Uniform decay of solutions for a quasilinear system of viscoelastic equations

Nonlinear Analysis 71 (2009) 2257–2267

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Uniform decay of solutions for a quasilinear system of viscoelastic equationsI Wenjun Liu ∗ College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China Department of Mathematics, Southeast University, Nanjing 210096, China

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a b s t r a c t

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Article history: Received 31 May 2008 Accepted 12 January 2009

In this paper, we consider a system of two coupled quasilinear viscoelastic equations in canonical form with Dirichlet boundary condition. We use the perturbed energy method to show that the dissipations given by the viscoelastic terms are strong enough to ensure uniform decay (with exponential and polynomial rates) of the solutions energy, which extends some existing results for a single equation to the case of a coupled system. © 2009 Elsevier Ltd. All rights reserved.

MSC: 35B37 35L55 93D15 93D20 Keywords: Uniform decay Coupled viscoelastic equations Exponential decay Polynomial decay Relaxation function

1. Introduction In this paper, we consider the following coupled system of quasilinear viscoelastic equations in canonical form

|ut | utt − ∆u − γ1 ∆utt +

Z

|vt |ρ vtt − ∆v − γ2 ∆vtt +

Z

ρ

t

g (t − τ )∆u(x, τ )dτ + f (u, v) = 0

in Ω × (0, ∞),

h(t − τ )∆v(x, τ )dτ + k(u, v) = 0

in Ω × (0, ∞),

0 t 0

u=v=0

(1.1)

on ∂ Ω × (0, ∞),

u(x, 0) = u0 (x), v(x, 0) = v0 (x),

ut (x, 0) = u1 (x)

in Ω , vt (x, 0) = v1 (x) in Ω ,

where Ω is a bounded domain in Rn (n ≥ 1) with a smooth boundary ∂ Ω , γ1 , γ2 ≥ 0 are constants and ρ is a real number such that 0 < ρ ≤ 2/(n − 2) if n ≥ 3 or ρ > 0 if n = 1, 2. The functions u0 , u1, v0 and v1 are given initial data. The relaxation functions g and h are continuous functions and the nonlinearities f (u, v) and k(u, v) will be specified later. Here

I This work was supported by the National Natural Science Foundation of China (10771032), the Natural Science Foundation of Jiangsu province (BK2006088) and the Science Research Foundation of Nanjing University of Information Science and Technology. ∗ Corresponding address: College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China. E-mail address: [email protected].

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.060

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W. Liu / Nonlinear Analysis 71 (2009) 2257–2267

u and v denote the transverse displacements of waves. This problem arises in the theory of viscoelasticity and describes the interaction of two scalar fields (see [1]). The motivation of our work is due to the initial boundary problem of the scalar equation

|ut |ρ utt − ∆u − ∆utt +

t

Z

g (t − τ )∆u(x, τ )dτ − γ ∆ut = b|u|p−2 u,

in Ω × (0, ∞),

0

u = 0,

(1.2)

on ∂ Ω × (0, ∞),

u(x, 0) = u0 (x),

ut (x, 0) = u1 (x),

in Ω ,

which has been discussed by many authors, and related results concerning R t existence, asymptotic behavior and blow-up have been recently established (see [2–10]). Here, we understand −∆utt , 0 g (t − τ )∆u(x, τ )dτ , −∆ut and |u|p−2 u to be the dispersion term, the viscoelasticity dissipative term, the viscosity dissipative term and the source term, respectively. The case of ρ = 0 in the absence of the dispersion term problem (1.2) has been extensively studied, and several results concerning existence, decay and blow up have been established. For example, the following equation utt − ∆u +

t

Z

g (t − τ )∆u(τ )dτ + a(x)ut + |u|γ u = 0,

in Ω × (0, ∞)

(1.3)

0

with the same initial and boundary conditions as that of (1.2) has been considered by Cavalcanti et al. in [11], where a : Ω → R+ is a function that may be null on a part of Ω . Under the condition that a(x) ≥ a0 > 0 on ω ⊂ Ω , with ω satisfying some geometry restrictions and

−ξ1 g (t ) ≤ g 0 (t ) ≤ −ξ2 g (t ),

t ≥ 0,

such that kg kL1 ((0,∞)) is small enough, the authors obtained an exponential rate of decay. This work extended the result of Zuazua [12], in which he considered (1.3) with g = 0 and the localized linear damping. Cavalcanti et al. [13] considered the equation utt − k0 ∆u +

t

Z

div[a(x)g (t − τ )∇ u(τ )]dτ + b(x)h(ut ) + f (u) = 0,

in Ω × (0, ∞),

0

with the same initial and boundary conditions as that of (1.2). Under the similar conditions on the relaxation function g as above, and a(x) + b(x) ≥ δ > 0 for all x ∈ Ω , they improved the results of [11] by establishing exponential stability for exponential decay function g and linear function h, and polynomial stability for polynomial decay function g and nonlinear function h, respectively. We should also mention that the reference [14] was the first paper in the literature that took into account the frictional versus viscoelastic effects, in which the energy decay rates for solutions of the wave equation with boundary dissipation of the memory type were given. The case of ρ > 0 in the absence of the source term (b = 0) (i.e. the equation

|ut |ρ utt − ∆u − ∆utt +

t

Z

g (t − τ )∆u(τ )dτ − γ ∆ut = 0, 0

with the same initial and boundary conditions as that of (1.2)) was studied by Cavalcanti et al. in [1], where a global existence result for γ ≥ 0 as well as an exponential decay for γ > 0 were established. These results have been extended by Messaoudi and Tatar [15] to a situation where a source term is competing with the dissipation terms induced by both the viscoelasticity and the viscosity. There, the authors combined well known methods with perturbation techniques to show that a solution with positive but small energy exist globally and decays to the rest state exponentially. Recently, Messaoudi and Tatar [16] studied problem (1.2) for the case of γ = 0, in which the source term competes with only the viscoelastic dissipation induced by the memory term. They showed that there exists an appropriate set S (called the stable set) such that if the initial data are in S the solution continues to live there forever. They also showed that the solution goes to zero with an exponential or polynomial rate, depending on the decay rate of the relaxation function. Our work is also motivated by the research of the well-known Klein–Gordon system utt − ∆u + m1 u + k1 uv 2 = 0,

vtt − ∆v + m2 v + k2 u2 v = 0,

(1.4)

which arises in the study of quantum field theory [17]. See also [18,19] for some generalizations of this system and references therein. As far as we know, the problem (1.1) with the viscoelastic effect described by the memory terms has not been well studied. In [20], Andrade and Mognon proved the well-posedness for the problem (1.1) in the case of ρ = 0 and γi = 0 (i = 1, 2) with the nonlinearities f (u, v) = |u|β−2 u|v|β ,

k(u, v) = |v|β−2 v|u|β .

More recently, Messaoudi and Tatar [21] obtained uniform decay results under weaker conditions than those in [20], and more general coupling functions f and k. For results of the same nature, we refer the reader to Santos [22], Sun and Wang [23,24].

W. Liu / Nonlinear Analysis 71 (2009) 2257–2267

2259

Motivated by the above research, we consider in the present work the coupled system (1.1) with only the viscoelastic dissipations induced by the memory terms. We use the perturbed energy method to show that the dissipations given by the viscoelastic terms are strong enough to ensure exponential or polynomial decay of the solutions energy, depending on the decay rate of the relaxation functions (see Theorem 3.7). For simplicity, we set γi = 1 (i = 1, 2) in this paper. Another coupled system with only the viscosity dissipations, i.e.

|ut |ρ utt − ∆u − ∆utt − ∆ut + f (u, v) = 0, in Ω × (0, ∞), |vt |ρ vtt − ∆v − ∆vtt − ∆vt + k(u, v) = 0, in Ω × (0, ∞), u = v = 0, on ∂ Ω × (0, ∞), u(x, 0) = u0 (x), ut (x, 0) = u1 (x), in Ω , v(x, 0) = v0 (x), vt (x, 0) = v1 (x), in Ω ,

(1.5)

was investigated recently in [25], where the exponential decay result of the solutions energy was obtained by using the multiplier method. This paper is organized as follows. In Section 2 we present some assumptions and definitions needed for our work. Section 3 is devoted to the proof of the uniform decay result. 2. Preliminaries In this section we present some assumptions and definitions needed in the proof of our main result. Firstly, we make the following assumptions as in [21]. (G1) g (t ), h(t ) : [0, ∞) → (0, ∞) are C 1 functions such that ∞

Z

g (s)ds = l1 > 0,

1−

∞

Z

h(s)ds = l2 > 0.

1−

0

0

(G2) There exist two positive constants ξ1 and ξ2 such that g 0 (t ) ≤ −ξ1 g p (t ),

t ≥ 0, 1 ≤ p < 3/2,

h (t ) ≤ −ξ2 h (t ),

t ≥ 0, 1 ≤ q < 3/2.

0

q

(G3) There exists a function F (u, v) ≥ 0 such that

∂F ∂F = f (u, v), = k(u, v). ∂u ∂v uf (u, v) + v k(u, v) − F (u, v) ≥ 0. And there exists a constant d > 0 such that

 |f (ξ , ς )| ≤ d min (|ξ |β1 + |ς |β2 ), |ξ |β−1 |ς |β ,  |k(ξ , ς )| ≤ d min (|ξ |β3 + |ς |β4 ), |ξ |β |ς |β−1 ,

∀ (ξ , ς ) ∈ R2 , ∀ (ξ , ς ) ∈ R2 ,

where 1 ≤ βi ≤

n n−2

,

i = 1, 2, 3, 4,

β > 1 if n = 1, 2;

1<β ≤

n−1 n−2

if n ≥ 3.

Remark 2.1. (G1) is necessary to guarantee R∞ R ∞ the hyperbolicity of the system (1.1). Conditions p < 3/2 and q < 3/2 are imposed so that 0 g 2−p (s)ds < ∞ and 0 h2−q (s)ds < ∞. Remark 2.2. As an example of functions satisfying (G3), we have (see also [21]) f (u, v) = |u|β−2 u|v|β

and

k(u, v) = |v|β−2 v|u|β ,

where β > 1 if n = 1, 2 and 1 < β ≤ (n − 1)/(n − 2) if n ≥ 3. We use the standard Lebesgue space Lp (Ω ) and the Sobolev space H01 (Ω ) with their usual scalar product and norms. We will use the embedding H01 (Ω ) ,→ Ls (Ω ) for 2 ≤ s ≤ 2n/(n − 2) if n ≥ 3 or s ≥ 2 if n = 1, 2. The same embedding constant, denoted by C∗ , will be used; i.e.,

kφks ≤ C∗ k∇φk2 .

(2.1)

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W. Liu / Nonlinear Analysis 71 (2009) 2257–2267

We introduce the generalized energy functional E (t ) := E (u(t )) =

1

ρ+2

2

0

1

ρ+2

1

1

1

1

kut kρ+2 + kvt kρ+2 + k∇ ut k22 + k∇vt k22 + (g ◦ ∇ u) + (h ◦ ∇v) ρ+2 ρ+2 2 2 2 2 Z Z t Z t     1 1 F (u, v)dx, g (s)ds k∇ uk22 + h(s)ds k∇vk22 + + 1− 1− 2

(2.2)

Ω

0

where the product ◦ is defined by

(as ◦ w)(t ) =

t

Z

as (t − τ )kw(t ) − w(τ )k22 dτ

(2.3)

0

for any locally integrable function a : R+ → R+ and w ∈ L∞ (0, T ; L2 (Ω )). It is easy to prove that Lemma 2.3. The modified energy functional satisfies, along solutions of (1.1), E 0 (t ) =

≤

1 2 1 2

1

1

1

2 1

2

2

(g 0 ◦ ∇ u)(t ) − g (t )k∇ u(t )k22 + (h0 ◦ ∇v)(t ) − h(t )k∇v(t )k22 (g 0 ◦ ∇ u)(t ) + (h0 ◦ ∇v)(t ) ≤ 0.

(2.4)

2

Remark 2.4. It follows from Lemma 2.3 that the energy is uniformly bounded (by E (0)) and decreasing in t, which implies that

k∇ ut k22 + k∇vt k22 + l1 k∇ uk22 + l2 k∇vk22 ≤ 2E (0).

(2.5)

3. Decay of the solutions energy In this section we shall prove the exponential or polynomial decay of the solutions energy depending on the decay rate of the relaxation function. The existence of a global solution will be discussed in another paper [26]. We use the perturbed energy method introduced by Cavalcanti et al. [14,1,11], Messaouli and Tatar [15,16,21] coupled with some technical lemmas and some technical ideas. To be precise, we define the functional L(t ) := ME (t ) + ε Ψ (t ) + χ (t ),

(3.1)

where ε and M are positive constants and 1

Ψ (t ) =

Z

1

|ut |ρ ut udx +

Z

|vt |ρ vt v dx +

Z

Z ∇ ut · ∇ udx +

ρ+1 Ω ρ+1 Ω Ω Z t Z  |ut |ρ ut χ (t ) = ∆ut − g (t − τ )(u(t ) − u(τ ))dτ dx ρ + 1 0 Z ΩZ  t |vt |ρ vt h(t − τ )(v(t ) − v(τ ))dτ dx. + ∆vt − ρ+1 0 Ω

Ω

∇vt · ∇v dx,

(3.2)

(3.3)

Remark 3.1. This functional was first introduced in [21] for the case of ρ = 0 and in the absence of the dispersion terms as far as (1.1) is concerned. Lemma 3.2. Assume that (G1) holds and let (u, v) be the solution of system (1.1), and we have t

Z Z Ω

g (t − τ )(u(t ) − u(τ ))dτ

dx ≤

C∗ρ+2 (1

ρ+1

− l1 )



t

h(t − τ )(v(t ) − v(τ ))dτ

ρ+2

0

dx ≤ C∗ρ+2 (1 − l2 )ρ+1

4E (0)

 ρ2

l1

0

Z Z Ω

ρ+2



4E (0) l2

 ρ2

(g ◦ ∇ u)(t ), (h ◦ ∇v)(t ),

where C∗ is the constant in (2.1). Proof. By noting that t

Z

g (t − s)(u(t ) − u(τ ))dτ = 0

t

Z 0

[g (t − τ )](ρ+1)/(ρ+2) [g (t − τ )]1/(ρ+2) (u(t ) − u(τ ))dτ

W. Liu / Nonlinear Analysis 71 (2009) 2257–2267

2261

and using Hölder’s inequality as well as (2.5), we get t

Z Z Ω

g (t − τ )(u(t ) − u(τ ))dτ

ρ+2

∞

Z

g (s)ds

dx ≤

ρ+1 Z

0

0

≤

t

g (t − τ ) 0

C∗ρ+2 (1

ρ+1

− l1 )

≤ The other inequality can be proved similarly.

ρ+1

− l1 )

Ω

|u(t ) − u(τ )|ρ+2 dxdτ

t

Z

ρ+2

g (t − τ )k∇ u(t ) − ∇ u(τ )k2 0

C∗ρ+2 (1

Z



4E (0)

 ρ2

l1

dτ

(a ◦ ∇ u)(t ).



Lemma 3.3. For ε small enough while M large enough, the relation

α1 L(t ) ≤ E (t ) ≤ α2 L(t )

(3.4)

holds for two positive constants α1 and α2 . Proof. By using Young’s inequality, Sobolev embedding theorem and (2.5), one can easily deduce that

Z 1 1 ρ+2 ρ ≤ 1 kut kρ+2 + | u | u udx kukρ+2 t t ρ+2 ρ + 1 ρ + 2 (ρ + 1 )(ρ + 2 ) Ω ≤ ≤

1

ρ+2 1

ρ+2

ρ+2

ρ+2

kut kρ+2 + ρ+2 kut kρ+2

C∗

(ρ + 1)(ρ + 2) ρ+2

+

C∗

(ρ + 1)(ρ + 2)

ρ+2

k∇ uk2 

2E (0) l1

 ρ2

k∇ uk22

and

Z Z t 1 ρ |ut | ut g (t − τ )(u(t ) − u(τ ))dτ dx ρ + 1 Ω 0 ρ+2 Z Z t 1 1 ρ+2 kut kρ+2 + g (t − τ )(u(t ) − u(τ ))dτ dx ≤ ρ+1 (ρ + 1)(ρ + 2) Ω 0  ρ ρ+2 1 C∗ 4E (0) 2 ρ+2 ≤ kut kρ+2 + (1 − l1 )ρ+1 (g ◦ ∇ u)(t ). ρ+1 (ρ + 1)(ρ + 2) l1 Therefore,

" #   ρ ρ+2  ε C∗ 2E (0) 2 1 ρ+2 ρ+2 L(t ) ≤ ME (t ) + + kut kρ+2 + kvt kρ+2 + ε k∇ uk22 + ρ+1 ρ+2 (ρ + 1)(ρ + 2) l1 2 " #  ρ ρ+2
1 ε+1 C∗ 2E (0) 2 +ε + k∇vk22 + k∇ ut k22 + k∇vt k22 (ρ + 1)(ρ + 2) l2 2 2 "   ρ2 # ρ+2 1 − l1 C∗ 4E (0) + + (1 − l1 )ρ+1 (g ◦ ∇ u)(t ) 2 (ρ + 1)(ρ + 2) l1 "   ρ2 # ρ+2 1 − l2 C∗ ρ+1 4E (0) + + (1 − l2 ) (h ◦ ∇v)(t ) 2 (ρ + 1)(ρ + 2) l2 

≤

1

α1

1

E (t )

and





ε + ρ+1 ρ+2 "

   ρ+2 ρ+2 L( t ) ≥ − kut kρ+2 + kvt kρ+2 ρ+2 ( #)  ρ ρ+2 M C∗ 2E (0) 2 1 + l1 − ε + k∇ uk22 2 (ρ + 1)(ρ + 2) l1 2 M

1

2262

W. Liu / Nonlinear Analysis 71 (2009) 2257–2267

( +

2

( +

M 2

( +

M 2

ρ+2

l2 − ε

" −

−

1 − l1

1 − l2 2

α2

 ρ2 +

l2

ρ+2

+

C∗

(ρ + 1)(ρ + 2)

+

(ρ + 1)(ρ + 2) 2 t 2

k∇ k + k∇v k




1

#) k∇vk22

2

ρ+1

(1 − l1 )

ρ+2

C∗

ut 22

2

1

2E (0)

(ρ + 1)(ρ + 2) 2

"



C∗

M − (ε + 1)

+ ≥

M

"

(1 − l2 )

+ Ω

4E (0)

 ρ2 #)

l1

ρ+1

Z





4E (0)

 ρ2 #)

l2

(g ◦ ∇ u)(t ) (h ◦ ∇v)(t )

F (u, v)dx

E (t )

for ε small enough while M large enough.



Lemma 3.4. Under the assumptions (G1)–(G3), the functional

Ψ (t ) =

Z

1

ρ+1

Ω

Z

1

|ut |ρ ut udx +

ρ+1

|vt |ρ vt v dx +

Ω

Z

Z Ω

∇ ut · ∇ udx +

∇vt · ∇v dx

Ω

satisfies, along solutions of (1.1), l1

Ψ (t ) ≤ − k∇ uk + 0

2 2

2

−

l2 2

t

Z

1

g

2l1

0

1

Z

k∇vk22 +

2l2

2 −p



(s)ds (g p ◦ ∇ u)(t ) + k∇ ut k22 +

t

1



h2−q (s)ds (hq ◦ ∇v)(t ) + k∇vt k22 + 0

ρ+2

kut kρ+2

ρ+1 1

ρ+2

ρ+1

Z

kvt kρ+2 −

Ω

F (u, v)dx.

(3.5)

Proof. Taking derivative of Ψ (t ), it follows from (1.1) that

Ψ (t ) = 0

1

ρ+2 kut kρ+2

Z

ρ

|ut | utt udx + k∇ k + ∇ u · ∇ utt dx Ω Z Z 1 ρ+2 + kvt kρ+2 + |vt |ρ vtt v dx + k∇vt k22 + ∇v · ∇vtt dx ρ+1 Ω Ω Z Z t 1 ρ+2 = kut kρ+2 − k∇ uk22 + ∇ u(t ) · g (t − τ )∇ u(τ )dτ dx + k∇ ut k22 ρ+1 Ω 0 Z Z t 1 ρ+2 2 kvt kρ+2 − k∇vk2 + ∇v(t ) · h(t − τ )∇v(τ )dτ dx + k∇vt k22 + ρ+1 Ω 0 Z Z − uf (u, v)dx − v k(u, v)dx. ρ+1

+

Z

ut 22

Ω

Ω

(3.6)

Ω

We now estimate the third term in the right side of (3.6) as follows (see also [21]):

Z Ω

∇ u( t ) ·

t

Z

g (t − τ )∇ u(τ )dτ dx ≤ 0

1 2

Thanks to Young’s inequality and the fact that

Z Ω

∇ u( t ) ·

≤

g (t − τ )∇ u(τ )dτ dx ≤ 0

+ ≤

t

Z

1 2

1 2

 1+

1

 Z Z

η1

Ω

k∇ uk22 +

Rt

1 2

0

2 − l1 2

2

Ω

R∞ 0

g (t − τ )(|∇ u(τ ) − ∇ u(t )| + |∇ u(t )|)dτ

dx.

(3.7)

g (τ )dτ = 1 − l we obtain, for η1 = l1 /(1 − l1 ) > 0,

k∇ uk + (1 + η1 ) 2

g (t − τ )|∇ u(τ ) − ∇ u(t )|dτ

2

0

1

2 2

t

t

Z Z Ω

g (t − τ )|∇ u(t )|dτ

2 dx

0

2 dx

0

1

2

2

k∇ u(t )k22 +

t

Z Z

g (τ )dτ ≤

k∇ uk + (1 + η1 )(1 − l1 ) k∇ u(t )k + 2 2

1

1 2l1

t

Z

2 2

1 2

 1+

1

η1

g

2−p



(s)ds (g p ◦ ∇ u)(t )

0



g 2−p (s)ds (g p ◦ ∇ u)(t ). 0

t

 Z

(3.8)

W. Liu / Nonlinear Analysis 71 (2009) 2257–2267

2263

Similar calculations also yield, for η2 = l2 /(1 − l2 ) > 0,

Z Ω

∇v(t ) ·

t

Z

h(t − τ )∇v(τ )dτ dx ≤

2 − l2 2

0

k∇v(t )k + 2 2

h

2l2

By combining (3.6)–(3.9) and (G3), estimate (3.5) is established.

t

Z

1

2 −q



(s)ds (hq ◦ ∇v)(t ).

(3.9)

0



Lemma 3.5. Under the assumptions (G1)–(G3), the functional

Z 

χ1 ( t ) =

Ω

|ut |ρ ut ρ+1

∆ut −

t

Z

g (t − τ )(u(t ) − u(τ ))dτ dx 0

satisfies, along solutions of (1.1) and for δ > 0,

( 2β1

χ1 (t ) ≤ δ 1 + 2(1 − l1 ) + 2d C∗ 0

2

2β2

+ 2 δ d C∗ 2



2

2E (0)



β2 −1

2E (0) l1 2 2

k∇vk −

l2

β1 −1 )

g (0) 4δ



k∇ uk + 2δ + 2 2

C∗2

 1+



ρ+1

t

 Z

3

g

4δ

2−p



(s)ds (g p ◦ ∇ u)(t )

0

(g 0 ◦ ∇ u)(t )

  Z t  Z t C 2δ 1 ρ+2 + δ + ∗ (2E (0))ρ − g (s)ds k∇ ut k22 − g (s)ds kut kρ+2 . ρ+1 ρ+1 0 0

(3.10)

Proof. Direct computations, using (1.1), yield

χ1 ( t ) =

Z

t

Z

ρ

Z

t

Z 

g (t − τ )(u(t ) − u(τ ))dτ dx + g (s)ds (∆utt − |ut | utt ) 0 0 Ω  Z  Z t |ut |ρ ut + ∆ ut − g 0 (t − τ )(u(t ) − u(τ ))dτ dx ρ+1 Ω 0  Z t Z g (t − τ )(∇ u(t ) − ∇ u(τ ))dτ dx = ∇ u( t ) · 0 Ω  Z t  Z Z t − g (t − τ )∇ u(τ )dτ · g (t − τ )(∇ u(t ) − ∇ u(τ ))dτ dx Ω 0 0 Z t  Z Z t + f (u, v) g (t − τ )(u(t ) − u(τ ))dτ dx − g (s)ds k∇ ut k22 Ω 0 0 Z Z t − ∇ ut · g 0 (t − τ )(∇ u(t ) − ∇ u(τ ))dτ dx

0

Ω

Ω

−

|ut |ρ ut ∆ ut − ρ+1

 ut dx

0

Z

1

ρ+1

Ω

|ut |ρ ut

t

Z

g 0 (t − τ )(u(t ) − u(τ ))dτ dx − 0

t

Z

1

ρ+1

0



ρ+2

g (s)ds kut kρ+2 .

(3.11)

Similarly to (3.6), we can estimate the right-side of (3.11). In fact, by applying Young’s inequality we get that, for any δ > 0,

Z Ω

∇ u(t ) ·

t

Z

g (t − τ )(∇ u(t ) − ∇ u(τ ))dτ



dx ≤ δk∇ uk22 +

0

1

t

Z

4δ



g 2−p (s)ds (g p ◦ ∇ u)(t ).

(3.12)

0

For the second term, by use of Young’s inequality we obtain t

Z Z Ω

g (t − τ )∇ u(τ )dτ 0

≤δ

0 t

Z Z



Ω

≤ 2δ +

 Z t  · g (t − τ )(∇ u(t ) − ∇ u(τ ))dτ dx

g (t − τ )(|∇ u(τ ) − ∇ u(t )| + |∇ u(t )|)dτ

2 dx +

0

1

 Z Z

4δ

Ω

t

g (t − τ )(|∇ u(t ) − ∇ u(τ )|)dτ

2

2 Z Z t g (t − τ )(|∇ u(t ) − ∇ u(τ )|)dτ dx 4δ Ω 0 1

dx + 2δ(1 − l1 )2 k∇ uk22

0

  Z t  1 ≤ 2δ + g 2−p (s)ds (g p ◦ ∇ u)(t ) + 2δ(1 − l1 )2 k∇ uk22 . 4δ 0

(3.13)

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W. Liu / Nonlinear Analysis 71 (2009) 2257–2267

For the third term, it follows from (G3) and (2.5) that [21]

Z Ω

t

Z

f (u, v)

g (t − τ )(u(t ) − u(τ ))dτ dx 0

2 β1

≤ 2δ d {C∗ k∇ uk 2

( ≤ 2δ d



2β1

2

C∗

2β1 2

2 β2

+ C∗ k∇vk

2E (0)

β1 −1

2 β2 2

}+

1

g

4δ

2β2

2 2

2−p



(s)ds (g p ◦ ∇ u)(t )

0



k∇ uk + C∗

l1

t

Z

2E (0)

β2 −1

l2

) k∇vk

2 2

+

t

Z

1 4δ

g

2−p



(s)ds (g p ◦ ∇ u)(t ).

(3.14)

0

The fifth term can be handled by t

Z

Z Ω

g 0 (t − τ )(∇ u(t ) − ∇ u(τ ))dτ dx ≤ δk∇ ut k22 +

∇ ut ·

g (0)

0

4δ

(−g 0 ◦ ∇ u)(t ).

(3.15)

By using Young’s inequality, Sobolev embedding theorem and (2.5), one can estimate the sixth term as 1

Z

ρ+1

Ω

g 0 (t − τ )(u(t ) − u(τ ))dτ dx

|ut | ut 0

δ ≤ ρ+1 ≤δ

t

Z

ρ

Z

C∗2

ρ+1

2(ρ+1)

| ut |

Ω

dx +

4δ(ρ + 1)

(2E (0))ρ k∇ ut k22 −

t

Z Z

1

Ω

g (0) 4δ(ρ + 1)

C∗2

g (t − τ )(u(t ) − (τ ))dτ 0

2 dx

0 t

Z Z Ω

g 0 (t − τ )|∇ u(t ) − ∇ u(τ )|2 dτ dx.

(3.16)

0

By combining the estimates (3.12)–(3.16) and (3.11), the assertion of the lemma is established.



Similar computations also yield the following. Lemma 3.6. Under the assumptions (G1)–(G3), the functional

χ2 ( t ) =

Z t Z  |vt |ρ vt ∆vt − h(t − τ )(v(t ) − v(τ ))dτ dx ρ+1 Ω 0

satisfies, along solutions of (1.1) and for δ > 0,

(

2E (0)

β4 −1 )

 Z t   3 k∇vk22 + 2δ + h2−q (s)ds (hq ◦ ∇v)(t ) l2 4δ 0  β3 −1   2 2E (0) C∗ h(0) + 2δ d2 C∗2β3 k∇ uk22 − 1+ (h0 ◦ ∇v)(t ) l1 4δ ρ+1   Z t  Z t C 2δ 1 ρ+2 + δ + ∗ (2E (0))ρ − h(s)ds k∇vt k22 − h(s)ds kvt kρ+2 . ρ+1 ρ+1 0 0 2β4

χ2 (t ) ≤ δ 1 + 2(1 − l2 ) + 2d C∗ 0

2

2



(3.17)

Now, we are ready to state and prove the uniform decay result. Theorem 3.7. Let (u0 , u1 ), (v0 , v1 ) ∈ H01 (Ω ) × H01 (Ω ) be given. Assume that (G1)–(G3) hold. Then, for each t0 > 0, there exist strictly positive constants K and κ such that the solution of (1.1) satisfies, for all t ≥ t0 , E (t ) ≤ K e−κ t ,

p = q = 1,

E (t ) ≤ K (1 + t )−1/(m−1) ,

(p, q) 6= (1, 1),

where m = max{p, q}. Proof. Since g and h are positive, we have that, for any t0 > 0 t

Z

g (s)ds ≥

Z

0

h(s)ds ≥ 0

g (s)ds ≥ a0 > 0,

t ≥ t0 ,

h(s)ds ≥ a0 > 0,

t ≥ t0 .

0 t

Z

t0

t0

Z 0

W. Liu / Nonlinear Analysis 71 (2009) 2257–2267

2265

By using (3.1), (3.5), (3.10) and (3.17), Lemma 2.3 and selecting M large enough so that M 2

−

g (0)

C∗2

 1+

4δ



ρ+1

M

> 0,

−

2

h(0)

C∗2

 1+

4δ



ρ+1

> 0,

we obtain, for t ≥ t0 , L (t ) ≤ − 0

−

a0 − ε 

ρ+1 ( ε l1 2

( −

ε l2 2

ρ+2 kut kρ+2

+

ρ+2 kvt kρ+2



−ε

Z Ω

F (u, v)dx

" 2β1

− δ 1 + 2(1 − l1 ) + 2d C∗ 2

2

2β4

− δ 1 + 2(1 − l2 ) + 2d C∗ g (0)

2

 β 1 −1

2E (0)



2

2 β3



+ 2d C∗

l1

" 2

2E (0)



 β 4 −1

2

2 β2

β3 −1 #)

l1



+ 2d C∗

l2

2E (0) 2E (0)

β2 −1 #)

l2

k∇ uk22 k∇vk22

ε

   Z t 3 g 2−p (s)ds (g p ◦ ∇ u)(t ) + 2δ + 2 4δ ρ+1 2l1 4δ 0      Z t   h(0) C∗2 ε 3 M 2 −q − 1+ ξ2 − + 2δ + h (s)ds (hq ◦ ∇v)(t ) − 2 4δ ρ+1 2l2 4δ 0   
C∗2 − (a0 − ε) − δ 1 + (2E (0))ρ k∇ ut k22 + k∇vt k22 . ρ+1 

−

M

−

C∗2



1+



ξ1 −



(3.18)

At this point, we choose ε so small that ε < a0 and (3.4) holds. Once ε is fixed, we select δ small enough satisfies

ε l1 2

ε l2 2

" 2 β1

− δ 1 + 2(1 − l1 ) + 2d C∗ 2

2

2 β4

− δ 1 + 2(1 − l2 ) + 2d C∗

 (a0 − ε) − δ 1 +

C∗2

ρ+1

2

2E (0)

 β 1 −1



2E (0)

2

2β3



+ 2d C∗

l1

" 2



 β 4 −1

l2

2

2β2

+ 2d C∗

2E (0)

β3 −1 #

l1



2E (0)

β2 −1 #

l2

> 0, > 0,

 (2E (0))ρ > 0.

As long as ε and δ are fixed, we choose M large enough such that (3.4) holds and g (0)

ε

  Z ∞  3 + 2δ + g 2−p (s)ds > 0, 2 4δ ρ+1 2l1 4δ 0       Z ∞  2 M h(0) C∗ ε 3 − 1+ ξ2 − + 2δ + h2−q (s)ds > 0. 2 4δ ρ+1 2l2 4δ 0 

M

−

C∗2



1+



ξ1 −



Thus, for all t ≥ t0 , we have, for some β 0 > 0, L (t ) ≤ −β 0

0



ρ+2 kut kρ+2

+

ρ+2 kvt kρ+2

Z + Ω

F (u, v)dx + k∇ uk22 + k∇vk22

 + (g p ◦ ∇ u)(t ) + (hq ◦ ∇v)(t ) + k∇ ut k22 + k∇vt k22 .

(3.19)

Case 1: p = q = 1. By virtue of the choice of ε , δ and M, estimate (3.19) yields, for some constant α > 0, L0 (t ) ≤ −α E (t ),

∀t ≥ t0 .

(3.20)

Hence, with the help of the left hand side inequality in (3.4) and (3.20), we find L0 (t ) ≤ −αα1 L(t ),

∀t ≥ t0 .

(3.21)

A simple integration of (3.21) over (t0 , t ) leads to L(t ) ≤ L(t0 )eαα1 t0 e−αα1 t ,

∀t ≥ t0 .

(3.22)

Thus (3.4) and (3.21) yield E (t ) ≤ α2 L(t0 )eαα1 t0 e−αα1 t = K e−κ t ,

∀t ≥ t 0 .

2266

W. Liu / Nonlinear Analysis 71 (2009) 2257–2267

Case 2: (p, q) 6= (1, 1). Without loss of generality, suppose that p ≥ q. Similar to the discussion in [21], we can obtain

(g p ◦ ∇ u) ≥ C1 (g ◦ ∇ u)p ,

(hq ◦ ∇v) ≥ C1 (h ◦ ∇v)q ,

(3.23)

for some constant C1 > 0. Consequently, a combination of (3.19) and (3.23) yields



ρ+2

ρ+2

L0 (t ) ≤ −C2 kut kρ+2 + kvt kρ+2 +

Z Ω

F (u, v)dx + k∇ uk22 + k∇vk22

+ (g ◦ ∇ u) + (h ◦ ∇v) + k∇ k + k∇v k p

q

ut 22

2 t 2



,

∀t ≥ t0 ,

(3.24)

for some constant C2 > 0. On the other hand, since E (t ) is uniformly bounded (by E (0)), we have





ρ+2

ρ+2

E p (t ) ≤ C3 E p−1 (0) kut kρ+2 + kvt kρ+2 +

Z Ω

F (u, v)dx + k∇ uk22 + k∇vk22 + k∇ ut k22 + k∇vt k22

 p q + (g ◦ ∇ u) + (h ◦ ∇v) ,



(3.25)

for all t ≥ t0 and some constant C3 > 0. By using the facts that p ≥ q and a combination of the last two inequalities and (3.4), we obtain L0 (t ) ≤ −C4 Lp (t ),

∀t ≥ t0 ,

(3.26)

for some constant C4 > 0. A simple integration of (3.26) over (t0 , t ) gives L(t ) ≤ K (1 + t )−1/(p−1) ,

∀t ≥ t0 .

By using (3.4) again, the proof is completed.

(3.27) 

Acknowledgment The author would like to express sincere gratitude to Professor Mingxin Wang for his enthusiastic guidance and constant encouragement. Thanks are also due to the anonymous referee for his/her constructive suggestions. References [1] M.M. Cavalcanti, V.N. Domingos Cavalcanti, J. Ferreira, Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Methods. Appl. Sci. 24 (2001) 1043–1053. [2] F. Alabau-Boussouira, P. Cannarsa, D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal. 254 (5) (2008) 1342–1372. [3] M.M. Cavalcanti, V.N. Domingos Cavalcanti, P. Martinez, General decay rate estimates for viscoelastic dissipative systems, Nonlinear Anal. 68 (1) (2008) 177–193. [4] V. Georgiev, G. Todorova, Existence of solutions of the wave equation with nonlinear damping and source terms, J. Differential Equations 109 (1994) 295–308. [5] X.S. Han, M.X. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.011. [6] A. Haraux, E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Ration. Mech. Anal. 150 (1988) 191–206. [7] H.A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form Putt = Au + F (u), Trans. Amer. Math. Soc. 192 (1974) 1–21. [8] W.J. Liu, The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity, J. Math. Pures Appl. 37 (1998) 355–386. [9] S.A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr. 260 (2003) 58–66. [10] E. Vitillaro, Global nonexistence theorems for a class of evolution equations with dissipation, Arch. Ration. Mech. Anal. 149 (1999) 155–182. [11] M.M. Cavalcanti, V.N. Domingos cavalcanti, J.A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations 2002 (44) (2002) 1–14. [12] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Parital Differential Equations 15 (1990) 205–235. [13] M.M. Cavalcanti, H.P. Oquendo, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim. 42 (4) (2003) 1310–1324. [14] M. Aassila, M.M. Cavalcanti, J.A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim. 38 (5) (2000) 1581–1602. [15] S.A. Messaoudi, N.-E Tatar, Global existence asymptotic behavior for a Nonlinear Viscoelastic Problem, Math. Methods Sci. Res. J. 7 (4) (2003) 136–149. [16] S.A. Messaoudi, N.-E Tatar, Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Methods Appl. Sci. 30 (2007) 665–680. [17] I.E. Segal, The global Cauchy problem for relativistic scalar field with power interactions, Bull. Soc. Math. France 91 (1963) 129–135. [18] L.A. Medeiros, M. Milla Miranda, Weak solutions for a system of nonlinear Klein–Gordon equations, Annal. Matem. Pura Appl. CXLVI (1987) 173–183. [19] J. Zhang, On the standing wave in coupled non-linear Klein–Gordon equations, Math. Methods Appl. Sci. 26 (2003) 11–25. [20] D. Andrade, A. Mognon, Global solutions for a system of Klein–Gordon equations with memory, Bol. Soc. Parana. Mat. Ser. 3 21 (1/2) (2003) 127–136. [21] S.A. Messaoudi, N.-E Tatar, Uniform stabilization of solutions of a nonlinear system of viscoelastic equations, Appl. Anal. 87 (3) (2008) 247–263. [22] M.L. Santos, Decay rates for solutions of a system of wave equations with memory, Electron. J. Differential Equations 2002 (38) (2002) 1–17.

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