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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.

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, 79], 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).

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)

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. □

Existence of the global attractor in \(\mathcal{H}\)

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)

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 \(\varPhi (\cdot ,\cdot )\in C(B_{1})\). □

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. □

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The project is supported by the National Natural Science Foundation of China (Grant No. 11872264, the role of the funding lies in the collection of data, the analysis of the paper and writing the manuscript).

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This paper is mainly completed by SZ, JZ and HW dealt with the nonlinear damping term as proving the existence of a bounded absorbing set. All authors read and approved the final manuscript.

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Zhang, S., Zhang, J. & Wang, H. Global attractors for Kirchhoff wave equation with nonlinear damping and memory. Bound Value Probl 2020, 116 (2020). https://doi.org/10.1186/s13661-020-01413-5

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Keywords

  • Memory
  • Nonlinear damping
  • Global attractors