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Existence of strong global attractors for hyperbolic equation with linear memory
Applied Mathematics and Computation 157 (2004) 745–758 www.elsevier.com/locate/amc
Existence of strong global attractors for hyperbolic equation with linear memory Qiaozhen Ma *, Chengkui Zhong Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PeopleÕs Republic of China
Abstract We prove that the existence of global attractors of strong solutions for the hyperbolic equations with linear memory using the semigroup approach. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Global attractor; Linear memory; Hyperbolic equation
1. Introduction Let X be an open bounded subset of Rn with sufficiently smooth boundary C. We consider the following semilinear hyperbolic equation with linear memory:
utt þ aut Kð0ÞDu
Z
1
K 0 ðsÞDuðt sÞ ds þ gðuÞ ¼ f ;
in X Rþ ;
0
uðx; tÞ ¼ 0; x 2 C; t 2 R; uðx; tÞ ¼ u0 ðx; tÞ; x 2 X; t 6 0 ð1:1Þ
* Corresponding author. Address: Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PeopleÕs Republic of China. E-mail addresses: [email protected] (Q. Ma), [email protected] (C. Zhong).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.080
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with a > 0 a given constant, Kð0Þ, Kð1Þ > 0 and K 0 ðsÞ 6 0 for every s 2 Rþ . This equation arise in the theory of isothermal viscoelastic, it describes a homogeneous and isotropic viscoelastic solid. In addition, if K 0 ðsÞ 0, on the one hand, (1.1) reduces to the wave equation, where g represents some displacement-dependent body force density; on the other hand, if gðuÞ ¼ sinðuÞ, then (1.1) is the sine-Gordon equation which is used to model, for instance, the dynamics of a Josephson junction derived by a current source, see [3,7]; and if c gðuÞ ¼ juj u, then (1.1) is the equation of relativistic quantum mechanics, see [3–5]. Giorgi et al. [1] gave the existence of global attractors for the above problem without damping term in R3 , but their results were obtained only in the weak Sobolev space H01 L2 L2l ðRþ ; H01 Þ. We will prove the existence of the attractor of (1.1) in the stronger space DðAÞ H01 L2l ðRþ ; DðAÞÞ. As in [1], we define gt ðx; sÞ ¼ uðx; tÞ uðx; t sÞ:
ð1:2Þ
Let lðsÞ ¼ K 0 ðsÞ, Kð1Þ ¼ 1. Eq. (1.1) transforms into the following system: Z 1 lðsÞDgt ðsÞ ds þ gðuÞ ¼ f ; ð1:3Þ utt þ aut Du 0
gt ¼ gs þ ut :
ð1:4Þ
Here (1.4) is obtained by differentiating (1.2). Initial-boundary value conditions are given by 8 uðx; tÞ ¼ 0; x 2 C; t P 0; > > > > < gt ðx; sÞ ¼ 0; ðx; sÞ 2 C Rþ ; t P 0; ð1:5Þ uðx; 0Þ ¼ u0 ðxÞ; x 2 X; > > u ðx; 0Þ ¼ v ðxÞ; x 2 X; > 0 > : t0 g ðx; sÞ ¼ g0 ðx; sÞ; ðx; sÞ 2 C Rþ ; having set 8 < u0 ðxÞ ¼ u0 ðx; 0Þ; v ðxÞ ¼ ot u0 ðx; tÞjt¼0 ; : 0 g0 ðx; sÞ ¼ u0 ðx; 0Þ u0 ðx; sÞ: The memory kernel l is required to satisfy the following hypotheses: (h1) (h2) (h3) (h4)
l 2 C 1 ðRþ Þ \ L1 ðRþ Þ, 8s 2 Rþ ; P 0, l0 ðsÞ 6 0, 8s 2 Rþ ; RlðsÞ 1 lðsÞ ds ¼ k0 > 0; 0 l0 ðsÞ þ dlðsÞ 6 0, 8s 2 Rþ , and some d > 0.
The function g is a C 2 function from R into R satisfying the following conditions:
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(g1) lim inf GðsÞ P 0, here GðsÞ ¼ s2 jsj!1
(g2) lim inf jsj!1
jg0 ðsÞj jsjc
Rs 0
747
gðsÞ ds;
¼ 0; with 0 6 c < 1 when n ¼ 1; 2, and 0 6 c 6
2 n2
when
n P 3; 1 GðsÞ (g3) there exists a constant C1 > 0 such that lim inf sgðsÞC P 0. s2
jsj!1
Now, we introduce the spaces L2 ðXÞ, H01 ðXÞ, DðAÞ ¼ H 2 ðXÞ \ H01 ðXÞ, and for every u 2 DðAÞ, Au ¼ Du. we endow these spaces with the usual scalar products and norms, ð; Þ, j j and ðð; ÞÞ, k k. Furthermore, we can define the s powers As of A for s 2 R. The space Vs ¼ DðA2 Þ turns out to be a Hilbert space with the inner product s s hu; vis ¼ A2 u; A2 v : We denote by k ks the norm on Vs and kuk2s ¼ hu; uis . In particular, H ¼ V0 ¼ L2 ðXÞ, V ¼ V1 ¼ H01 ðXÞ, and DðAÞ ¼ V2 ¼ H 2 ðXÞ \ H01 ðXÞ. We have DðAÞ V H V . Here V is the dual of V , and each space is dense in the following one and the injections are continuous. In line with (h1), let L2l ðRþ , Vs Þ be the Hilbert space of Vs -valued functions on þ R , endowed with the inner product Z 1 lðrÞhuðrÞ; wðrÞis dr hu; wil;s ¼ 0
and the norm kukl;s ¼ hu; uil;s ¼
Z
1
2
lðrÞkuðrÞks dr: 0
2 kukl;V
2
2
2
In particular, ¼ kukl;1 , kukl;DðAÞ ¼ kukl;2 . Finally, we introduce the following Hilbert spaces H0 ¼ H01 ðXÞ L2 ðXÞ L2l ðRþ ; H01 ðXÞÞ ¼ V H L2l ðRþ ; V Þ and H1 ¼ DðAÞ V L2l ðRþ ; DðAÞÞ:
2. Notation and preliminaries We infer from (g1), (g3) that for any g > 0, there exists Cg , Cg0 such that for any s 2 R, GðsÞ þ gs2 P Cg ;
ð2:1Þ
sgðsÞ C1 GðsÞ þ gs2 P Cg0 :
ð2:2Þ
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In order to obtain the global attractor of the problem (1.3)–(1.5), we need the following theorem of existence, uniqueness of solution and continuous dependence to the initial data. Theorem 2.1 [1,3]. Let (h1)–(h2) and (g1)–(g3) hold. If f 2 L2 , z0 ¼ ðu0 ; v0 ; g0 Þ 2 H0 , then there exists a unique solution z ¼ ðu; ut ; gÞ of (1.3)– (1.5), such that z 2 Cð½0; T ; H0 Þ 8T > 0: If, furthermore, z0 2 H1 , then z satisfies z 2 Cð½0; T ; H1 Þ 8T > 0: Thus we can define for every t 2 R, the mapping SðtÞ : z0 ! z:
ð2:3Þ
They map H0 ¼ V H L2l ðRþ ; V Þ into itself and H1 ¼ DðAÞ V L2l ðRþ ; DðAÞÞ into itself and they enjoy the group properties Sðt þ sÞ ¼ SðtÞ SðsÞ Sð0Þ ¼ I:
8s; t; 2 R;
Hence SðtÞ is the inverse of SðtÞ 8t 2 R. Moreover, these operators are homeomorphisms in H0 , H1 , respectively. Since the group SðtÞ associated with Eqs. (1.3) and (1.4) is not uniform compact, we cannot expect to obtain the existence of the attractor of the group by the use of the usual theorem about the existence of attractor. Fortunately, in [6], the authors have given equivalent conditions of the existence for the global attractor. Applying these conditions, we can easily obtain the existence of the global attractor of the group SðtÞ in stronger Sobolev space H1 . Let us recapitulate the abstract result in [6] which will be used in our consideration. Definition 2.1. A C 0 semigroup fSðtÞgt P 0 in a complete metric space M is called x-limit compact, if for any > 0 and for every bounded subset B of M, there exists a t0 > 0 such that ! [ c SðtÞB 6 ; t P t0
where c is the measure of noncompactness defined by cðAÞ ¼ inffdj A admits finite cover by subsets of M whose diameter 6 dg:
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Definition 2.2. A C 0 semigroup fSðtÞgt P 0 in a Banach space X is said to satisfy condition (C) if for any > 0 and for any bounded set B of X , there exists tðBÞ > 0 and a finite dimensional subspace X1 of X , such that fkPSðtÞBkg is bounded and fkðI P ÞSðtÞxkg <
for t P tðBÞ; x 2 B;
where P : X ! X1 is a bounded projector.
Theorem 2.2. Let fSðtÞgt P 0 be a C 0 semigroup in a complete metric space X . Then SðtÞ has a global attractor if and only if (1) fSðtÞgt P 0 is x-limit compact; (2) there exists a bounded absorbing subset B of X .
Theorem 2.3. Let X be Banach space and fSðtÞgt P 0 be a C 0 semigroup in X . Assume that fSðtÞgt P 0 satisfies condition (C), then fSðtÞgt P 0 is x-limit compact. In fact, if X is a uniform convex Banach space, especially, Hilbert space, fSðtÞgt P 0 is x-limit compact if and only if condition (C) holds true.
3. Absorbing sets in H0 and H1 3.1. Absorbing set in H0 We formally take the scalar product in H of Eq. (1.3) with v ¼ ut þ ru, after a computation, we find 1 d 2 2 2 ðkuk þ jvj Þ þ rkuk þ ða rÞðut ; vÞ þ ðg; ut Þl;V þ rðg; uÞl;V 2 dt þ ðgðuÞ; vÞ ¼ ðf ; vÞ: Exploiting (1.4), (h3)–(h4) and H€ older inequality, we have ða rÞðut ; vÞ ¼ ða rÞjvj2 rða rÞðu; vÞ;
ð3:1Þ
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Z 1 1 d 2 kgkl;V þ ðg; ut Þl;V ¼ ðg; gt þ gs Þl;V ¼ lðsÞðgðsÞ; gs ðsÞÞV ds 2 dt 0 Z 1 d 1 1 kgk2l;V þ ¼ lðsÞ dkgðsÞk2V 2 dt 2 0 Z 1 d 1 1 0 2 kgkl;V ¼ l ðsÞkgðsÞk2V ds 2 dt 2 0 Z 1 d d 1 2 2 kgkl;V þ P lðsÞkgðsÞkV ds 2 dt 2 0 1 d d kgk2l;V þ kgk2l;V ; ¼ 2 dt 2 d k0 r2 2 2 kuk : rðg; uÞl;V P kgkl;V 4 d Hence we conclude from (3.1) that
1 d k0 r ðkuk2 þ jvj2 þ kgk2l;V Þ þ r 1 kuk2 þ ða rÞjvj2 2 dt d d 2 rða rÞðu; vÞ þ kgkl;V þ ðgðuÞ; vÞ 6 ðf ; vÞ: 4 2
ð3:2Þ
2
Moreover, using kuk P k1 juj , here k1 is the first eigenvalue of D in H01 ðXÞ, when r small enough, such that 1
k0 r ar P 1 r; d 2k1
a a rP : 2 4
ð3:3Þ
Then we obtain
k0 r 2 2 r 1 kuk þ ða rÞjvj rða rÞðu; vÞ d
k0 r ar Pr 1 kuk2 þ ða rÞjvj2 pffiffiffiffiffi kuk jvj d k1
2
k0 r ar a kuk2 þ jvj2 kuk2 þ ða rÞjvj2 Pr 1 d 2 2k1
k0 r ar a a 2 2 2 2 ¼r 1 r jvj P rð1 rÞkuk þ jvj ð3:4Þ kuk þ d 2k1 2 4 and ðgðuÞ; vÞ ¼ ðgðuÞ; ut þ ruÞ ¼
d dt
Z
GðuÞ dx þ rðgðuÞ; uÞ: X
ð3:5Þ
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Combining with (3.4) and (3.5), we deduce from (3.2)
Z 1 d a 2 2 2 2 2 kuk þ jvj þ kgkl;V þ 2 GðuÞ dx þ rð1 rÞkuk þ jvj 2 dt 4 X d þ kgk2l;V þ rðgðuÞ; uÞ 6 ðf ; vÞ: 4
ð3:6Þ
According to (2.1) and (2.2), there exist two constant K1 , K2 , such that Z 1 2 GðuÞ dx þ kuk þ K1 P 0 8u 2 V ; ð3:7Þ 4 X Z 1 2 ðu; gðuÞÞ C1 GðuÞ dx þ kuk þ K2 P 0 8u 2 V : ð3:8Þ 4 X From (3.6) entails
Z 1 d a 2 2 2 2 2 kuk þ jvj þ kgkl;V þ 2 GðuÞ dx þ rð1 rÞkuk þ jvj 2 dt 4 X Z d r þ kgk2l;V þ C1 r GðuÞ dx kuk2 rK2 6 jf j jvj: 4 4 X Namely
Z 1 d 3 a 2 2 2 2 2 kuk þ jvj þ kgkl;V þ 2 GðuÞ dx þ r r kuk þ jvj 2 dt 4 4 X Z d 2 2 a 2 2 þ kgkl;V þ C1 r GðuÞ dx 6 rK2 þ jf j jvj 6 rK2 þ jf j þ jvj : 4 a 8 X ð3:9Þ Setting
3 a d a1 ¼ min 2r r ; ; ; C1 r : 4 4 2
We conclude that d 4 W ðtÞ þ a1 W ðtÞ 6 2rK2 þ jf j2 þ 2a1 K1 :¼ C2 : dt a Here 2
2
2
W ðtÞ ¼ kuk þ jvj þ kgkl;V þ 2
Z
GðuÞ dx þ 2K1 X
P
1 2 2 2 kuk þ jvj þ kgkl;V P 0: 2
ð3:10Þ
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By the Gronwall Lemma, we get C2 W ðtÞ 6 W ð0Þ expða1 tÞ þ ð1 expða1 tÞÞ 8t P t0 a1 and C2 lim supW ðtÞ ¼ :¼ q20 : a1 t!1
ð3:11Þ
Let q00 > q0 be fixed and assume that W ð0Þ 6 R. It readily follows from (3.11) that for t P t0 ¼ t0 ðR; q00 Þ, 1 R : t0 ðR; q00 Þ ¼ log 0 a1 q0 2 q20 We have W ðtÞ 6 q00 2 and
2 d 2 2 2 2 2 2 kuðtÞk þ jut ðtÞj þ kgt ðsÞkl;V 6 1 þ pffiffiffiffiffi ðkuðtÞk þ jvðtÞj þ kgt ðsÞkl;V Þ k1
2
2 d d 6 2 1 þ pffiffiffiffiffi W ðtÞ 6 2 1 þ pffiffiffiffiffi q00 2: k1 k1
We observe thatRif B is a bounded subset of H0 ¼ V H L2l , then by the condition (g2), f X GðuÞ dxg is bounded, too. Hence Z 2 2 2 ku0 k þ jv0 þ ru0 j þ kg0 kl;V þ 2 Gðu0 Þ dx þ 2K1 R ¼ RðBÞ ¼ sup fu0 ;v0 ;g0 g
X
< 1:
2 We set 2 1 þ pdffiffiffi q00 2 ¼ q21 and we end up to k1
Lemma 3.1. The ball of H0 , B0 ¼ BH0 ð0; q1 Þ, centered at 0 of radius q1 , is an absorbing set in H0 for the group SðtÞ. For any bounded subset B in H0 , SðtÞB B0 for t P t0 . 3.2. Absorbing set in H1 We take the scalar product in H of system (1.3) and (1.4) with Av ¼ Aut þ rAu, we have
1 d k0 r 2 2 2 ðjAuj þ kvk þ kgkl;DðAÞ Þ þ r 1 jAuj2 þ ða rÞkvk2 2 dt d d 2 rða rÞðAu; vÞ þ kgkl;DðAÞ þ ððgðuÞ; vÞÞ 6 ðf ; AvÞ: ð3:12Þ 4 2 2 As the same as (3.4), exploiting jAuj P k2 kuk , when r small enough, such that 1
k0 r ar P 1 r; d 2k2
a a rP : 2 4
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We have
k0 r a r 1 jAuj2 þ ða rÞkvk2 rða rÞðAu; vÞ P rð1 rÞjAuj2 þ kvk2 d 4 ð3:13Þ and ðf ; AvÞ ¼
d d r ðf ; AuÞ þ rðf ; AuÞ 6 ðf ; AuÞ þ jAuj2 þ 2rjf j2 : dt dt 8
ð3:14Þ
It follows from (g2) that for > 0, there exists a constant C > 0, such that jg0 ðsÞj 6 jsjc þ C
8s 2 R:
Hence jððgðuÞ; vÞÞj Z 0 ¼ g ðuÞru rv dx Z X 6 jg0 ðuÞj jruj jrvj dx Z ZX c 6 juj jruj jrvj dx þ C jruj jrvj dx X
Z
12
X
6 juj2c jruj2 dx kvk þ C kuk kvk 8 X 2q1 R 2p1 R > > < X juj2cq dx X jruj2p dx kvk þ C kuk kvk; n ¼ 1; 2 6 n2 R 1n R > 2n 2n cn > n2 : juj dx jruj dx kvk þ C kuk kvk; n P 3 X X c
6 kuk C32 jAuj kvk þ
a 4C 2 2 2 kvk þ kuk : 16 a
ð3:15Þ
Here C3 is the positive constant satisfying 8 1 > < R juj2cq dx 2q ; n ¼ 1; 2 c X C3 kuk P > : R jujcn dx1n n P 3; 8 X 2p1 R > > < X jruj2p dx ; n ¼ 1; 2 C3 jAuj P n2 R > 2n 2n > n2 : jruj dx ; n P 3: X If ðu0 ; v0 ; g0 Þ belongs to a bounded set B of H1 , then B is also bounded in H0 , and for t P t0 , we have
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kuk2 þ jut j2 þ kgk2l;V 6 q21 : t0 , q1 are given in Lemma 3.1. We choose 0 < 2 <
ar : 4 8q4c 1 C3
From (3.15) entails a 4C 2 2 2 kvk þ kuk 16 a 4 42 q2c a 4C 2 q2 2 2 1 C3 jAuj þ kvk þ 1 6 8 a a 2 2 r a 4C q 2 2 6 jAuj þ kvk þ 1 ; t P t0 : 4 8 a c
jððgðuÞ; vÞÞj 6 kuk C32 jAuj kvk þ
ð3:16Þ
Combining with (3.13) and (3.14), and (3.16), we deduce from (3.12) that
d 5 2 2 2 2 jAuj þ kvk 2ðf ; AuÞ þ kgkl;DðAÞ þ 2r r jAuj dt 8 a d 8C 2 q2 þ kvk2 þ kgk2l;DðAÞ 6 4rjf j2 þ 1 ; t P t0 : 4 2 a Hence
d 5 a 2 2 2 2 2 ðjAu f j þ kvk þ kgkl;DðAÞ Þ þ 2r r jAu f j þ kvk dt 8 4 d 2 þ kgkl;DðAÞ 6 C4 ; t P t0 : 2
ð3:17Þ
Here C4 ¼ 2r
21 8C 2 q2 2 r jf j þ 1 : 8 a
Setting
5 a d a2 ¼ min 2r r ; ; : 8 4 2 By the Gronwall Lemma, we obtain 2
2
2
jAuðtÞ f j þ kvðtÞk þ kgt ðsÞkl;DðAÞ 2
2
2
6 ðjAuðt0 Þ f j þ kvðt0 Þk þ kgt0 ðsÞkl;DðAÞ Þ expða2 ðt t0 ÞÞ þ t P t0 :
C4 ; a2
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Thus we have the following conclusion: Theorem 3.1. Let X be an open bounded set of Rn with smooth boundary C, and let g be a C 2 function from R to R satisfying (g1)–(g3). Then the ball BððA1 f ; 0Þ; q2 Þ in H1 , centered at fA1 f ; 0g of radius q2 > Ca24 is absorbing in H1 for the semigroup SðtÞ associated with the problem (1.3)–(1.5). 4. Attractor in H1 In order to obtain the global attractor in H1 , we first need the following Lemmas of the properties of compactness about the nonlinear operator g. Lemma 4.1 [2]. Let g be C 2 function from R into R satisfying (g2). Then g : H01 ðXÞ ! L2 ðXÞ is continuously compact. Lemma 4.2 [2]. Let g be C 2 function from R into R satisfying (g2), and g : DðAÞ ! H01 ðXÞ be defined by Z Z ððgðuÞ; vÞÞ ¼ rgðuÞrv dx ¼ g0 ðuÞru rv dx 8u 2 DðAÞ; v 2 H01 ðXÞ: X
X
Then g is continuous compact. Our main result: Theorem 4.1. The group {S(t)} associated with the system (1.3) and (1.4) has a global attractor A, which is compact, connected and maximal in H1 . It attracts all bounded subsets of H1 in the norm of H1 . Proof. Let fwk g be an orthonormal basis of L2 ðXÞ which consists of eigenvectors of A. The corresponding eigenvalues are denoted by kk , k ¼ 1; 2; . . . and k1 < k2 6 k3 6 ; kj ! 1, as j ! 1. Then fwk g is also an orthogonal basis of V . We write Vm ¼ spanfw1 ; . . . ; wm g: Since f 2 V , and g : DðAÞ ! V are compact operators verified in Lemma 4.2, for any > 0, there exists some m such that ð4:1Þ kðI Pm Þf k 6 ; 4 8u 2 BDðAÞ ðA1 f ; q2 Þ; kðI Pm ÞgðuÞk 6 ð4:2Þ 4 where Pm : V ! Vm is the orthogonal projector, and q2 is given by Theorem 3.1. For any ðu; ut ; gt Þ 2 H1 , we write
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ðu; ut ; gt Þ ¼ ðu1 ; ut1 ; gt1 Þ þ ðu2 ; ut2 ; gt2 Þ: Here ðu1 ; ut1 ; gt1 Þ ¼ ðPm u; Pm ut ; Pm gt Þ: Take the scalar product in H of Eq. (1.3) with Av2 ¼ Aut2 þ rAu2 , combining with (1.4), we find
1 d k0 r 2 2 2 2 2 ðjAu2 j þ kv2 k þ kg2 kl;DðAÞ Þ þ r 1 jAu2 j þ ðd rÞkv2 k 2 dt d d rða rÞðAu2 ; v2 Þ þ kg2 k2l;DðAÞ þ ððgðuÞ; v2 ÞÞ ¼ ðf ; Av2 Þ: ð4:3Þ 4 It is the same as (3.13)
k0 r r 1 jAu2 j2 þ ðd rÞkv2 k2 rða rÞðAu2 ; v2 Þ d a P rð1 rÞjAu2 j2 þ kv2 k2 4
ð4:4Þ
and by (4.1) we obtain ðf ; Av2 Þ 6
d r 2 r 2 ðf2 ; Au2 Þ þ jAu2 j þ : dt 8 8
ð4:5Þ
Here f2 ¼ ðI Pm Þf . Combining with (4.2), (4.4) and (4.5), from (4.3) entails 1 d a 2 2 2 2 2 ðjAu2 j þ kv2 k þ kg2 kl;DðAÞ Þ þ rð1 rÞjAu2 j þ kv2 k 2 dt 4 d r d 2 r : þ kg2 k2l;DðAÞ 6 jAu2 j2 þ kv2 k þ ðf2 ; Au2 Þ þ 4 8 4 dt 8 It follows that
d 7 2 2 2 2 ðjAu2 j þ kv2 k þ kg2 kl;DðAÞ 2ðf2 ; Au2 ÞÞ þ 2r r jAu2 j dt 8 a d 2 2 r 2 2 : þ þ kv2 k þ kg2 kl;DðAÞ 6 4 2 4 4a Setting a3 ¼ minf2rð78 rÞ; a4 ; d2g. Hence d ðjAu2 f2 j2 þ kv2 k2 þ kg2 k2l;DðAÞ Þ þ a3 ðjAu2 f2 j2 þ kv2 k2 þ kg2 k2l;DðAÞ Þ dt
1 r a3 2 6 þ þ :¼ C5 2 : 4a 4 16
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757
Using the Gronwall Lemma, we end up to 2
2
2
jAu2 f2 j þ kv2 k þ kg2 kl;DðAÞ 2
2
2
6 ðjAu2 ðt1 Þ f2 j þ kv2 ðt1 Þk þ kgt21 ðsÞkl;DðAÞ Þ expða3 ðt t1 ÞÞ þ
C5 2 a3
for t P t1 P t0 : Namely 2
2
2
jAu2 f2 j þ kv2 k þ kg2 kl;DðAÞ 6 q22 expða3 ðt t1 ÞÞ þ
C5 2 a3
for t P t1 :
Here q2 is given by Theorem 3.1. Take t2 large enough such that t2 t1 P
1 q2 log 22 : a3
Then
C5 2 2 2 2 jAu2 f2 j þ kv2 k þ kg2 kl;DðAÞ 6 1 þ a3
for t P t2 :
Since jAu2 j2 þ kut 2k2 þ kg2 k2l;DðAÞ 6 2ðjAu2 f2 j2 þ jf2 j2 Þ þ 2kv2 k2 þ 2r2 ku2 k2
2C5 2 2 2 þ 2kg2 kl;DðAÞ 6 3 þ þ ku2 k a3 and the embedding from DðAÞ into V is compact, so choose m large enough such that ku2 k2 6 2 . Hence jAu2 j2 þ kut2 k2 þ kg2 k2l;DðAÞ 6 C6 2 : Here C6 ¼ 4 þ
2C5 : a3
This shows that fSðtÞg is x-limit compact in H1 . h
Acknowledgement The work was supported by the National Science Foundation of China under grant 19971036.
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References [1] C. Giorgi, J.E. Munoz Rivera, V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl. 260 (2001) 83–99. [2] S.H. Wang, C.K. Zhong, Existence of strong global attractors for linearly damped wave equations, submitted for publication. [3] R. Teman, Infinite Dimensional Dynamical System in Mechanics and Physics, second ed., Spring-Verlag, 1997. [4] J.M. Ghidaglia, R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl. 66 (9) (1987) 273–319. [5] J.K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. [6] Q.F. Ma, S.H. Wang, C.K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J. 51 (6) (2002) 1541–1559. [7] A.R. Bishop, K. Fesser, P.S. Lonmdahl, S.E. Trullinger, Influence of solutions in the state on chaos in the driven damped sine-Gordon system, Physica D 7 (1983) 259–279.
Existence of strong global attractors for hyperbolic equation with linear memory Qiaozhen Ma *, Chengkui Zhong Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PeopleÕs Republic of China
Abstract We prove that the existence of global attractors of strong solutions for the hyperbolic equations with linear memory using the semigroup approach. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Global attractor; Linear memory; Hyperbolic equation
1. Introduction Let X be an open bounded subset of Rn with sufficiently smooth boundary C. We consider the following semilinear hyperbolic equation with linear memory:
utt þ aut Kð0ÞDu
Z
1
K 0 ðsÞDuðt sÞ ds þ gðuÞ ¼ f ;
in X Rþ ;
0
uðx; tÞ ¼ 0; x 2 C; t 2 R; uðx; tÞ ¼ u0 ðx; tÞ; x 2 X; t 6 0 ð1:1Þ
* Corresponding author. Address: Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PeopleÕs Republic of China. E-mail addresses: [email protected] (Q. Ma), [email protected] (C. Zhong).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.080
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with a > 0 a given constant, Kð0Þ, Kð1Þ > 0 and K 0 ðsÞ 6 0 for every s 2 Rþ . This equation arise in the theory of isothermal viscoelastic, it describes a homogeneous and isotropic viscoelastic solid. In addition, if K 0 ðsÞ 0, on the one hand, (1.1) reduces to the wave equation, where g represents some displacement-dependent body force density; on the other hand, if gðuÞ ¼ sinðuÞ, then (1.1) is the sine-Gordon equation which is used to model, for instance, the dynamics of a Josephson junction derived by a current source, see [3,7]; and if c gðuÞ ¼ juj u, then (1.1) is the equation of relativistic quantum mechanics, see [3–5]. Giorgi et al. [1] gave the existence of global attractors for the above problem without damping term in R3 , but their results were obtained only in the weak Sobolev space H01 L2 L2l ðRþ ; H01 Þ. We will prove the existence of the attractor of (1.1) in the stronger space DðAÞ H01 L2l ðRþ ; DðAÞÞ. As in [1], we define gt ðx; sÞ ¼ uðx; tÞ uðx; t sÞ:
ð1:2Þ
Let lðsÞ ¼ K 0 ðsÞ, Kð1Þ ¼ 1. Eq. (1.1) transforms into the following system: Z 1 lðsÞDgt ðsÞ ds þ gðuÞ ¼ f ; ð1:3Þ utt þ aut Du 0
gt ¼ gs þ ut :
ð1:4Þ
Here (1.4) is obtained by differentiating (1.2). Initial-boundary value conditions are given by 8 uðx; tÞ ¼ 0; x 2 C; t P 0; > > > > < gt ðx; sÞ ¼ 0; ðx; sÞ 2 C Rþ ; t P 0; ð1:5Þ uðx; 0Þ ¼ u0 ðxÞ; x 2 X; > > u ðx; 0Þ ¼ v ðxÞ; x 2 X; > 0 > : t0 g ðx; sÞ ¼ g0 ðx; sÞ; ðx; sÞ 2 C Rþ ; having set 8 < u0 ðxÞ ¼ u0 ðx; 0Þ; v ðxÞ ¼ ot u0 ðx; tÞjt¼0 ; : 0 g0 ðx; sÞ ¼ u0 ðx; 0Þ u0 ðx; sÞ: The memory kernel l is required to satisfy the following hypotheses: (h1) (h2) (h3) (h4)
l 2 C 1 ðRþ Þ \ L1 ðRþ Þ, 8s 2 Rþ ; P 0, l0 ðsÞ 6 0, 8s 2 Rþ ; RlðsÞ 1 lðsÞ ds ¼ k0 > 0; 0 l0 ðsÞ þ dlðsÞ 6 0, 8s 2 Rþ , and some d > 0.
The function g is a C 2 function from R into R satisfying the following conditions:
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(g1) lim inf GðsÞ P 0, here GðsÞ ¼ s2 jsj!1
(g2) lim inf jsj!1
jg0 ðsÞj jsjc
Rs 0
747
gðsÞ ds;
¼ 0; with 0 6 c < 1 when n ¼ 1; 2, and 0 6 c 6
2 n2
when
n P 3; 1 GðsÞ (g3) there exists a constant C1 > 0 such that lim inf sgðsÞC P 0. s2
jsj!1
Now, we introduce the spaces L2 ðXÞ, H01 ðXÞ, DðAÞ ¼ H 2 ðXÞ \ H01 ðXÞ, and for every u 2 DðAÞ, Au ¼ Du. we endow these spaces with the usual scalar products and norms, ð; Þ, j j and ðð; ÞÞ, k k. Furthermore, we can define the s powers As of A for s 2 R. The space Vs ¼ DðA2 Þ turns out to be a Hilbert space with the inner product s s hu; vis ¼ A2 u; A2 v : We denote by k ks the norm on Vs and kuk2s ¼ hu; uis . In particular, H ¼ V0 ¼ L2 ðXÞ, V ¼ V1 ¼ H01 ðXÞ, and DðAÞ ¼ V2 ¼ H 2 ðXÞ \ H01 ðXÞ. We have DðAÞ V H V . Here V is the dual of V , and each space is dense in the following one and the injections are continuous. In line with (h1), let L2l ðRþ , Vs Þ be the Hilbert space of Vs -valued functions on þ R , endowed with the inner product Z 1 lðrÞhuðrÞ; wðrÞis dr hu; wil;s ¼ 0
and the norm kukl;s ¼ hu; uil;s ¼
Z
1
2
lðrÞkuðrÞks dr: 0
2 kukl;V
2
2
2
In particular, ¼ kukl;1 , kukl;DðAÞ ¼ kukl;2 . Finally, we introduce the following Hilbert spaces H0 ¼ H01 ðXÞ L2 ðXÞ L2l ðRþ ; H01 ðXÞÞ ¼ V H L2l ðRþ ; V Þ and H1 ¼ DðAÞ V L2l ðRþ ; DðAÞÞ:
2. Notation and preliminaries We infer from (g1), (g3) that for any g > 0, there exists Cg , Cg0 such that for any s 2 R, GðsÞ þ gs2 P Cg ;
ð2:1Þ
sgðsÞ C1 GðsÞ þ gs2 P Cg0 :
ð2:2Þ
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In order to obtain the global attractor of the problem (1.3)–(1.5), we need the following theorem of existence, uniqueness of solution and continuous dependence to the initial data. Theorem 2.1 [1,3]. Let (h1)–(h2) and (g1)–(g3) hold. If f 2 L2 , z0 ¼ ðu0 ; v0 ; g0 Þ 2 H0 , then there exists a unique solution z ¼ ðu; ut ; gÞ of (1.3)– (1.5), such that z 2 Cð½0; T ; H0 Þ 8T > 0: If, furthermore, z0 2 H1 , then z satisfies z 2 Cð½0; T ; H1 Þ 8T > 0: Thus we can define for every t 2 R, the mapping SðtÞ : z0 ! z:
ð2:3Þ
They map H0 ¼ V H L2l ðRþ ; V Þ into itself and H1 ¼ DðAÞ V L2l ðRþ ; DðAÞÞ into itself and they enjoy the group properties Sðt þ sÞ ¼ SðtÞ SðsÞ Sð0Þ ¼ I:
8s; t; 2 R;
Hence SðtÞ is the inverse of SðtÞ 8t 2 R. Moreover, these operators are homeomorphisms in H0 , H1 , respectively. Since the group SðtÞ associated with Eqs. (1.3) and (1.4) is not uniform compact, we cannot expect to obtain the existence of the attractor of the group by the use of the usual theorem about the existence of attractor. Fortunately, in [6], the authors have given equivalent conditions of the existence for the global attractor. Applying these conditions, we can easily obtain the existence of the global attractor of the group SðtÞ in stronger Sobolev space H1 . Let us recapitulate the abstract result in [6] which will be used in our consideration. Definition 2.1. A C 0 semigroup fSðtÞgt P 0 in a complete metric space M is called x-limit compact, if for any > 0 and for every bounded subset B of M, there exists a t0 > 0 such that ! [ c SðtÞB 6 ; t P t0
where c is the measure of noncompactness defined by cðAÞ ¼ inffdj A admits finite cover by subsets of M whose diameter 6 dg:
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Definition 2.2. A C 0 semigroup fSðtÞgt P 0 in a Banach space X is said to satisfy condition (C) if for any > 0 and for any bounded set B of X , there exists tðBÞ > 0 and a finite dimensional subspace X1 of X , such that fkPSðtÞBkg is bounded and fkðI P ÞSðtÞxkg <
for t P tðBÞ; x 2 B;
where P : X ! X1 is a bounded projector.
Theorem 2.2. Let fSðtÞgt P 0 be a C 0 semigroup in a complete metric space X . Then SðtÞ has a global attractor if and only if (1) fSðtÞgt P 0 is x-limit compact; (2) there exists a bounded absorbing subset B of X .
Theorem 2.3. Let X be Banach space and fSðtÞgt P 0 be a C 0 semigroup in X . Assume that fSðtÞgt P 0 satisfies condition (C), then fSðtÞgt P 0 is x-limit compact. In fact, if X is a uniform convex Banach space, especially, Hilbert space, fSðtÞgt P 0 is x-limit compact if and only if condition (C) holds true.
3. Absorbing sets in H0 and H1 3.1. Absorbing set in H0 We formally take the scalar product in H of Eq. (1.3) with v ¼ ut þ ru, after a computation, we find 1 d 2 2 2 ðkuk þ jvj Þ þ rkuk þ ða rÞðut ; vÞ þ ðg; ut Þl;V þ rðg; uÞl;V 2 dt þ ðgðuÞ; vÞ ¼ ðf ; vÞ: Exploiting (1.4), (h3)–(h4) and H€ older inequality, we have ða rÞðut ; vÞ ¼ ða rÞjvj2 rða rÞðu; vÞ;
ð3:1Þ
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Z 1 1 d 2 kgkl;V þ ðg; ut Þl;V ¼ ðg; gt þ gs Þl;V ¼ lðsÞðgðsÞ; gs ðsÞÞV ds 2 dt 0 Z 1 d 1 1 kgk2l;V þ ¼ lðsÞ dkgðsÞk2V 2 dt 2 0 Z 1 d 1 1 0 2 kgkl;V ¼ l ðsÞkgðsÞk2V ds 2 dt 2 0 Z 1 d d 1 2 2 kgkl;V þ P lðsÞkgðsÞkV ds 2 dt 2 0 1 d d kgk2l;V þ kgk2l;V ; ¼ 2 dt 2 d k0 r2 2 2 kuk : rðg; uÞl;V P kgkl;V 4 d Hence we conclude from (3.1) that
1 d k0 r ðkuk2 þ jvj2 þ kgk2l;V Þ þ r 1 kuk2 þ ða rÞjvj2 2 dt d d 2 rða rÞðu; vÞ þ kgkl;V þ ðgðuÞ; vÞ 6 ðf ; vÞ: 4 2
ð3:2Þ
2
Moreover, using kuk P k1 juj , here k1 is the first eigenvalue of D in H01 ðXÞ, when r small enough, such that 1
k0 r ar P 1 r; d 2k1
a a rP : 2 4
ð3:3Þ
Then we obtain
k0 r 2 2 r 1 kuk þ ða rÞjvj rða rÞðu; vÞ d
k0 r ar Pr 1 kuk2 þ ða rÞjvj2 pffiffiffiffiffi kuk jvj d k1
2
k0 r ar a kuk2 þ jvj2 kuk2 þ ða rÞjvj2 Pr 1 d 2 2k1
k0 r ar a a 2 2 2 2 ¼r 1 r jvj P rð1 rÞkuk þ jvj ð3:4Þ kuk þ d 2k1 2 4 and ðgðuÞ; vÞ ¼ ðgðuÞ; ut þ ruÞ ¼
d dt
Z
GðuÞ dx þ rðgðuÞ; uÞ: X
ð3:5Þ
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Combining with (3.4) and (3.5), we deduce from (3.2)
Z 1 d a 2 2 2 2 2 kuk þ jvj þ kgkl;V þ 2 GðuÞ dx þ rð1 rÞkuk þ jvj 2 dt 4 X d þ kgk2l;V þ rðgðuÞ; uÞ 6 ðf ; vÞ: 4
ð3:6Þ
According to (2.1) and (2.2), there exist two constant K1 , K2 , such that Z 1 2 GðuÞ dx þ kuk þ K1 P 0 8u 2 V ; ð3:7Þ 4 X Z 1 2 ðu; gðuÞÞ C1 GðuÞ dx þ kuk þ K2 P 0 8u 2 V : ð3:8Þ 4 X From (3.6) entails
Z 1 d a 2 2 2 2 2 kuk þ jvj þ kgkl;V þ 2 GðuÞ dx þ rð1 rÞkuk þ jvj 2 dt 4 X Z d r þ kgk2l;V þ C1 r GðuÞ dx kuk2 rK2 6 jf j jvj: 4 4 X Namely
Z 1 d 3 a 2 2 2 2 2 kuk þ jvj þ kgkl;V þ 2 GðuÞ dx þ r r kuk þ jvj 2 dt 4 4 X Z d 2 2 a 2 2 þ kgkl;V þ C1 r GðuÞ dx 6 rK2 þ jf j jvj 6 rK2 þ jf j þ jvj : 4 a 8 X ð3:9Þ Setting
3 a d a1 ¼ min 2r r ; ; ; C1 r : 4 4 2
We conclude that d 4 W ðtÞ þ a1 W ðtÞ 6 2rK2 þ jf j2 þ 2a1 K1 :¼ C2 : dt a Here 2
2
2
W ðtÞ ¼ kuk þ jvj þ kgkl;V þ 2
Z
GðuÞ dx þ 2K1 X
P
1 2 2 2 kuk þ jvj þ kgkl;V P 0: 2
ð3:10Þ
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By the Gronwall Lemma, we get C2 W ðtÞ 6 W ð0Þ expða1 tÞ þ ð1 expða1 tÞÞ 8t P t0 a1 and C2 lim supW ðtÞ ¼ :¼ q20 : a1 t!1
ð3:11Þ
Let q00 > q0 be fixed and assume that W ð0Þ 6 R. It readily follows from (3.11) that for t P t0 ¼ t0 ðR; q00 Þ, 1 R : t0 ðR; q00 Þ ¼ log 0 a1 q0 2 q20 We have W ðtÞ 6 q00 2 and
2 d 2 2 2 2 2 2 kuðtÞk þ jut ðtÞj þ kgt ðsÞkl;V 6 1 þ pffiffiffiffiffi ðkuðtÞk þ jvðtÞj þ kgt ðsÞkl;V Þ k1
2
2 d d 6 2 1 þ pffiffiffiffiffi W ðtÞ 6 2 1 þ pffiffiffiffiffi q00 2: k1 k1
We observe thatRif B is a bounded subset of H0 ¼ V H L2l , then by the condition (g2), f X GðuÞ dxg is bounded, too. Hence Z 2 2 2 ku0 k þ jv0 þ ru0 j þ kg0 kl;V þ 2 Gðu0 Þ dx þ 2K1 R ¼ RðBÞ ¼ sup fu0 ;v0 ;g0 g
X
< 1:
2 We set 2 1 þ pdffiffiffi q00 2 ¼ q21 and we end up to k1
Lemma 3.1. The ball of H0 , B0 ¼ BH0 ð0; q1 Þ, centered at 0 of radius q1 , is an absorbing set in H0 for the group SðtÞ. For any bounded subset B in H0 , SðtÞB B0 for t P t0 . 3.2. Absorbing set in H1 We take the scalar product in H of system (1.3) and (1.4) with Av ¼ Aut þ rAu, we have
1 d k0 r 2 2 2 ðjAuj þ kvk þ kgkl;DðAÞ Þ þ r 1 jAuj2 þ ða rÞkvk2 2 dt d d 2 rða rÞðAu; vÞ þ kgkl;DðAÞ þ ððgðuÞ; vÞÞ 6 ðf ; AvÞ: ð3:12Þ 4 2 2 As the same as (3.4), exploiting jAuj P k2 kuk , when r small enough, such that 1
k0 r ar P 1 r; d 2k2
a a rP : 2 4
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753
We have
k0 r a r 1 jAuj2 þ ða rÞkvk2 rða rÞðAu; vÞ P rð1 rÞjAuj2 þ kvk2 d 4 ð3:13Þ and ðf ; AvÞ ¼
d d r ðf ; AuÞ þ rðf ; AuÞ 6 ðf ; AuÞ þ jAuj2 þ 2rjf j2 : dt dt 8
ð3:14Þ
It follows from (g2) that for > 0, there exists a constant C > 0, such that jg0 ðsÞj 6 jsjc þ C
8s 2 R:
Hence jððgðuÞ; vÞÞj Z 0 ¼ g ðuÞru rv dx Z X 6 jg0 ðuÞj jruj jrvj dx Z ZX c 6 juj jruj jrvj dx þ C jruj jrvj dx X
Z
12
X
6 juj2c jruj2 dx kvk þ C kuk kvk 8 X 2q1 R 2p1 R > > < X juj2cq dx X jruj2p dx kvk þ C kuk kvk; n ¼ 1; 2 6 n2 R 1n R > 2n 2n cn > n2 : juj dx jruj dx kvk þ C kuk kvk; n P 3 X X c
6 kuk C32 jAuj kvk þ
a 4C 2 2 2 kvk þ kuk : 16 a
ð3:15Þ
Here C3 is the positive constant satisfying 8 1 > < R juj2cq dx 2q ; n ¼ 1; 2 c X C3 kuk P > : R jujcn dx1n n P 3; 8 X 2p1 R > > < X jruj2p dx ; n ¼ 1; 2 C3 jAuj P n2 R > 2n 2n > n2 : jruj dx ; n P 3: X If ðu0 ; v0 ; g0 Þ belongs to a bounded set B of H1 , then B is also bounded in H0 , and for t P t0 , we have
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kuk2 þ jut j2 þ kgk2l;V 6 q21 : t0 , q1 are given in Lemma 3.1. We choose 0 < 2 <
ar : 4 8q4c 1 C3
From (3.15) entails a 4C 2 2 2 kvk þ kuk 16 a 4 42 q2c a 4C 2 q2 2 2 1 C3 jAuj þ kvk þ 1 6 8 a a 2 2 r a 4C q 2 2 6 jAuj þ kvk þ 1 ; t P t0 : 4 8 a c
jððgðuÞ; vÞÞj 6 kuk C32 jAuj kvk þ
ð3:16Þ
Combining with (3.13) and (3.14), and (3.16), we deduce from (3.12) that
d 5 2 2 2 2 jAuj þ kvk 2ðf ; AuÞ þ kgkl;DðAÞ þ 2r r jAuj dt 8 a d 8C 2 q2 þ kvk2 þ kgk2l;DðAÞ 6 4rjf j2 þ 1 ; t P t0 : 4 2 a Hence
d 5 a 2 2 2 2 2 ðjAu f j þ kvk þ kgkl;DðAÞ Þ þ 2r r jAu f j þ kvk dt 8 4 d 2 þ kgkl;DðAÞ 6 C4 ; t P t0 : 2
ð3:17Þ
Here C4 ¼ 2r
21 8C 2 q2 2 r jf j þ 1 : 8 a
Setting
5 a d a2 ¼ min 2r r ; ; : 8 4 2 By the Gronwall Lemma, we obtain 2
2
2
jAuðtÞ f j þ kvðtÞk þ kgt ðsÞkl;DðAÞ 2
2
2
6 ðjAuðt0 Þ f j þ kvðt0 Þk þ kgt0 ðsÞkl;DðAÞ Þ expða2 ðt t0 ÞÞ þ t P t0 :
C4 ; a2
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755
Thus we have the following conclusion: Theorem 3.1. Let X be an open bounded set of Rn with smooth boundary C, and let g be a C 2 function from R to R satisfying (g1)–(g3). Then the ball BððA1 f ; 0Þ; q2 Þ in H1 , centered at fA1 f ; 0g of radius q2 > Ca24 is absorbing in H1 for the semigroup SðtÞ associated with the problem (1.3)–(1.5). 4. Attractor in H1 In order to obtain the global attractor in H1 , we first need the following Lemmas of the properties of compactness about the nonlinear operator g. Lemma 4.1 [2]. Let g be C 2 function from R into R satisfying (g2). Then g : H01 ðXÞ ! L2 ðXÞ is continuously compact. Lemma 4.2 [2]. Let g be C 2 function from R into R satisfying (g2), and g : DðAÞ ! H01 ðXÞ be defined by Z Z ððgðuÞ; vÞÞ ¼ rgðuÞrv dx ¼ g0 ðuÞru rv dx 8u 2 DðAÞ; v 2 H01 ðXÞ: X
X
Then g is continuous compact. Our main result: Theorem 4.1. The group {S(t)} associated with the system (1.3) and (1.4) has a global attractor A, which is compact, connected and maximal in H1 . It attracts all bounded subsets of H1 in the norm of H1 . Proof. Let fwk g be an orthonormal basis of L2 ðXÞ which consists of eigenvectors of A. The corresponding eigenvalues are denoted by kk , k ¼ 1; 2; . . . and k1 < k2 6 k3 6 ; kj ! 1, as j ! 1. Then fwk g is also an orthogonal basis of V . We write Vm ¼ spanfw1 ; . . . ; wm g: Since f 2 V , and g : DðAÞ ! V are compact operators verified in Lemma 4.2, for any > 0, there exists some m such that ð4:1Þ kðI Pm Þf k 6 ; 4 8u 2 BDðAÞ ðA1 f ; q2 Þ; kðI Pm ÞgðuÞk 6 ð4:2Þ 4 where Pm : V ! Vm is the orthogonal projector, and q2 is given by Theorem 3.1. For any ðu; ut ; gt Þ 2 H1 , we write
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ðu; ut ; gt Þ ¼ ðu1 ; ut1 ; gt1 Þ þ ðu2 ; ut2 ; gt2 Þ: Here ðu1 ; ut1 ; gt1 Þ ¼ ðPm u; Pm ut ; Pm gt Þ: Take the scalar product in H of Eq. (1.3) with Av2 ¼ Aut2 þ rAu2 , combining with (1.4), we find
1 d k0 r 2 2 2 2 2 ðjAu2 j þ kv2 k þ kg2 kl;DðAÞ Þ þ r 1 jAu2 j þ ðd rÞkv2 k 2 dt d d rða rÞðAu2 ; v2 Þ þ kg2 k2l;DðAÞ þ ððgðuÞ; v2 ÞÞ ¼ ðf ; Av2 Þ: ð4:3Þ 4 It is the same as (3.13)
k0 r r 1 jAu2 j2 þ ðd rÞkv2 k2 rða rÞðAu2 ; v2 Þ d a P rð1 rÞjAu2 j2 þ kv2 k2 4
ð4:4Þ
and by (4.1) we obtain ðf ; Av2 Þ 6
d r 2 r 2 ðf2 ; Au2 Þ þ jAu2 j þ : dt 8 8
ð4:5Þ
Here f2 ¼ ðI Pm Þf . Combining with (4.2), (4.4) and (4.5), from (4.3) entails 1 d a 2 2 2 2 2 ðjAu2 j þ kv2 k þ kg2 kl;DðAÞ Þ þ rð1 rÞjAu2 j þ kv2 k 2 dt 4 d r d 2 r : þ kg2 k2l;DðAÞ 6 jAu2 j2 þ kv2 k þ ðf2 ; Au2 Þ þ 4 8 4 dt 8 It follows that
d 7 2 2 2 2 ðjAu2 j þ kv2 k þ kg2 kl;DðAÞ 2ðf2 ; Au2 ÞÞ þ 2r r jAu2 j dt 8 a d 2 2 r 2 2 : þ þ kv2 k þ kg2 kl;DðAÞ 6 4 2 4 4a Setting a3 ¼ minf2rð78 rÞ; a4 ; d2g. Hence d ðjAu2 f2 j2 þ kv2 k2 þ kg2 k2l;DðAÞ Þ þ a3 ðjAu2 f2 j2 þ kv2 k2 þ kg2 k2l;DðAÞ Þ dt
1 r a3 2 6 þ þ :¼ C5 2 : 4a 4 16
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757
Using the Gronwall Lemma, we end up to 2
2
2
jAu2 f2 j þ kv2 k þ kg2 kl;DðAÞ 2
2
2
6 ðjAu2 ðt1 Þ f2 j þ kv2 ðt1 Þk þ kgt21 ðsÞkl;DðAÞ Þ expða3 ðt t1 ÞÞ þ
C5 2 a3
for t P t1 P t0 : Namely 2
2
2
jAu2 f2 j þ kv2 k þ kg2 kl;DðAÞ 6 q22 expða3 ðt t1 ÞÞ þ
C5 2 a3
for t P t1 :
Here q2 is given by Theorem 3.1. Take t2 large enough such that t2 t1 P
1 q2 log 22 : a3
Then
C5 2 2 2 2 jAu2 f2 j þ kv2 k þ kg2 kl;DðAÞ 6 1 þ a3
for t P t2 :
Since jAu2 j2 þ kut 2k2 þ kg2 k2l;DðAÞ 6 2ðjAu2 f2 j2 þ jf2 j2 Þ þ 2kv2 k2 þ 2r2 ku2 k2
2C5 2 2 2 þ 2kg2 kl;DðAÞ 6 3 þ þ ku2 k a3 and the embedding from DðAÞ into V is compact, so choose m large enough such that ku2 k2 6 2 . Hence jAu2 j2 þ kut2 k2 þ kg2 k2l;DðAÞ 6 C6 2 : Here C6 ¼ 4 þ
2C5 : a3
This shows that fSðtÞg is x-limit compact in H1 . h
Acknowledgement The work was supported by the National Science Foundation of China under grant 19971036.
758
Q. Ma, C. Zhong / Appl. Math. Comput. 157 (2004) 745–758
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