (v0(s) R ( :y)2 dy + v1(s): R y( :y)2 dy) + e j j248 R vnull( :y)2 dy= v0(s)O(j j2) + e j j248 R (yTA(s)y 2trA(s))( :y)2 dy= O j j2jsj1+ + e j j248 4pjsj R (yTA0y 2trA0)( :y)2 dyfor some > 0, according to i) of Proposition 3.7, withA0 = Q IN k 00 0 Q 1:With the change of variable y = Q 1z (Q is an orthonormal matrix) we write:(II) = O j j2jsj1+ + e j j248 4pjsj R N kXi=1 (z2i 2) (Q :z)2 (z)dz, Connections between critical points of equation (3)209therefore,(II) = 4pjsj N kXi=1 Z (z2i 2)(Q :z)2 dz +O j j2jsj1+ +O j j4jsj :(52)Gathering (50), (51) and (52), we write:R w(y + ; s) dy= + 4pjsj N kXi=1 Z (z2i 2)(Q :z)2 dz +O( 1s2 ) +O j j2jsj1+ +O j j3jsj :Now, if we take = (s) = 1jsj1=4Q 1e1 where e1 = (1; 0; :::; 0), thenZ w(y + (s); s) dy = + 4pjsj3=2 8 +O1jsj3=2+ :If we take s large enough, and a(s) = e s=2 (s), thenZ w(y + (s)es=2; s) > :This concludes the proof of ii) of Proposition 3.7 and the proof of Theorem2. We now prove Corollaries 1 and 2:Proof of Corollary 2:We consider w a nonnegative solution of (3) de ned for (y; s) 2 RN( 1; s ) where s 2 R [ f+1g. We assume that there is a constant C0 suchthat8a 2 RN ; 8s s ; Ea(w(s)) C0(53)where Ea is de ned in (10).Through some geometrical transformations, we de ne below ŵ, a solution of(3) de ned on RN R, which satis es the hypotheses of Theorem 2. Then, wededuce the characterization of w from the one given in Theorem 2 for ŵ.Let us de ne u(t) a solution of (1) by:y = xp t ; s = log( t); u(x; t) = ( t) 1p 1w(y; s)(54)where (x; t) 2 RN ( 1; T ) with T = e s if s is nite and T = 0 ifs = +1. Then we introduce ŵ a solution of (3):y = xpT t ; s = log(T t); ŵ(y; s) = (T t) 1p 1u(x; t)(55)de ned for (y; s) 2 RN R. We have then 8(y; s) 2 RN ( 1; s ),w(y; s) = (1 + T es) 1p 1 ŵ( yp1 + T es ; s log(1 + T es)):(56)We claim that ŵ 2 L1(RN R). Indeed, from (53), (54) and i) of Proposition2.2, we have 8(x; t) 2 RN ( 1; T ), ju(x; t)j M(C0)(T t) 1p 1 . Hence,(55) implies that 8(y; s) 2 RN R, jw(y; s)j M(C0). 210Optimal estimates for blow-up rate and behaviorSince w is nonnegative, ŵ is also nonnegative, and then, by Theorem 2 wehave:either ŵ 0, or ŵor ŵ(y; s) = '(s s0) for some s0 2 R, where '(s) = (1 + es) 1p 1 .Therefore, by (56), we have:either w 0, or w(y; s) = (1 es s ) 1p 1or w(y; s) = (1 es s ) 1p 1 1 + exp(s log(1 es s s0))1p 1= 1 + es(e s0 e s )1p 1 .Since s0 is arbitrary in R, this concludes the proof of Corollary 2.Proof of Corollary 3:Let u(x; t) be a nonnegative solution of (1) de ned for (x; t) 2 RN ( 1; T )which satis es ju(x; t)j C(T t) 1p 1 . We introduce w(y; s) = w0(y; s) wherew0 is de ned in (2). Then, it is easy to see that w satis es all the hypothesesof Theorem 2. Therefore, either w 0 of there exists t0 0 such that 8(y; s) 2RN+1 , w(y; s) = (1 +t0es) 1p 1 . Thus, either u 0 or u(x; t) = (T + t0t) 1p 1 . This concludes the proof of Corollary 3.A Proof of Proposition 3.2We proceed in 3 steps: In Step 1, we give a new version of an ODE lemmaby Filippas and Kohn [3] which will be applied in Step 2 in order to show thateither vnull or v+ is predominant in L2 as s! 1. In Step 3, we show that inthe case where v+ is predominant, then either v0(s) or v1(s) predominates theother.Step 1: An ODE lemmaLemma A.1 Let x(s), y(s) and z(s) be absolutely continuous, real valued func-tions which are non negative and satisfyi) (x; y; z)(s)! 0 as s! 1, and 8s s , x(s) + y(s) + z(s) 6= 0,ii) 8 > 0, 9s0 2 R such that 8s s08<: _zc0z (x+ y)j _xj(x+ y + z)_yc0y + (x+ z):(57)Then, either x+ y = o(z) or y + z = o(x) as s! 1.Proof: Filippas and Kohn showed in [3] a slightly weaker version of this lemma(with in the conclusion x; y; z ! 0 exponentially fast instead of x + y = o(z)).We adapt here their proof to get the proof of lemma A.1.By rescaling in time, we may assume c0 = 1.Part 1: Let > 0. We show in this part that either:9s2( ) such that 8s s2; z(s) + y(s) C x(s);(58)or 9s2( ) such that 8s s2; x(s) + y(s) C z(s):(59)We rst show that 8s s0( ), (s) 0 where = y 2 (x+ z). Proof of Proposition 3.2211We argue by contradiction and suppose that there exists s s0( ) suchthat (s ) > 0. Then, if s s and (s) > 0, we have form (57)_(s) = _y 2 ( _x + _z)y + (x + z) + 2 2(x + y + z) 2 (z (x + y))(1 4 8 2)x (3 2 8 2)z 0.Therefore, 8s s , (s) (s ) > 0, which contradicts (s)! 0 as s! 1 .Thus8s s0( ); y 2 (x+ z):(60)Therefore, (57) yields_z12z 2 xj _xj2 (x+ z)(61)Let (s) = 8 x(s) z(s). Two cases arise then:Case 1: 9s2 s0( ) such that (s2) > 0.Suppose then (s) = 0 and compute _(s)._(s) = 8 _x _z 16 2(x+ z) 12z + 2 x = z(s)( 14 2 16 2).Since z(s) > 0 (otherwise z(s) = 0, x(s) = 0 and then y(s) = 0 by (60), whichis excluded by the hypothesis), we have(s) = 0 =) _(s) < 0:Since (s2) > 0, this implies 8s s2, (s) > 0, i.e. 8 x(s) > z(s). Together with(60), this yields (58).Case 2: 8s s0( ), (s) 0 i.e. 8 x z(s).In this case, (61) yields8s s0( ); _z 14z; and _x (2 + 14)z:Therefore, we get by integration:z(s) 14 Z s1 z(t)dt and x(s) (2 +14) Z s1 z(t)dt;which yields x(s) (8 + 1)z(s). We inject this in (61) and get_x(s) 2 (x+ z) 2 z(2 + 8 ). Again, by integration:x(s) 2 (2 + 8 ) R s1 z(t)dt 8 (2 + 8 )z(s). Together with (60), this yields(59).Part 2: Let < 1C . Then either (58) or (59) occurs.For example, (58) occurs, that is 9s2( ) s0 such that 8s s2, z+y C x.Let 0be an arbitrary positive number. Then, according to Part 1, either8ss02, z + y C 0x for somes02( 0),or 8ss02, y + x C 0z for somes02( 0).Only the rst case occurs. Indeed, if not, then for s min(s2;s02), xC 0z C 0C x C2 2x since 0 . Since (C )2 < 1, we have x 0 andz y 0 for s min(s2;s02), which is excluded by the hypotheses.Do the same if (59) occurs.This concludes the proof of lemma A.1.Step 2: Competition between v+, vnull and vIn this step we show that either kv(s)kL2 +kv+(s)kL2 =o(kvnull(s)kL2 )(which is case iii) of Proposition 3.2) or 212Optimal estimates for blow-up rate and behaviorkv(s)kL2 +kvnull(s)kL2 =o(kv+(s)kL2 ) (which yields case i) or ii) of Propo-sition 3.2) in Step 3).This situation is exactly symmetric to the one in section 4 in Filippas andKohn's paper [3]. Indeed, we are treating the same equation (38), but we havekv(s)kL1loc ! 0 as s ! 1 whereas in [3],kv(s)kL1loc ! 0 as s ! +1. Never-theless, the derivation of the di erential inequalities satis ed by v , vnull andv+ in [3] is still valid here with the changes: \s ! +1" becomes s ! 1and \s large enough" becomes \ s large enough". Therefore, we claim that [3]implies:Lemma A.2 8 > 0, 9s0 2 R such that for a.e. s s0:_z( 12 )z (x+ y)j _xj(x+ y + z)_y( 12 )y + (x+ z)where z(s) =kv+(s)kL2 , x(s) =kvnull(s)kL2 andy(s) = kv(s)kL2 + kjyj k2 v2(s)kL2 for a xed integer k.Now, sincekv(s)kL1loc ! 0 as s! 1, we have (x; y; z)(s)! 0 as s! 1.We can not have x(s1)+y(s1)+ z(s1) = 0 for some s1 2 R, because this impliesthat 8y 2 RN , v(y; s1) = 0, and from the uniqueness of the solution to theCauchy problem of equation (38) and v(s1) = 0, we have 8(y; s) 2 RN R,v(y; s) = 0, which contradicts + v ! 0 as s! +1. Applying lemma A.1 withc0 =14 , we get:either kv(s)kL2 +kv+(s)kL2 =o(kvnull(s)kL2 )or kv(s)kL2 +kvnull(s)kL2 =o(kv+(s)kL2 ).Step 3: Competition between v0 and v1In this step, we focus on the case where kv(s)kL2 +kvnull(s)kL2=o(kv+(s)kL2 ). We will show that it leads either to case i) or case ii) ofProposition 3.2.Let us rst remark that lemma A.1 implies in this case that8 > 0, z(s) =kv+(s)kL2 = O(e( 12)) as s! 1:(62)Now, we want to derive from (38) the equations satis ed by v0 and v1. Wemust estimate R f(v(y; s))km(yi) (y)dy for m = 0; 1 and i = 1; ::N (see (41) forkm). Let us give this crucial estimate:Lemma A.3 There exists 0 > 0 and an integer k0 > 4 such that for all 2(0; 0), 9s0 2 R such that 8s s0, R v2jyjk0 dyc0(k0) 4k0z(s)2.Proof: Let I(s) = R v2jyjk0 dy 1=2. We rst derive a di erential inequalitysatis ed by I(s). If we multiply (38) by vjyjk0 and integrate over RN , we obtain:12 dds (I(s)2) = Z v:Lvjyjk0 dy + Z vf(v)jyjk0 dy:Since v is bounded by M , we get R vf(v)jyjk0 dy MC R v2jyjk0 dy.After some calculations, we show thatR v:Lvjyjk0 dyk2 (k +N 2) R jyjk0 2v2 dy + (1k4 )I(s)2. Proof of Proposition 3.2213Using Schwartz's inequality, we nd:R v2jyjk0 2 dy I(s) R v2jyjk0 4 dy 1=2.Let us bound R v2jyjk0 4 dy 1=2. If k0 > 4 and > 0, thenR v2jyjk0 4 dy 1=2 Rjyj1 v2jyjk0 4 dy 1=2 +Rjyj1 v2jyjk0 4 dy 1=22 k0=2 R v2 dy 1=2 + 2I2 2k0=2z(s) + 2I since R v2 dy 1=2 R v2+ dy 1=2 = z(s) as s! 1.Combining all the previous bounds, we obtain:I 0(s)I + d 2k0=2z with = k04 1 MC k02 (k0 + N 2) 2 and d =k0(k0 +N 2).We claim that there exist an integer k0 > 4 and 0 > 0 such that 8 2 (0; 0),1. Hence,I 0(s) I(s) + d 2k0=2z(s):(63)Now, we will derive a di erential inequality satis ed by z in order to couple itwith (63), and then prove lemma A.3.We project (38) onto the positive subspace of L, we multiply the result byv+ and then, we integrate over RN to get:12 dds (z(s)2) = Z Lv+:v+ dy + Z P+(f(v))v+ dy:Since (Spec L) \ R + = f1;12g, we have R Lv+:v+ dy 12z(s)2.Using Schwartz's inequality, we obtain:R P+(f(v))v+ dy R P+(f(v))2 dy 1=2 R v2+ dy 1=2R f(v)2 dy 1=2 z(s).Since v ! 0 as s! 1 uniformly on compact sets, we have:R f(v)2 dy C2 R v4 dy = C2Rjyj1 v4 dy + C2Rjyj1 v4 dy2 R v2 dy+C2M2 k0 R v2jyjk0 4 2z2+C2M2 k0I2 for all > 0, providedthat s s0( ; ).Thus, R f(v)2 dy 1=2 2 z + CM k0=2I .Combining all the previous estimates, we obtain:z0(s) 12z(s) 2 z CM k0=2I(s):(64)With = 1=8, (63) and (64) yield:8s s0 z0(s)14z(s) CM k0=2I(s)I 0(s)I(s) + d 2k0=2z(s):Now, we are ready to conclude the proof of lemma A.3:Let (s) = I(s) 2d 2k0=2z(s). Let us assume (s) > 0 and show that0(s) < 0.0(s) = I 0 2d 2k0=2z0 ( I + d 2k0=2z) 2d 2k0=2( 14z CM k0=2I)I( 1 +14 + 2CMd 2) = I( 34 + 2CM 2d)If we choose 0 such that 8 2 (0; 0), 34 + 2CM 2d < 0, then (s) > 0implies I(s) > 0 and 0(s) < 0. Since (s) ! 0 as s ! 1 (because v ! 0 214Optimal estimates for blow-up rate and behavioruniformly on compact sets), we conclude that for some s1 2 R, 8s s1, (s) 0.Since d = k0(k0 +N 2), lemma A.3 is proved.Using lemma A.3, we try to estimate R f(v)km(yi) dy.Since jf(v) p2 v2j C(M)v3, we write:Z f(v) dy = p2 Z v2 dy +O(Z v3 dy):(65)For all > 0, > 0 and s s0, we write:j R v3 dyj jRjyj1 v3 dyj+ jRjyj1 v3 dyjjRjyj1 v3 dyj+M k0 R v2jyjk0 dy jRjyj1 v3 dyj+Mc0(k0) 4z(s)2.We x > 0 small enough such thatMc0(k0) 4 2 . Then, we take s s1( )such that jRjyj1 v3 dyj4Rjyj1 v2 dy4 R v2 dy (because v ! 0 inL1(B(0; ))).Since R v2 dy z(s)2 as s! 1, we get for s s2( ), j R v3 dyj z(s)2.Therefore, equation (38) and (65) yield:v00(s) = v0(s) + p2 z(s)2(1 + (s))(66)where (s)! 0 as s! 1.Using the same type of calculations as for R v3 dy, we can prove thatR v2k1(yi) dy = O(z(s)2). Therefore, (38) yields the following vectorial equa-tion:v01(s) = 12v1(s) + (s)z(s)2(67)where is bounded.From (62), (66), (67) and standard ODE techniques, we get:8 > 0; v0(s) = O(e(1 )s) and v1(s) = C1e s2 + O(e(1 )s):Since z(s)2 =v0(s)2 +2jv1(s)j2, we write (66) asv00(s) = v0(s) + p2 jC1j2es(1 + (s)) + (s)where (s) = O(e2(1 )s). Therefore,8 > 0; v0(s) = pjC1j2ses(1 + o(1)) +C0es +O(e2(1 )s)(68)as s! 1.Two cases arise:i) If C1 6= 0, then v1(s) C1e s2pjC1j2ses v0(s). This is case ii) ofProposition 3.2.ii) If C1 = 0, then jz(s)j Ce(1 )s, and (67) yields v1 = O(e(2 )s). From(68), we have v0(s) =C0es +O(e(2 )s).We claim that C0 < 0 (If not, then the function F (s) = esv0(s) goes to C0 0as s ! 1 and is increasing if s s0. Therefore, 8s s0, v0(s) C0es 0.Since v is bounded and + v 0, we have from Proposition 3.5 8s 2 R,R ( + v(y; s)) dy , that is v0(s) 0. Proof of Proposition 3.4215Hence, 8s s0, v0(s) = 0 and z(s) =p2jv1(s)j. Then, (67) implies that8s s0, v1(s) = 0 and z(s) = 0. Since R v2 dy z(s), we have v 0 andwin a neighborhood of 1 and then on RN R which contradicts w ! 0as s! +1).Thus, v0(s) C0es Ce(2 )s jv1(s)j. This is Case i) of Proposition 3.2.This concludes the proof of Proposition 3.2.B Proof of Proposition 3.4Let us recall Proposition 3.4:Proposition B.1 Let V be an L1 solution of@V@s = (L+ l(s))V + F (V )(69)de ned for (y; s) 2 RN R, where L =12y:r+ 1, l(s) =pes(p 1)(1+es) andF (V ) = j'+ V jp 1('+ V ) 'p p'p 1V .Assume that V ! 0 as s! 1 uniformly on compact sets of RN ,jV0(s)j+ jV1(s)j = O(e(2 )s) andkVnull(s)kL2 = o(es) as s! 1:(70)Then V 0.In order to show that V 0 in RN+1 , we proceed in three steps: in Step 1,we do an L2 analysis for V as s ! 1 , similarly as in Part 2 of section 2 toshow that either kV(s)kL2 kV+(s)kL2 or kV(s)kL2 kVnull(s)kL2 . Then, wetreat these two cases successively in Steps 2 and 3 to show that V 0.Step 1: L2 analysis for V as s! 1Lemma B.1 As s! 1 , eitheri) kV(s)kL2 +kVnull(s)kL2 =o(kV+(s)kL2 )or ii) kV(s)kL2 +kV+(s)kL2 =o(kVnull(s)kL2 ).Proof: One can adapt easily the proof of Filippas and Kohn in [3] here. Indeed,V satis es almost the same type of equation (because l(s)! 0 as s! 1 , andjF (V )j CV 2), and V ! 0 as s! 1 uniformly on compact sets. Therefore,we claim that up to the change of \s! 1 " into \s! +1", section 4 of [3]impliesLemma B.2 8 > 0, 9s0 2 R such that for a.e. s s0:_Z( 12 )Z (X + Y )j _Xj(X + Y + Z)_Y( 12 )Y + (X + Z)where Z(s) =kV+(s)kL2 , X(s) =kVnull(s)kL2 andY (s) = kV(s)kL2 + kjyj k2 V2(s)kL2 for a xed integer k. 216Optimal estimates for blow-up rate and behaviorSince kV(s)kL1loc ! 0 as s ! 1 and V is bounded in L1, we have(X;Y; Z)(s)! 0 as s! 1. Similarly as in Step 2 of Appendix A, we can nothave X(s) + Y (s) + Z(s) = 0 for some s 2 R. Therefore, the conclusion followsfrom lemma A.1, in the same way as in Step 2 of Appendix A.Step 2: Case kV(s)kL2 +kVnull(s)kL2 =o(kV+(s)kL2 )Since (69) and (38) are very similar (the only real di erence is the presence in(69) of l(s) which goes to zero as s! 1 ), one can adapt without di culty allthe Step 3 of Appendix A and show that V0 and V1 satisfy equations analogousto (66) and (67): 8s s0V 00(s) = V0(s)(1 + l(s)) +a0(s)(V0(s)2 +2jV1(s)j2)V 01(s) = V1(s)( 12 + l(s)) +a1(s)(V0(s)2 +2jV1(s)j2)(71)where a0 and a1 are bounded.According to (70), there exist B > 0 and s1 s0 such that 8s s1ja0(s)j B; ja1(s)j B; jV0(s)j e 3s2 and jV1(s)j e 3s2 :(72)We claim then that the following lemma yields V 0:Lemma B.3 8n 2 N, 8s s1, jVm(s)j ( 32e(s1)B)2n 1e3 2n 1s for m = 0and m = 1, where e(s1) = e R s11l(t)dt.Indeed, the lemma yields that 8s s2 V0(s) = V1(s) = 0 for some s2 s1. SincekV(s)kL2 kV+(s)kL2 as s! 1 , we have 8s s3, 8y 2 RN , V (y; s) = 0 forsome s3 s2. The uniqueness of the solution of the Cauchy problem: 8s s3,V satis es equation (69) and V (s3) = 0 yields V 0 in RN+1 .Proof of lemma B.3: We proceed by induction:n = 0, the hypothesis is true by (72).We suppose that for n 2 N, we have8s s1, jVm(s)j ( 32e(s1)B)2n 1e3 2n 1s for m = 0; 1. Let us prove that8s s1, jVm(s)j (32e(s1)B)2n+1 1e3 2ns for m = 0; 1.Let Fm(s) = Vm(s)e (1m2 )s R s1l(t)dt. From (71) and the induction hypothesis,we have: 8s s1,jF 0m(s)j e (1m2 )s R s11l(t)dtB 3( 32e(s1)B)2(2n 1)e3 2ns. By the inductionhypothesis, lims! 1Fm(s) = 0. Hence, 8s s1,jFm(s)j = j R s1 F 0m( )d j R s1 jF 0m( )jd3e(s1)B(32e(s1)B)2n+1 2 R s1 e(3 2n (1m2 )) d=23 2n (1m2 ) (32e(s1)B)2n+1 1e(3 2n (1m2 ))s.Since 3 2n (1m2 ) 2 and l(s) 0, this yields8s s1, jVm(s)j (32e(s1)B)2n+1 1e3 2ns for m = 0; 1. This concludes theproof of lemma B.3.Step 3: Case kV(s)kL2 +kV+(s)kL2 =o(kVnull(s)kL2 )In order to show that V 0, it is enough to show that Vnull 0 or equiva-lently that 8i; j 2 f1; ::; Ng, V2;ij 0. Proof of Proposition 3.4217For this purpose, we derive form (69) an equation satis ed by V2;ij as s !1 :V 02;ij(s) = l(s)V2;ij(s) + Z F (V ) H2;ijkH2;ijk2L2 dy:(73)We have to estimate the last term of (73):if i = j, then H2;ij(y) = y2i 2 andj Z F (V )H2;ii dyj C Z V 2 dy + C Z V 2jyj2 dy;(74)if i 6= j, then H2;ij(y) = yiyj andj Z F (V )H2;ij dyj C Z V 2jyj2 dy:(75)The hypothesis of this step implies thatZ V 2 dy 2 Z V 2null dy:(76)It remains then to bound R V 2jyj2 dy. This will be done through this lemma,which is analogous to lemma A.3:Lemma B.4 There exists 0 > 0 and an integer k0 > 5 such that for all 2(0; 0), 9s0 2 R such that 8s s0, R V 2jyjk0 dyc0(k0) 4 k0 R V 2null dy.Proof: We will argue similarly as in the proof of lemma A.3.Let I(s) = R V 2jyjk0 dy 1=2 and use the notationX(s) = R V 2nulljyjk0 dy1=2.From (69), we derive the following equation for I(s):12 dds (I(s)2) = Z V:LV jyjk0 dy + l(s)I(s)2 + Z V F (V )jyjk0 dy:Since v is bounded by M , we can assume jV j M + 1 =M 0 and getR V F (V )jyjk0 dy M 0C R V 2jyjk0 dy. We can also assume that jl(s)j 112 .As for lemma A.3, we can show that for all > 0R V:LV jyjk0 dy k02 (k0+N 2)I(s)( 2 k0=2 R V 2 dy 1=2+ 2I)+(1k4 )I(s)2.Combining these bounds with (76), we get:I 0(s)I + d 2k0=2z with = k04 1 112 M 0C k02 (k0 +N 2) 2 andd = k0(k0 +N 2).It is clear that there exist an integer k0 > 5 and 0 > 0 such that 8 2 (0; 0),1. Hence,I 0(s) I(s) + d 2k0=2X(s):(77)Let us derive a di erential equation satis ed by X .From (69), we obtain:12 dds (X(s)2) = l(s)X(s)2 + Z Pnull(F (V ))Vnull dy:By Schwartz's inequality, we have:R Pnull(F (V ))Vnull dy R Pnull(F (V ))2 dy 1=2 R V 2null dy 1=2 218Optimal estimates for blow-up rate and behaviorR F (V )2 dy 1=2X(s).Since V ! 0 as s! 1 uniformly on compact sets, we have:R F (V )2 dy C2 R V 4 dy = C2Rjyj1 V 4 dy + C2Rjyj1 V 4 dy2 R V 2 dy + C2M 02 k0 R V 2jyjk0 4 2X2 + C2M 02k0I2 for all > 0, pro-vided that s s0( ; ).Thus, R F (V )2 dy 1=2 2 X + CM 0k0=2I .Since jl(s)j 112 , we combine all the previous bounds to get:jX 0(s)j (2 + 112)X(s) + CM 0k0=2I(s):(78)With = 1=12, (77) and (78) yield:8s s1 jX 0(s)j14X(s) + CM 0k0=2I(s)I 0(s)I(s) + d 2k0=2X(s):Now, we conclude the proof of lemma A.3:Let (s) = I(s) 2d 2k0=2X(s). Let us assume (s) > 0 and show that0(s) < 0.0(s) = I 0 2d 2k0=2X 0( I + d 2k0=2X) + 2d 2k0=2( 14X(s) + CM 0k0=2I)I( 1 + 12 + 2CM 0d 2 +14 ) = I(14 + 2CM 0 2d)If we choose 0 such that 8 2 (0; 0),14 + 2CM 0 2d < 0, then (s) > 0implies I(s) > 0 and 0(s) < 0. Since (s) ! 0 as s ! 1 (because V ! 0uniformly on compact sets), we conclude that for some s2 2 R, 8s s1, (s) 0.Since d = k0(k0 +N 2), lemma B.4 is proved.Lemma B.4 allows us to bound R V 2jyj2 dy. Indeed, for xed 2 (0; 0) ands s0, we have:R V 2jyj2 dyRjyj1 V 2jyj2 dy +Rjyj1 V 2jyj2 dy2Rjyj1 V 2 dy + k0 2Rjyj1 V 2jyjk0 dy2 R V 2 dy +c0(k0) 2 R V 2 dy = C( ; k0) R V 2 dy.With this bound, (74) and (75), equation (69) yields: 8s s0,V 02;ij(s) = l(s)V2;ij(s) +a2;ij(s)kVnull(s)k2L2where a2;ij is bounded.According to (70), there exist then B > 0 and s1 s0 such that 8s s1,8i; j 2 f1; :::; Ng,ja2;ij(s)j B; jV2;ij(s)j es:We claim that the following lemma yields V 0:Lemma B.5 8n 2 N, 8s s1, 8i; j 2 f1; :::; Ng,jV2;ij(s)j (8N2(N +1)2e(s1)B)2n 1e2ns where e(s1) = e R s11l(t)dt.Indeed, this lemma yields 8s s1, 8i; j 2 f1; :::; Ng, V2;ij(s) = 0 for somes2 s1. Hence, 8s s2, 8y 2 RN , Vnull(y; s) = 0, and by the hypothesis ofthis step, 8s s3, 8y 2 RN , V (y; s) = 0 for some s3 s3. The uniquenessof the solutions to the Cauchy problem: 8s s3, V satis es equation (69) andV (s3) = 0 yields V 0 in RN+1 .We escape the proof of lemma B.5 since it is completely analogous to theproof of lemma B.3. Proof of i) of Proposition 3.7219C Proof of i) of Proposition 3.7We proceed in 4 steps: in Step 1, we derive form the fact thatkv(s)kL2kvnull(s)kL2 an equation satis ed by vnull(s) as s! 1 . Then, we nd in Step2 c > 0, C > 0 and s0 2 R such that cjsj 1 kv(s)kL2 Cjsj 1 for s s0. InStep 3, we use this estimate to derive a more accurate equation for vnull. Weuse this equation in Step 4 to get the asymptotic behaviors of vnull(y; s), v0(s)and v1(s).Step 1: An ODE satis ed by vnull(y; s) as s! 1This step is very similar to Step 3 in Appendix B where we handled theequation (69) instead of (38) as in the present context.From (38) we have by projection:v02;ij(s) = Z f(v) H2;ij(y)kH2;ijk2L2 (y)dy(79)We will prove the following proposition here:Proposition C.1 i) 8i; j 2 f1; :::; Ng,v02;ij(s) = p2 Z v2null(y; s) H2;ij(y)kH2;ijk2L2 (y)dy +o(kvnull(s)k2L2 ):(80)as s! 1 .ii) There exists a symmetric N N matrix A(s) such that 8s 2 R,vnull(y; s) = yTA(s)y 2tr(A(s))(81)and c0kA(s)kkvnull(s)kL2 C0kA(s)k(82)for some positive constants c0 and C0. Moreover,A0(s) = 4pA2(s) + o(kA(s)k2) as s! 1:(83)Remark: kAk stands for any norm on the space of N N symmetric matrices.Remark: The interest of the introduction of the matrix A(s) is that it genera-lizes to N 2 the situation of N = 1. Indeed, if N = 1, then it is obvious thatvnull(y; s) = yv2(s)y 2v2(s) and that (80) implies v02(s) = 4pv2(s)2+o(v2(s)2).Let us remark that in the case N = 1, we get immediately v2(s)4ps ass ! 1, which concludes the proof of Proposition 3.7. Unfortunately, we cannot solve the system (83) so easily if N 2. Nevertheless, the intuition givenby the case N = 1 will guide us in next steps in order to re ne the system (83)and reach then a similar result (see Step 2).Proof of Proposition C.1:Let us remark that ii) follows directly form i). Indeed, we have by de nitionof H2;ij and v2;ij (see (44) and (45)):vnull(y; s) =Xij v2;ij(s)H2;ij(y) = NXi=1 v2;ii(s)(y2i 2)+Xide ne A(s) = (aij(s))i;j byaii(s) = v2;ii(s); and for i < j; aij(s) = aji(s) = 12v2;ij(s);(84) 220Optimal estimates for blow-up rate and behaviorthen (81) follows. (82) follows form the equivalence of norms in nite dimensionN(N+1)2 . (83) follows from (80) by simple but long calculations which we escapehere.Now, we focus on the proof of i). For this purpose, we try to estimate theright-hand side of equation (79).As in Step 3 of Part 3, this will be possible thanks to the following lemma:Lemma C.1 There exists 0 > 0 and an integer k0 > 4 such that for all 2(0; 0), 9s0 2 R such that 8s s0, R v2jyjk0 dyc0(k0) 4 k0 R v2null dy.Proof: The proof of lemma B.4 holds for lemma C.1 with the changes V ! v,F ! f and l(s)! 0.Now we estimate R f(v)H2;ij dy:Since f(v) = p2 v2 + g(v) where jg(v)j Cjvj3, we write:Z f(v)H2;ij dy = p2 Z v2nullH2;ij dy + (I) + (II)(85)where(I) = p2 Z (v2 v2null)H2;ij dy(86)and (II) = Z g(v)H2;ij dy:(87)The proof of Proposition C.1 will be complete if we show that (I) and (II) areo(kvnull(s)kL2 ). Since H2;ij(y) = y2i 2 if i = j and H2;ij(y) = yiyj if i 6= j, it isenough to show that for all > 0, I1, I2, II1 and II2 are lower thatkvnull(s)kL2for all s s0( ), whereI1 = R jv2 v2nullj dy; I2 = R jv2 v2nulljjyj2 dy;II1 = R jg(v)j dy;II2 = R jg(v)jjyj2 dy:We start with I1: Since R v2 dy R v2null dy,I1 = R (v2+ + v2) dy R v2null dy if s s1( ).For I2, we consider 2 (0; 0), and write:I2Rjyj1 jv2 v2nulljjyj2 dy +Rjyj1 jv2 v2nulljjyj2 dy : = I21 + I22.We rst estimate I21:Since v = v +vnull+v+, we have v2 v2null = (v++v )2+2vnull(v++v ).Hence,I21Rjyj1(v+ + v )2jyj2 dy + 2Rjyj1 jvnull(v+ + v )jjyj2 dy2 R (v+ + v )2 dy + 2 R v2nulljyj4 dy 1=2 R (v+ + v )2 dy 1=2.Since R v2 dy R R v2null dy, we haveR (v+ + v )23 R v2null dy if s s2( ).Since the null subspace of L in nite dimensional, all the norms on it are equi-valent, therefore, there exists C4(N) such that:R v2nulljyj4 dyC4(N)2 R v2null dy.Therefore, I21 ( + 2C4(N) 3=2) R v2null dy if s s2( ).For I22, we write:I22Rjyj1 jv2 v2nulljjyj2 dy k0 2 R v2jyjk0 dy + k0 2 R v2nulljyjk0 dyc0(k0) 2 R v2null dy + k0 2Ck0(N)2 R v2null dy Proof of i) of Proposition 3.7221by lemma C.1 and the equivalence of norms for vnull. Collecting all the aboveestimates, we getI2 ( +2C4(N) 3=2+c0(k0)+ k0 2Ck0 (N)2) R v2null dy for s s2( ). If = ( )is small enough, thenI2 R v2null dy for s s3( ).Now, we handle II1 and II2 in the same time: we consider 2 (0; 0) andwrite for m = 0 or m = 2:j R jg(v)jjyjm dy C R jvj3jyjm dyCRjyj1 jvj3jyjm dy + CRjyj1 jvj3jyjm dyC 0 mRjyj1 v2 dy + CM k0 mRjyj1 v2jyjk0 dy(C 0 m +CMc0(k0) 4 m)2 R v2null dywhere we used the fact that v ! 0 as s! 1 in L1(B(0; 1)), jv(y; s)j M ,lemma C.1 and R v2 dy R v2null dy.Now, we can choose = ( ) and then 0 = 0( ) such that for s s5( )R jg(v)jjyjm dy R v2null dy.Setting s0( ) = min(s1( ); s3( ); s5( )), we have: 8 > 0, 8s s0( ),I1 + I2 + II1 + II2 4 R v2null dy. Therefore (I) + (II) =o(kvnull(s)kL2 ) ass! 1 .Thus, combining this with (79) and (85) concludes the proof of PropositionC.1.Step 2:kv(s)kL2 behaves like 1jsj as s! 1In this step, we show that although we can not derive directly from (80)the asymptotic behavior of vnull(s) (and then the one of v(s)), we can use it toshow thatkv(s)kL2 behaves like 1jsj as s ! 1 . More precisely, we have thefollowing Proposition:Proposition C.2 If kv(s)kL2 +kv+(s)kL2 =o(kvnull(s)kL2 ), then for slarge enough, we havecjsj kv(s)kL2 Cjsjfor some positive constants c and C.Proof: Sincekv(s)kL2 kvnull(s)kL2 , and because of (82), it is enough toshow thatcjsj kA(s)k Cjsj(88)for s large. The proof is completely parallel to section 3 of Filippas and Liu[4]. Therefore, we give only its main steps.We rst give a result from the perturbation theory of linear operators whichasserts that A(s) has continuously di erentiable eigenvalues:Lemma C.2 Suppose that A(s) is a N N symmetric and continuously dif-ferentiable matrix-function in some interval I. Then, there exist continuouslydi erentiable functions 1(s); :::; N (s) in I, such that for all j 2 f1; ::; Ng,A(s) (j)(s) = j(s) (j)(s);for some (properly chosen) orthonormal system of vector-functions(1)(s); :::; (N)(s). 222Optimal estimates for blow-up rate and behaviorThe proof of this lemma is contained (for instance) in Kato [10] or Rellich [13].We consider then 1(s); :::; N (s) the eigenvalues of A(s). It is well-knownthat NXi=1 j ij is a norm on the space ofN N symmetric matrices. We choose thisnorm to prove (88). From (83), we can derive an equation satis ed by ( i(s))i:Lemma C.3 The eigenvalues of A(s) satisfy 8i 2 f1; :::; Ng0i(s) = 4p2i (s) + o NXi=1 2i (s)! :The proof of lemma 3.3 in [4] holds here with the slight change: s ! +1becomes s! 1 and s large enough becomes s large enough.Now, we claim that with the introduction of i( ) = i( ), we have:8i 2 f1; :::; Ng0i( ) = 4p2i ( ) + o NXi=1 2i ( )! as ! +1;-80,Xi j i( )j 6= 0 (Indeed, if not, then for all i, i 0, i 0, andthen A(s), vnull(s) and v(s) are identically zero.)Section 3 of [4] yields (directly and without any adaptations) that for all1,cXi j i( )j C :Since kA(s)k=Xi j i(s)j=Xi j i( s)j, this concludes the proof of (88) andthe proof of Proposition C.2.Step 3: A new ODE satis ed by vnull(y; s)In this step, we show that sincekvkL2 behaves like 1jsj , then all the Lqnorms are in some sense equivalent as s ! 1 for this particular v. Then,we will do a kind of center-manifold theory for this particular v to show thatkv+(s)kL2 + kv(s)kL2 is in factO(kvnull(s)k2L2 ) and not onlyo(kvnull(s)kL2 ).These two estimates are used then to rederive a more accurate equation satis edby vnull(y; s).Lemma C.4 Ifkv+(s)kL2 + kv(s)kL2 =o(kvnull(s)kL2 ), theni) for every r > 1, q > 1, there exists C = C(r; q) such thatZ vr(y; s) dy 1=r C Z vq(y; s) dy 1=qfor s large enough.ii)kv+(s)kL2 + kv(s)kL2 =O(kvnull(s)k2L2 ) as s! 1 . Proof of i) of Proposition 3.7223Proof of i) of lemma C.4: The crucial estimate is an a priori estimate of solutionsof (38) shown by Herrero and Velazquez in [9]. This a priori estimate is a versionof i) holding for all bounded (in L1) solutions of (38), but with a delay time;although they proved their result in the case N = 1 for solutions de ned fors 2 [0;+1), their proof holds in higher dimensions with s 2 R.Lemma C.5 (Herrero-Velazquez) Assume that v solves (38) and jvj M <1. Then for any r > 1, q > 1 and L > 0, there exist s 0 = s 0(q; r) andC = C(r; q; L) > 0 such thatZ vr(y; s+ s ) dy 1=r C Z vq(y; s) dy 1=qfor any s 2 R and any s 2 [s 0; s 0 + L].Set s 1 = s 0(2; r) and s 2 = s 0(q; 2). For s large enough, we write accordingto lemma C.5 and Proposition C.2:R vr(y; s) dy 1=r C1 R v2(y; s s 1) dy 1=2 C2=(s s 1)C3=(s+ s 2) C4 R v2(y; s+ s 2) dy 1=2 C5 R vq(y; s) dy 1=q . Thus, i) oflemma C.4 follows .Proof of ii) of lemma C.4: We argue as in Step 2 of Appendix A, and usethe same notations: x(s) =kvnull(s)kL2 , y(s) = kv(s)kL2 , z(s) =kv+(s)kL2and N(s) = kV2kL2 . We have already derived (in the proofs of lemmas A.3 andB.4) two di erential inequalities satis ed by x and z. By the same techniques(see also [3]), we can show thatz012z CNjx0jCNy012y + CN:By i) of lemma C.4, we have N(s)Ckv(s)k2L2 = C(x2(s) + y2(s) + z2(s)) forlarge s.Since x; y; z ! 0 as s! 1 , we can write for s large:z013z C(x + y)2jx0jC(x + y + z)2y013y + C(x + z)2:The conclusion then follows form the following ODE lemma by Filippas andLiu:Lemma C.6 (Filippas-Liu) Let x(s), y(s) and z(s) be absolutely continuous,real valued functions which are non negative and satisfyi) (x; y; z)(s)! 0 as s! 1,ii) 8s s08<: _zc0z c1(x+ y)2j _xjc1(x+ y + z)2_yc0y + c1(x+ z)2: 224Optimal estimates for blow-up rate and behaviorfor some positive constants c0 and c1. Then,either (i) x; y; z ! 0 exponentially fast as s! 1,or (ii) for all s s1, y + z b(c0; c1)x2 for some s1 s0.Proof: see lemma 4.1 in [4].Now, using lemma C.4, we derive a new equation satis ed by vnull:Proposition C.3 8i; j 2 f1; :::; Ng,v02;ij(s) = p2 Z v2null(y; s) H2;ij(y)kH2;ijk2L2 (y)dy+O(kvnull(s)k3L2 ):as s! 1 .Moreover,A0(s) = 4pA2(s) +O( 1s3 ) as s! 1:The proof of Proposition 4.1 in [4] holds here with the usual changes: s! +1becomes s! 1 .Step 4: Asymptotic behavior of vnull(y; s), v0(s) and v1(s)Setting A( ) = A( ), we see thatA0( ) = 4pA2( ) +O( 13 ) as ! +1:Therefore, Proposition 5.1 in [4] yields (directly and without any adapta-tions) the existence of > 0 and a N N orthonormal matrix Q such thatA( ) = 4p A0 +O( 11+ )whereA0 = Q IN k 00 0 Q 1for some k 2 f0; 1; :::; N 1g. Together with (81), this yields the behavior ofvnull(y; s) announced in i) of Proposition 3.7.It also yieldskvnull(s)kL2 = Z v2null(y; s) (y)dy 1=2= Z (yTA(s)y 2trA(s))2 (y)dy 1=2= 4ps Z (yTA0y 2trA0)2 (y)dy 1=2 +O 1jsj1+ :With the change of variables, y = Qz, we get since Q is orthonormal:kvnull(s)kL2 = 4ps 0@Z N kXi=1 (z2i 2)!2 (z)dz1A1=2 +O 1jsj1+ Proof of i) of Proposition 3.7225= 4ps N kXi=1 Z (z2i 2)2 (z)dz!1=2 +O 1jsj1+=psrN k2 +O 1jsj1+where we used the fact that (y2i 2)i is an orthogonal system with respect tothe measure dy.Sincekv(s)kL2 =kvnull(s)kL2 +Okvnull(s)k2L2 (ii) of lemma C.4), we getkv(s)kL2 =psrN k2 +O 1jsj1+ :(89)Integrating (38) with respect to dy, we ndv00(s) = v0(s) + Z f(v) dy:Since jf(v)j Cv2, we get from (89)v00(s) = v0(s) +O( 1s2 ) as s! 1:Therefore, it follows that v0(s) = O( 1s2 ) as s! 1:Using lemma C.1, we have: for all 2 (0; 0),Z v2jyjk0 dyc0(k0) 4 k0 Z v2null dy2c0(k0) 4 k0 Z v2 dy:Therefore,R v2jyj dyRjyj1 v2jyj dy +Rjyj1 v2jyj dy1 R v2 dy + k0 1 R v2jyjk0 dy( 1 +2c0(k0) 3) R v2 dy.If we x > 0, thenZ v2jyj dy C( ; k0) Z v2 dy:(90)Integrating (38) with respect to yi dy, we ndv01;i(s) =12v1;i(s) + Z f(v)yi2 dy:Since jf(v)j Cv2, we get from (90) and (89)v1(s) = O( 1s2 ) as s! 1:This concludes the proof of i) of Proposition 3.7.AcknowledgmentThe second author is thankful for the invitation of the Institute for AdvancedStudy where part of this work was done. Bibliography227Bibliography[1] Berestycki, H., and Nirenberg, L., On the method of moving planes andthe sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22, 1991, pp. 1-37.[2] Bricmont, J., and Kupiainen, A., Universality in blow-up for nonlinearheat equations, Nonlinearity 7, 1994, pp. 539-575.[3] Filippas, S., and Kohn, R., Re ned asymptotics for the blowup of utu = up, Comm. Pure Appl. Math. 45, 1992, pp. 821-869.[4] Filippas, S., and Liu, W., On the blow-up of multidimensional semilinearheat equations, Ann. Inst. H. Poincar e Anal. Non Lin eaire 10, 1993, pp.313-344.[5] Gidas, B., Ni, W.M., and Nirenberg, L., Symmetry and related propertiesvia the maximum principle, Comm. Math. Phys. 68, 1979, pp. 209-243.[6] Giga, Y., and Kohn, R., Asymptotically self-similar blowup of semilinearheat equations, Comm. Pure Appl. Math. 38, 1985, pp. 297-319.[7] Giga, Y., and Kohn, R., Characterizing blowup using similarity variables,Indiana Univ. Math. J. 36, 1987, pp. 1-40.[8] Giga, Y., and Kohn, R., Nondegeneracy of blow-up for semilinear heatequations, Comm. Pure Appl. Math. 42, 1989, pp. 845-884.[9] Herrero, M.A, et Velazquez, J.J.L., Blow-up behavior of one-dimensionalsemilinear parabolic equations, Ann. Inst. H. Poincar eAnal. Non Lin eaire10, 1993, pp. 131-189.[10] Kato, T., Perturbation theory for linear operators, Springer-Verlag, Ber-lin, 1980.[11] Merle, F., Solution of a nonlinear heat equation with arbitrary givenblow-up points, Comm. Pure Appl. Math. 45, 1992, pp. 263-300.[12] Merle, F., and Zaag, H., Re ned uniform estimates at blow-up and ap-plications for nonlinear heat equations, Geom. Funct. Anal., to appear.[13] Rellich, F., Perturbation theory for eigenvalue problems, Lecture Notes,New York University, 1953.[14] Zaag, H., Blow-up results for vector valued nonlinear heat equations withno gradient structure, Ann. Inst. H. Poincar e Anal. Non Lin eaire 15,1998, to appear. 228Optimal estimates for blow-up rate and behaviorAddress:D epartement de math ematiques, Universit e de Cergy-Pontoise, 2 avenue Adol-phe Chauvin, Pontoise, 95 302 Cergy-Pontoise cedex, France.D epartement de math ematiques et informatique, Ecole Normale Sup erieure, 45rue d'Ulm, 75 230 Paris cedex 05, France.e-mail: [email protected], [email protected] Chapitre 3Re ned uniform estimatesat blow-up and applicationsfor nonlinear heat equations 230Re ned uniform estimates at blow-up and applicationsRe ned uniform estimates at blow-up andapplications for nonlinear heat equations yFrank MerleUniversite de Cergy-PontoiseHatem ZaagEcole Normale Superieure and Universite de Cergy-Pontoise