Nonlinear Analysis 69 (2008) 3802–3810www.elsevier.com/locate/na

Global attractor for plate equation with nonlinear dampingI

Lu Yang∗, Cheng-Kui Zhong

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China

Received 19 March 2007; accepted 11 October 2007

Abstract

In this paper, we study the long-time behavior of plate equation with nonlinear damping and critical nonlinearity. We prove theexistence of a global attractor in the space H2

0 (Ω)× L2(Ω).c© 2007 Elsevier Ltd. All rights reserved.

MSC: 35L70; 35B40; 35B41

Keywords: Global attractor; Nonlinear damping; Critical exponent

1. Introduction

In this paper, we consider the following plate equation with nonlinear damping and critical nonlinearity:

ut t + a(x)g(ut )+∆2u + λu + f (u) = h(x), x ∈ Ω , (1.1)

with boundary and initial conditions of the form

u|∂Ω =∂

∂νu|∂Ω = 0, u|t=0 = u0(x), ut |t=0 = u1(x). (1.2)

Here Ω ⊂ Rn is a bounded domain with a sufficiently smooth boundary, λ > 0, h(x) ∈ L2(Ω). The function a(x)satisfies:

a(x) ∈ L∞(Ω), a(x) ≥ α0 > 0 in Ω , (1.3)

where α0 is a constant. The function f ∈ C1(R) satisfies

| f ′(s)| ≤ C1(1+ |s|p), (1.4)

lim inf|s|→∞

f (s)

s> −λ1, (1.5)

I The project is supported by the NNSF of China (No. 10471056).∗ Corresponding author.

E-mail address: yanglu03@lzu.cn (L. Yang).

0362-546X/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2007.10.016

L. Yang, C.-K. Zhong / Nonlinear Analysis 69 (2008) 3802–3810 3803

where 0 < p < ∞ if n ≤ 4, and 0 < p ≤ 4n−4 if n > 4, λ1 is the best constant in the Poincare type inequality∫

Ω |∇u|2 ≥ λ121

∫Ω |u|

2. The damping function g ∈ C1(R) satisfies

g(0) = 0, g is strictly increasing, and lim inf|s|→∞

g′(s) > 0, (1.6)

|g(s)| ≤ C2(1+ |s|q), (1.7)

with 1 ≤ q < ∞ if n ≤ 4, and 1 ≤ q ≤ n+4n−4 if n > 4. The number 4

n−4 is called the critical exponent, since thenonlinearity f is not compact in this case, which is an essential difficulty in proving the existence of global attractor,see, for example, [2–4] for the linear damping and [5–8,12,15,16,19–21] for the nonlinear damping.

In recent years the research of wave equations have attracted considerable attention (see, e.g., [1–8,12,15,16,19–21]and references therein). In the case of linear damping, the problem has been dealt with rather successfully in [2–4].For the nonlinear damping, there is a lot of work devoted to this topic. The existence of attractors for wave equationwith pure interior damping was established in [5,7,12] and the references therein. Attractors and their structure, inthe context of wave equation with full boundary damping were obtained in [15,16]. In [8], the authors showedthe existence of global attractor for the wave equation with localized nonlinear damping a(x)g(ut ). The coupledwave-plate system with Neumann boundary conditions were considered in [6]. The wave dynamics with nonlinearinterior/boundary dissipation was studied in [19], and also the paper [20] deals with localized interior damping and asource term of critical exponent. The recent work [21] has developed a rather comprehensive theory with the focus oncritical exponents and nonlinear damping.

The plate equation arises in the nonlinear theory of oscillations. This problem has been treated in many papers.When g is linear, in [24] the existence of global attractor has been solved for (1.1) and (1.2) with damping a(x)ut andcritical nonlinearity. If a(x) ≡ α0 > 0, the existence of global attractor for plate equation was discussed in [1] for thesubcritical exponent, and [23] for the critical exponent. In the case of nonlinear damping, to our knowledge the mostgeneral treatment for plate equation is given in [21]. One of the most important plate models is von Karman equation,which describes nonlinear oscillations of a plate accounting for large displacements (see [10–14,21,22,25] for moredetails). The long-time behavior of weak solutions of von Karman equation was investigated by many authors, see,for example, [10–12,21,22,25] for the nonlinear interior damping, and [13] for the nonlinear boundary damping. Thelong-time dynamics of von Karman equation with a mixed nonlinear boundary-interior damping was studied in [14].More complicated plate models can be found in [17,18] and the references therein.

In present paper, our goal is to prove the existence of a global attractor for an n-dimensional plate equationwith nonlinear damping and critical nonlinearity. Our ideas and methods are mainly from [4,12,14,19,21,22]. Then-dimensional plate equation can be regarded as a generalization of two-dimensional plate equation in mathematics.Unlike the two-dimensional case, which has strong backgrounds in mathematical physics (see, e.g., [6,10–14,17,18,21–25]), maybe we cannot temporarily give the physical background for the n-dimensional case (n ≥ 3). However,from the mathematical viewpoint, we point out that it requires more complicated techniques to consider an n-dimensional problem compared with the two-dimensional case, and our results include the relevant ones in the two-dimensional case.

This paper is organized as follows: in Section 2, we give some preparations for our consideration; in Section 3, wegive the existence of bounded absorbing set in H2

0 (Ω)× L2(Ω); in Section 4, we derive the asymptotic compactnessfor the corresponding semigroup S(t)t≥0 generated by problems (1.1)–(1.7).

2. Preliminaries

Hereafter, the norm and scalar product in L2(Ω) are denoted by ‖ · ‖ and (, ), respectively. Let C denotes a generalpositive constant, which may be different in different estimates. It is known that under condition (1.3)–(1.7) thesolution operator S(t)(u0, u1) = (u(t), ut (t)), t ≥ 0, of problems (1.1) and (1.2) generate a C0-semigroup on theenergy space H2

0 (Ω)× L2(Ω) (see [7,24] etc).The existence and uniqueness results (in the space H2

0 (Ω)× L2(Ω)) are given in the following theorem.

Theorem 2.1 ([7,24]). Let Ω be a bounded domain of Rn with smooth boundary, under assumptions (1.3)–(1.7),then for any initial data u0 ∈ H2

0 (Ω) and u1 ∈ L2(Ω), the problems (1.1) and (1.2) have a unique global solution(u, ut ) ∈ C([0, T ]; H2

0 (Ω)× L2(Ω)) for any T > 0, and (u, ut ) depends continuously on (u0, u1).

3804 L. Yang, C.-K. Zhong / Nonlinear Analysis 69 (2008) 3802–3810

Next, we recall the simple compactness criterion stated as [14, Proposition 3.2] (see also [4], [6, Proposition5.2], [21, Chap.2], [19, Proposition 4.2]). The initial idea of the method occurred in [22], and the result stated in [14,Proposition 3.2] is an abstract version of [22, Theorem 2] that can be derived from the arguments given in [22].

Definition 2.2 ([4,14]). Let X be a Banach space and B be a bounded subset of X . We call a function φ(·, ·) whichdefined on X × X is a contractive function on B × B if for any sequence xn

∞

n=1 ⊂ B, there is a subsequencexnk

∞

k=1 ⊂ xn∞

n=1 such that

limk→∞

liml→∞

φ(xnk , xnl ) = 0.

Denote all such contractive functions on B × B by C(B).

Theorem 2.3 ([4,14]). Let S(t)t≥0 be a semigroup on a Banach space (X, ‖ · ‖) and has a bounded absorbing setB0. Moreover, assume that for any ε ≥ 0 there exist T = T (B0, ε) and φT (·, ·) ∈ C(B0) such that

‖S(T )x − S(T )y‖ ≤ ε + φT (x, y) for all x, y ∈ B0,

where φT depends on T . Then S(t)t≥0 is asymptotically compact in X, i.e., for any bounded sequence yn∞

n=1 ⊂ Xand tn with tn →∞, S(tn)yn

∞

n=1 is precompact in X.

In the following, we will recall some simple properties of the nonlinear damping function g.

Lemma 2.4 ([7,22]). Let g(·) satisfy condition (1.6). Then for any δ > 0 there exists Cδ > 0, such that

|u − v|2 ≤ δ + Cδ(g(u)− g(v))(u − v) for u, v ∈ R.

3. Absorbing set in H20 (Ω) × L2(Ω)

In this section, we prove the existence of bound absorbing set in H20 (Ω)× L2(Ω). Its proof is inspired by [7,9].

Lemma 3.1. Under assumptions (1.3)–(1.7), the semigroup S(t)t≥0 corresponding to problems (1.1) and (1.2) havea bounded absorbing set in H2

0 (Ω)× L2(Ω).

Proof. We set

E(t) =12

∫Ω(|ut |

2+ |∆u|2 + λ|u|2)dx +

∫Ω(F(u)− hu)dx .

Multiplying (1.1) by ut and integrating over Ω , we get

ddt

E(t)+∫Ω

a(x)g(ut )ut = 0, (3.1)

so from (1.3) and (1.6) we have

E(t) ≤ E(0), ∀t ≥ 0. (3.2)

It is obviously that (1.3) and (1.6) imply that: there are δ > 0 and Cδ > 0 such that

(a(x)g(ut ), ut ) ≥ 2δ‖ut‖2− Cδmes(Ω),

(a(x)g(ut )− δut , ut ) ≥ δ‖ut‖2− Cδmes(Ω), (3.3)

and from (1.5) we know that there are λ1 > λ′ > 0 and C0 such that

( f (u), u) > −λ′‖u‖2 − C0mes(Ω), (3.4)∫Ω

F(u)dx > −λ′

2‖u‖2 − C0mes(Ω). (3.5)

From (3.5), using Holder inequality and Young inequality we obtain that

−C1(mes(Ω)+ ‖h‖2) ≤12‖ut‖

2+ C‖∆u‖2 − C1(mes(Ω)+ ‖h‖2) ≤ E(t) ≤ E(0). (3.6)

L. Yang, C.-K. Zhong / Nonlinear Analysis 69 (2008) 3802–3810 3805

So from (3.1) and (3.6), we have∫ t

0

∫Ω

a(x)g(ut (s))ut (s)dxds = E(0)− E(t) ≤ E(0)+ C1(mes(Ω)+ ‖h‖2), ∀t ≥ 0. (3.7)

From (1.7) and (1.3), using Holder inequality and Young inequality we get∣∣∣∣∫Ω

a(x)g(ut )u

∣∣∣∣ ≤ C∫Ω

a(x)|u| + C

(∫Ω

a(x)g(ut )ut

) qq+1

(∫Ω

a(x)|u|q+1) 1

q+1

≤ C∫Ω

a(x)|u| + Cη‖∆u‖q−1

q

∫Ω

a(x)g(ut )ut + η‖∆u‖2, (3.8)

where η is a constant, which will be determined later.Multiplying (1.1) by v = ut + δu and integrating over Ω , where δ comes from (3.3), we get

ddt

(∫Ω

12(|v|2 + |∆u|2 + λ|u|2 − δ2

|u|2)+ F(u)− hu

)+ (a(x)g(ut )− δut , ut )+ δ(a(x)g(ut ), u)

+ δλ‖u‖2 + δ‖∆u‖2 + δ( f (u), u)− δ(h(x), u) = 0. (3.9)

Set

Eδ(t) =12

∫Ω(|v|2 + |∆u|2 + λ|u|2 − δ2

|u|2)dx +∫Ω(F(u)− hu)dx,

and

I (t) = (a(x)g(ut )− δut , ut )+ δ(a(x)g(ut ), u)+ δλ‖u‖2 + δ‖∆u‖2 + δ( f (u), u)− δ(h(x), u),

so we have

ddt

Eδ(t)+ I (t) = 0. (3.10)

From (3.5), using Holder inequality and Young inequality, we can choose δ small enough, such that

Eδ(t) ≥∫Ω

(12|v|2 + C |∆u|2 − C0

)dx − C‖h‖2

≥ C(‖v‖2 + ‖∆u‖2

)− C1(mes(Ω)+ ‖h‖2). (3.11)

Similarly, from (3.3), (3.4) and (3.8), we get

I (t) ≥ δ‖ut‖2− Cδmes(Ω)− δ

(C∫Ω

a(x)|u| + Cη‖∆u‖q−1

q

∫Ω

a(x)g(ut )ut + η‖∆u‖2)

+Cδ‖∆u‖2 − δC1(mes(Ω)+ ‖h‖2)

= δ‖ut‖2− Cδmes(Ω)− δ

(C∫Ω

a(x)|u| + Cη‖∆u‖q−1

q

∫Ω

a(x)g(ut )ut

)+Cδ

(1−

η

C

)‖∆u‖2 − δC1(mes(Ω)+ ‖h‖2)

≥ Cδ(‖ut‖2+ ‖∆u‖2)− C ′δ(mes(Ω)+ ‖h‖2)− δCη‖∆u‖

q−1q

∫Ω

a(x)g(ut )ut

≥ Cδ(‖ut‖2+ ‖∆u‖2)− C ′δ(mes(Ω)+ ‖h‖2)− CE(0)

∫Ω

a(x)g(ut )ut , (3.12)

where we can choose η so small that 1 − ηC > 0. CE(0) is a constant which depends on δ, Cη and E(0), here E(0)

depends only on the initial data, C ′δ is a constant depending on δ and C1 but independent of the initial data.

3806 L. Yang, C.-K. Zhong / Nonlinear Analysis 69 (2008) 3802–3810

Integrating (3.10), from (3.7), (3.11) and (3.12) we have

C(‖v‖2 + ‖∆u‖2

)− C1(mes(Ω)+ ‖h‖2)− Eδ(0)− CE(0)(E(0)+ C1(mes(Ω)+ ‖h‖2))

≤ −

∫ t

0

(Cδ(‖ut (s)‖

2+ ‖∆u(s)‖2)− C ′δ(mes(Ω)+ ‖h‖2)

)ds, (3.13)

therefore, for any ρ >C ′δ(mes(Ω)+‖h‖2)

Cδ, there exists t0, such that

‖ut (t0)‖2+ ‖∆u(t0)‖

2≤ ρ.

Set B0 = (u0, v0) ∈ H20 (Ω)× L2(Ω)|‖∆u0‖

2+ ‖v0‖

2≤ ρ, so we have that B0 is a bounded absorbing set.

Define

B1 =⋃t≥0

S(t)B0,

therefore, B1 is also a bounded absorbing set, moreover, B1 is positive invariant.

4. Asymptotic compactness in H20 (Ω) × L2(Ω)

In this section, we will first give some a priori estimates about the energy inequalities on account of theidea presented in [4,6,12,14,19,21,22]. Then we use Theorem 4.1 to establish the asymptotic compactness inH2

0 (Ω)× L2(Ω). For convenience, we always denote by B1 a bounded absorbing set obtained in Lemma 3.1.Hereafter, we will use the following notations:

Ew(t) =12

∫Ω|wt (t)|

2+

12

∫Ω|∆w(t)|2.

4.1. A priori estimates

The main purpose of this part is to establish (4.13)–(4.15), which will be used to obtain the asymptotic compactness.Without loss of generality, we deal only with the strong solutions in the following, the generalized solution case thenfollows easily by a density argument. The following process is derived from the standard energy method given in [4,6,12,14,19,21,22].

Let (ui (t), uit (t))(i = 1, 2) be the corresponding solution to (ui0, v

i0) ∈ B1, and let w(t) = u1(t) − u2(t). Then

w(t) satisfying

wt t + a(x)(g(u1t )− g(u2t ))+∆2w + λw + f (u1(t))− f (u2(t)) = 0, (4.1)

with the initial condition (w(0), wt (0)) = (u10, v

10)− (u

20, v

20).

At first, multiplying (4.1) by w and integrating over [0, T ] × Ω , we get∫ T

0

∫Ω|∆w(s)|2dxds =

∫Ωwt (0)w(0)dx −

∫Ωwt (T )w(T )dx − λ

∫ T

0

∫Ω|w(s)|2dxds

−

∫ T

0

∫Ω( f (u1(s))− f (u2(s)))w(s)dxds +

∫ T

0

∫Ω|wt (s)|

2dxds

−

∫ T

0

∫Ω

a(x)(g(u1t (s))− g(u2t (s)))w(s)dxds. (4.2)

Then we have∫ T

0Ew(s)ds =

12

∫Ωwt (0)w(0)dx −

12

∫Ωwt (T )w(T )dx −

12λ

∫ T

0

∫Ω|w(s)|2dxds

−12

∫ T

0

∫Ω( f (u1(s))− f (u2(s)))w(s)dxds +

∫ T

0

∫Ω|wt (s)|

2dxds

−12

∫ T

0

∫Ω

a(x)(g(u1t (s))− g(u2t (s)))w(s)dxds. (4.3)

L. Yang, C.-K. Zhong / Nonlinear Analysis 69 (2008) 3802–3810 3807

Secondly, multiplying (4.1) by wt and integrating over [s, T ] × Ω , we get

Ew(T )+∫ T

s

∫Ω

a(x)(g(u1t (τ ))− g(u2t (τ )))wt (τ )dxdτ +12λ‖w(T )‖2

= Ew(s)−∫ T

s

∫Ω( f (u1(τ ))− f (u2(τ )))wt (τ )dxdτ +

12λ‖w(s)‖2, (4.4)

where 0 ≤ s ≤ T . Integrating (4.4) over [0, T ] with respect to s, we obtain that

T Ew(T ) ≤∫ T

0Ew(s)ds +

12λ

∫ T

0

∫Ω|w(s)|2dxds −

∫ T

0

∫ T

s

∫Ω( f (u1(τ ))− f (u2(τ )))wt (τ )dxdτds, (4.5)

from (4.4) we also have∫ T

0

∫Ω

a(x)(g(u1t (τ ))− g(u2t (τ )))wt (τ )dxdτ ≤ Ew(0)−∫ T

0

∫Ω( f (u1(τ ))

− f (u2(τ )))wt (τ )dxdτ +12λ‖w(0)‖2. (4.6)

Combining with (1.3) and Lemma 2.4, we get that, for any δ > 0,∫ T

0

∫Ω|wt (τ )|

2dxdτ ≤ δT mes(Ω)+ CδEw(0)+12

Cδλ‖w(0)‖2

−Cδ

∫ T

0

∫Ω( f (u1(τ ))− f (u2(τ )))wt (τ )dxdτ, (4.7)

where Cδ is a constant which depends on δ and α0. So we have∫ T

0Ew(s)ds ≤ δT mes(Ω)+ CδEw(0)+

12

Cδλ‖w(0)‖2 +12

∫Ωwt (0)w(0)dx

−12

∫Ωwt (T )w(T )dx − Cδ

∫ T

0

∫Ω( f (u1(τ ))− f (u2(τ )))wt (τ )dxdτ

−12λ

∫ T

0

∫Ω|w(s)|2dxds −

12

∫ T

0

∫Ω( f (u1(s))− f (u2(s)))w(s)dxds

−12

∫ T

0

∫Ω

a(x)(g(u1t (s))− g(u2t (s)))w(s)dxds. (4.8)

Next, we need to tackle with∫ T

0

∫Ω a(x)(g(u1t )− g(u2t ))w.

The following estimate can be easily derived by using similar arguments as in [21, Chap. 5]. However, for the sakeof completeness we give the proof.

From (1.7), we have

|g(s)|1q ≤ C(1+ |s|),

and

|g(s)|q+1

q = |g(s)|1q · |g(s)| ≤ C(1+ |s|)|g(s)|,

combining (1.6), we get

|g(s)|q+1

q ≤

C, |s| ≤ 1,2Cg(s)s, |s| ≥ 1,

(4.9)

3808 L. Yang, C.-K. Zhong / Nonlinear Analysis 69 (2008) 3802–3810

where C is a constant which is independent of s. Multiplying (1.1) by uit (t), we obtain

12

ddt

∫Ω

(|uit |

2+ |∆ui |

2+ λ|ui |

2+ 2F(ui )

)+

∫Ω

a(x)g(uit )uit =

∫Ω

h(x)uit ,

which, combining with the existence of bounded absorbing set, implies that∫ T

0

∫Ω

a(x)g(uit )uit ≤ Cρ, (4.10)

where Cρ is a constant which depends on the size of B1 in H20 (Ω) × L2(Ω), but is independent of T . Therefore,

following derivation in [19] on page 169 (also included in [6,12,21]), we have, from (4.9) and (4.10)∣∣∣∣∫ T

0

∫Ω

a(x)g(uit )w

∣∣∣∣ ≤ ∫ T

0

∫Ω(|uit |≤1)

|a(x)g(uit )w| +

∫ T

0

∫Ω(|uit |≥1)

|a(x)g(uit )w|

≤

∫ T

0

∫Ω(|uit |≤1)

C |a(x)w| +∫ T

0

∫Ω(|uit |≥1)

|a(x)g(uit )||w|

≤

∫ T

0

∫Ω(|uit |≤1)

C |a(x)w| +

(∫ T

0

∫Ω(|uit |≥1)

a(x)|g(uit )|q+1

q

) qq+1

×

(∫ T

0

∫Ω(|uit |≥1)

|a(x)||w|q+1

) 1q+1

≤

∫ T

0

∫Ω(|uit |≤1)

C |a(x)w| + 2C

(∫ T

0

∫Ω(|uit |≥1)

a(x)g(uit )uit

) qq+1

×

(∫ T

0

∫Ω(|uit |≥1)

|a(x)||w|q+1

) 1q+1

≤

∫ T

0

∫Ω(|uit |≤1)

C |a(x)w| + Cρ

(∫ T

0

∫Ω(|uit |≥1)

|a(x)||w|q+1

) 1q+1

≤

∫ T

0

∫Ω(|uit |≤1)

C |a(x)w| + CρT1

q+1 . (4.11)

Combining (4.5), (4.8) and (4.11), we obtain that

T Ew(T ) ≤ δT mes(Ω)+ CδEw(0)+12

Cδλ‖w(0)‖2 +12

∫Ωwt (0)w(0)dx

−12

∫Ωwt (T )w(T )dx − Cδ

∫ T

0

∫Ω( f (u1(s))− f (u2(s)))wt (s)dxds

+C∫ T

0

∫Ω|a(x)w|dxds −

12

∫ T

0

∫Ω( f (u1(s))− f (u2(s)))w(s)dxds

+CρT1

q+1 −

∫ T

0

∫ T

s

∫Ω( f (u1(τ ))− f (u2(τ )))wt (τ )dxdτds. (4.12)

Set

CM = δT mes(Ω)+ CδEw(0)+12

Cδλ‖w(0)‖2 + CρT1

q+1

+12

∫Ωwt (0)w(0)dx −

12

∫Ωwt (T )w(T )dx, (4.13)

L. Yang, C.-K. Zhong / Nonlinear Analysis 69 (2008) 3802–3810 3809

φδ,T ((u10, υ

10 ), (u

20, υ

20 )) = C

∫ T

0

∫Ω|a(x)w|dxds −

12

∫ T

0

∫Ω( f (u1(s))− f (u2(s)))w(s)dxds

−Cδ

∫ T

0

∫Ω( f (u1(s))− f (u2(s)))wt (s)dxds

−

∫ T

0

∫ T

s

∫Ω( f (u1(τ ))− f (u2(τ )))wt (τ )dxdτds. (4.14)

Then we have

Ew(T ) ≤CM

T+

1Tφδ,T ((u

10, υ

10 ), (u

20, υ

20 )). (4.15)

4.2. Asymptotic compactness

In this subsection, following similar arguments as in [4,6,19,21], we shall prove the asymptotic compactness of thesemigroup S(t)t≥0 in H2

0 (Ω)× L2(Ω), which is given in the following theorem.

Theorem 4.1. Under assumptions (1.3)–(1.7), then the semigroup S(t)t≥0 corresponding to problems (1.1) and(1.2) is asymptotically compact in H2

0 (Ω)× L2(Ω).

Proof. Since the semigroup S(t)t≥0 has a bounded absorbing set, for any fixed ε > 0, we can choose firstδ ≤ ε

2mes(Ω) , and then let T so large that

CM

T≤ ε.

Hence, thanks to Theorem 2.3, we only need to verify that the function φδ,T (·, ·) defined in (4.14) belongs to C(B1)

for each fixed T . Let (un, utn ) be the corresponding solution of (un0, v

n0 ) ∈ B1, n = 1, 2, . . . . Then, since B1 is a

bounded positively invariant set in H20 (Ω)× L2(Ω), without loss of generality (at most by passing subsequences), we

assume that

un → u weakly star in L∞(0, T ; H20 (Ω)), (4.16)

unt → ut weakly star in L∞(0, T ; L2(Ω)), (4.17)

un → u strongly in L2(0, T ; L2(Ω)), (4.18)

un → u strongly in Lk(0, T ; Lk(Ω)), (4.19)

for k < 2nn−4 , where we use the compact embedding H2

0 → Lk .Now, we will deal with each term corresponding to that in (4.14) one by one.At first, from (4.18), we have

limn→∞

limm→∞

∫ T

0

∫Ω( f (un(s))− f (um(s)))(un(s)− um(s))dxds = 0. (4.20)

Secondly, from (4.19), noticing the condition (1.3), we obtain that

limn→∞

limm→∞

∫ T

0

∫Ω

a(x)|un(s)− um(s)|dxds = 0. (4.21)

Finally, following the similar argument given in [19, Lemma 4.4], we get

limn→∞

limm→∞

∫ T

0

∫Ω( f (un(s))− f (um(s)))(unt (s)− umt (s))dxds = 0, (4.22)

limn→∞

limm→∞

∫ T

0

∫ T

s

∫Ω( f (un(τ ))− f (um(τ )))(unt (τ )− umt (τ ))dxdτds = 0. (4.23)

Hence, combining (4.20)–(4.23) we get φδ,T (·, ·) ∈ C(B1) immediately.

3810 L. Yang, C.-K. Zhong / Nonlinear Analysis 69 (2008) 3802–3810

4.3. Existence of global attractor

Theorem 4.2. Under assumptions (1.3)–(1.7), then problems (1.1) and (1.2) have a global attractor in H20 (Ω) ×

L2(Ω), which is invariant and compact.

Proof. Lemma 3.1 and Theorem 4.1 imply the existence of global attractor immediately.

Remark 4.3. We should point out that, our result, Theorem 4.2, can be also obtained by applying a more generalabstract result given in [21, Theorem 3.40] by Chueshov and Lasiecka.

Acknowledgments

The authors wish to thank the referee for his/her valuable comments and suggestions, especially forbringing [10,15,16,19,21] to our attention. In addition, we are grateful to Professor Chueshov for providing us valuablereferences [19,21].

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