On uniform decay of the solution for a damped nonlinear coupled system of wave equations with nonlinear boundary damping and memory term

Applied Mathematics and Computation 148 (2004) 207–223 www.elsevier.com/locate/amc

On uniform decay of the solution for a damped nonlinear coupled system of wave equations with nonlinear boundary damping and memory term Jeong Ja Bae Department of Mathematics, Pusan National University, Pusan 609-735, South Korea

Abstract In this paper we are concerned with the existence and energy decay of the solution to the initial boundary value problem for the nonlinear coupled wave equations with nonlinear boundary damping and memory term. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Existence of solution; Uniform decay; Wave equation; Boundary value problem; A priori estimates

1. Introduction In this paper, we are concerned with the existence and uniform decay of the solution to the initial boundary value problem for the nonlinear coupled wave equations with nonlinear boundary damping and memory term of the form: u00 þ D2 u þ av þ g1 ðu0 Þ ¼ 0 v00  Dv þ au þ g2 ðv0 Þ ¼ 0 u¼

ou ¼0 om

on C
ð0; 1Þ

on Q ¼ X
ð0; 1Þ; on Q ¼ X
ð0; 1Þ; with

C ¼ C0 [ C1 ;

E-mail address: [email protected] (J.J. Bae). 0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00838-X

ð1:1Þ ð1:2Þ ð1:3Þ

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J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

v¼0

on R1 ¼ C1
ð0; 1Þ;

ð1:4Þ

ou þ v þ v0 þ gðtÞjv0 jq v0 ¼ g  jvjc v om

on R0 ¼ C0
ð0; 1Þ;

uðx; 0Þ ¼ u0 ðxÞ;

u0 ðx; 0Þ ¼ u1 ðxÞ;

vðx; 0Þ ¼ v0 ðxÞ;

v0 ðx; 0Þ ¼ v1 ðxÞ on x 2 X;

ð1:5Þ ð1:6Þ

where X is a bounded domain in Rn with C 2 boundary C :¼ oX such that C0 , C1 have positive measures, g  u ¼ RCt ¼ C0 [ C1 , C0 \ C1 ¼ ;n and gðt  rÞuðrÞ dr, Du ¼ Ri¼1 ðo2 u=ox2i Þ and m denotes the unit outer normal 0 vector pointing towards X. Assuming the kernel g provides a damping effect, we prove existence of strong solution u ¼ uðx; tÞ. Moreover, when q ¼ c, the uniform decay of the energy 1 1 1 1 2 2 2 2 EðtÞ ¼ ku0 ðtÞk þ kDuðtÞk þ kv0 ðtÞk þ krvðtÞk 2 2 Z 2 2 1 2 þ kvðtÞkC0 þ auðtÞvðtÞ dx 2 X is proved. There exists a large body of literature regarding viscoelastic problems with the memory term acting in the domain. Among the numerous works in this direction, we can cite Jiang and Munoz Rivera [7]. Related to blow up of the solutions of equations with nonlinear damping and source term acting in the domain we can cite the work of Georgiev and Todorova [5]. Guesmia [6] investigated the existence and asymptotic behavior of solutions of (1.1)–(1.3) with Dirichlet boundary conditions. Cavalcanti [4] considered the existence and uniform decay of solutions of wave equation of the form: Kðx; tÞu00 þ K2 u0  Du ¼ 0 on Q ¼ X
ð0; 1Þ; u¼0

on R1 ¼ C1
ð0; 1Þ;

ou c þ u þ u0 þ gðtÞju0 jq u0 ¼ g  juj u om uðx; 0Þ ¼ u0 ðxÞ;

u0 ðx; 0Þ ¼ u1 ðxÞ

on R0 ¼ C0
ð0; 1Þ;

on x 2 X:

Authors [8,9] have studied the existence and uniform decay of strong solutions of Kirchhoff type wave equations with nonlinear boundary conditions (1.5). For the existence of solutions of Kirchhoff type wave equations with Dirichlet boundary conditions, see [2,3]. In this paper, we will study the existence and uniform decay of solutions of coupled wave equations with nonlinear boundary damping and memory source term. It is important to observe that as far as we are concerned it has never

J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

209

been considered nonlinear memory terms acting in the boundary in the literature. To obtain the existence of solutions we make use of Faedo–GalerkinÕs approximation and also to show the uniform stabilization we use the perturbed energy method. Our paper is organized as follows: In Section 2, we give some notations, assumptions and main result. In Section 3, we prove the existence of solutions of the problems (1.1)–(1.6) and the uniform decay of energy is given in Section 4.

2. Assumptions and main result Throughout this paper we define 1

V ¼ fu 2 H ðXÞ; u ¼ 0 ðu; vÞC0 ¼

on C1 g;

Z uðxÞvðxÞ dC; C0

kuk ¼ kukL2 ðXÞ

and

ðu; vÞ ¼

p

kukp;C0 ¼

Z

Z uðxÞvðxÞ dx; X p

juðxÞj dx;

C0

kuk1 ¼ kukL1 ðXÞ :

The variational formulation associated with problems (1.1)–(1.6) are given by, respectively, ðu00 ; wÞ þ ðDu; DwÞ þ ðav; wÞ þ ðg1 ðu0 Þ; wÞ ¼ 0; 00

0

w 2 H02 ðXÞ;

ð2:1Þ

0

ðv ; wÞ þ ðrv; rwÞ þ ðau; wÞ þ ðg2 ðv Þ; wÞ þ ðv; wÞC0 þ ðv ; wÞC0 q

þ ðgðtÞjv0 j v0 ; wÞC0 Z t c ¼ gðt  rÞðjvðrÞj vðrÞ; wÞC0 dr;

w2V:

ð2:2Þ

0

(A1 ) Let us consider g 2 W 1;1 ð0; 1Þ \ W 1;1 ð0; 1Þ; gðtÞ P 0, 8 t P 0 verifying  m0 gðtÞ 6 g0 ðtÞ 6  m1 gðtÞ 8t P t0 ; gð0Þ ¼ 0; jg0 ðtÞj 6 m2 gðtÞ

8t 2 ½0; t0 

R1 for some m0 , m1 , m2 > 0 and l ¼ 1  0 gðrÞ dr > 0. (A2 ) The functions gi , i ¼ 1; 2 are nondecreasing C 1 functions and gi ð0Þ ¼ 0. Furthermore, there exists positive constants ai such that a1 jxj 6 jgi ðxÞj 6 a2 jxj 8 x 2 R;

i ¼ 1; 2:

pffiffiffiffiffiffi (A3 ) The function a belongs to L1 ðXÞ and kakL1 ðXÞ < 1= kl; where k is a pos2 2 itive number such that kuk 6 kkruk 8u 2 H01 ðXÞ and l is a positive number such that kuk2 6 lkDuk2 8u 2 H02 ðXÞ.

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Remark 1. Note that the energy is nonnegative. In fact, we have Z Z auv dx P  kakL1 ðXÞ uv dx X X pffiffiffiffiffiffi P  klkakL1 ðXÞ kDukkrvk pffiffiffiffiffiffi kl 2 2 kakL1 ðXÞ ðkDuk þ krvk Þ: P 2 (2.3) and assumption ðA3 Þ imply 1 1 1 1 2 2 2 2 EðtÞ ¼ ku0 ðtÞk þ kDuðtÞk þ kv0 ðtÞk þ krvðtÞk 2 2 Z 2 2 1 2 þ kvðtÞkC0 þ auðtÞvðtÞ dx 2 X 1 0 1 1 1 2 2 2 2 P ku ðtÞk þ kDuðtÞk þ ku0 ðtÞk þ krvðtÞk 2 2 pffiffiffiffiffiffi 2 2 kl 1 kakL1 ðXÞ ðkDuðtÞk2 þ krvðtÞk2 Þ þ kvðtÞk2C0  2 2 pffiffiffiffiffiffi 1 1 ¼ ku0 ðtÞk2 þ ð1  klkakL1 ðXÞ ÞðkDuðtÞk2 þ krvðtÞk2 Þ 2 2 1 0 1 2 þ kv ðtÞk þ kvðtÞk2C0 P 0: 2 2

ð2:3Þ

Now we are in position to state our main result. Theorem 2.1. Let us consider u0 , u1 2 H02 ðXÞ and v0 , v1 2 H01 ðXÞ \ H 2 ðXÞ satisfy ðov0 =omÞ þ v0 þ v1 ¼ 0 on C0 . Under the assumptions ðA1 Þ–ðA3 Þ, suppose that c, m1 1 q satisfy 0 < c < q 6 ðn2 Þ if n P 3; or c; q > 0 if n ¼ 1; 2 and ðcþ2 Þ > 2. Then 2 problems (1.1)–(1.6) have a unique solution ðu; vÞ : X
X ! R such that ðu; vÞ 2 L1 ð0; 1; H02 ðXÞÞ
L1 ð0; 1; H01 ðXÞÞ, ðu0 ; v0 Þ 2 L1 ð0; 1; H02 ðXÞÞ
L1 ð0; 1; H01 ðXÞÞ, ðu00 ; v00 Þ 2 L2 ð0; 1; L2 ðXÞÞ
L2 ð0; 1; L2 ðXÞÞ. Moreover, if q ¼ c, then there exist positive constants C1 and C2 such that EðtÞ 6 C1 Eð0Þ expðC2
tÞ:

3. Proof of Theorem 2.1 In this section we are going to show the existence of solution of problems (1.1)–(1.6) using Faedo–GalerkinÕs approximation. For this end we represent by fwj gj2N a basis in H02 ðXÞ which is orthonormal in L2 ðXÞ, by VmP the subspace of H02 ðXÞ generated by the first m vectors. Next we define um ðtÞ ¼ mj¼1 gjm ðtÞwj , where ðum ðtÞ; vm ðtÞÞ is a solution of the following Cauchy problems: ðu00m ; wÞ þ ðDum ; DwÞ þ ðavm ; wÞ þ ðg1 ðu0m Þ; wÞ ¼ 0;

ð3:1Þ

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211

ðv00m ; wÞ þ ðrvm ; rw Þ þ ðaum ; wÞ þ ðg2 ðv0m Þ; wÞ þ ðvm ; wÞC0 þ ðv0m ; wÞC0 þ ðgðtÞjv0m jq v0m ; wÞC0 Z t c ¼ gðt  rÞðjvm ðrÞj vm ðrÞ; wÞC0 dr;

w 2 H02 ðXÞ

ð3:2Þ

0

with the initial conditions, m X um ð0Þ ¼ u0m ¼ ðu0 ; wj Þwj ! u0

in H02 ;

j¼1

vm ð0Þ ¼ v0m

m X ¼ ðv0 ; wj Þwj ! u0

u0m ð0Þ ¼ u1m ¼

j¼1 m X

in H01 ðXÞ \ H 2 ðXÞ; ð3:3Þ

ðu1 ; wj Þwj ! u1

in H02 ðXÞ;

m X ¼ ðv1 ; wj Þwj ! u1

in H01 ðXÞ:

j¼1

v0m ð0Þ ¼ v1m

j¼1

Note that we can solve the system (3.1)–(3.3) by PicardÕs iteration method. In fact, the system (3.1)–(3.3) have a unique solution on some interval ½0; Tm Þ. The extension of the solution to the whole interval ½0; 1Þ is a consequence of the first estimate which we are going to prove below. 3.1. A priori estimate I Replacing w by u0m ðtÞ in (3.1) and by v0m ðtÞ in (3.2), respectively and adding the results,  assumption ðA1 Þ yield  Z t d 1 cþ2 cþ2 Em ðtÞ þ gðtÞkvm ðtÞkcþ2;C0 þ gðt  rÞkvm ðrÞkcþ2;C0 dr dt cþ2 0 2 qþ2 þ kv0m ðtÞkC0 þ gðtÞkv0m ðtÞkqþ2;C0 þ ðg1 ðu0m ðtÞÞ; u0m ðtÞÞ þ ðg2 ðv0m ðtÞÞ; v0m ðtÞÞ Z t 1 c cþ2 g0 ðtÞkvm ðtÞkcþ2;C0 gðt  rÞðjvm ðrÞj vm ðrÞ; v0m ðtÞÞC0 dr þ ¼ c þ 2 0 Z t c cþ2 þ gðtÞðjvm ðtÞj vm ðtÞ; v0m ðtÞÞC0 þ g0 ðt  rÞkvm ðrÞkcþ2;C0 dr 0 Z t m2 c cþ2 6 gðtÞkvm ðtÞkcþ2;C0 gðt  rÞðjvm ðrÞj vm ðrÞ; v0m ðtÞÞC0 dr þ c þ2 0 Z t þ gðtÞðjvm ðtÞjc vm ðtÞ; v0m ðtÞÞC0 þ m2 gðt  rÞkvm ðrÞkcþ2 ð3:4Þ cþ2;C0 dr; 0

where 1 1 1 1 1 Em ðtÞ ¼ ku0m ðtÞk2 þ kDum ðtÞk2 þ kv0m ðtÞk2 þ krvm ðtÞk2 þ kvm ðtÞk2C0 2 2 2 2 2 Z þ aum ðtÞvm ðtÞ dx: X

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J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

Note that H€ olderÕs inequality and YoungÕs inequality [1] give us c

cþ1

ðjvm ðrÞj vm ðrÞ; v0m ðtÞÞC0 6 kvm ðrÞkcþ2;C0 kv0m ðtÞkcþ2;C0 cþ2

cþ2

6 C1 ðgÞkvm ðrÞkcþ2;C0 þ gkv0m ðtÞkcþ2;C0 :

ð3:5Þ

Thus we have Z t Z t c cþ2 0 gðt  rÞðjvm ðrÞj vm ðrÞ; vm ðtÞÞC0 dr 6 C1 ðgÞ gðt  rÞkvm ðrÞkcþ2;C0 dr 0 0 Z t cþ2 0 þ gkvm ðtÞkcþ2;C0 gðrÞ dr: ð3:6Þ 0

qþ2

cþ2

ðC0 Þ,!L ðC0 Þ and therefore we can obtain Z t Z t Z t cþ2 qþ2 gðrÞ dr 6 C2 ðgÞ gðrÞ dr þ g gðrÞ drkv0m ðtÞkqþ2;C0 : gkv0m ðtÞkcþ2;C0

Since q P c, L

0

0

0

ð3:7Þ Therefore (3.6) and (3.7) yield Z t Z t c cþ2 gðt  rÞðjvm ðrÞj vm ðrÞ; v0m ðtÞÞC0 dr 6 C1 ðgÞ gðt  rÞkvm ðrÞkcþ2;C0 dr 0 0 Z t Z t qþ2 0 þ C2 ðgÞ gðrÞ dr þ g gðrÞ drkvm ðtÞkqþ2;C0 : ð3:8Þ 0

0

Similarly applying H€ olderÕs inequality, YoungÕs inequality and the result Lqþ2 ðC0 Þ,!Lcþ2 ðC0 Þ, we have c

cþ1

gðtÞðjvm ðtÞj vm ðtÞ; v0m ðtÞÞC0 6 gðtÞkvm ðtÞkcþ2;C0 kv0m ðtÞkcþ2;C0 cþ2 6 C3 ðgÞgðtÞkvm ðtÞkcþ2;C þ ggðtÞkv0m ðtÞkcþ2 cþ2;C0 0 cþ2

qþ2

6 C3 ðgÞgðtÞkvm ðtÞkcþ2;C0 þ gðtÞC4 ðgÞ þ ggðtÞkv0m ðtÞkqþ2;C0 :

ð3:9Þ

Therefore (3.4), (3.8) and (3.9) give   Z t d 1 cþ2 cþ2 Em ðtÞ þ gðtÞkvm ðtÞkcþ2;C0 þ gðt  rÞkvm ðrÞkcþ2;C0 dr dt cþ2 0 2

þ ðg1 ðu0m ðtÞÞ; u0m ðtÞÞ þ ðg2 ðv0m ðtÞÞ; v0m ðtÞ þ kv0m ðtÞkC0 qþ2

þ ðð1  gÞgðtÞ  gkgkL1 ð0;1Þ Þkv0m ðtÞkqþ2;C0   Z t m2 6 ðC1 ðgÞ þ m2 Þ gðt  rÞkvm ðrÞkcþ2 dr þ C ðgÞ þ 3 cþ2;C0 cþ2 0 Z t cþ2 þ C4 ðgÞgðtÞ þ C2 ðgÞ gðrÞ dr:
gðtÞkvm ðtÞkcþ2;C 0 0

ð3:10Þ

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213

Note that we can choose g > 0 sufficiently small such that ð1  gÞgðtÞ  gkgkL1 ð0;1Þ > C0 gðtÞ for some constant C0 , which can be from assumption ðA1 Þ. Moreover, since g1 and g2 are nondecreasing and g1 ð0Þ ¼ g2 ð0Þ ¼ 0, ðg1 ðu0m ðtÞÞ; u0m ðtÞÞ þ ðg2 ðv0m ðtÞÞ; v0m ðtÞÞ P 0. Integrating it over ½0; t, choosing g > 0 sufficiently small and employing GronwallÕs lemma we obtain the first estimate: Z t 1 cþ2 cþ2 gðtÞkvm ðtÞkcþ2;C0 þ gðt  rÞkvm ðrÞkcþ2;C0 dr cþ2 0 Z t Z t 2 0 þ kvm ðrÞkC0 dr þ C0 gðrÞkv0m ðrÞkqþ2 qþ2;C0 dr 6 L1 ;

Em ðtÞ þ

0

0

where L1 > 0 is a constant independent of m; that is, we get 1 0 1 1 1 1 kum ðtÞk2 þ kDum ðtÞk2 þ kv0m ðtÞk2 þ krvm ðtÞk2 þ kvm ðtÞk2C0 2 2 2 2 2 Z Z t 1 cþ2 2 gðtÞkvm ðtÞkcþ2;C0 þ þ aum ðtÞvm ðtÞ dx þ kv0m ðrÞkC0 dr c þ 2 X 0 Z t Z t cþ2 þ gðt  rÞkvm ðrÞkcþ2;C0 dr þ C0 gðrÞkv0m ðrÞkqþ2 qþ2;C0 dr 6 L1 : 0

ð3:11Þ

0

3.2. A priori estimate II First of all we are estimating v00m ð0Þ in the L2 -norm. Considering w ¼ v00m ð0Þ in (3.2), from Green theorem   ov0 00 2 00 00 ; v ð0Þ þ ðau0 ; v00m ð0ÞÞ þ ðg2 ðv1 Þ; v00m ð0ÞÞ kvm ð0Þk  ðDv0 ; vm ð0ÞÞ  om m þ ðv0 ; v00m ð0ÞÞC0 þ ðv1 ; v00m ð0ÞÞC0 ¼ 0 Since

ov0 om

in H01 ðXÞ \ H 2 ðXÞ:

þ v0 þ v1 ¼ 0 on C0 ,

kv00m ð0Þk2  ðDv0 ; v00m ð0ÞÞ þ ðau0 ; v00m ð0ÞÞ þ ðg2 ðv1 Þ; v00m ð0ÞÞ ¼ 0: Assumption ðA2 Þ and SchwarzÕs inequalities imply kv00m ð0Þk2 6 kDv0 kkv00m ð0Þk þ and so kv00m ð0Þk 6 L2 ;

pffiffiffi lkakL1 ðXÞ kDu0 kkv00m ð0Þk þ a2 kv1 kkv00m ð0Þk

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J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

where L2 is a positive constant independent of m. Now, differentiating (3.1) and (3.2), substituting w by u00m ðtÞ and v00m ðtÞ, respectively and then adding the results, assumption ðA1 Þ yields d 2 2 2 Fm ðtÞ þ ðg10 ðu0m ðtÞÞ; ju00m ðtÞj Þ þ ðg20 ðv0m ðtÞÞ; jv00m ðtÞj Þ þ kv00m ðtÞkC0 dt q 2 þ ðq þ 1ÞgðtÞðjv0m ðtÞj ; jv00m ðtÞj ÞC0 Z t q 0 c 0 0 00 ¼ g ðtÞðjvm ðtÞj vm ðtÞ; vm ðtÞÞC0 þ g0 ðt  rÞðjvm ðrÞj vm ðrÞ; v00m ðtÞÞC0 dr 0 Z t c 6 m2 gðt  rÞðjvm ðrÞj vm ðrÞ; v00m ðtÞÞC0 dr 0

q

þ m0 gðtÞðjv0m ðtÞj v0m ðtÞ; v00m ðtÞÞC0 ;

ð3:12Þ

where 1 1 1 1 2 2 2 2 Fm ðtÞ ¼ ku00m ðtÞk þ kDu0m ðtÞk þ kv00m ðtÞk þ krv0m ðtÞk 2 2 2 2 Z 1 þ au0m ðtÞv0m ðtÞ dx þ kv0m ðtÞk2C0 : 2 X Now, SchwarzÕs inequality, YoungÕs inequality and first estimate give us q

jm0 gðtÞðjv0m ðtÞj v0m ðtÞ; v00m ðtÞÞC0 j 6 m0 gðtÞ

Z

q

q

þ1

jv0m ðtÞj2 jv0m ðtÞj2 jv00m ðtÞj dC

C0

m2 q 2 0 00 6 0 gðtÞkv0m ðtÞkqþ2 qþ2;C0 þ ggðtÞðjvm ðtÞj ; jvm ðtÞj ÞC0 : 4g

ð3:13Þ

Now, taking into account that ððc þ 1Þ=ð2c þ 2ÞÞ þ ð1=2Þ ¼ 1, using the generalized H€ older inequality and the continuity of the trace operator c0 : H 1 ðXÞ ,! L2 ðCÞ for 1 6 q 6 ðð2n  2Þ=ðn  2ÞÞ, we obtain c

00

ðjvm ðrÞj vm ðrÞ; v ðtÞÞC0 dr 6

Z

jvm ðrÞj

2cþ2

cþ1  Z 2cþ2 12 2 00 dC jvm ðtÞj dC

C0

C0

6 CðT ; gÞkrvm ðrÞk 6 CðT ; gÞð2L1 Þ

cþ1

2cþ2

2

þ gkv00m ðtÞkC0 2

þ gkv00m ðtÞkC0 : ð3:14Þ

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215

Thus from (3.14), we get Z t c m2 gðt  rÞðjvm ðrÞj vm ðrÞ; v00m ðtÞÞC0 dr 0 Z t cþ1 2 6 m2 gðt  rÞðCðT ; gÞð2L1 Þ þ gkv00m ðtÞkC0 Þ dr 0

6 m2 CðT ; gÞð2L1 Þcþ1 kgkL1 ð0;1Þ þ gm2 kv00m ðtÞk2C0 kgkL1 ð0;1Þ :

ð3:15Þ

Combining the estimates (3.13) and (3.15), we get d Fm ðtÞ þ kv00m ðtÞk2C0 þ ðq þ 1  gÞgðtÞðjv0m ðtÞjq ; jv00m ðtÞj2 ÞC0 dt m2 qþ2 cþ1 2 6 0 gðtÞkv0m ðtÞkqþ2;C0 þ ðm2 CðT ; gÞð2L1 Þ þ gm2 kv00m ðtÞkC0 ÞkgkL1 ð0;1Þ ; 4g ð3:16Þ 2

2

where we have used the fact that ðg10 ðu0m ðtÞÞ; ju00m ðtÞj Þ þ ðg20 ðv0m ðtÞÞ; jv00m ðtÞj Þ P 0 since gi , i ¼ 1; 2 are nondecreasing. Integrating (3.16) over ½0; t, choosing g > 0 sufficiently small and employing (3.11) and GronwallÕs lemma we obtain the second estimate: Z t 2 kv00m ðsÞkC0 ds 6 L3 ; Fm ðtÞ þ 0

where L3 > 0 is independent of m, that is, 1 00 1 1 1 1 ku ðtÞk2 þ kDu0m ðtÞk2 þ kv00m ðtÞk2 þ krv0m ðtÞk2 þ kv0m ðtÞk2C0 2 m 2 2 2 2 Z Z t 2 þ au0m ðtÞv0m ðtÞ dx þ kv00m ðsÞkC0 ds 6 L3 :

ð3:17Þ

0

X

The estimates above are sufficient to pass to the limit in the linear terms of problems (3.1) and (3.2). Next we are going to consider the nonlinear ones of the problem (3.2). Analysis of the nonlinear terms. From the above estimates we have that 1

ðvm Þ is bounded in L2 ð0; T ; H 2 ðC0 ÞÞ; 1

ð3:18Þ

ðv0m Þ is bounded in L2 ð0; T ; H 2 ðC0 ÞÞ;

ð3:19Þ

ðv00m Þ is bounded in L2 ð0; T ; L2 ðC0 ÞÞ:

ð3:20Þ

From (3.18)–(3.20), taking into consideration that the immersion 1 H 2 ðCÞ ,! L2 ðCÞ is continuous and compact and using Aubin compactness theorem, we can extract a subsequence ðvl Þ of ðvm Þ such that vl ! v a:e: on R0

and

v0l ! v0 a:e: on R0

ð3:21Þ

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J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

and therefore c

c

q

q

jv0l j v0l ! jv0 j v0 a:e: on R0 :

jvl j vl ! jvj v;

ð3:22Þ

On the other hand, from the first and second estimate we obtain ðg  jvl jc vl Þ is bounded in L2 ðR0 Þ;

ð3:23Þ

ðgjv0l jq v0l Þ is bounded in L2 ðR0 Þ:

ð3:24Þ

Combining (3.22)–(3.24), we deduce that c

c

g  jvl j vl ! g  jvj v weakly in L2 ðR0 Þ; q

q

gjv0l j v0l ! gjv0 j v0 weakly in L2 ðR0 Þ: The last convergences are sufficient to pass to the limit in the nonlinear terms of problem (3.2).


4. Uniform decay We define the energy EðtÞ of the problems (1.1)–(1.6) by 1 1 1 1 EðtÞ ¼ ku0 ðtÞk2 þ kDuðtÞk2 þ kv0 ðtÞk2 þ krvðtÞk2 2 2 2 2 Z 1 2 þ kvðtÞkC0 þ auðtÞvðtÞ dx: 2 X

ð4:1Þ

Then the derivative of the energy is given by 2

E0 ðtÞ ¼ ðg1 ðu0 ðtÞÞ; u0 ðtÞÞ  ðg2 ðv0 ðtÞÞ; v0 ðtÞÞ  kv0 ðtÞkC0 Z t qþ2 c  gðtÞkv0 ðtÞkqþ2;C0 þ gðt  rÞðjvðrÞj vðrÞ; v0 ðtÞÞC0 dr:

ð4:2Þ

0

Defining ðg
vÞðtÞ :¼

Z

t c

0

2

gðt  rÞkvðrÞj vðrÞ  vðtÞjC0 dr;

a simple computation gives us

ð4:3Þ

J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

ðg
vÞ0 ðtÞ ¼

Z

217

t

g0 ðt  rÞkvðrÞjc vðrÞ  vðtÞj2C0 dr 0  Z t Z t d þ kvðtÞk2C0 gðrÞ dr  2 gðt  rÞðjvðrÞjc vðrÞ; v0 ðtÞÞC0 dr dt 0 0 Z t c ¼ ðg0
vÞðtÞ  2 gðt  rÞðjvðrÞj vðrÞ; v0 ðtÞÞC0 dr 0   Z t d 2 kvðtÞkC0 gðrÞ dr  gðtÞkvðtÞk2C0 : ð4:4Þ þ dt 0

Thus we have Z

t 0

gðt  rÞðjvðrÞjc vðrÞ; v0 ðtÞÞC0 dr   Z t 1 1 0 1 d 0 2 kvðtÞkC0 ¼  ðg
vÞ ðtÞ þ ðg
vÞðtÞ þ gðrÞ dr 2 2 2 dt 0 1  gðtÞkvðtÞk2C0 : 2

ð4:5Þ

Define the modified energy by 1 1 1 1 2 2 2 2 eðtÞ ¼ ku0 ðtÞk þ kDuðtÞk þ kv0 ðtÞk þ krvðtÞk 2Z 2 2 2 1 þ auðtÞvðtÞ dx þ ðg
vÞðtÞ 2 X   Z t 1 1 2 cþ2 þ 1 gðtÞkvðtÞkcþ2;C0 : gðrÞ dr kvðtÞkC0 þ 2 c þ 2 0

ð4:6Þ

Then 2

e0 ðtÞ ¼ ðg1 ðu0 ðtÞÞ; u0 ðtÞÞ  ðg2 ðv0 ðtÞÞ; v0 ðtÞÞ  kv0 ðtÞkC0 1 g0 ðtÞkvðtÞkcþ2 cþ2;C0 cþ2 1 1 c 2 þ gðtÞðjvðtÞj vðtÞ; v0 ðtÞÞC0  gðtÞkvðtÞkC0 þ ðg0
vÞðtÞ: 2 2  gðtÞkv0 ðtÞkqþ2 qþ2;C0 þ

ð4:7Þ

Considering YoungÕs inequality, we get c

0

gðtÞðjvðtÞj vðtÞ; v ðtÞÞC0 6 gðtÞ 6 ggðtÞkv

0

cþ2 ðtÞkcþ2;C0

þg

Z

1 cþ1

jvðtÞj

cþ2

1 cþ1 Z cþ2 cþ2 cþ2 0 dC jv ðtÞj dC

C0 cþ2 gðtÞkvðtÞkcþ2;C0 :

C0

ð4:8Þ

218

J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

Thus for c ¼ q, assumption ðA1 Þ implies 2

e0 ðtÞ 6  ðg1 ðu0 ðtÞÞ; u0 ðtÞÞ  ðg2 ðv0 ðtÞÞ; v0 ðtÞÞ  kv0 ðtÞkC0 1 m1 qþ2 2  ð1  gÞgðtÞkv0 ðtÞkqþ2;C0  gðtÞkvðtÞkC0  ðg
vÞðtÞ 2 2   1 m1 cþ2 cþ1  gðtÞkvðtÞkcþ2;C0 : g cþ2

ð4:9Þ

Choosing g ¼ 2ðcþ1Þ then 1  g > 12 and so 1 e0 ðtÞ 6  ðg1 ðu0 ðtÞÞ; u0 ðtÞÞ  ðg2 ðv0 ðtÞÞ; v0 ðtÞÞ  kv0 ðtÞk2C0  gðtÞkv0 ðtÞkqþ2 qþ2;C0 2 1 m1 2 ðg
vÞðtÞ: ð4:10Þ  bgðtÞkvðtÞkcþ2 cþ2;C0  gðtÞkvðtÞkC0  2 2 On the other hand we note that from assumption ðA1 Þ 1 1 1 2 2 2 EðtÞ ¼ ku0 ðtÞk þ kDuðtÞk þ kv0 ðtÞk 2 2 2 Z 1 1 þ krvðtÞk2 þ kvðtÞk2C0 þ auðtÞvðtÞ dx 2 2 X 1 0 1 1 0 1 2 2 2 2 6 ku ðtÞk þ kDuðtÞk þ kv ðtÞk þ krvðtÞk 2  2 2 2  Z t Z 1 2 1 þ gðrÞ dr kvðtÞkC0 þ auðtÞvðtÞ dx 6 l1 eðtÞ 2l X 0

ð4:11Þ

and therefore it is enough to obtain the desired exponential decay for the modified energy eðtÞ which will be done below. For this purpose let k, l be the positive numbers such that 2

2

8 v 2 H02 ðXÞ;

2

2

8 v 2 H01 ðXÞ

kvk 6 lkDvk

kvk 6 kkrvk

and for every
> 0 let us define the perturbed modified energy by e
ðtÞ ¼ eðtÞ þ
wðtÞ; wherewðtÞ ¼ ðu0 ðtÞ; uðtÞÞ þ ðv0 ðtÞ; vðtÞÞ: Proposition 4.1. We have the inequality for each
> 0 pffiffiffi pffiffiffi pffiffiffi je
ðtÞ  eðtÞj 6
maxf k; lgeðtÞ 
k
eðtÞ 8 t P 0:

J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

219

Proof. Applying Cauchy SchwarzÕs inequality pffiffiffi pffiffiffi jwðtÞj 6 lku0 ðtÞkkDuðtÞk þ kkv0 ðtÞkkvðtÞk   pffiffiffi pffiffiffi 1 0 1 1 0 1 2 2 2 2 ku ðtÞk þ kDuðtÞk þ kv ðtÞk þ krvðtÞk 6 maxf k; lg 2 2 2 2 pffiffiffi pffiffiffi pffiffiffi 6 maxf k; lgeðtÞ 
k
eðtÞ: ð4:12Þ Thus we have

je
ðtÞ  eðtÞj ¼
jwðtÞj 6


pffiffiffi k
eðtÞ:

ð4:13Þ




Proposition 4.2. There exist C1 > 0 and
1 such that for
2 ð0;
1  e0
ðtÞ 6 
C1 eðtÞ:

Proof. Using the problem (1.1) we have

2

2

2

w0 ðtÞ ¼ ku0 ðtÞk þ kv0 ðtÞk  kDuðtÞk  2

Z

auðtÞvðtÞ dx  krvðtÞk

2

X

 ðg1 ðu0 ðtÞÞ; uðtÞÞ  ðg2 ðv0 ðtÞÞ; vðtÞÞ  kvðtÞk2C0  ðv0 ðtÞ; vðtÞÞC0 Z t  ðgðtÞjv0 ðtÞjq v0 ðtÞ; vðtÞÞC0 þ gðt  rÞðjvðrÞjc vðrÞ; vðtÞÞC0 dr 0

3 3 1 1 2 2 2 2 ¼ eðtÞ þ ku0 ðtÞk þ kv0 ðtÞk  kDuðtÞk  krvðtÞk 2 2 2 2 Z 1 þ ðg
vÞðtÞ  auðtÞvðtÞ dx  ðg1 ðu0 ðtÞÞ; uðtÞÞ  ðg2 ðv0 ðtÞÞ; vðtÞÞ 2 X Z t 1 1 1 2 2 cþ2 gðtÞkvðtÞkcþ2;C0 gðrÞ drkvðtÞkC0 þ  kvðtÞkC0  2 2 0 cþ2  ðv0 ðtÞ; vðtÞÞC0  ðgðtÞjv0 ðtÞjq v0 ðtÞ; vðtÞÞC0 Z t þ gðt  rÞðjvðrÞjc vðrÞ; vðtÞÞC0 dr: 0

ð4:14Þ

220

J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

Note that SchwarzÕs inequality implies Z t c gðt  rÞðjvðrÞj vðrÞ; vðtÞÞC0 dr 0 Z t Z t c 2 ¼ gðt  rÞðjvðrÞj vðrÞ  vðtÞ; vðtÞÞC0 dr þ gðt  rÞkvðtÞkC0 dr 0 0 Z Z t 1 t 3 c 2 2 6 gðt  rÞjjvðrÞj vðrÞ  vðtÞjC0 dr þ kvðtÞkC0 gðrÞ dr 2 0 2 0 Z t 1 3 2 ¼ ðg
vÞðtÞ þ kvðtÞkC0 gðrÞ dr: ð4:15Þ 2 2 0 Also applying Sobolev imbedding, we have jðvðtÞ; v0 ðtÞÞC0 j 6 kvðtÞkC0 kv0 ðtÞkC0 6 l
krvðtÞkkv0 ðtÞkC0

ð4:16Þ

l
2 2 6 gkrvðtÞk þ kv0 ðtÞkC0 ; 4g 2

where l
is the positive number such that kvkC0 6 l
krvk; H€ olderÕs inequality and Young inequality imply

8v 2 H01 ðXÞ. Also

jðgðtÞjv0 ðtÞjq v0 ðtÞ; vðtÞÞj 6 gðtÞkv0 ðtÞkqþ1 qþ2;C0 kvðtÞkqþ2;C0 qþ2

qþ2

6 hðgÞgðtÞkv0 ðtÞkqþ2;C0 þ ggðtÞkvðtÞkqþ2;C0

ð4:17Þ

and Z    pffiffiffiffiffiffi  auðtÞvðtÞ dx 6 klkak 1 kDuðtÞkkrvðtÞk L ðXÞ   X pffiffiffiffiffiffi kl 2 2 6 kakL1 ðXÞ ðkDuðtÞk þ krvðtÞk Þ: 2

ð4:18Þ

Also assumption ðA3 Þ implies j  ðg1 ðu0 ðtÞÞ; uðtÞÞ  ðg2 ðv0 ðtÞÞ; vðtÞÞj Z Z g g l k g1 ðu0 ðtÞÞ2 dx þ g2 ðv0 ðtÞÞ2 dx 6 kuðtÞk2 þ kvðtÞk2 þ l k 4g X 4g X Z ðk þ lÞa2 u0 ðtÞg1 ðu0 ðtÞÞ þ v0 ðtÞg2 ðv0 ðtÞÞ dx 6 gðkDuðtÞk2 þ krvðtÞk2 Þ þ 4g X  ðk þ lÞa2 1 2 2 2 qþ2 6 gðkDuðtÞk þ krvðtÞk Þ þ  e0 ðtÞ  kvðtÞkC0  gðtÞkv0 ðtÞkqþ2;C0 4g 2  1 m1 2  bgðtÞkvðtÞkcþ2   gðtÞkvðtÞk ðg
vÞðtÞ ð4:19Þ cþ2;C0 C0 2 2

J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

221

and

3 2

Z

2

2

ju0 ðtÞj þ jv0 ðtÞj dx 6 X

3 2a1

Z

u0 ðtÞgðu0 ðtÞÞ þ v0 ðtÞgðv0 ðtÞÞ dx

X

 3 1 2 qþ2 cþ2 6  e0 ðtÞ  kvðtÞkC0  gðtÞkv0 ðtÞkqþ2;C0  bgðtÞkvðtÞkcþ2;C0 2a1 2  1 m1 2 ð4:20Þ  gðtÞkvðtÞkC0  ðg
vÞðtÞ : 2 2

Combining these inequalities, we have 

 1 2 qþ2 e0 ðtÞ þ kvðtÞkC0 þ gðtÞkv0 ðtÞkqþ2;C0 2  1 m1 2 gðtÞkvðtÞk ðg
vÞðtÞ þ bgðtÞkvðtÞkcþ2 þ þ cþ2;C0 C0 2 2 pffiffiffiffiffiffi pffiffiffiffiffiffi     kl kl 1 1 2 2  kakL1 ðXÞ kDuðtÞk  kakL1 ðXÞ krvðtÞk g  2g  2 2 2 2 Z t 1 1 2 2 cþ2 gðtÞkvðtÞkcþ2;C0 þ ðg
vÞðtÞ  kvðtÞkC0 þ gðrÞ drkvðtÞkC0 þ 2 c þ 2 0

w0 ðtÞ 6  eðtÞ 

3 ðk þ lÞa2 þ 4g 2a1

l
2 0 2 qþ2 qþ2 kv ðtÞkC0 þ hðgÞgðtÞkv0 ðtÞkqþ2;C0 þ ggðtÞkvðtÞkqþ2;C0 4g     3 ðk þ lÞa2 0 1 3 ðk þ lÞa2 2 ¼ eðtÞ  þ þ e ðtÞ  kvðtÞkC0 þ 4g 4g 2a1 2 2a1   3 ðk þ lÞa2  hðgÞ gðtÞkv0 ðtÞkqþ2 þ  qþ2;C0 4a1 8g   3b ðk þ lÞa2 b 1 cþ2   þ gðtÞkvðtÞkcþ2;C0 2a1 4g cþ2    Z t 3 ðk þ lÞa2 2 gðtÞ   gðrÞ dr kvðtÞkC0 þ 8g 4a1 0   3m1 ðk þ lÞa2 m1 þ   1 ðg
vÞðtÞ 4a1 8g pffiffiffiffiffiffi   1 kl 2 2 2 kakL1 ðXÞ ðkDuðtÞk þ krvðtÞk Þ  gkDuðtÞk   2g  2 2 þ

þ

l
2 0 2 qþ2 kv ðtÞkC0 þ ggðtÞkvðtÞkqþ2;C0 : 4g

ð4:21Þ

222

J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

Combining (4.10) and (4.21) and considering q ¼ c, we get e0
ðtÞ ¼ e0 ðtÞ þ
w0 ðtÞ "

#  

l2 1 2 0 kv ðtÞkC0  
hðgÞ kv0 ðtÞkcþ2 6 
eðtÞ  1  C1
 cþ2;C0 4g 2     Z t
1  1  gðrÞ dr kvðtÞk2C0 
g gðtÞkvðtÞkcþ2 gðtÞ 
cþ2;C0 cþ2 2 0   hm i 1 1 2 
ðg
vÞðtÞ  g
kDuðtÞk2 
þ C1 kvðtÞkC0  2 2 pffiffiffiffiffiffi   1 kl kakL1 ðXÞ ðkDuðtÞk2 þ krvðtÞk2 Þ; 
ð4:22Þ  2g  2 2

where C1 ¼ ð3=2a1 Þ þ ððk þ lÞa2 =4gÞ. Defining ( ) 4g 1 cþ2 m1 kgkL1 ð0;1Þ ; ; ;
1 ¼ min ; ; 4gC1 þ l
2 2hðgÞ ðc þ 2Þg þ 1 2 2kgkL1 ð0;1Þ choosing
2 ð0;
1 , then e0
ðtÞ 6 
C2 eðtÞ for some constant C2 > 0:




ð4:23Þ

n o Continuing the proof of Theorem 2.1. Let
0 ¼ min ð1=2k
1=2 Þ;
1 and let us consider
2 ð0;
0 . As we have
< ð1=2k
1=2 Þ, we conclude from Proposition 4.1 ð1 
k
1=2 ÞeðtÞ < e
ðtÞ < ð1 þ
k
1=2 ÞeðtÞ and so 1 3 eðtÞ < e
ðtÞ < eðtÞ: 2 2

ð4:24Þ

Thus we have 2 e0
ðtÞ 6  C2
e
ðtÞ 3 and    d 2 e
ðtÞ exp C2
t 6 0: dt 3 Integrating (4.25), inequality (4.24) implies   2 eðtÞ 6 3eð0Þ exp  C2
t : 3

ð4:25Þ

ð4:26Þ

J.J. Bae / Appl. Math. Comput. 148 (2004) 207–223

Hence from (4.11) and (4.26) we get   2 EðtÞ 6 l1 eðtÞ 6 3eð0Þl1 exp  C2
t ; 3 This concludes the proof of Theorem 2.1.

223

t P t0 :




Acknowledgement The work was supported by the Korea Research Foundation Grant.

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