PARABOLIC SYSTEMS AND REGULARIZEDSEMIGROUPSQuan

PARABOLIC SYSTEMS 185 such that u = F(F 1u). We de ne u(A) 2 B(X) by (1:3) u(A)x = ZRn(F 1u)( )e i( ;A)x d for x 2 X: De ne MN (FL1) = f(ujk); ujk 2 FL1g. Similarly, MN(Lp) and so on. If u = (ujk) 2 MN(FL1) then u(A) (ujk(A)) 2 B(XN ). It is known that MN (FL1) is a (non-commutative) Banach algebra under matrix pointwise multiplication and addition with norm kukFL1 kF 1ukL1 , where F 1u = (F 1ujk). Furthermore, u 7! u(A) is an algebra homomorphism from MN(FL1) into B(XN), and ku(A)k MkukFL1 for all u 2 MN(FL1) and some M > 0. The following lemma (cf. [15, 19, 24]) will play an important role in our proofs. Lemma 1.1. (a) Let E = f (A)x; 2 S and x 2 Xg. Then E D(A1), E = X, P (A)jEN = P (A), and (A)P (A) P (A) (A) = (P )(A) for 2 S. (b) (Bernstein) If n=2 < j 2 N, then Hj(Rn) FL1 and there exists M > 0 such that kukFL1 Mkuk1 n=2j L2 X j j=j kD ukn=2j L2 for u 2 Hj(Rn): (c) Let ut 2 C1 for t 0. If there exist constants M , L, a > 0 and b 2a n 1 such that jDkut( )j M(1 + tjkj)j jbjkj a for j j L, t 0, and k 2 Nn0 (jkj [n2 ] + 1), then there exist M 0 > 0 and 2 C1 c such that ut(1 ) 2 FL1 and kut(1 )kFL1 M 0(1 + tn=2) for t 0. Finally, let jAj2 =Pnj=1 A2j and C = (1+ jAj2) =2 ( 2 R) as fractional powers. Then C = (1 + j j2) =2(A) 2 B(X) for > 0 (see [5]), and (R(C ))N D(P (A)) for > m (cf. [15, 24]). Relating to P (A) as the generator of a C -regularized semigroup (see x3 below), we have the following result of wellposedness of (1.1) (cf. [4, 24]). Lemma 1.2. Let a 0, and M , ! be suitable constants depending on . (a) If P (A) generates an (exponentially bounded) C -regularized semigroup for every > a, then for every > a and ~u0 2 (R(C +m))N , (1.1) has a unique solution ~u(t) (i.e., ~u 2 C([0;1);XN )\C1((0;1);XN )) with k~u(t)k M e! tkC ~u0k for t 0. (b) If P (A) generates an (exponentially bounded) di erentiable C -regularized semigroup for some > a, then for every ~u0 2 (R(C ))N , (1.1) has a unique solution ~u(t) with k~u(t)k M e! tkC ~u0k for t 0. 186 QUAN ZHENG AND YONGSHENG LI 2. Parabolic P ( ) in the sense of Petrovskij. In the sequel, we set Pm( ) = Pj j=m P (the principal part of P ( )) and (P ( )) = sup1 j N Re j( ) (the spectral bound of P ( )), where j( ) (1 j N) are the eigenvalues of P ( ). From Friedman [8, p. 171], there exists a constant M > 0 such that (2:1) et (P ( )) ketP ( )k M(1 + t+ tj jm)N 1et (P ( )) for 2 Rn and t 0: If (Pm( )) < 0 for 2 Rn n f0g then P ( ) is said to be parabolic in the sense of Petrovskij [9]. It is known that m is always even for a parabolic system P ( ) in the sense of Petrovskij. In particular, the following characterizations hold. Lemma 2.1. Let = arctan , where = supj j=1 (Pm( )) and = supj j=1 j (iPm( ))j. Then the following statements are equivalent. (a) P ( ) is parabolic in the sense of Petrovskij. (b) There exist constants ! > 0 and !0 2 R such that (P ( )) !j jm+ !0 for 2 Rn. (c) There exist constants !, M > 0 such that keP ( )k M exp( !j jm) for 2 Rn. (d) For every " 2 (0; ), there exist constants !", M", L" > 0 such that ketP ( )k M"exp( !"j jmRe t) for j j L" and t 2 ". Proof. The equivalence of (a) and (b) follows from [8, p. 191]. (d))(c) is obvious. (c))(b) follows easily from the rst inequality in (2.1). It remains to show (a))(d). For any " 2 (0; ), choose !" = ( tan( "))=3. Then !" > 0. Since the eigenvalues of Pm( ) are all homogeneous, the parabolicity of P ( ) in the sense of Petrovskij implies that (Pm( )) j jm and j (iPm( ))j j jm for 2 Rn. It thus follows from the second inequality in (2.1) that k exp(tPm( ))k k exp(Pm( )Re t)k k exp(iPm( ) Im t)k M(1 + Re t+Re tj jm)N 1exp( (Pm( ))Re t) (1 + j Im tj+ j Im tjj jm)N 1exp(j (iPm( ))jj Im tj) M 0 "(1 + Re t+Re tj jm)N 1exp( 3!"j jmRe t) for 2 Rn and t 2 ". Also, there exists a constant ! > 0 such that ktP ( ) tPm( )k !jtjj jm 1 !"j jmRe t for j j L" and t 2 "; ABSTRACT PARABOLIC SYSTEMS 187 where L" = ! sec( ")=!". Combining these two estimates yields ketP ( )k k exp(tPm( ))k k exp(tP ( ) tPm( ))k M 0 "(1 + Re t+Re tj jm)N 1exp( 2!"j jmRe t) M" exp( !"j jmRe t) for j j L" and t 2 ": Therefore the statement (d) holds. We now can start with the main result of this section as follows. Theorem 2.2. Let P ( ) be parabolic in the sense of Petrovskij and be dened in Lemma 2.1. Then P (A) generates an analytic semigroup (T (t))t2 on XN , which satis es the following statements. (a) kT (t)k M(1 + tN 1+n=2)e!t for all t 0 and some M > 0, where ! = sup 2Rn (P ( )). (b) D(P (A)) (R(C ))N for 2 (0;m). (c) T ( ) 2 H( ; (B(A1))N ). Proof. By induction with respect to jkj (k 2 Nn0 ), we have that (2:2) Dk(P ( )letP ( )) = jkj X j=j0 tjQj( )etP ( ) for 2 Rn; t 2 C; and l 2 N0; where Qj( ) is an N N matrix of polynomials of degree m(j + l) jkj, and j0 is the least nonnegative integer such that j0 jkj=m l. For any " 2 (0; ), let t 2 " and l 2 N0. Then, by (2.2) and Lemma 2.1(b), there exists a constant !0 " > 0 such that kDk(P ( )letP ( ))k (M"(Re t)jkj=m l exp( !0 "j jmRe t) for j j L" M"(Re t)jkj=m l exp(!0 "Re t) for j j < L" where, and in the sequel, M" denotes a generic constant depending on ". Consequently kDk(P letP )kL2 M"(Re t)jkj=m l(exp(!0 "Re t) + (Re t) n=2m): Thus, by Bernstein's theorem, P letP 2MN(FL1) and (2:3) kP letP kFL1 M"(Re t) l exp(2!0 "Re t) for t 2 " and l 2 N0: De ne T (t) = (etP )(A) for t 2 and T (0) = IN . Then (2.3) (with l = 0) implies that kT (t)k M" exp(2!0 "Re t) (t 2 ") and T (t + s) =PARABOLIC SYSTEMS 187 where L" = ! sec( ")=!". Combining these two estimates yields ketP ( )k k exp(tPm( ))k k exp(tP ( ) tPm( ))k M 0 "(1 + Re t+Re tj jm)N 1exp( 2!"j jmRe t) M" exp( !"j jmRe t) for j j L" and t 2 ": Therefore the statement (d) holds. We now can start with the main result of this section as follows. Theorem 2.2. Let P ( ) be parabolic in the sense of Petrovskij and be dened in Lemma 2.1. Then P (A) generates an analytic semigroup (T (t))t2 on XN , which satis es the following statements. (a) kT (t)k M(1 + tN 1+n=2)e!t for all t 0 and some M > 0, where ! = sup 2Rn (P ( )). (b) D(P (A)) (R(C ))N for 2 (0;m). (c) T ( ) 2 H( ; (B(A1))N ). Proof. By induction with respect to jkj (k 2 Nn0 ), we have that (2:2) Dk(P ( )letP ( )) = jkj X j=j0 tjQj( )etP ( ) for 2 Rn; t 2 C; and l 2 N0; where Qj( ) is an N N matrix of polynomials of degree m(j + l) jkj, and j0 is the least nonnegative integer such that j0 jkj=m l. For any " 2 (0; ), let t 2 " and l 2 N0. Then, by (2.2) and Lemma 2.1(b), there exists a constant !0 " > 0 such that kDk(P ( )letP ( ))k (M"(Re t)jkj=m l exp( !0 "j jmRe t) for j j L" M"(Re t)jkj=m l exp(!0 "Re t) for j j < L" where, and in the sequel, M" denotes a generic constant depending on ". Consequently kDk(P letP )kL2 M"(Re t)jkj=m l(exp(!0 "Re t) + (Re t) n=2m): Thus, by Bernstein's theorem, P letP 2MN(FL1) and (2:3) kP letP kFL1 M"(Re t) l exp(2!0 "Re t) for t 2 " and l 2 N0: De ne T (t) = (etP )(A) for t 2 and T (0) = IN . Then (2.3) (with l = 0) implies that kT (t)k M" exp(2!0 "Re t) (t 2 ") and T (t + s) = 188 QUAN ZHENG AND YONGSHENG LI T (t)T (s) (t; s 2 ). Also, (2.3) (with l = 1) implies that t 7! etP 2 H( ;MN(FL1)), and so T ( ) 2 H( ; B(XN )). Since for 2 S kT (t) (A) (A)k M"jtj exp(2!0"Re t)kP kFL1 ! 0 ( " 3 t! 0); the strong continuity of T (t) (t 2 " [ f0g) follows from E = X and the estimate of T (t) (t 2 "). Thus (T (t))t2 is an analytic semigroup. To show that P (A) is its generator, let L for large be the Laplace transform of (T (t))t 0. Then from Lemma 1.1(a) one has P (A)T (t) (A) = T (t)P (A) (A). Also, by Fubini's theorem, L ( P (A)) (A) = Z 1 0 e tetPdt( P ) (A) = (A): Thus Lemma 1.1(a) leads to L = R( ; P (A)) for large . The claim now follows from [5, p. 627]. (a) Let !1 < !2 < !. Then by Lemma 2.1(b) there exist constants ; L > 0 such that (P ( )) 2 j jm+!1 for j j L. It thus follows from (2.1) and (2.2) (with l = 0) that (2:4) kDketP ( )k (Mtjkj=m exp( j jmt+ !2t) for j j L and t 0 M(tj0 + tjkj)(1 + tN 1)e!t for j j < L and t 0: From this we obtain that kDketPkL2 M(tj0 + tjkj)(1 + tN 1)e!t +Mt(2jkj n)=2me!0t for t > 0: Thus, by Bernstein's theorem, etP 2 MN(FL1) and ketP kFL1 M(1 + tN 1+n=2)e!t for t > 0, and so (a) follows. (b) Let w( ) = (1 + j j2) =2(!0 P ( )) 1 for some !0 > !. Since (2.4) (with k = 0) and the Hille-Yosida theorem lead to k!0 P ( )k M j jm for j j L, it follows that kDkw( )k M j j jkj+ m for j j L and M for j j < L. Hence w(A) 2 B(XN). Noting that (a) implies !0 2 (P (A)), one has D(P (A)) (R(C ))N . (c) We rst show by induction that D(P (A)j( )) (D(A ))N for = ( 1; ; n) 2 Nn0 , where j( ) = [ j j m ] + 1. Let v ( ) = (!0 P ( )) j( ) for 2 Rn. As seen above, we have v (A) 2 B(XN). If j j = 0, then the statement is obvious. Let ~ = ( 1; ; j+1; ; n) and, for any x 2 XN , choose EN 3 xk ! x. Then, by Lemma 1.1(a), AjA R(!0; P (A))j(~ )xk ! v~ (A)x, Since Aj is closed, the induction assumption yields A R(!0; P (A))j(~ )x 2 (D(Aj))N , and thus the claim follows. Now we deduce from the properties of analytic semigroups that A T (t) = v (A) j( ) Xk=0 j( ) k !(!0)j( ) k( 1)kT (k)(t) for 2 Nn0 and t 2 ABSTRACT PARABOLIC SYSTEMS 189 and therefore (c) is proved. Let X be some function space on which translations are uniformly bounded and strongly continuous. Then the above result can be applied to P (D) (i.e., take A = D) onX, immediately. In the sequel, we assume that all partial differential operators (PDOs) have the maximal domains in the distributional sense, and so they are closed and densely de ned on X. X can be chosen as, for example, Lp (1 p <1), ff 2 C(Rn); f is bounded and uniformly continuousg, ff 2 C(Rn); limjxj!1 f(x) = 0g, ff 2 C(Rn); f(x) exists as jxj ! 1g, ff 2 C(Rn); f is 1-periodicg, or ff 2 C(Rn); f is almost periodicg with sup-norms. Moreover, let W ;X ( 0) be the completion of S under the norm kuk ;X kukX + kF 1((1 + j j2) =2Fu)kX for u 2 S: When X = Lp (1 < p <1) and 2 N0, W ;p W ;X is the usual Sobolev space. Corollary 2.3. Let P ( ) be parabolic in the sense of Petrovskij and be dened in Lemma 2.1. Then P (D) generates an analytic semigroup (T (t))t2 on XN , which satis es Theorem 2.2(a) and (Wm+ ;X)N D(P (D)) (Wm ;X)N for any 2 (0;m). We remark that by Mihlin's multiplier theorem (see, e.g., [20, p. 96]) D(P (D)) = (Wm;p)N if X = Lp (1 < p < 1). We also remark that Theorem 2.2 can be applied to some PDOs with space-dependent coe cients and de ned on bounded domains, such as on Lp([0; 1]n) (1 p < 1), ff 2 C([0; 1]n); f jxj=0 = f jxj=1 = 0g, or ff 2 C([0; 1]n); f jxj=0 = f jxj=1g. For details we refer to [15, 24]. Example 2.4. Consider the following equation (2:5) utt 2a ut b u+ c 2u = 0 where a; b; c are positive constants. Then we can reduce it into the rst order system associated with the matrix of polynomials P ( ) = 0 j j2 b+ cj j2 2aj j2! : Since its principal part, i.e., P2( ) has eigenvalues ( a pa2 c)j j2, it follows from Corollary 2.3 that P (D) generates an analytic semigroup (T (t))t2 on X2, where = ( =2 for a2 c arcsin(a=pc) for a2 < c:PARABOLIC SYSTEMS 189 and therefore (c) is proved. Let X be some function space on which translations are uniformly bounded and strongly continuous. Then the above result can be applied to P (D) (i.e., take A = D) onX, immediately. In the sequel, we assume that all partial differential operators (PDOs) have the maximal domains in the distributional sense, and so they are closed and densely de ned on X. X can be chosen as, for example, Lp (1 p <1), ff 2 C(Rn); f is bounded and uniformly continuousg, ff 2 C(Rn); limjxj!1 f(x) = 0g, ff 2 C(Rn); f(x) exists as jxj ! 1g, ff 2 C(Rn); f is 1-periodicg, or ff 2 C(Rn); f is almost periodicg with sup-norms. Moreover, let W ;X ( 0) be the completion of S under the norm kuk ;X kukX + kF 1((1 + j j2) =2Fu)kX for u 2 S: When X = Lp (1 < p <1) and 2 N0, W ;p W ;X is the usual Sobolev space. Corollary 2.3. Let P ( ) be parabolic in the sense of Petrovskij and be dened in Lemma 2.1. Then P (D) generates an analytic semigroup (T (t))t2 on XN , which satis es Theorem 2.2(a) and (Wm+ ;X)N D(P (D)) (Wm ;X)N for any 2 (0;m). We remark that by Mihlin's multiplier theorem (see, e.g., [20, p. 96]) D(P (D)) = (Wm;p)N if X = Lp (1 < p < 1). We also remark that Theorem 2.2 can be applied to some PDOs with space-dependent coe cients and de ned on bounded domains, such as on Lp([0; 1]n) (1 p < 1), ff 2 C([0; 1]n); f jxj=0 = f jxj=1 = 0g, or ff 2 C([0; 1]n); f jxj=0 = f jxj=1g. For details we refer to [15, 24]. Example 2.4. Consider the following equation (2:5) utt 2a ut b u+ c 2u = 0 where a; b; c are positive constants. Then we can reduce it into the rst order system associated with the matrix of polynomials P ( ) = 0 j j2 b+ cj j2 2aj j2! : Since its principal part, i.e., P2( ) has eigenvalues ( a pa2 c)j j2, it follows from Corollary 2.3 that P (D) generates an analytic semigroup (T (t))t2 on X2, where = ( =2 for a2 c arcsin(a=pc) for a2 < c: 190 QUAN ZHENG AND YONGSHENG LI Moreover, noting that aj j2 pa2j j4 bj j2 cj j4 are the eigenvalues of P ( ) one has that kT (t)k M(1 + t1+nX ) for t 0. 3. Parabolic P ( ) in the sense of Shilov. This section is concerned with the parabolic P (A) generating a regularized semigroup. Let r 2 (0;m], P ( ) is said to be r-parabolic in the sense of Shilov [9] if there exist constants ! > 0 and !0 2 R such that (P ( )) !j jr+!0 for 2 Rn. In particular, P ( ) is said to be Petrovskij correct [9] if r = 0, i.e., sup 2Rn (P ( )) <1. By Lemma 2.1, P ( ) is parabolic in the sense of Petrovskij if and only if it is m-parabolic in the sense of Shilov. The de nitions of (exponentially bounded) regularized semigroups can be given by using Laplace transforms [4, 23]. Let C 2 B(X) be injective. An exponentially bounded and strongly continuous family (T (t))t 0 B(X) is called a C-regularized semigroup generated by a linear operator B if C 1BC = B, B is injective (for large 2 R), R(C) R( B), and ( B) 1C is the Laplace transform of (T (t))t 0. If, in addition, T ( ) 2 C([0;1); B(X)) \ C1((0;1); B(X)) then (T (t))t 0 is said to be di erentiable. Theorem 3.1. Let P ( ) be r-parabolic in the sense of Shilov for some r 2 (0;m), and > (m r)(N 1 + n=2). Then P (A) generates a differentiable C -regularized semigroup (T (t))t 0 on XN , which satis es the following statements. (a) kT (l)(t)k MM l 0((l!t l)m=r + tN 1+n=2)e!t for t 0 and l 2 N0, where M and M0 are constants independent of t and l, and ! = sup 2Rn (P ( )). (b) D(P (A)) (R(C ))N , where = 0 if r nm n+2 and 0 < < r n(m r)=2 if r > nm n+2 . (c) T ( ) 2 C1((0;1); (B(A1))N ). Proof. By our assumption on P ( ), for !00 < ! there exist constants !0, L > 0 such that (3:1) (P ( )) 2!0j jr + !00 for j j L: Let jkj [n=2]+1. Then, by induction, there exists a constant M1 0 such that (3:2) kDkP ( )lk M l 1j jml jkj for j j L and l 2 N: ABSTRACT PARABOLIC SYSTEMS 191 From (2.1) and (3.1) we have that kDketP ( )k M jkj Xj=0 tj j jmj jkj(1 + t+ tj jm)N 1 exp( 2!0j jrt+ !00t) M j j(m r 1)jkj+(m r)(N 1) exp( !0j jrt+ !t) (3.3) for j j L and t 0, where, and in the sequel,M denotes a generic constant independent of l, t and . Set ut = (1+ j j2) =2etP for t 0. It then follows from (3.2), (3.3), and Leibniz's formula that kDk(P ( )lut( ))k MM l 1j j(m r 1)jkj+(m r)(N 1)+lm exp( !0j jrt+ !t) MM l 2(l!t l)m=rj j(m r 1)jkj+(m r)(N 1) e!t for j j L, t > 0 and l 2 N0, where M2 = M1(r!0=m) m=r. Write P lut = (ul;t ij ), then we obtain by Lemma 1.1(c) that ul;t ij (1 ij) 2 FL1 for some ij 2 C1 c , and (3:4) kul;t ij (1 ij)kFL1 MM l 2(l!t l)m=re!t for t > 0 and l 2 N0: On the other hand, let K > 0 such that supp ij f 2 Rn; j j Kg for 1 i; j N . Since an induction implies that (3:5) kDkP ( )lk M l 3 for j j K and l 2 N; it follows from (2.2) and (2.1) that kDk(P ( )lut( ))k MM l 3(1 + tN 1+jkj)e!t for j j K; t 0 and l 2 N0: Thus Leibniz's formula and Bernstein's theorem lead to ul;t ij ij 2 FL1 and (3:6) kul;t ij ijkFL1 MM l 3(1 + tN 1+jkj)e!t for t 0 and l 2 N0: Combining (3.4) and (3.6) one nds that P lut 2MN (FL1) and (3:7) kP lutkFL1 MM l 0((l!t l)m=r + tN 1+n=2)e!t for t > 0 and l 2 N0; where M0 = 2max(M2;M3). De ne T (t) = ut(A) for t 0. Here we note that when l = 0, (3.4) is yet true for t = 0. Furthermore, observing carefully the proof of (3.7) and using Lebesgue's dominated convergence theorem, one nds that ut (t 0) is continuous in the norm k kFL1 , and so T ( ) 2 C([0;1); B(X)). Obviously, (3.7) implies (a) and T ( ) 2 C1((0;1); B(X)), while C 1 P (A)C = P (A) can be deduced from Lemma 1.1(a). The remainder of the proof may bePARABOLIC SYSTEMS 191 From (2.1) and (3.1) we have that kDketP ( )k M jkj Xj=0 tj j jmj jkj(1 + t+ tj jm)N 1 exp( 2!0j jrt+ !00t) M j j(m r 1)jkj+(m r)(N 1) exp( !0j jrt+ !t) (3.3) for j j L and t 0, where, and in the sequel,M denotes a generic constant independent of l, t and . Set ut = (1+ j j2) =2etP for t 0. It then follows from (3.2), (3.3), and Leibniz's formula that kDk(P ( )lut( ))k MM l 1j j(m r 1)jkj+(m r)(N 1)+lm exp( !0j jrt+ !t) MM l 2(l!t l)m=rj j(m r 1)jkj+(m r)(N 1) e!t for j j L, t > 0 and l 2 N0, where M2 = M1(r!0=m) m=r. Write P lut = (ul;t ij ), then we obtain by Lemma 1.1(c) that ul;t ij (1 ij) 2 FL1 for some ij 2 C1 c , and (3:4) kul;t ij (1 ij)kFL1 MM l 2(l!t l)m=re!t for t > 0 and l 2 N0: On the other hand, let K > 0 such that supp ij f 2 Rn; j j Kg for 1 i; j N . Since an induction implies that (3:5) kDkP ( )lk M l 3 for j j K and l 2 N; it follows from (2.2) and (2.1) that kDk(P ( )lut( ))k MM l 3(1 + tN 1+jkj)e!t for j j K; t 0 and l 2 N0: Thus Leibniz's formula and Bernstein's theorem lead to ul;t ij ij 2 FL1 and (3:6) kul;t ij ijkFL1 MM l 3(1 + tN 1+jkj)e!t for t 0 and l 2 N0: Combining (3.4) and (3.6) one nds that P lut 2MN (FL1) and (3:7) kP lutkFL1 MM l 0((l!t l)m=r + tN 1+n=2)e!t for t > 0 and l 2 N0; where M0 = 2max(M2;M3). De ne T (t) = ut(A) for t 0. Here we note that when l = 0, (3.4) is yet true for t = 0. Furthermore, observing carefully the proof of (3.7) and using Lebesgue's dominated convergence theorem, one nds that ut (t 0) is continuous in the norm k kFL1 , and so T ( ) 2 C([0;1); B(X)). Obviously, (3.7) implies (a) and T ( ) 2 C1((0;1); B(X)), while C 1 P (A)C = P (A) can be deduced from Lemma 1.1(a). The remainder of the proof may be 192 QUAN ZHENG AND YONGSHENG LI carried out by modifying the corresponding parts of the proof of Theorem 2.2. We now turn to a stronger condition on P ( ). To this end, put e (P ) = supfRe(Py; y); y 2 RN and kyk = 1g for P 2MN(C); where ( ; ) is the inner product in CN and kyk = (y; y)1=2. Note that we also can write e (P ) = supfRe z; z 2 n: r:(P )g, where n.r.(P ) is the numerical range of P . Theorem 3.2. Assume there exist constants r 2 (0;m), !0 > 0 and !00 2 R such that (3:8) e (P ( )) !0j jr + !00 for 2 Rn: Let > n(m r)=2. Then P (A) generates a di erentiable C -regularized semigroup (T (t))t 0 on XN satisfying kT (l)(t)k MM l 0((l!t l)m=r + tn=2)e!t for t 0 and l 2 N0; where M and M0 are constants independent of t and l, and ! = sup 2Rn e (P ( )). Proof. By (3.8) there exist constants , L > 0 such that e (P ( )) j jr+! for j j L. Thus the Lumer-Phillips theorem implies that (3:9) ketP ( )k (expf 2!1j jrt+ !tg for j j L and t 0 e!t for j j K and t 0; where K > 0 is chosen as in the proof of Theorem 3.1. Set ut = (1+ j j2) =2etP for t 0. Then combining (2.2), (3.2), (3.5) and (3.9) yields thatkDk(P ( )lut( ))k (MM l 2(l!t l)m=rj j(m r 1)jkj e!t for j j L MM l 3(1 + tjkj)e!t for j j K; where t > 0 and l 2 N0. Now, the remainder of the proof may be copied from that of the proof of Theorem 3.1. From the proof of Theorem 3.1 and 3.2 one nds that in the case r = 0 the following result holds, which sharpens Theorem 3.1 and 4.2 in [2]. Theorem 3.3. (a) Let ! sup 2Rn (P ( )) < 1 and > m(N 1 + n=2). Then P (A) generates a C -regularized semigroup (T (t))t 0 on XN satisfying T ( ) 2 C([0;1); (B(X))N ) and kT (t)k M(1 + tN 1+n=2)e!t for t 0. ABSTRACT PARABOLIC SYSTEMS 193 (b) Let ! sup 2Rn e (P ( )) <1 and > nm=2. Then P (A) generates a C -regularized semigroup (T (t))t 0 on XN satisfying T ( ) 2 C([0;1); (B(X))N ) and kT (t)k M(1 + tn=2)e!t for t 0. Let X be chosen as in the end of Section 2, and de ne nX (= nj 1 2 1 p j if X = Lp (1 < p <1) > n=2 if X = L1 or the space of continuous functions: Then the following holds. Corollary 3.4. (a) Let P ( ) be r-parabolic in the sense of Shilov for some r 2 (0;m) (resp. satisfy (3.8)). Then P (D) generates a di erentiable R(1; ) regularized semigroup on XN , where = (m r)(N 1+nX)=2 (resp. = nX(m r)=2): (b) Let ! < 1, where ! is de ned in Theorem 3.3(a) (resp. (b)). Then P (D) generates a norm-continuous, R(1; ) -regularized semigroup on XN , where = m(N 1 + nX)=2 (resp. = mnX=2). In particular, when X = L2 we can choose = m(N 1)=2 (resp. = 0): When X is a space of continuous functions or L1, Corollary 3.4 follows from Theorem 3.1-3.3, immediately. When X = Lp (1 < p <1), Corollary 3.4 can be deduced by modifying the proofs of Theorem 3.1-3.3. The main points are using the Riesz-Thorin convexity theorem andMiyachi's multiplier theorem G in [17] and noting u(D) = F 1(uF ) for u 2 FL1 and 2 S. We refer to [11, 24] for the details. When X = L2, the result (with = m(N 1)=2) was shown in [12], while the result (with = 0) can be shown by modifying its proof (cf. the proof of Theorem 3.2). We remark that Corollary 3.4(b) is essentially due to [11, 12]. Moreover, when X = ff 2 C(Rn); f is boundedg or L1, one can show that Corollary 3.4 is still true for nX > n=2 (cf. [16]). Example 3.5. (a) Consider the following linear system (cf. [1]) 8>><>>:ut = 2auxx + bvx cvxxx vt = ux u(0) = u0; v(0) = v0 where a; b; c are positive constants. The corresponding matrix of polynomials is P ( ) = 2a 2 ib + ic 3 i 0 ! :PARABOLIC SYSTEMS 193 (b) Let ! sup 2Rn e (P ( )) <1 and > nm=2. Then P (A) generates a C -regularized semigroup (T (t))t 0 on XN satisfying T ( ) 2 C([0;1); (B(X))N ) and kT (t)k M(1 + tn=2)e!t for t 0. Let X be chosen as in the end of Section 2, and de ne nX (= nj 1 2 1 p j if X = Lp (1 < p <1) > n=2 if X = L1 or the space of continuous functions: Then the following holds. Corollary 3.4. (a) Let P ( ) be r-parabolic in the sense of Shilov for some r 2 (0;m) (resp. satisfy (3.8)). Then P (D) generates a di erentiable R(1; ) regularized semigroup on XN , where = (m r)(N 1+nX)=2 (resp. = nX(m r)=2): (b) Let ! < 1, where ! is de ned in Theorem 3.3(a) (resp. (b)). Then P (D) generates a norm-continuous, R(1; ) -regularized semigroup on XN , where = m(N 1 + nX)=2 (resp. = mnX=2). In particular, when X = L2 we can choose = m(N 1)=2 (resp. = 0): When X is a space of continuous functions or L1, Corollary 3.4 follows from Theorem 3.1-3.3, immediately. When X = Lp (1 < p <1), Corollary 3.4 can be deduced by modifying the proofs of Theorem 3.1-3.3. The main points are using the Riesz-Thorin convexity theorem andMiyachi's multiplier theorem G in [17] and noting u(D) = F 1(uF ) for u 2 FL1 and 2 S. We refer to [11, 24] for the details. When X = L2, the result (with = m(N 1)=2) was shown in [12], while the result (with = 0) can be shown by modifying its proof (cf. the proof of Theorem 3.2). We remark that Corollary 3.4(b) is essentially due to [11, 12]. Moreover, when X = ff 2 C(Rn); f is boundedg or L1, one can show that Corollary 3.4 is still true for nX > n=2 (cf. [16]). Example 3.5. (a) Consider the following linear system (cf. [1]) 8>><>>:ut = 2auxx + bvx cvxxx vt = ux u(0) = u0; v(0) = v0 where a; b; c are positive constants. The corresponding matrix of polynomials is P ( ) = 2a 2 ib + ic 3 i 0 ! : 194 QUAN ZHENG AND YONGSHENG LI By Theorem 4.14(e) in [7] P ( ) is not the generator of a strongly continuous semigroup on (L1(R))2. Since the eigenvalues of P ( ) are a 2 pa2 4 b 2 c 4, P ( ) is 2-parabolic in the sense of Shilov. It thus follows from Corollary 3.4(a) that P (D) generates a di erentiable R(1; ) regularized semigroup on (L1(R))2, where > 3=4. (b) We now reduce (2.5) with c = 0 (cf. [2]) into the rst order system (1.2) with P ( ) = 0 1 bj j2 2aj j2! : Since the eigenvalues of P ( ) are aj j2 pa2j j4 bj j2, P ( ) is Petrovskij correct and satis es that (P ( )) 0 for 2 Rn. Thus Corollary 3.4(b) implies that P (D) generates a norm-continuous, R(1; ) -regularized semigroup onX2, where = 1+nX . Particularly, in the case n = 1 we can choose > 3=2, so that we obtain an improvement of the result in [2, Example 4.5], in which > 7=4 is required. Example 3.6. Consider the higher order Cauchy problem (3:10) (u(N)(t) =PNk=1 pk(A)u(k 1)(t) for t > 0 u(k 1)(0) = uk for 1 k N on X, where pk( ) (1 k N) are polynomials of 2 Rn. Let ( ) = sup1 j N Re j( ), where j( ) (1 j N) are the roots of the characteristic equation N = PNk=1 pk( ) k 1. Then (3.10) is said to be r-parabolic in the sense of Shilov [7, p. 218] if there exist constants ! > 0 and !0 2 R such that ( ) !j jr + !0 for 2 Rn. Write (3.10) as the system (1.1), in which~u(0) = 0B@ u1 ... uN1CA and P ( ) = 0BBBBBBB@ 0 I 0 0 ... . . . . . . . . . ... ... . . . . . . 0 0 0 I p1( ) pN( ) 1CCCCCCCA : Noting that det( P ( )) = N N Xk=1 pk( ) k 1; the r-parabolicity of (3.10) in the sense of Shilov is equivalent to that of P ( ). If (3.10) is r-parabolic in the sense of Shilov, then Theorem 3.1 and Lemma 1.2 imply that (3.10) has a unique solution u 2 CN 1([0;1);X) \ ABSTRACT PARABOLIC SYSTEMS 195 CN((0;1);X) for every ~u0 2 (R(C ))N , where > (m r)(N 1 + n=2) and m = maxfdegree of pkg. From this (2.5) has a unique solution u 2 C1([0;1);X) \ C2((0;1);X) for every initial value pair (u(0; ); ut(0; )) 2 (W ;X)2, where > 2 + n. For general higher order Cauchy problems, we refer to, e.g., [18, 22]. 4. Systems with time-dependent coe cients. In this section we consider P (t; ) Pj j m P (t) , where P 2 C([0; T ];MN (C)) for j j m. Let , e be some convex neighborhoods of [0; T ] in C. Then we write T = f(t; s) 2 R2; 0 s < t Tg and = f(t; s) 2 2; t 6= s; j arg(t s)j < g, where 2 (0; =2], and denote by T (resp. ) the closure of T (resp. ). Moreover, Dt = @=@t. Let C 2 B(X) be injective. A two parameter family U(t; s) 2 B(X), (t; s) 2 T , is called a C-regularized evolution system if U(t; r)U(r; s) = U(t; s)C for 0 s r t T , U(t; t) = C for 0 t T , and U( ; )x 2 C(T ;X) for x 2 X. In particular, an I-regularized evolution system is called an evolution system. Theorem 4.1. Let P (t; ) be parabolic in the sense of Petrovskij for every t 2 [0; T ]. Then there exists a unique evolution system (U(t; s))(t;s)2T on XN such that (a) U( ; ) 2 C1(T ; (B(A1))N ), DtU(t; s) = P (t; A)U(t; s) and DsU(t; s) = P (s;A)U(t; s) for (t; s) 2 T . (b) P 2Cj([0; T ];MN (C)) (j j m) for some j 2 N implies U( ; ) 2 Cj+1(T ; (B(A1))N ). In particular P 2C1([0; T ];MN (C)) (j j m) implies U( ; ) 2 C1(T ; (B(A1))N ). (c) P 2H(e ;MN(C)) (j j m) for some e implies U( ; )2 H( ; (B(A1))N ) for some . Proof. Since our assumptions on P (t; ) imply that supf (P (t; )); 2 Rn; t 2 [0; T ]g < 1 and that there exist constants , L > 0 such that (P (t; )) j jm for j j L and t 2 [0; T ], the same argument as in the proof of Theorem 2.2 leads to v t;s 2 FL1 and (4:1) kv t;skFL1 M(t s) j j=m for (t; s) 2 T ; where v t;s( ) = ut;s( ) ( 2 Nn0 ) and ut;s( ) = expfR t s P ( ; )d g. De ne U(t; s) = ut;s(A) for (t; s) 2 T and U(t; t) = I for t 2 [0; T ]. It easily follows from the properties of our functional calculus and (4.1) with j j = 0 that (U(t; s))(t;s)2T is an evolution system on XN , while the uniqueness follows from (a).PARABOLIC SYSTEMS 195 CN((0;1);X) for every ~u0 2 (R(C ))N , where > (m r)(N 1 + n=2) and m = maxfdegree of pkg. From this (2.5) has a unique solution u 2 C1([0;1);X) \ C2((0;1);X) for every initial value pair (u(0; ); ut(0; )) 2 (W ;X)2, where > 2 + n. For general higher order Cauchy problems, we refer to, e.g., [18, 22]. 4. Systems with time-dependent coe cients. In this section we consider P (t; ) Pj j m P (t) , where P 2 C([0; T ];MN (C)) for j j m. Let , e be some convex neighborhoods of [0; T ] in C. Then we write T = f(t; s) 2 R2; 0 s < t Tg and = f(t; s) 2 2; t 6= s; j arg(t s)j < g, where 2 (0; =2], and denote by T (resp. ) the closure of T (resp. ). Moreover, Dt = @=@t. Let C 2 B(X) be injective. A two parameter family U(t; s) 2 B(X), (t; s) 2 T , is called a C-regularized evolution system if U(t; r)U(r; s) = U(t; s)C for 0 s r t T , U(t; t) = C for 0 t T , and U( ; )x 2 C(T ;X) for x 2 X. In particular, an I-regularized evolution system is called an evolution system. Theorem 4.1. Let P (t; ) be parabolic in the sense of Petrovskij for every t 2 [0; T ]. Then there exists a unique evolution system (U(t; s))(t;s)2T on XN such that (a) U( ; ) 2 C1(T ; (B(A1))N ), DtU(t; s) = P (t; A)U(t; s) and DsU(t; s) = P (s;A)U(t; s) for (t; s) 2 T . (b) P 2Cj([0; T ];MN (C)) (j j m) for some j 2 N implies U( ; ) 2 Cj+1(T ; (B(A1))N ). In particular P 2C1([0; T ];MN (C)) (j j m) implies U( ; ) 2 C1(T ; (B(A1))N ). (c) P 2H(e ;MN(C)) (j j m) for some e implies U( ; )2 H( ; (B(A1))N ) for some . Proof. Since our assumptions on P (t; ) imply that supf (P (t; )); 2 Rn; t 2 [0; T ]g < 1 and that there exist constants , L > 0 such that (P (t; )) j jm for j j L and t 2 [0; T ], the same argument as in the proof of Theorem 2.2 leads to v t;s 2 FL1 and (4:1) kv t;skFL1 M(t s) j j=m for (t; s) 2 T ; where v t;s( ) = ut;s( ) ( 2 Nn0 ) and ut;s( ) = expfR t s P ( ; )d g. De ne U(t; s) = ut;s(A) for (t; s) 2 T and U(t; t) = I for t 2 [0; T ]. It easily follows from the properties of our functional calculus and (4.1) with j j = 0 that (U(t; s))(t;s)2T is an evolution system on XN , while the uniqueness follows from (a). 196 QUAN ZHENG AND YONGSHENG LI (a) As shown in the proof of Theorem 2.2(b), one can deduce from (4.1) that A U(t; s) = v t;s(A) 2 C(T ; B(XN )). Similarly, from Dtv t;s = P (t; )v t;s andDsv t;s = P (s; )v t;s for (t; s) 2 T one has thatDtA U(t; s) = P (t; A)A U(t; s) 2 C(T ; B(XN )) and DsA U(t; s) = P (s;A)A U(t; s) 2 C(T ; B(XN)), respectively. Therefore we have the claim. (b) We will show (b) by induction on j 2 N0. When j = 0, the statement has been showed in (a). Assume the statement is true for k j. Then for any j1, j2 2 N0 with j1 + j2 = j + 1, it follows from the rst equation in (a) and our assumptions that Dj1+1 t Dj2 s U(t; s) = X k1+k2=j1 j1 k1!Dk1 t P (t; A)Dk2 t Dj2 s U(t; s) 2 C(T ; (B(A1))N): Similarly Dj1 t Dj2+1 s U(t; s) 2 C(T ; (B(A1))N ), and so U( ; ) 2 Cj+2(T ; (B(A1))N ). (c) By the assumptions on P (t; ) we have supf (P (t; )); 2 Rn; t 2 g < 1 for some with e . Then it follows from Theorem 2.2 that for every xed s 2 , P (s;A) generates an analytic semigroup (T (t; s))t2 satisfying (4:2) kT (t; s)k Me!jtj for t 2 and s 2 ; where constants ! and 2 (0; =2] can be chosen to be independent of s (see the proof of Theorem 2.2). On the other hand, set v t ( ) = (!0 P (t; )) j( ) for t 2 and 2 Nn0 , where !0 > ! and j( ) = [ j j m ] + 1. Then a direct computation yields that t 7! v t 2 H( ;MN (FL1)). Thus, as proved in Theorem 2.2(c), we have (4:3) A R(!0; P ( ; A))j( ) 2 H( ; B(XN )) for 2Nn0 : In particular, R(!0; P ( ; A)) 2 H( ; B(XN )). It now follows from this, (4.2), and [14, Theorem 1] (note also [21, Theorem 5.7.2]) that U( ; ) 2 H( ; B(XN)). Combining (4.3) with this we nd U( ; ) 2 H( ; (B(A1))N). The subsequent theorem can be showed by combining the proofs of Theorem 3.1, 3.2 and 4.1. Theorem 4.2. Assume there exist constants > 0, ! 2 R and r 2 (0;m) such that (P (t; )) (resp. e (P (t; ))) j jr+! for 2 Rn and t 2 [0; T ]. Let > (m r)(N 1 + n=2) (resp. > n(m r)=2). Then there exists a ABSTRACT PARABOLIC SYSTEMS197unique C -regularized evolution system (U(t; s))(t;s)2T on XN such that theconclusions (a) and (b) of Theorem 4.1 are still true.Corresponding to Theorem 3.3 we have the following theorem, in whichthe numerical range part is related to an result in [4, Example 31.4].Theorem 4.3. Let supf (P (t; )) (resp: e(P (t; ))); 2 Rn and t 2[0; T ]g < 1 and > m(N 1 + n=2) (resp. > nm=2). Then there existsa unique C -regularized evolution system (U(t; s))(t;s)2T on XN such thatU(t; s) :(R(Cm))N ! (R(C ))N for some > m, U( ; )x 2C1(T ;XN ),DtU(t; s)x = P (t; A)U(t; s)x and DsU(t; s)x = P (s;A)U(t; s)x for x 2(R(Cm))N and (t; s) 2 T .Proof. As seen in the proof of Theorem 3.1 (resp. 3.2), we have thatu t;s (1 + j j2) =2 exp Z ts P ( ; )d2MN(FL1) for (t; s) 2 T :It then is easy to check that (U(t; s))(t;s)2T is a C-regularized evolutionsystem on XN , where U(t; s) = ut;s(A) for (t; s) 2 T .On the other hand, choose m < < m + m(N 1 + n=2) (resp. m(N 1+n=2) (resp. > mn=2). It thusfollows from U(t; s)Cm = C ut;s(A) that U(t; s) :(R(Cm))N ! (R(C ))N .Also, noting R(C ) D(A ) for j j < , one has that U(t; s) :(R(Cm))N !D(P (t; A)) for (t; s) 2 T . The desired equations now follow from this.Finally, the uniqueness of (U(t; s))(t;s)2T can be proved by the standardmethod (cf. the proof of [15, Corollary 5.4]).We now may apply Theorem 4.1-4.2 to the time-dependent system(4:4) (~ut(t; x) = P (t;D)~u(t; x) for x 2 Rn and 0 < t T~u(0; x) = ~u0(x)for x 2 Rn;on some function space X, for example, one of the spaces listed at the endof Section 2.Corollary 4.4. Assume there exist constants > 0, ! 2 R and r 2 [0;m]such that (P (t; )) (resp. e(P (t; ))) j jr+! for 2 Rn and t 2 [0; T ].Then (4.4) has a unique solution ~u(t; x) for every ~u0 2 (W ;X)N , where= ((m r)(N 1 + nX) (resp: nX(m r)) if r 2 (0;m]m(N + nX) (resp: m(1 + nX))if r = 0:PARABOLIC SYSTEMS197unique C -regularized evolution system (U(t; s))(t;s)2T on XN such that theconclusions (a) and (b) of Theorem 4.1 are still true.Corresponding to Theorem 3.3 we have the following theorem, in whichthe numerical range part is related to an result in [4, Example 31.4].Theorem 4.3. Let supf (P (t; )) (resp: e(P (t; ))); 2 Rn and t 2[0; T ]g < 1 and > m(N 1 + n=2) (resp. > nm=2). Then there existsa unique C -regularized evolution system (U(t; s))(t;s)2T on XN such thatU(t; s) :(R(Cm))N ! (R(C ))N for some > m, U( ; )x 2C1(T ;XN ),DtU(t; s)x = P (t; A)U(t; s)x and DsU(t; s)x = P (s;A)U(t; s)x for x 2(R(Cm))N and (t; s) 2 T .Proof. As seen in the proof of Theorem 3.1 (resp. 3.2), we have thatu t;s (1 + j j2) =2 exp Z ts P ( ; )d2MN(FL1) for (t; s) 2 T :It then is easy to check that (U(t; s))(t;s)2T is a C-regularized evolutionsystem on XN , where U(t; s) = ut;s(A) for (t; s) 2 T .On the other hand, choose m < < m + m(N 1 + n=2) (resp. m(N 1+n=2) (resp. > mn=2). It thusfollows from U(t; s)Cm = C ut;s(A) that U(t; s) :(R(Cm))N ! (R(C ))N .Also, noting R(C ) D(A ) for j j < , one has that U(t; s) :(R(Cm))N !D(P (t; A)) for (t; s) 2 T . The desired equations now follow from this.Finally, the uniqueness of (U(t; s))(t;s)2T can be proved by the standardmethod (cf. the proof of [15, Corollary 5.4]).We now may apply Theorem 4.1-4.2 to the time-dependent system(4:4) (~ut(t; x) = P (t;D)~u(t; x) for x 2 Rn and 0 < t T~u(0; x) = ~u0(x)for x 2 Rn;on some function space X, for example, one of the spaces listed at the endof Section 2.Corollary 4.4. Assume there exist constants > 0, ! 2 R and r 2 [0;m]such that (P (t; )) (resp. e(P (t; ))) j jr+! for 2 Rn and t 2 [0; T ].Then (4.4) has a unique solution ~u(t; x) for every ~u0 2 (W ;X)N , where= ((m r)(N 1 + nX) (resp: nX(m r)) if r 2 (0;m]m(N + nX) (resp: m(1 + nX))if r = 0: 198QUAN ZHENG AND YONGSHENG LIHere we note that Miklin's multiplier theorem implies (Wm;p)ND(P (t;D)) for t 2 [0; T ] and 1 < p <1.Example 4.5. We consider the iterated evolution equation(4:5) (QNj=1(Dtipj(t;D))u(t; x) = 0 for x 2 Rn and 0 < t TDj 1t u(0; x) = uj(x)for x 2 Rn and 1 j Non Lp (1 < p < 1), where pj(t; ) =Pj j mj pj (t) and pj ( ) 2C([0; T ];R) for 1 j N . Then the roots of the characteristic equa-tion of (4.5) are j(t; ) = j j2 + ipj(t; ), and so Re j(t; ) = j j2for t 2 [0; T ], 2 Rn and 1 j N . The same way as in Exam-ple 3.6 and Corollary 4.4 yield now that (4.5) has a unique solution u 2CN 1([0; T ]; Lp) \ CN((0; T ]; Lp) for every (u1; ; uN) 2 (W ;p)N , where= (N 1 + nj 12 1p j)(PNj=1max(2;mj) 2):By Corollary 4.4 we easily generalize Example 3.6 to the case that thehigher order Cauchy problem (3.10) has time-dependent cpe cients.References[1] J.L. Boldrini, Asymptotic behavior of traveling wave solutions of the equations forthe ow of a uid with small viscosity and capillarity, Quart. Appl. Math., 44(1987), 697-708.[2] R. deLaubenfels, Matrices of operators and regularized semigroups, Math. 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Tokyo, 28(1981), 267-315.[18] F. Neubrander, Wellposedness of higher order abstract Cauchy problems, Trans.Amer. Math. Soc., 295 (1986), 257-290.[19] S. Sjostrand, On the Riesz means of the solutions of the Schrodinger equation, Ann.Scuola Norm. Sup. Pisa, 24 (1970), 331-348.[20] E.M. Stein, Singular Integrals and Di erentiability Properties of Functions, Prince-ton Univ. Press, New Jersey, 1970.[21] H. Tanabe, Equations of evolution, Pitman, London, 1979.[22] Q. Zheng, A Hille-Yosida theorem for the higher-order abstract Cauchy problem,Bull. London Math. Soc., 24 (1992), 531-539.[23], Controllability of a class of linear systems in Banach space, Proc. Amer.Math. Soc., 123 (1995), 1241-1251.[24], Cauchy problems for polynomials of generators of bounded C0-groups andfor di erential operators, preprint, 1995.Received May 20, 1996 and revised February 27, 1997. This project was supported by theNational Science Foundation of China.Huazhong Univ. of Science & TechnologyWuhan 430074, P.R. of ChinaE-mail address: [email protected] address of the second author:Institute of Applied Physics and Computational MathematicsP.O. Box 8009-26, Beijing 100088P.R. of ChinaPARABOLIC SYSTEMS199[12] M. Hieber, A. Holderrieth and F. Neubrander, Regularized semigroups and systemsof linear partial di erential equations, Ann. Scuola Norm. Sup. Pisa, 19 (1992),363-379.[13] L. Hormander, Estimates for translation invariant operators in Lp spaces, ActaMath., 104 (1960), 93-140.[14] H. Komatsu, Abstract analyticity in time and unique continuation property of solu-tions of a parabolic equation, J. Fac. Sci. Univ. Tokyo, 9 (1961), 1-11.[15] Y. Lei, W. Yi and Q. Zheng, Semigroups of operators and polynomials of generatorsof bounded strongly continuous groups, Proc. London Math. Soc., 69 (1994), 144-170.[16] Y. Lei and Q. Zheng, The application of C-semigroups to di erential operators inLp(Rn), J. Math. Anal. Appl., 188 (1994), 809-818.[17] A. Miyachi, On some singular Fourier multipliers, J. Fac. Sci. Univ. Tokyo, 28(1981), 267-315.[18] F. Neubrander, Wellposedness of higher order abstract Cauchy problems, Trans.Amer. Math. Soc., 295 (1986), 257-290.[19] S. Sjostrand, On the Riesz means of the solutions of the Schrodinger equation, Ann.Scuola Norm. Sup. Pisa, 24 (1970), 331-348.[20] E.M. Stein, Singular Integrals and Di erentiability Properties of Functions, Prince-ton Univ. Press, New Jersey, 1970.[21] H. Tanabe, Equations of evolution, Pitman, London, 1979.[22] Q. Zheng, A Hille-Yosida theorem for the higher-order abstract Cauchy problem,Bull. London Math. Soc., 24 (1992), 531-539.[23], Controllability of a class of linear systems in Banach space, Proc. Amer.Math. Soc., 123 (1995), 1241-1251.[24], Cauchy problems for polynomials of generators of bounded C0-groups andfor di erential operators, preprint, 1995.Received May 20, 1996 and revised February 27, 1997. This project was supported by theNational Science Foundation of China.Huazhong Univ. of Science & TechnologyWuhan 430074, P.R. of ChinaE-mail address: [email protected] address of the second author:Institute of Applied Physics and Computational MathematicsP.O. Box 8009-26, Beijing 100088P.R. of China