Dynamic Analysis of a Nonlinear Timoshenko Equation

and Applied Analysis 3 Here, S t denotes the corresponding semigroup on H, generated by problem 1.1 , and ·, · 2 is the inner product in L2 Ω . The following energy equation holds: E0 E t ∫ t 0 δ‖v τ ‖λdτ, 2.3 where E t ≡ E u t , v t ≡ 1 2 ‖v t ‖2 J u t , 2.4 J u ≡ 1 2 a u 1 2 ( γ 1 )c u − 1 r b u , 2.5 with a u ≡ ‖u‖B, b u ≡ μ‖u‖r , c u ≡ β‖∇u‖ γ 1 2 . 2.6 Here, E0 ≡ E u0, v0 is the initial energy, and ‖ · ‖q denotes the norm in the L Ω space. If the maximal time of existence TM < ∞, then S t u0, v0 → ∞ as t ↗ TM, in the norm of H: ‖ u, v ‖H ≡ ‖u‖B ‖v‖2 ≡ ‖Δu‖2 α‖∇u‖2 ‖v‖2. 2.7 In that case, from 2.3 – 2.6 , ‖u t ‖r → ∞ as t ↗ TM. Now, we define, respectively, the stable potential well and unstable sets: W ≡ I u > 0 ∪ {0} ∩ J u < d , V ≡ I u < 0 ∩ J u < d , 2.8 where I u ≡ a u c u − b u . 2.9 Here, I u < 0 denotes the set of u ∈ B with that property, and the depth of the potential well is defined as follows: d ≡ r − 2 2r S r−2 , 2.10 4 Abstract and Applied Analysis where √ S ≡ inf 0/ u∈B b̂ u >0 ( â u 1/2 b̂ u 1/r ) , 2.11 â u ≡ a u κ1c u , b̂ u ≡ b u κ2c u . 2.12 with κ1 ≡ r − 2(γ 1) r − 2 (γ 1) , κ2 ≡ κ1 − 1 −rγ r − 2 (γ 1) . 2.13 We assume that r ≥ 2 γ 1 , and since γ ≥ 1, then κ1 ∈ 0, 1/2 , and κ2 ∈ −1,−1/2 . Also note that if r 2 γ 1 , then κ1 0, κ2 −1, and we have the following characterization of the depth of the potential well 2.10 2.11 : d inf 0/ u∈B sup λ≥0 J λu , 2.14 which is the definition given in 6 , where a nondissipative nonlinear wave equation is studied. Consider any u ∈ B, r > 2, and r ≤ 2n/ n − 4 if n ≥ 5, then â u ≥ C Ω b u 2/r κ1c u ≥ C Ω b u 2/r , 2.15 where C Ω > 0, is any constant in the Sobolev-Poincaré’s inequality ( ‖Δu‖2 α‖∇u‖2 )1/2 ≥ √ C Ω μ‖u‖r . 2.16 Moreover, if b̂ u > 0, from 2.15 and since b u ≥ b̂ u , â u ≥ C Ω b̂ u 2/r . 2.17 Hence, S ≥ C Ω , and d ≥ D ≡ r − 2 /2r C Ω r/ r−2 > 0. If ue denotes any nonzero equilibria of equation 1.1 , E ≡ [ 0/ ue ∈ B : Δue −M ( ‖u‖2 ) Δue f ue ] , 2.18 then, by 2.1 in Theorem 2.1 with u t ue w, we get that ue belongs to the Nehari manifold, N, that is, E ⊂ N ≡ [ 0/ u ∈ B : Î u I u 0 ] , 2.19 where Î u ≡ â u − b̂ u . Abstract and Applied Analysis 5 Consequently, b̂ ue â ue > 0. Furthermore, from 2.17 which is an equality when C Ω S, we conclude that the Nehari manifold can be represented by the line: y x, in the plane with axes x b̂ u and y â u , beginning at the point: y x S r−2 2r/ r−2 d. We also note thatand Applied Analysis 5 Consequently, b̂ ue â ue > 0. Furthermore, from 2.17 which is an equality when C Ω S, we conclude that the Nehari manifold can be represented by the line: y x, in the plane with axes x b̂ u and y â u , beginning at the point: y x S r−2 2r/ r−2 d. We also note that J u 1 2 â u − 1 r b̂ u . 2.20 From these facts it follows that the depth of the potential well 2.10 is characterized by d inf u∈N J u r − 2 2r , 2.21 where 0 < ≡ inf u∈N â u inf u∈N b̂ u . 2.22 Hence, any equilibrium is such that ue ∈ J u ≥ d . Moreover, like in 6 , the set of extremals of 2.21 is characterized by set of equilibria with least energy, that is the ground state N∗ ≡ ue ∈ E : J ue d [ ue ∈ E : â ue b̂ ue ] . 2.23 Observe that J u d is a tangent line to the curve defined by the equality in 2.17 with C Ω S, at the point N∗, which holds if b̂ u > 0. On the other hand, we notice that κ1b̂ u − κ2â u κ1b u − κ2a u > 0, 2.24 and is equal to zero if and only if a u 0 b u . Hence, if b̂ u < 0, then â u > r − 2(γ 1) −rγ b̂ u . 2.25 Therefore, next results about the stable and unstable sets follow. Lemma 2.2. The following properties of V and W hold: i W is a neighborhood of 0 ∈ B. ii 0 / ∈ I u < 0 (closure in B), in particular 0 / ∈ V . 6 Abstract and Applied Analysis iii W W ∪W− ∪ {0}, where W ≡ W ∩ [ b̂ u > 0 ] [ r−2 /r b̂ u 2/r ≤ â u < 2 r b̂ u r − 2 r , 0 < b̂ u < ] , W− ≡ W ∩ [ b̂ u < 0 ] ⊂ [ r − 2(γ 1) −rγ b̂ u < â u < 2 r b̂ u r − 2 r , −γ < b̂ u < 0 ] . 2.26 iv V [ r−2 /r b̂ u 2/r ≤ â u < 2 r b̂ u r − 2 r , b̂ u > ] . v N∗ W ∩ V W ∩ V ue ∈ N, â ue b̂ ue . vi W I u < 0 c ∩ J u < d , V I u > 0 ∪ {0} c ∩ J u < d . The following result follows easily like in 23 . Lemma 2.3. One has that J u > r − 2 2r â u > r − 2 2r b̂ u , 2.27 J u > r − 2 2r a u r − 2(γ 1) 2r ( γ 1 ) c u > γ 2 ( γ 1 )a u r − 2 ( γ 1 ) 2r ( γ 1 ) b u , 2.28 for any u ∈ B such that I u > 0, in particular if 0/ u ∈ W , and d < r − 2 2r â u < r − 2 2r b̂ u , 2.29 d < r − 2 2r a u r − 2(γ 1) 2r ( γ 1 ) c u < γ 2 ( γ 1 )a u r − 2 ( γ 1 ) 2r ( γ 1 ) b u , 2.30 for any u ∈ B, such that I u < 0, in particular if u ∈ V . A set V ⊂ H is positive invariant, with respect to problem 1.1 , if the corresponding generated semigroup S t on H is such that S t V ⊂ V. 2.31 Lemma 2.4. Let u, v denote any solution of 1.1 , given by Theorem 2.1. Then, the sets S ≡ E u, v < d ∩ u, v ∈ H : u ∈ W , 2.32 U ≡ E u, v < d ∩ u, v ∈ H : u ∈ V , 2.33 are positive invariant. Abstract and Applied Analysis 7 Proof. First, we show that S is positive invariant. In order to do that, we take u0, v0 ∈ S. Then, by 2.4 , J u t ≤ E u t , v t ≤ E0 < d, for any t ≥ 0. Now, if S is not positive invariant, there exists some t̂ > 0, such that I u t̂ 0, with u t̂ / 0. Then, by 2.21 , d ≤ J u t̂ . But this is impossible because J u t̂ < d. The proof of the positive invariance of U is quite similar. Indeed, if this is not true there exists some t̂ > 0, such that I u t̂ 0. From ii of Lemma 2.2 u t̂ / 0, and this implies the same contradiction as before. Next result gives an interpretation of sets S and U and follows from Lemma 2.2. Lemma 2.5. The sets S and U have the properties S ⊂ E u, v < d ∩ [ u, v ∈ H : r−2 /r b̂ u 2/r ≤ â u < 2 r b̂ u r − 2 r , 0 < b̂ u < ] ,and Applied Analysis 7 Proof. First, we show that S is positive invariant. In order to do that, we take u0, v0 ∈ S. Then, by 2.4 , J u t ≤ E u t , v t ≤ E0 < d, for any t ≥ 0. Now, if S is not positive invariant, there exists some t̂ > 0, such that I u t̂ 0, with u t̂ / 0. Then, by 2.21 , d ≤ J u t̂ . But this is impossible because J u t̂ < d. The proof of the positive invariance of U is quite similar. Indeed, if this is not true there exists some t̂ > 0, such that I u t̂ 0. From ii of Lemma 2.2 u t̂ / 0, and this implies the same contradiction as before. Next result gives an interpretation of sets S and U and follows from Lemma 2.2. Lemma 2.5. The sets S and U have the properties S ⊂ E u, v < d ∩ [ u, v ∈ H : r−2 /r b̂ u 2/r ≤ â u < 2 r b̂ u r − 2 r , 0 < b̂ u < ] , ∪ E u, v < d ∩ [ u, v ∈ H : r − 2 ( γ 1 ) −rγ b̂ u < â u < 2 r b̂ u r − 2 r , −γ < b̂ u < 0 ] U E u, v < d ∩ [ u, v ∈ H : r−2 /r b̂ u 2/r ≤ â u < 2 r b̂ u r − 2 r , b̂ u > ] , , 2.34 S ∩ U ue, 0 ∈ H : ue ∈ N∗ [ ue, 0 ∈ H : ue ∈ N, â ue b̂ ue ] . 2.35 The following result is a direct consequence of vi in Lemma 2.2 and Lemma 2.4. Lemma 2.6. For every solution of 1.1 , only one of the following holds: i there exists some t0 ≥ 0 such that u t0 , v t0 ∈ S, and remains there for every t > t0, ii there exists some t0 ≥ 0 such that u t0 , v t0 ∈ U, and remains there for every t > t0, iii u t , v t ∈ E u, v ≥ d for every t ≥ 0. Hence, we notice that the sets S and U play an important role in the dynamics of 1.1 . Moreover, we will prove that any solution eventually contained in S converges to the zero equilibrium. If enters in U, either blowups in a finite time or it is global but without a uniform bound in H for every t ≥ 0, in the case that λ > 2, in 1.6 . Also, we will prove that any solution with u t , v t ∈ E u, v ≥ d , for every t ≥ 0, is bounded and converges to the set of nonzero equilibria E. We will need the following inequalities to show blowup and convergence to the zero equilibrium, respectively, in the dissipative case. Lemma 2.7. Let F ∈ W loc R be a nonnegative function such that Ḟ t ≥ CFa t a.e. for t ≥ 0, 2.36 with a > 1 and C > 0. Then, there exists some T ∗ > 0 such that limt↗T∗F t ∞. 8 Abstract and Applied Analysis Proof. Define G t ≡ F1−a t , then Ġ t ≤ 1 − a C < 0 a.e. for t ≥ 0. 2.37 Hence, 0 < G 0 1 − a Ct, which is only possible if t < T ∗ ≡ 1/C a − 1 F1−a 0 . Lemma 2.8. Let F ∈ W loc R be a nonnegative function such that Ḟ t ≤ −CFa t a.e. for t ≥ 0, 2.38 with a ≥ 1 and C > 0. Then, for t ≥ 0, if a > 1 F t ≤ F0 { 1 tC a − 1 Fa−1 0 }−1/ a−1 , 2.39 and, if a 1 F t ≤ F0e−Ct. 2.40 Proof. Consider a > 1, and notice that F 1−a ̇ t ≥ a − 1 C. Then, we integrate and obtain the first inequality. Now, let a → 1, and the second one follows. 3. Timoshenko Equation Due to our assumptions on r and γ , we restrict our analysis to dimensions n ≤ 5. Indeed, since γ ≥ 1, 2 γ 1 < r and r ≤ 2 n − 2 / n − 4 , if n ≥ 5, then our analysis considers, n 5 whenever γ < 2. We also notice that in any case we do not consider the interval 2 < r ≤ 4. Moreover, r ≤ 6 whenever n 5. We begin with a characterization of blowup when δ > 0 and λ ≥ 2. Theorem 3.1. Let u t , v t S t u0, v0 be a solution of problem 1.1 , and suppose that r > 2 γ 1 . A necessary and sufficient condition for nonglobality, blowup by Theorem 2.1, is that λ < r and there exists t0 ≥ 0 such that u t0 , v t0 ∈ U. Proof. Sufficiency By Lemma 2.4, u t , v t ∈ U for all t > t0. Now, we consider the function defined, along the solution, by V t ≡ d − E t , 3.1 and notice that because of energy equation 2.3 , V t ≥ d − E0 ≡ V0 > 0, 3.2 where, now E0 ≡ E u t0 , v t0 . Abstract and Applied Analysis 9 Notice that from 2.29 in Lemma 2.3, V t ≤ d − J u t ≤ d − r r − 2 1 r b̂ u t − 2 r − 2 1 r b u t − γ r − 2 (γ 1)c u t . 3.3and Applied Analysis 9 Notice that from 2.29 in Lemma 2.3, V t ≤ d − J u t ≤ d − r r − 2 1 r b̂ u t − 2 r − 2 1 r b u t − γ r − 2 (γ 1)c u t . 3.3 We will need some estimates. First, we notice that from energy equation in terms of V t and 3.3 , ∣∣δ ( u t , v t |v t |λ−2 ) 2 ∣∣ ≤ δ‖u t ‖λ‖v t ‖λ−1 λ ≤ C Ω δ‖u t ‖r‖v t ‖λ−1 λ ≤ C Ω δ‖u t ‖1−k r ‖u t ‖r ‖v t ‖λ−1 λ ≤ C Ω δ‖u t ‖1−k r [ ν‖u t ‖ r 1 C ν ‖v t ‖λλ ] < CV 1−k /r t [ νδ‖u t ‖ r 1 C ν V̇ t ] , 3.4 where k ∈ 1, r/λ , C ≡ C Ω r/μ 1−k /r , C Ω > 0 is the constant in the continuous embedding L Ω ⊂ L Ω , C ν > 0, and ν > 0 will be chosen later. Consider a positive number q to be chosen later, from 3.2 3.3 , we obtain −I u t −qJ u t q − 2 2 a u t q − 2(γ 1) 2 ( γ 1 ) c u t r − q r b u t ≥ q V0 − d q − 2 2 a u t q − 2(γ 1) 2 ( γ 1 ) c u t r − q r b u t . 3.5 If V0 ≥ d, we choose q ≡ 2 γ 1 , and from 3.5 we get −I u t ≥ r − 2 ( γ 1 ) r b u t . 3.6 If V0 < d, then we notice that from 3.2 3.3 , V0 − d ≥ r − 2 V0 − d r − 2 V0 2d ( 1 r b u t − γ r − 2 (γ 1)c u t ) . 3.7 10 Abstract and Applied Analysis Hence and from 3.5 , we have the estimate −I u t ≥ q − 2 2 a u t q 2 ( γ 1 ) ( q − 2(γ 1) q 2γ d − V0 r − 2 V0 2d ) c u t q r ( r − q q − r − 2 d − V0 r − 2 V0 2d ) b u t . 3.8 In this case, we choose the number q so that the coefficient of c u t in 3.8 be equal to zero, then q ≡ 2 ( γ 1 ) r − 2 V0 2d ( r − 2(γ 1V0 2 ( γ 1 ) d . 3.9 We note that 2 < q < 2 γ 1 , and we get −I u t ≥ γra u t ( r − 2(γ 1))b u t ( r − 2(γ 1V0 2 ( γ 1 ) d V0. 3.10 Therefore, from 3.6 and 3.10 , −I u t ≥ Ĉb u t , 3.11