Global attractors for Kirchhoff wave equation with nonlinear damping and memory

Abstract

In this paper, we prove that the existence of global attractors for a Kirchhoff wave equation with nonlinear damping and memory.

1 Introduction

Let Ω be an open bounded subset of $\mathbb{R}^{N}$ with sufficiently smooth boundary Γ, we consider the following Kirchhoff wave equation with nonlinear damping and linear memory:

$$\begin{aligned}& \begin{aligned}[b] &u_{tt}-M\bigl( \Vert \nabla u \Vert ^{2}\bigr)\Delta u+a(x)g(u_{t})-k(0) \triangle u- \int _{0}^{\infty }k'(s)\Delta u(t-s)\,ds \\ &\quad{}+f(u)=h(x), \quad \text{in } \varOmega \times \mathbb{R}^{+}, \end{aligned} \end{aligned}$$

(1)

$$\begin{aligned}& u|_{x\in \varGamma }(x,t)=0,\qquad u(x,0)=u_{0}(x), \qquad u_{t}(x,0)=u_{1}(x), \end{aligned}$$

(2)

$$\begin{aligned}& u(x,t)=u_{0}(x,t),\quad x\in \varOmega ,t\leq 0, \end{aligned}$$

(3)

where $M(s)=1+s^{\frac{m}{2}}$, $m\geq 1$, $k(0)$, $k(\infty )>0$ and $k'(s)\leq 0$ for every $s\in \mathbb{R}^{+}$, and the assumptions on nonlinear functions $f(u)$, $g(u_{t})$, $a(x)$ and external force term $h(x)$ will be specified later.

This kind of wave models goes back to Kirchhoff. In 1883, Kirchhoff [1] firstly introduced the following equation to describe small vibrations of an elastic stretch string:

$$ u_{tt}-M\bigl( \Vert \nabla u \Vert ^{2}\bigr)\Delta u=h, $$

where $M(s)=a+bs$. There has been much research on global attractors; Lazo studied the existence for the IBVP of the Kirchhoff equation with memory term [2]

$$ u_{tt}-M\bigl( \Vert \nabla u \Vert ^{2}\bigr)\Delta u+ \int _{0}^{t}g(t-\tau )\Delta u(x, \tau )\,d\tau =0. $$

Chueshov [3] studied the well-posedness and the global attractors of the Kirchhoff equation with strong nonlinear damping

$$ u_{tt}-\sigma \bigl( \Vert \nabla u \Vert ^{2}\bigr) \Delta u_{t}-\phi \bigl( \Vert \nabla u \Vert ^{2}\bigr) \Delta ^{\theta } u+g(u)=h(x), \quad \frac{1}{2}\leq \theta < 1. $$

Next, Chueshov [4] also studied the Kirchhoff equation with strong nonlinear damping in nature space $\mathcal{H}=H_{0}^{1}(\varOmega )\cap L^{p+1}(\varOmega )\times L^{2}( \varOmega )$ as $\theta =1$. For related work on the Kirchhoff wave equations with strong damping, see [5, 6] and the references therein.

When $M(s)=0$, Eq. (1) become the well-known wave equation. Ma and Zhong [7] showed the existence of global attractors for the hyperbolic equation with memory

$$ u_{tt}+\alpha u_{t}-K(0)\triangle u- \int _{0}^{\infty }K'(s)\Delta u(t-s) \,ds+g(u)=f. $$

Recently, Park and Kang [8] studied the existence of global attractors for the semilinear hyperbolic with nonlinear damping and memory

$$ u_{tt}+a(x)g(u_{t})+\lambda u-K(0)\triangle u- \int _{0}^{\infty }K'(s) \Delta u(t-s) \,ds+f(u)=h(x). $$

In [9], Kang and Rivera showed the existence of global attractors for the beam equation localized nonlinear damping and memory

$$ u_{tt}+a(x)g(u_{t})+\triangle ^{2} u-K(0) \bigl(1+ \Vert \nabla u \Vert ^{2}\bigr) \triangle u- \int _{0}^{\infty }K'(s)\Delta u(t-s) \,ds+f(u)=h. $$

Motivated by [5, 7–9], we will prove the existence of global attractors for Eq. (1).

Following the framework proposed in [7], we shall add a new variable η to the system, which corresponds to the relative displacement history. Let us define

$$ \eta =\eta ^{t}(x,s)=u(x,t)-u(x,t-s). $$

(4)

By differentiation, we have

$$ \eta ^{t}_{t}(x,s)=-\eta ^{t}_{s}(x,s)+u_{t}(x,t). $$

(5)

Let $\mu (s)=-k'(s)$, $k(\infty )=1$, (1) transforms into the following system:

$$\begin{aligned}& u_{tt}-\bigl(1+ \Vert \nabla u \Vert ^{m}\bigr)\Delta u+a(x)g(u_{t})- \int _{0}^{\infty } \mu (s)\Delta \eta (x,s)\,ds+f(u)=h, \end{aligned}$$

(6)

$$\begin{aligned}& \eta _{t}=-\eta _{s}+u_{t}, \end{aligned}$$

(7)

with boundary condition

$$ u=0 ,\quad \text{on } \varGamma \times \mathbb{R}^{+},\qquad \eta =0 , \quad \text{on } \partial \varOmega \times \mathbb{R}^{+}\times \mathbb{R}^{+}, $$

(8)

and initial conditions

$$ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x),\qquad \eta ^{t}(x,0)=0,\qquad \eta ^{0}(x,s)= \eta _{0}(x,s). $$

(9)

This paper is organized as follows. In Sect. 2, we introduce some preliminaries. In Sect. 3, we show the existence of a bounded absorbing set in $\mathcal{H}$. In Sect. 4, we give the existence of global attractors of problems (6)–(9).

2 Preliminaries

We first state some assumptions, which will be used in this paper.

Assumption (1)

The memory kernel μ is required to satisfy the following hypotheses:

(h1):

$\mu (s)\in C^{1}(\mathbb{R})\cap L^{1}(\mathbb{R})$, $\forall s\in \mathbb{R}^{+}$;

(h2):

$\int ^{\infty }_{0}\mu (s)\,ds=k(0)$;

(h3):

$\mu (s)\geq 0$, $\mu '(s)\leq 0$;

(h4):

$\mu '(s)+k_{1}\mu (s)\leq 0$, $\forall s\in \mathbb{R}^{+}$, for some $k_{1}>0$.

Assumption (2)

The function $a(x)$ satisfies

$$ a(x)\in L^{\infty }(\varOmega ),\qquad a(x)\geq \alpha _{0}>0, $$

(10)

where $\alpha _{0}$ is a constant.

Assumption (3)

The function $f\in C^{1}(\mathbb{R})$ satisfies

$$\begin{aligned}& \bigl\vert f'(s) \bigr\vert \leq C_{1}\bigl(1+ \vert s \vert ^{p}\bigr), \end{aligned}$$

(11)

$$\begin{aligned}& \lim_{ \vert s \vert \rightarrow \infty }\inf \frac{f(s)}{s}>-\lambda _{1}, \end{aligned}$$

(12)

where $0< p<\infty$, if $n\leq 2$, and $0< p\leq \frac{2}{n-2}$ if $n\leq 2$. $\lambda _{1}$ is the constant in the Poincáre type inequality $\|\nabla u\|^{2}\geq \lambda _{1}\|u\|^{2}$.

Assumption (4)

The damping function $g\in C^{1}(\mathbb{R})$ satisfies

$$\begin{aligned}& g(0)=0,\qquad g \text{ is strictly increasing},\quad \text{and}\quad \liminf _{ \vert s \vert \rightarrow \infty } g'(s)>0, \end{aligned}$$

(13)

$$\begin{aligned}& \bigl\vert g(s) \bigr\vert \leq C_{2}\bigl(1+ \vert s \vert ^{q}\bigr), \end{aligned}$$

(14)

with $1\leq q<\infty$ if $n\leq 2$, and $1\leq q\leq \frac{n+2}{n-2}$ if $n>2$.

In order to consider the relative displacement η as a new variable, we introduce the weighted $L^{2}$-space

$$ \mathcal{M}=L^{2}_{\mu }\bigl(\mathbb{R}^{+};H^{1}_{0} \bigr)=\biggl\{ \xi :\mathbb{R}^{+} \rightarrow H^{1}_{0}( \varOmega )\Big| \int ^{\infty }_{0}\mu (s) \bigl\Vert \nabla \xi (s) \bigr\Vert ^{2}_{2}\,ds< \infty \biggr\} , $$

which is a Hilbert space endowed with inner product and norm

$$ (\xi ,\zeta )_{\mathcal{M}}= \int ^{\infty }_{0}\mu (s) \biggl( \int _{ \varOmega }\nabla \xi (s)\nabla \zeta (s)\,dx\biggr)\,ds \quad \text{and}\quad \Vert \xi \Vert ^{2}_{\mathcal{M}}= \int ^{\infty }_{0}\mu (s) \Vert \nabla \xi \Vert ^{2}_{2}\,ds, $$

respectively.

Our analysis is given on the phase space

$$ \mathcal{H}=H^{1}_{0}(\varOmega )\times L^{2}(\varOmega )\times \mathcal{M}, $$

which is equipped with the norm

$$ \bigl\Vert (u,v,\eta ) \bigr\Vert ^{2}_{\mathcal{H}}= \Vert \nabla u \Vert ^{2}+ \Vert v \Vert ^{2}+ \Vert \eta \Vert ^{2}_{ \mathcal{M}}. $$

In order to obtain the global attractors of the problems (6)–(9), we need the following theorem of existence, uniqueness of solution and continuous dependence on the initial data.

Theorem 2.1

([9])

Let assumptions(1)–(4)hold, if$z_{0}=(u_{0},v_{0},\eta _{0})\in \mathcal{H}$, then there exists a unique solution$z=(u,u_{t},\eta )$of (6)–(9) such that

$$ z\in C\bigl([0,T],\mathcal{H}\bigr) \quad \textit{for all } T>0. $$

Next,we recall the simple compactness criterion stated in [9, 10].

Definition 2.1

([9, 10])

Let X be a Banach space and B be a bounded subset of X, we call a function $\varPhi (\cdot ,\cdot )$ which defined on $X\times X$, is a contractive on $B\times B$ if for any sequence $\{x_{n}\}^{\infty }_{n=1}\subset B$, there is a subsequence $\{x_{n_{k}}\}^{\infty }_{k=1}\subset \{x_{n}\}^{\infty }_{n=1}$ such that

$$ \lim_{k\rightarrow \infty }\lim_{l\rightarrow \infty }\varPhi _{T}(x_{n_{k}},x_{n_{l}})=0. $$

(15)

Denote all such contractive functions on $B\times B$ by $C(B)$.

Theorem 2.2

([9, 10])

Let$\{s(t)\}_{t\geq 0}$be a semigroup on a Banach space$(X,\| \cdot \|)$and has a bounded absorbing set$B_{0}$. Moreover, assume that for any$\varepsilon \geq 0$there exist$T=T(B_{0},\varepsilon )$and$\varPhi (\cdot ,\cdot )\in C(B)$such that

$$ \bigl\Vert S(T)x-S(T)y \bigr\Vert \leq \varepsilon +\varPhi _{T}(x,y)\quad \textit{for all } x,y \in B_{0}, $$

where$\varPhi _{T}$depends onT. Then$\{s(t)\}_{t\geq 0}$is asymptotically compact inX, i.e., for any bounded sequence$\{y_{n}\}_{n}^{\infty }\subset X$and$\{t_{n}\}$with$t_{n}\rightarrow \infty$, $\{S(t_{n})y_{n}\}_{n=1}^{\infty }$is compact inX.

Lemma 2.1

([11])

Let$g(\cdot )$satisfy condition (13). Then for any$\delta > 0$there exists$c(\delta )>0$, such that

$$ \vert u-v \vert ^{2}\leq \delta +C(\delta ) \bigl(g(u)-g(v)\bigr) (u-v),\quad \textit{for all } u,v \in \mathbb{R}. $$

(16)

3 Absorbing set in $\mathcal{H}$

In this section, we prove the existence of the bounded absorbing set in $\mathcal{H}$. We use $C_{i}$ to denote several positive constants.

Lemma 3.1

Under assumptions(1)–(4), the semigroup$\{S(t)\}_{t\geq 0}$corresponding to problems (6)–(9) has a bounded absorbing set in$\mathcal{H}$.

Proof

we take the scalar product in $L^{2}$ of system (6) with $u_{t}$ and (7) with η, respectively, we have

$$ \begin{aligned}[b] &\frac{d}{dt}\biggl(\frac{1}{2} \Vert u_{t} \Vert ^{2}+\frac{1}{2} \Vert \nabla u \Vert ^{2}+\frac{1}{m+2} \Vert \nabla u \Vert ^{m+2}+ \frac{1}{2} \Vert \eta \Vert _{ \mathcal{M}}^{2}+ \int _{\varOmega }\bigl(F(u)-hu\bigr)\,dx\biggr) \\ &\quad{}+(\eta ,\eta _{s})_{\mathcal{M}}+\bigl(a(x)g(u_{t}),u_{t} \bigr)=0, \end{aligned} $$

(17)

where $F(u)=\int _{0}^{u}f(s)\,ds$. As in [7]

$$ \begin{aligned}[b] (\eta ,\eta _{s})_{\mathcal{M}}&= \frac{1}{2} \int _{0}^{ \infty }\mu (s)\frac{d}{ds} \bigl\Vert \nabla \eta (s) \bigr\Vert ^{2} \,ds \\ &=-\frac{1}{2} \int _{0}^{\infty }\mu '(s) \bigl\Vert \nabla \eta (s) \bigr\Vert ^{2}\,ds \geq \frac{k_{1}}{2} \Vert \eta \Vert _{\mathcal{M}}^{2}. \end{aligned} $$

(18)

We set

$$ E(t)=\frac{1}{2} \Vert u_{t} \Vert ^{2}+ \frac{1}{2} \Vert \nabla u \Vert ^{2}+ \frac{1}{m+2} \Vert \nabla u \Vert ^{m+2}+\frac{1}{2} \Vert \eta \Vert _{\mathcal{M}}^{2}+ \int _{\varOmega }\bigl(F(u)-hu\bigr)\,dx. $$

Then from (17) and (18) we obtain

$$ \frac{d}{dt}E(t)+\frac{k_{1}}{2} \Vert \eta \Vert _{\mathcal{M}}^{2}+ \int _{ \varOmega }\bigl(a(x)g(u_{t})u_{t}\bigr)\,dx \leq 0. $$

(19)

From (10), (13) we obtain

$$ E(t)\leq E(0), \quad t\geq 0. $$

(20)

By the hypothesis (12) we know that there are $\lambda >\lambda _{1}> 0$ and $C_{0}$ such that

$$ \bigl(f(u),u\bigr)>-\frac{\lambda }{2} \Vert u \Vert ^{2}-C_{0} \operatorname{mes}(\varOmega ), \qquad \int _{\varOmega }F(u)\,dx>-\frac{\lambda }{4} \Vert u \Vert ^{2}-C_{0}\operatorname{mes}(\varOmega ). $$

(21)

Using the Young inequality, we have

$$ - \int _{\varOmega }hudx\geq -\varepsilon \Vert u \Vert ^{2}- \frac{1}{4\varepsilon } \Vert h \Vert ^{2}, $$

we choose proper λ and ε small enough so that $\frac{1}{2}-\frac{\lambda }{4\lambda _{1}}-\varepsilon >\frac{1}{8}$, and we have

$$ \begin{aligned}[b] E(0)&\geq E(t)\geq \frac{1}{2} \Vert u_{t} \Vert ^{2}+ \frac{1}{m+2} \Vert \nabla u \Vert ^{m+2}+\frac{1}{8} \Vert \nabla u \Vert ^{2}+ \frac{1}{2} \Vert \eta \Vert _{\mathcal{M}}^{2}-C_{1} \bigl(\operatorname{mes}(\varOmega )+ \Vert h \Vert ^{2}\bigr) \\ &\geq -C_{1}\bigl(\operatorname{mes}(\varOmega )+ \Vert h \Vert ^{2}\bigr), \end{aligned} $$

(22)

combining (19) with (22), we have

$$ \int _{0}^{t} \int _{\varOmega }\bigl(a(x)g(u_{t})u_{t}\bigr)\,dx \leq E(0)-E(t)\leq E(0)+C_{1}\bigl(\operatorname{mes}( \varOmega )+ \Vert h \Vert ^{2}\bigr) ,\quad \forall t \geq 0. $$

(23)

Taking the scalar product in $L^{2}$ of (6) with $v=u_{t}+\varepsilon u$, we obtain

$$ \begin{aligned}[b] &\frac{d}{dt}\biggl(\frac{1}{2} \Vert v \Vert ^{2}+\frac{1}{2} \Vert \nabla u \Vert ^{2}+ \frac{1}{m+2} \Vert \nabla u \Vert ^{m+2}+ \frac{1}{2} \Vert \eta \Vert _{\mathcal{M}}^{2}- \frac{\varepsilon ^{2}}{2} \Vert u \Vert ^{2}+\varepsilon \Vert \nabla u \Vert ^{2} \\ &\quad{}+ \int _{\varOmega }\bigl(F(u)-hu\bigr)\,dx\biggr)+\varepsilon \Vert \nabla u \Vert ^{m+2}+ \frac{k_{1}}{2} \Vert \eta \Vert _{\mathcal{M}}^{2}+\varepsilon \bigl(f(u),u\bigr) \\ &\quad{}+\bigl(a(x)g(u_{t})-\varepsilon u_{t},u_{t} \bigr)+\varepsilon \bigl(a(x)g(u_{t}),u\bigr)- \varepsilon (h,u)\leq \varepsilon (\eta ,u)_{\mathcal{M}}. \end{aligned} $$

(24)

Let

$$\begin{aligned}& F(t)=\frac{1}{2} \Vert v \Vert ^{2}+\frac{1}{2} \Vert \nabla u \Vert ^{2}+\frac{1}{m+2} \Vert \nabla u \Vert ^{m+2}+\frac{1}{2} \Vert \eta \Vert _{\mathcal{M}}^{2}- \frac{\varepsilon ^{2}}{2} \Vert u \Vert ^{2}+ \int _{\varOmega }\bigl(F(u)-hu\bigr)\,dx, \\& \begin{aligned}[b] G(t)&=\varepsilon \Vert \nabla u \Vert ^{2}+\varepsilon \Vert \nabla u \Vert ^{m+2}+ \frac{k_{1}}{2} \Vert \eta \Vert _{\mathcal{M}}^{2}+\varepsilon \bigl(f(u),u\bigr)- \varepsilon (h,u)-\varepsilon (\eta ,u)_{\mathcal{M}} \\ &\quad{}+\bigl(a(x)g(u_{t})-\varepsilon u_{t},u_{t} \bigr)+\varepsilon \bigl(a(x)g(u_{t}),u\bigr), \end{aligned} \end{aligned}$$

so

$$ \frac{d}{dt}F(t)+G(t)\leq 0. $$

(25)

Similarly, using (21), the Poincáre inequality and the Young inequality, choosing proper λ and ε small enough so that $\frac{1}{2}-\frac{\varepsilon ^{2}}{2\lambda _{1}}- \frac{\lambda }{4\lambda _{1}}-\varepsilon >\frac{1}{8}$, we have

$$ F(t)\geq \frac{1}{2} \Vert v \Vert ^{2}+\frac{1}{8} \Vert \nabla u \Vert ^{2}+ \frac{1}{m+2} \Vert \nabla u \Vert ^{m+2}+\frac{1}{2} \Vert \eta \Vert _{\mathcal{M}}^{2}-C\bigl(\operatorname{mes}( \varOmega )+ \Vert h \Vert ^{2}\bigr). $$

(26)

It is obvious that (10) and (13) imply that there are $\varepsilon >0$ and $C>0$ such that

$$\begin{aligned}& \begin{gathered} \bigl(a(x)g(u_{t}),u_{t}\bigr)\geq 2\varepsilon \Vert u_{t} \Vert ^{2}-C_{\varepsilon } \operatorname{mes}( \varOmega ), \\ \bigl(a(x)g(u_{t})-\varepsilon u, u_{t}\bigr)\geq \varepsilon \Vert u_{t} \Vert ^{2}-C( \varepsilon ) \operatorname{mes}(\varOmega ). \end{gathered} \end{aligned}$$

(27)

Due to the Young inequality we have

$$ \varepsilon (\eta ,u)_{\mathcal{M}}\geq -\frac{k_{1}}{4} \Vert \varepsilon \Vert ^{2}_{\mathcal{M}}-\frac{k(0)\delta ^{2}}{k_{1}} \Vert \nabla u \Vert ^{2}. $$

(28)

Using (13) and (14) yields

$$ \bigl\vert g(s) \bigr\vert ^{\frac{q+1}{q}}= \bigl\vert g(s) \bigr\vert ^{\frac{1}{q}} \bigl\vert g(s) \bigr\vert \leq C\bigl(1+ \vert s \vert \bigr) \bigl\vert g(s) \bigr\vert , $$

so

$$ \textstyle\begin{cases} \vert g(s) \vert ^{\frac{q+1}{q}}\leq C, &\vert s \vert \leq 1; \\ \vert g(s) \vert ^{\frac{q+1}{q}}\leq 2Cg(s)s, & \vert s \vert \geq 1, \end{cases} $$

(29)

where C is a constant which is independent of s.

Then from (29), using the Hölder inequality, the Young inequality and the Sobolev embedding $H^{1}_{0}(\varOmega )\hookrightarrow L^{q+1}(\varOmega )$, we obtain

$$ \begin{aligned}[b] & \biggl\vert \int _{\varOmega }a(x)g(u_{t})u\,dx \biggr\vert \\ &\quad \leq \int _{\varOmega ( \vert u_{t} \vert \leq 1)} \bigl\vert a(x)g(u_{t})u \bigr\vert \,dx+ \int _{\varOmega ( \vert u_{t} \vert \geq 1)} \bigl\vert a(x)g(u_{t})u \bigr\vert \,dx \\ &\quad \leq \int _{\varOmega ( \vert u_{t} \vert \leq 1)}C \bigl\vert a(x)u \bigr\vert \,dx\\ &\qquad {}+\biggl( \int _{\varOmega ( \vert u_{t} \vert \geq 1)}a(x) \bigl\vert g(u_{t}) \bigr\vert ^{\frac{q+1}{q}}\,dx\biggr)^{\frac{q}{q+1}}\biggl( \int _{ \varOmega ( \vert u_{t} \vert \geq 1)}a(x) \vert u \vert ^{q+1}\,dx \biggr)^{\frac{1}{q+1}} \\ &\quad \leq \int _{\varOmega ( \vert u_{t} \vert \leq 1)}C \bigl\vert a(x)u \bigr\vert \,dx\\ &\qquad {}+2C\biggl( \int _{\varOmega ( \vert u_{t} \vert \geq 1)}a(x)g(u_{t})u_{t}\,dx \biggr)^{\frac{q}{q+1}}\biggl( \int _{\varOmega ( \vert u_{t} \vert \geq 1)}a(x) \vert u \vert ^{q+1}\,dx \biggr)^{\frac{1}{q+1}} \\ &\quad \leq \frac{C}{4\gamma } \int _{\varOmega } \biggl\vert \frac{a(x)}{a_{0}} \biggr\vert ^{2}\,dx\\ &\qquad {}+C \gamma a_{0}^{2} \Vert u \Vert ^{2}+C_{\gamma }\biggl( \int _{\varOmega ( \vert u_{t} \vert \geq 1)}a(x)g(u_{t})u_{t}\,dx\biggr) \Vert u \Vert _{q+1}^{ \frac{q-1}{q}}+\eta \Vert u \Vert _{q+1}^{2} \\ &\quad \leq \frac{C}{4\gamma }\operatorname{mes}(\varOmega )+C\gamma a_{0}^{2} \Vert u \Vert ^{2}+C_{s}C_{ \gamma } \Vert \nabla u \Vert ^{\frac{q-1}{q}} \int _{\varOmega }a(x)g(u_{t})u_{t}\,dx+ \gamma C_{s} \Vert \nabla u \Vert ^{2}, \end{aligned} $$

(30)

where $a_{0}=\sup_{x\in \varOmega }{a(x)}$, and γ is a constant. From (21), (27), (28), (30) we have

$$ \begin{aligned}[b] G(t)&\geq \varepsilon \Vert u_{t} \Vert ^{2}+\varepsilon \Vert \nabla u \Vert ^{m+2}+ \frac{k_{1}}{4} \Vert \eta \Vert ^{2}_{\mathcal{M}}\\&\quad{}+ \varepsilon \biggl(\frac{1}{2}-\frac{k(0)\varepsilon ^{2}}{k_{1}}-C\biggr) \Vert \nabla u \Vert ^{2}-\biggl(\varepsilon C\gamma a_{0}^{2}+ \frac{\varepsilon \lambda }{4}\biggr) \Vert u \Vert ^{2} \\ &\quad{}-\varepsilon C \Vert \nabla u \Vert ^{\frac{q-1}{q}} \int _{\varOmega }a(x)g(u_{t})u_{t} \,dx-C'_{ \varepsilon }\bigl(\operatorname{mes}(\varOmega )+ \Vert h \Vert ^{2}\bigr), \end{aligned} $$

we choose ε and C small enough so that $\frac{1}{2}-\frac{k(0)\varepsilon ^{2}}{k_{1}}-C>\frac{1}{4}$, we get

$$ \begin{aligned}[b] G(t)&\geq \frac{\varepsilon }{4}\bigl( \Vert u_{t} \Vert ^{2}+ \Vert \nabla u \Vert ^{2}\bigr)+ \frac{k_{1}}{4} \Vert \eta \Vert ^{2}_{\mathcal{M}}\\&\quad{}-C_{E(0)} \int _{\varOmega }a(x)g(u_{t})u_{t} \,dx-C'_{ \varepsilon }\bigl(\operatorname{mes}(\varOmega )+ \Vert h \Vert ^{2}\bigr), \end{aligned} $$

(31)

where $C_{E(0)}$ is a constant which depends on ε, γ, C and $E(0)$, $C'_{\varepsilon }$ is a constant depending on ε, $C_{\delta }$ and C.

We have

$$\begin{aligned} \Vert u_{t} \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+ \Vert \eta \Vert _{\mathcal{M}}^{2}&= \Vert u_{t}+\delta u-\delta u \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+ \Vert \eta \Vert _{\mathcal{M}}^{2} \\ &\leq 2 \Vert v \Vert ^{2}+\biggl(\frac{2\delta ^{2}}{\lambda _{1}}+1\biggr) \Vert \nabla u \Vert ^{2}+ \Vert \eta \Vert _{\mathcal{M}}^{2} \\ &\leq C_{0}\bigl( \Vert v \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+ \Vert \eta \Vert _{\mathcal{M}}^{2}\bigr), \end{aligned}$$

(32)

where $C_{0}=\max \{2,1+\frac{2\delta ^{2}}{\lambda _{1}}\}$.

Integrating (25), combining with (23), (26), (31), yields

$$ \begin{aligned}[b] & \Vert u_{t} \Vert ^{2}+ \Vert \nabla u \Vert ^{2}+ \Vert \eta \Vert _{\mathcal{M}}^{2}-4C \bigl(\operatorname{mes}( \varOmega )+ \Vert h \Vert ^{2}\bigr) \\ &\qquad{}-4C_{0}F(0)-4C_{0}C_{E(0)}(E(0)+C\bigl( \operatorname{mes}(\varOmega )+ \Vert h \Vert ^{2}\bigr) \\ & \quad \leq - \int _{0}^{t}\bigl(\delta ' C_{0} \bigl( \bigl\Vert u_{t}(s) \bigr\Vert ^{2}+ \bigl\Vert \nabla u(s) \bigr\Vert ^{2}+ \bigl\Vert \eta ^{s}(\tau ) \bigr\Vert ^{2}_{\mathcal{M}} \bigr)\\ &\qquad {}-4C_{0}C'_{\varepsilon }\bigl(\operatorname{mes}( \varOmega )+ \Vert h \Vert ^{2}\bigr)\bigr)\,ds, \end{aligned} $$

(33)

where $\delta '=\min \{\delta ,k_{1}\}$. Therefore, for any $\rho >\frac{4C'_{\varepsilon }(\operatorname{mes}(\varOmega )+\|h\|^{2})}{\delta '}$ there exists $t_{0}$ such that

$$ \bigl\Vert u_{t}(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \nabla u(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \eta ^{t_{0}}(\tau ) \bigr\Vert ^{2}_{ \mathcal{M}}\leq \rho . $$

(34)

Set

$$ B_{0}=\bigl\{ (u_{0},v_{0},\eta _{0}) \in \mathcal{H}\mid \Vert \nabla u_{0} \Vert ^{2}+ \Vert v_{0} \Vert ^{2}+ \Vert \eta _{0} \Vert ^{2}_{\mathcal{M}}\leq \rho \bigr\} , $$

then we see $B_{0}$ is a bounded absorbing set. Define

$$ B_{1}=\bigcup_{{t\geq 0}}S(t)B_{0}, $$

so $B_{1}$ is also a bounded absorbing set. □

4 Existence of the global attractor in $\mathcal{H}$

4.1 A priori estimate

Firstly, we use the prior estimates to obtain the asymptotic compactness following the standard energy method. In this section, $C_{i}$ are positive constants.

Let $(u,u_{t},\eta )$ and $(v,v_{t},\xi )$ be two solution to systems (6)–(9), and $(u,u_{t},\eta )$ and $(v,v_{t},\xi )\in B_{1}$, $\omega (t)=u(t)-v(t)$, $\zeta =\eta -\xi$. Then $\omega (t)$, ζ satisfy

$$\begin{aligned}& \begin{aligned}[b] &\omega _{tt}- \Vert \nabla u \Vert ^{m}\triangle u+ \Vert \nabla v \Vert ^{m} \triangle v- \triangle \omega - \int _{0}^{\infty }\mu (s)\Delta \zeta (s)\,ds \\ &\quad{}+a(x)g(u_{1t})-a(x)g(u_{2t})+f(u_{1})-f(u_{2})=0, \end{aligned} \end{aligned}$$

(35)

$$\begin{aligned}& \zeta _{t}=-\zeta _{s}+\omega _{t}, \end{aligned}$$

(36)

firstly, taking the scalar product in $L^{2}$ of (35) with ω and integrating over $[0,T]$, we get

$$ \begin{aligned}[b] \int _{0}^{T} \bigl\Vert \nabla \omega (s) \bigr\Vert ^{2}\,ds&= \int _{\varOmega } \omega _{t}(0)\omega (0)\,dx- \int _{\varOmega }\omega _{t}(T)\omega (T)\,dx+ \int _{0}^{T} \bigl\Vert \omega _{t}(s) \bigr\Vert ^{2}\,ds \\ &\quad{}- \int _{0}^{T} \bigl\Vert \nabla u(s) \bigr\Vert ^{m} \bigl\Vert \nabla \omega (s) \bigr\Vert ^{2}\,ds- \int _{0}^{T}( \zeta ,\omega )_{\mathcal{M}}\,ds \\ &\quad{}- \int _{0}^{T} \int _{\varOmega }\bigl( \bigl\Vert \nabla u(s) \bigr\Vert ^{m}- \bigl\Vert \nabla v(s) \bigr\Vert ^{m}\bigr) \nabla v(s)\nabla \omega (s)\,dx \,ds \\ &\quad{}- \int _{0}^{T} \int _{\varOmega }a(x) \bigl(g\bigl(u_{t}(s)\bigr)-g \bigl(v_{t}(s)\bigr)\bigr)\omega (s)\,dx \,ds \\ &\quad{}- \int _{0}^{T} \int _{\varOmega }\bigl(f\bigl(u(s)\bigr)-f\bigl(v(s)\bigr)\bigr)\omega (s)\,dx \,ds. \end{aligned} $$

(37)

Using the Young inequality and $(h3)$, we obtain

$$ (\zeta ,\omega )_{\mathcal{M}}\geq -\frac{1}{2} \Vert \nabla \omega \Vert ^{2}- \frac{k(0)}{2} \Vert \zeta \Vert _{\mathcal{M}}^{2}. $$

(38)

Secondly, taking the scalar product in $L^{2}$ of (35), (36) with $\omega _{t}$ and integrating over $[0,T]$, we get

$$ \begin{aligned}[b] &\frac{d}{dt}\biggl(\frac{1}{2} \Vert \omega _{t} \Vert ^{2}+\frac{1}{2} \Vert \nabla \omega \Vert ^{2}+\frac{1}{2} \Vert \zeta \Vert _{\mathcal{M}}^{2}\biggr)+ \int _{ \varOmega }\bigl( \Vert \nabla u \Vert ^{m}- \Vert \nabla v \Vert ^{m}\bigr)\nabla v\nabla \omega _{t} \,dx \\ &\quad{}+(\zeta ,\zeta _{s})_{\mathcal{M}}+ \int _{\varOmega }\bigl(f(u)-f(v)\bigr) \omega _{t}\,dx+ \int _{\varOmega }a(x) \bigl(g(u_{t})-g(v_{t}) \bigr)\omega _{t}\,dx=0. \end{aligned} $$

(39)

Let

$$ E_{\omega }(t)=\frac{1}{2} \Vert \omega _{t} \Vert ^{2}+\frac{1}{2} \Vert \nabla\omega \Vert ^{2}+ \frac{1}{2} \Vert \zeta \Vert _{\mathcal{M}}^{2}. $$

Integrating (39) over $(s,T]$ and combining with (38), where $s\in [0,T]$, we have

$$ \begin{aligned}[b] &E_{\omega }(t)+\frac{k_{1}}{2} \int _{s}^{T} \Vert \zeta \Vert ^{2}_{ \mathcal{M}}+ \int _{s}^{T} \int _{\varOmega }a(x) \bigl(g\bigl(u_{t}(\tau )\bigr)-g \bigl(v_{t}( \tau )\bigr)\bigr)\omega _{t}(\tau )\,dx \,d\tau \\ &\qquad{}+\frac{1}{2} \int _{\varOmega } \bigl\Vert \nabla u(T) \bigr\Vert ^{m} \bigl\Vert \nabla \omega (T) \bigr\Vert ^{2}\,dx \\ &\quad \leq E_{\omega }(s)+\frac{1}{2} \int _{\varOmega } \bigl\Vert \nabla u(s) \bigr\Vert ^{m} \bigl\Vert \nabla \omega (s) \bigr\Vert ^{2}\,dx \\ &\qquad{}+\frac{m}{2} \int _{s}^{T} \int _{\varOmega } \bigl\Vert \nabla \omega (\tau ) \bigr\Vert ^{2} \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m-1}\nabla u_{t}(\tau )\,dx \,d\tau \\ &\qquad{}- \int _{s}^{T} \int _{\varOmega }\bigl( \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m}- \bigl\Vert \nabla v( \tau ) \bigr\Vert ^{m}\bigr) \nabla v(\tau )\nabla \omega _{t}(\tau )\,dx \,d\tau \\ &\qquad{}- \int _{s}^{T} \int _{\varOmega }\bigl(f\bigl(u(\tau )\bigr)-f\bigl(v(\tau )\bigr) \bigr)\omega _{t}( \tau )\,dx \,d\tau . \end{aligned} $$

(40)

Integrating (40) over $[0,T]$ with respect to s, we get

$$ \begin{aligned}[b] T E_{\omega }(t)&\leq \int _{0}^{T}E_{\omega }(s)\,ds+ \frac{1}{2} \int _{0}^{T} \int _{\varOmega } \bigl\Vert \nabla u(s) \bigr\Vert ^{m} \bigl\Vert \nabla\omega (s) \bigr\Vert ^{2}\,dx \\ &\quad{}+ \frac{m}{2} \int _{0}^{T} \int _{s}^{T} \int _{\varOmega } \bigl\Vert \nabla\omega (\tau ) \bigr\Vert ^{2} \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m-1}\nabla u_{t}(\tau )\,dx\,d \tau \,ds \\ &\quad{}- \int _{0}^{T} \int _{s}^{T} \int _{\varOmega }\bigl( \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m}- \bigl\Vert \nabla v(\tau ) \bigr\Vert ^{m}\bigr) \nabla v(\tau )\nabla \omega _{t}(\tau )\,dx\,d \tau \,ds \\ &\quad{}- \int _{0}^{T} \int _{s}^{T} \int _{\varOmega }\bigl(f\bigl(u(\tau )\bigr)-f\bigl(v(\tau )\bigr) \bigr) \omega _{t}(\tau )\,dx \,d\tau \,ds. \end{aligned} $$

(41)

Due to (10), (40), and Lemma 2.1, we obtain, for any $\delta >0$,

$$ \begin{aligned}[b] & \int _{0}^{T} \bigl\Vert \zeta ^{\tau } \bigr\Vert ^{2}_{\mathcal{M}}\,d\tau + \int _{0}^{T} \bigl\Vert \omega _{t}( \tau ) \bigr\Vert ^{2}\,d\tau \\ &\quad \leq C_{2}E_{\omega }(0) -C_{2} \int _{0}^{T} \int _{\varOmega }\bigl(f\bigl(u(\tau )\bigr)-f\bigl(v( \tau )\bigr) \bigr)\omega _{t}(\tau )\,dx \,d\tau \\ &\qquad{}+\delta T \operatorname{mes}(\varOmega )-\frac{C_{2}}{2} \int _{\varOmega } \bigl\Vert \nabla u(T) \bigr\Vert ^{m} \bigl\Vert \nabla \omega (T) \bigr\Vert ^{2}\,dx \\ &\qquad{}-C_{2} \int _{0}^{T} \int _{\varOmega }\bigl( \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m}- \bigl\Vert \nabla v(\tau ) \bigr\Vert ^{m}\bigr) \nabla v(\tau )\nabla \omega _{t}(\tau )\,dx \,d\tau \\ &\qquad{}-C_{2} \int _{0}^{T} \int _{\varOmega }\bigl(f\bigl(u(\tau )\bigr)-f\bigl(v(\tau )\bigr) \bigr)\omega _{t}( \tau )\,dx \,d\tau \\ &\qquad{}+\frac{mC_{2}}{2} \int _{0}^{T} \int _{\varOmega } \bigl\Vert \nabla \omega (\tau ) \bigr\Vert ^{2} \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m-1}\nabla u_{t}(\tau )\,dx \,d\tau , \end{aligned} $$

(42)

where $C_{2}$ is a constant which depends on δ, $\alpha _{0}$ and $k_{1}$.

Thus, from (37), (38) and (42) we have

$$\begin{aligned} \int _{0}^{T}E_{\omega }(t)\,dt&\leq C_{3}\delta T \operatorname{mes}( \varOmega )+C_{2}C_{3}E_{\omega }(0)- \frac{C_{2}C_{3}}{2} \int _{\varOmega } \bigl\Vert \nabla u(T) \bigr\Vert ^{m} \bigl\Vert \nabla \omega (T) \bigr\Vert ^{2}\,dx \\ &\quad{}+\frac{C_{2}C_{3}}{2} \int _{\varOmega } \bigl\Vert \nabla u(0) \bigr\Vert ^{m} \bigl\Vert \nabla\omega (0) \bigr\Vert ^{2}\,dx \\ &\quad{}-C_{2}C_{3} \int _{0}^{T} \int _{\varOmega }\bigl( \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m}- \bigl\Vert \nabla v(\tau ) \bigr\Vert ^{m}\bigr) \nabla v(\tau )\nabla \omega _{t}(\tau )\,dx\,d \tau \\ &\quad{}-C_{2}C_{3} \int _{0}^{T} \int _{\varOmega }\bigl(f\bigl(u(\tau )\bigr)-f\bigl(v(\tau )\bigr) \bigr) \omega _{t}(\tau )\,dx \,d\tau \\ &\quad{}+\frac{mC_{2}C_{3}}{2} \int _{0}^{T} \int _{\varOmega } \bigl\Vert \nabla \omega (\tau ) \bigr\Vert ^{2} \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m-1}\nabla u_{t}(\tau )\,dx \,d\tau \\ &\quad{}+ \int _{\varOmega }\omega _{t}(0)\omega (0)\,dx- \int _{\varOmega }\omega _{t}(T) \omega (T)\,dx \\ &\quad{}- \int _{0}^{T} \bigl\Vert \nabla u(s) \bigr\Vert ^{m} \bigl\Vert \nabla \omega (s) \bigr\Vert ^{2}\,ds \\ &\quad{}- \int _{0}^{T} \int _{\varOmega }\bigl( \bigl\Vert \nabla u(s) \bigr\Vert ^{m}- \bigl\Vert \nabla v(s) \bigr\Vert ^{m}\bigr) \nabla v(s)\nabla \omega (s)\,dx \,ds \\ &\quad{}- \int _{0}^{T} \int _{\varOmega }a(x) \bigl(g\bigl(u_{t}(s)\bigr)-g \bigl(v_{t}(s)\bigr)\bigr)\omega (s)\,dx \,ds \\ &\quad{}- \int _{0}^{T} \int _{\varOmega }\bigl(f\bigl(u(s)\bigr)-f\bigl(v(s)\bigr)\bigr)\omega (s)\,dx \,ds, \end{aligned}$$

(43)

where $C_{3}=\max \{\frac{3}{2},\frac{k(0)+1}{2}\}$. From (23) and the existence of the absorbing set, we get

$$\begin{aligned}& \int _{0}^{T} \int _{\varOmega }a(x) \bigl(g(u_{t})\bigr)u_{t} \,dx \,ds\leq C_{\rho }, \end{aligned}$$

(44)

$$\begin{aligned}& \int _{0}^{T} \int _{\varOmega }a(x) \bigl(g(v_{t})\bigr)v_{t} \,dx \,ds\leq C_{\rho }, \end{aligned}$$

(45)

where $C_{\rho }$ is a constant which depends on $\operatorname{mes}(\varOmega )$, $\|h\|^{2}$ and the size of $B_{0}$. By a similar method to that of (30) and (43), (44), we have

$$ \begin{aligned}[b] & \biggl\vert \int _{0}^{T} \int _{\varOmega }a(x)g\bigl(u_{t}(s)\bigr)\omega (s)\,dx \,ds \biggr\vert \\ &\quad \leq C^{\frac{q}{q+1}} \int _{0}^{T} \int _{\varOmega ( \Vert u_{t} \Vert \leq 1)} \bigl\vert a(x) \omega \bigr\vert \,dx \,ds \\ &\qquad{}+(2C)^{\frac{q}{q+1}}\biggl( \int _{0}^{T} \int _{\varOmega ( \vert u_{t} \vert \geq 1)}a(x)g(u_{t})u_{t}\,dx \,ds \biggr)^{ \frac{q}{q+1}}\\ &\qquad {}\times\biggl( \int _{0}^{T} \int _{\varOmega ( \vert u_{t} \vert \geq 1)}a(x) \vert \omega \vert ^{q+1}\,dx \,ds \biggr)^{\frac{1}{q+1}} \\ &\quad \leq C^{\frac{q}{q+1}} \int _{0}^{T} \int _{\varOmega }a(x) \vert \omega \vert \,dx \,ds+C_{ \rho }T^{\frac{1}{q+1}}, \end{aligned} $$

(46)

similarly

$$ \biggl\vert \int _{0}^{T} \int _{\varOmega }a(x)g\bigl(v_{t}(s)\bigr)\omega (s)\,dx \,ds \biggr\vert \leq C^{ \frac{q}{q+1}} \int _{0}^{T} \int _{\varOmega }a(x) \vert \omega \vert \,dx \,ds+C_{\rho }T^{ \frac{1}{q+1}}, $$

(47)

combining (41), (43), (46), (47), we have

$$ TE_{\omega }(T)\leq C_{B}+\varPhi _{T} \bigl(z_{0}^{1},z_{0}^{2}\bigr), $$

(48)

where

$$\begin{aligned}& \begin{aligned}[b] C_{B}&=C_{3}\delta T \operatorname{mes}(\varOmega )+C_{2}C_{3}E_{\omega }(0)+ \int _{\varOmega }\omega _{t}(0)\omega (0)\,dx- \int _{\varOmega }\omega _{t}(T) \omega (T) \,dx+2C_{\rho }T^{\frac{1}{q+1}} \\ &\quad{}+\frac{C_{2}C_{3}}{2} \int _{\varOmega } \bigl\Vert \nabla u(0) \bigr\Vert ^{m} \bigl\Vert \nabla\omega (0) \bigr\Vert ^{2}\,dx- \frac{C_{2}C_{3}}{2} \int _{\varOmega } \bigl\Vert \nabla u(T) \bigr\Vert ^{m} \bigl\Vert \nabla \omega (T) \bigr\Vert ^{2}\,dx, \end{aligned} \end{aligned}$$

(49)

$$\begin{aligned}& \begin{aligned}[b] \varPhi _{T}\bigl(z_{0}^{1},z_{0}^{2} \bigr)&= -C_{2}C_{3} \int _{0}^{T} \int _{\varOmega }\bigl( \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m}- \bigl\Vert \nabla v(\tau ) \bigr\Vert ^{m}\bigr) \nabla v(\tau )\nabla \omega _{t}(\tau )\,dx \,d\tau \\ &\quad{}-C_{2}C_{3} \int _{0}^{T} \int _{\varOmega }\bigl(f\bigl(u(\tau )\bigr)-f\bigl(v(\tau )\bigr) \bigr) \omega _{t}(\tau )\,dx \,d\tau \\ &\quad{}+\frac{mC_{2}C_{3}}{2} \int _{0}^{T} \int _{\varOmega } \bigl\Vert \nabla \omega (\tau ) \bigr\Vert ^{2} \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m-1}\nabla u_{t}(\tau )\,dx \,d\tau \\ &\quad{}- \int _{0}^{T} \int _{\varOmega }\bigl( \bigl\Vert \nabla u(s) \bigr\Vert ^{m}- \bigl\Vert \nabla v(s) \bigr\Vert ^{m}\bigr) \nabla v(s)\nabla \omega (s)\,dx \,ds \\ &\quad{}- \int _{0}^{T} \int _{\varOmega }\bigl(f\bigl(u(s)\bigr)-f\bigl(v(s)\bigr)\bigr)\omega (s)\,dx \,ds - \int _{0}^{T} \bigl\Vert \nabla u(s) \bigr\Vert ^{m} \bigl\Vert \nabla \omega (s) \bigr\Vert ^{2}\,ds \\ &\quad{}+2C^{\frac{q}{q+1}} \int _{0}^{T} \int _{\varOmega }a(x) \bigl\vert \omega (s) \bigr\vert \,dx \,ds + \frac{1}{2} \int _{0}^{T} \int _{\varOmega } \bigl\Vert \nabla u(s) \bigr\Vert ^{m} \bigl\Vert \nabla\omega (s) \bigr\Vert ^{2}\,dx \,ds \\ &\quad{}+\frac{m}{2} \int _{0}^{T} \int _{s}^{T} \int _{\varOmega } \bigl\Vert \nabla\omega (\tau ) \bigr\Vert ^{2} \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m-1}\nabla u_{t}(\tau )\,dx\,d \tau \,ds \\ &\quad{}- \int _{0}^{T} \int _{s}^{T} \int _{\varOmega }\bigl( \bigl\Vert \nabla u(\tau ) \bigr\Vert ^{m}- \bigl\Vert \nabla v(\tau ) \bigr\Vert ^{m}\bigr) \nabla v(\tau )\nabla \omega _{t}(\tau )\,dx\,d \tau \,ds \\ &\quad{}- \int _{0}^{T} \int _{s}^{T} \int _{\varOmega }\bigl(f\bigl(u(\tau )\bigr)-f\bigl(v(\tau )\bigr) \bigr) \omega _{t}(\tau )\,dx \,d\tau \,ds. \end{aligned} \end{aligned}$$

(50)

Then we have

$$ E_{\omega }(T)\leq \frac{C_{B}}{T}+\frac{1}{T}\varPhi _{T}\bigl(z_{0}^{1},z_{0}^{2} \bigr). $$

(51)

4.2 Asymptotic compactness

In this subsection, following the argument in [9, 10], we will prove the asymptotic compactness of the semigroup $\{S(t)\}_{t\geq 0}$ in $\mathcal{H}$, which is given in the following theorem.

Theorem 4.1

Under assumptions(1)–(4), the semigroup$\{S(t)\}_{t\geq 0}$to systems (6)–(9) is asymptotically compact in$\mathcal{H}$.

Proof

since the semigroup $\{S(t)\}_{t\geq 0}$ has a bounded absorbing set, for every fixed $\varepsilon >0$, we can choose that $\delta \leq \frac{\varepsilon }{2C_{3}\operatorname{mes}(\varOmega )}$, and then let T become so large that

$$ \frac{C_{B}}{T}\leq \varepsilon . $$

(52)

Hence, thanks to Theorem 2.2, we only need to verify that the function $\varPhi _{T}(z_{0}^{1},z_{0}^{2})$ defined in (50) belongs to $C(B_{1})$ for each fixed T. and we claim that

$$ \bigl\Vert S(t)z_{0}^{1}-S(t)z_{0}^{2} \bigr\Vert _{\mathcal{H}}\leq \varepsilon +\varPhi _{T} \bigl(z_{0}^{1},z_{0}^{2}\bigr),\quad \forall z_{0}^{1},z_{0}^{2}\in B. $$

(53)

Here $(u(t),u_{t}(t),\eta )=S(t)z_{0}^{1}$ and $(v(t),v_{t}(t),\xi )=S(t)z_{0}^{1}$ are the solutions of (6)–(9) with respect to initial $z_{0}^{1},z_{0}^{2}\in B_{1}$. Then, since $C(B_{1})$ is a bounded positively invariant set in $\mathcal{H}$, it follows that $(u_{n},u_{n_{t}},\eta ^{n})$ is uniformly bounded in $\mathcal{H}$. We have

$$\begin{aligned}& u_{n}\rightarrow u \quad \text{weakly star in } L^{\infty } \bigl(0,T;H_{0}^{1}( \varOmega )\bigr), \end{aligned}$$

(54)

$$\begin{aligned}& u_{n_{t}}\rightarrow u_{t} \quad \text{weakly star in } L^{\infty }\bigl(0,T;L^{2}( \varOmega )\bigr). \end{aligned}$$

(55)

Then, by the compact embedding $H_{0}^{1}(\varOmega )\hookrightarrow L^{k}(\varOmega )$, we have

$$\begin{aligned}& u_{n}\rightarrow u \quad \text{strongly in } L^{2} \bigl(0,T;L^{2}(\varOmega )\bigr), \end{aligned}$$

(56)

$$\begin{aligned}& u_{n}\rightarrow u \quad \text{strongly in } L^{k} \bigl(0,T;L^{k}(\varOmega )\bigr), \end{aligned}$$

(57)

where $k\leq \frac{2n}{n-2}$, therefore from (56) we have

$$\begin{aligned}& \lim_{l\rightarrow \infty }\lim_{k\rightarrow \infty } \int _{0}^{T} \int _{\varOmega }\bigl(f\bigl(u_{l}(\tau )\bigr)-f \bigl(u_{k}(\tau )\bigr)\bigr) \bigl(u_{l_{t}}(\tau )-u_{k_{t}}( \tau )\bigr)\,dx \,d\tau =0, \end{aligned}$$

(58)

$$\begin{aligned}& \lim_{l\rightarrow \infty }\lim_{k\rightarrow \infty } \int _{0}^{T} \int _{\varOmega }\bigl(f\bigl(u_{l}(\tau )\bigr)-f \bigl(u_{k}(\tau )\bigr)\bigr) \bigl(u_{l}(\tau )-u_{k}( \tau )\bigr)\,dx \,d\tau =0, \end{aligned}$$

(59)

then from (57) and (10), we obtain

$$ \lim_{l\rightarrow \infty }\lim_{k\rightarrow \infty } \int _{0}^{T} \int _{\varOmega }a(x) \bigl\vert u_{l}(s)-u_{k}(s) \bigr\vert \,dx \,ds=0. $$

(60)

Finally, we follow a similar argument to the ones given in [9, 10]. We have

$$\begin{aligned}& \lim_{l\rightarrow \infty }\lim_{k\rightarrow \infty } \int _{0}^{T} \int _{\varOmega } \bigl\Vert \nabla u_{l}(\tau )-\nabla u_{k}(\tau ) \bigr\Vert ^{2}\nabla u_{l}( \tau )\nabla u_{k_{t}}(\tau )\,dx \,d\tau =0, \end{aligned}$$

(61)

$$\begin{aligned}& \lim_{l\rightarrow \infty }\lim_{k\rightarrow \infty } \int _{0}^{T} \int _{\varOmega }\bigl( \bigl\Vert \nabla u_{l}(\tau ) \bigr\Vert ^{2}- \bigl\Vert \nabla u_{k}(\tau ) \bigr\Vert ^{2}\bigr) \nabla u_{l}(\tau ) (\nabla u_{l}-\nabla u_{k})\,dx \,d\tau =0, \end{aligned}$$

(62)

$$\begin{aligned}& \lim_{l\rightarrow \infty }\lim_{k\rightarrow \infty } \int _{0}^{T} \bigl\Vert \nabla u_{l}(t) \bigr\Vert ^{2} \bigl\Vert \nabla u_{l}(t)-\nabla u_{k}(t) \bigr\Vert ^{2}\,dt=0, \end{aligned}$$

(63)

$$\begin{aligned}& \lim_{l\rightarrow \infty }\lim_{k\rightarrow \infty } \int _{0}^{T} \bigl\Vert u_{l}(t)-u_{k}(t) \bigr\Vert ^{2}\,dt=0, \end{aligned}$$

(64)

$$\begin{aligned}& \lim_{l\rightarrow \infty }\lim_{k\rightarrow \infty } \int _{0}^{T} \bigl\Vert \nabla u_{l}(t)-\nabla u_{k}(t) \bigr\Vert ^{2}\,dt=0. \end{aligned}$$

(65)

Finally, combining (58)–(65) we get $\varPhi (\cdot ,\cdot )\in C(B_{1})$. □

4.3 Existence of global attractor

Theorem 4.2

Under assumptions(1)–(4), then problems (6)–(9) have a global attractor in$\mathcal{H}$, which is invariant and compact.

Proof

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