\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 278, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/278\hfil  Non-autonomous Berger equation]
{Existence of attractors for the non-autonomous Berger equation
with nonlinear damping}

\author[L. Yang, X. Wang \hfil EJDE-2017/278\hfilneg]
{Lu Yang, Xuan Wang}

\address{Lu Yang (corresponding author)\newline
School of Mathematics and Statistics,
Lanzhou University,
Lanzhou, 730000, China. \newline
Key Laboratory of Applied Mathematics and Complex Systems,
 Gansu Province, China}
\email{yanglu@lzu.edu.cn}

\address{Xuan Wang \newline
College of Mathematics and Statistics,
Northwest Normal University,
 Lanzhou, 730070, China}
\email{wangxuan@nwnu.edu.cn}

\dedicatory{Communicated by Zhaosheng Feng}

\thanks{Submitted September 23, 2016. Published November 8, 2017.}
\subjclass[2010]{35B40, 35B41, 35L70}
\keywords{Uniform attractor; Berger
 equation; nonlinear damping}

\begin{abstract}
 In this article, we study the long-time behavior of the non-autonomous
 Berger equation with nonlinear damping. We prove the existence of a
 compact uniform attractor for the Berger equation with nonlinear
 damping in the space $(H^2(\Omega)\cap H_0^1(\Omega))\times L^2(\Omega)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In this article, we consider the non-autonomous Berger
equation with nonlinear damping,
\begin{equation}\label{1.1.1}
\begin{gathered}
 u_{tt}+\gamma g(u_t)+\Delta^2 u+(\Gamma-\int_{\Omega}|\nabla u|^2dx)\Delta u=p(x,t),
\quad  x\in \Omega, \\
 u|_{\partial\Omega}=\Delta u|_{\partial\Omega}=0,  \\
 u(x,\tau)=u_{\tau}^{0}(x),\quad u_t(x,\tau)=u_{\tau}^{1}(x).
 \end{gathered}
\end{equation}
Here $\Omega\subset \mathbb{R}^n$ is a bounded domain with a
sufficiently smooth boundary; $\gamma>0$, and $\Gamma$ are constants. 
The damping function $g\in C^1(\mathbb{R})$ satisfies
\begin{gather}\label{1.1.2}
 g(0)=0,\quad g \text{ is strictly increasing},\quad \liminf_{|s|\to\infty} g'(s)>0,\\
\label{1.1.3}
|g(s)|\le  C(1+|s|^q),
\end{gather}
with $1\leq q<\infty$ if $n\leq4$, and $1\leq q< \frac{n+4}{n-4}$ if
$n>4$. The external force $p(x,t)$ satisfies
\begin{gather}\label{1.1.4}
 p(x,t)\in L^{\infty}(\mathbb{R};L^2(\Omega)), \\
\label{1.1.5}
 \partial_{t}p\in L_{b}^{r}(\mathbb{R};L^r(\Omega)) \text{ with }
 r>\frac{2n}{n+4}.
\end{gather}

Equation \eqref{1.1.1} describes the nonlinear oscillation of a
plate. The function $u(x,t)$ measures the deflection of the plate at
the point $x$ and the moment of time $t$. The boundary condition
implies that the edges of the plate are hinged. The function
$p(x,t)$ describes the transverse load on the plate. The parameter
$\Gamma$ is proportional to the value of compressive force acting in
the plane of the plate. The value $\gamma$ describes the environment
resistance.

In this paper, we consider the non-autonomous system \eqref{1.1.1}
via the uniform attractor of the corresponding family of processes
$\{U_\sigma (t,\tau)\}$, $\sigma\in \Sigma$. For the Berger
equation, the feature of the model \eqref{1.1.1} is that: (i) the
equation does not account for rotational inertia, (i.e., $\Delta
u_{tt}$), (ii) the damping is nonlinear, and (iii) the external forcing
$p(x,t)$ is not translation compact in
$L_{loc}^{2}(\mathbb{R};L^2(\Omega))$.

For the autonomous case, if $n=1$,  equation \eqref{1.1.1}
becomes a well-known beam equation which was treated by many
authors, see, for example, \cite{ball3,BM2,SYD} for the linear
damping and \cite{Chow} for the nonlinear damping. In
\cite{Marzocchi}, Marzocchi obtained the global attractor of beam
equation with linear strong damping (i.e. $u_{xxxxt}$). Sell and You
\cite{SY} showed the existence of the
global attractor for \eqref{1.1.1} with linear
damping in the one-dimensional case. In \cite{Naboka}, Naboka
considered the existence of the
global attractor of two coupling berger plate
equations with linear damping. Later, Lasiecka and Chueshov
\cite{CHL1} gave a detailed discussion about the existence of the
global attractor for the equation \eqref{1.1.1} in the space
$(H^2(\Omega)\cap H_0^1(\Omega))\times L^2(\Omega)$. Ma and Narciso
\cite{T.F. Ma} established the global attractor for the nonlinear
beam equation with nonlinear damping and source terms. The existence
of the exponential attractor for the plate equation was proved in
\cite{maqiaozhen}.

In the case of non-autonomous system, for the wave equation, Sun et
al  \cite{SCD} discussed the dynamical behavior of the
non-autonomous wave equation. The random wave equation has been
studied in \cite{yangmeihua}. The asymptotic behavior of the
solution for the non-autonomous viscoelastic equation was considered
in \cite{Qin1,Qin2}.

The non-autonomous wave equation has attracted much attention in
recent years. However, the non-autonomous plate equation with
nonlinear damping is less discussed, especially for the Berger
equation. This paper is devoted to the dynamical behavior of the
solution of the equation \eqref{1.1.1}.

In this article, inspired by the ideas in \cite{CHL1,Khanmamedov,SCD},
we prove the existence of a compact uniform attractor for
problem \eqref{1.1.1} in the apace $(H^2(\Omega)\cap
H_0^1(\Omega))\times L^2(\Omega)$. The main emphasis is placed on
the external force and the nonlinear dissipation.

This article is organized as follows: In Section 2, we recall some
results about function space and uniform attractor we will use in
this paper; In Section 3, we give the existence of uniformly
absorbing set in $(H^2(\Omega)\cap H_0^1(\Omega))\times
L^2(\Omega)$; In the last Section, we derive uniform asymptotic
compactness of the corresponding family of processes
 $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$
 generated by the problem \eqref{1.1.1}.

\section{Preliminaries}

In this section, we recall some fundamental concepts
about the non-autonomous dynamical system, see more details in
\cite{CV}.

Let $X$ be a Banach space, and $\Sigma$ be a parameter set.
The operator $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ is said to be
a family of processes in $X$ with symbol space $\Sigma$ if for any
$\sigma\in\Sigma$,
\begin{gather}\label{4.4.1}
 U_{\sigma}(t,s)\circ
 U_{\sigma}(s,\tau)=U_{\sigma}(t,\tau),\quad \forall  t\ge s\ge
 \tau,\;\tau\in \mathbb{R}, \\
\label{4.4.2}
 U_{\sigma}(\tau,\tau)=\operatorname{Id},\quad \forall \tau\in
 \mathbb{R},
\end{gather}
where $\operatorname{Id}$ is the identity.
Let $\{T(s)\}_{s\ge0}$ be the translation semigroup on $\Sigma$, we
say that a family of processes
$\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ satisfies the translation
identity if
\begin{gather}\label{4.4.3}
 U_{\sigma}(t+s,\tau+s)=U_{T(s)\sigma}(t,\tau),\quad \forall \sigma\in\Sigma,\;
t\ge\tau,\,\tau\in  \mathbb{R},\; s\ge0, \\
\label{4.4.4}
 T(s)\Sigma=\Sigma,\quad \forall  s\ge0.
\end{gather}
By $\mathcal{B}(X)$ we denote the collection of all bounded sets of
$X$, and $\mathbb{R}_{\tau}=\{t\in \mathbb{R},t\ge\tau\}$.


\begin{definition}[\cite{CV}] \rm
A bounded set $B_0 \in \mathcal{B}(X)$ is said to be a bounded uniformly 
(w.r.t. $\sigma\in\Sigma$) absorbing set for
$\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ if for any
$\tau\in\mathbb{R}$ and $B \in \mathcal{B}(X)$ there exists
$T_0=T_0(B,\tau)$ such that
$\cup_{\sigma\in\Sigma}U_{\sigma}(t;\tau)B\subset B_0$ for all
$t\ge T_0$.
\end{definition}


\begin{definition}[\cite{CV}] \rm
A set $\mathcal{A} \subset X$ is said
to be uniformly (w.r.t $\sigma\in\Sigma$) attracting for the family
of processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ if for any
fixed $\tau\in\mathbb{R}$ and any $B \in \mathcal{B}(X)$
$$
\lim_{t\to+\infty}\Big(\sup_{\sigma\in\Sigma}\operatorname{dist}(U_{\sigma}
(t;\tau)B;\mathcal{A})\Big)=0,
$$
$\operatorname{dist}(\cdot,\cdot)$ is the usual Hausdorff semidistance in $X$
between two sets.
\end{definition}

\begin{definition}[\cite{CV}] \rm
A closed set $\mathcal{A}_\Sigma
\subset X$ is said to be the uniform (w.r.t $\sigma\in\Sigma$)
attractor of the family of processes
$\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ if it is uniformly (w.r.t
$\sigma\in\Sigma$) attracting (attracting property) and contained in
any closed uniformly (w.r.t $\sigma\in\Sigma$) attracting set
$\mathcal{A}'$ of the family of processes
$\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$: $\mathcal{A}_\Sigma
\subseteq \mathcal{A}'$ (minimality property).
\end{definition}

\begin{definition}[\cite{CV}] \rm
A function $\varphi$ is said to be
translation bounded in $L_{loc}^r(\mathbb{R};X)$, if
$$
\|\varphi\|_b^r=\sup_{t\in\mathbb{
R}}\int_t^{t+1}\|\varphi\|_X^rds<+\infty.
$$
Denote by $L_{b}^r(\mathbb{R};X)$ the set of all translation bounded
functions in $L_{loc}^r(\mathbb{R};X)$.
\end{definition}

Next we recall some properties of the nonlinear damping
function $g$.

\begin{lemma}[\cite{Feireisl,Khanmamedov}]\label{yingli1}
Let $g(\cdot)$ satisfy condition \eqref{1.1.3}. Then for any
$\delta>0$ there exists $C_{\delta}>0$, such that
$$
|u-v|^2\le\delta+C_{\delta}(g(u)-g(v))(u-v)\,\,\quad\quad\quad\text{
for }u,v\in R.
$$
\end{lemma}

Hereafter, the norm in $L^2(\Omega)$ is denoted by $\|\cdot\|$.
$H^s(\Omega)$ stands for the usual Sovolev space when $s\ge0$ with
the form $\|u\|_s$. $C,$ $C_i$ denote a general positive constant,
$i=1,\dots$, which will be different in different estimates.


\section{Existence of uniformly absorbing set}

\subsection{Setting of the problem}

Similar to the autonomous case (e.g., see \cite{CHL1}), we can
obtain the following existence and uniqueness results and the
time-dependent terms make no essential complications.

\begin{theorem}\label{dingli0.1}
Let $\Omega$ be a bounded domain of $\mathbb{R}^n$ with smooth
boundary, $g$ satisfies \eqref{1.1.2}-\eqref{1.1.3}, $p(x,t)\in
L^{\infty}(\mathbb{R};L^2(\Omega))$. Then for any initial data
$(u_{\tau}^0,u_{\tau}^1)\in(H^2(\Omega)\cap H_0^1(\Omega))\times
L^2(\Omega)$, the problem \eqref{1.1.1} has an unique solution
$u(t)$ which satisfies $(u(t),u_t(t))\in
C(\mathbb{R}_{\tau};(H^2(\Omega)\cap H_0^1(\Omega))\times
L^2(\Omega))$ and $\partial_{tt}u(t)\in
L^2_{loc}(\mathbb{R}_{\tau};H^{-2}(\Omega))$.
\end{theorem}


Let $y(t)=(u(t),u_t(t))$, $y_{\tau}=(u_{\tau}^0,u_{\tau}^1),
E_0=(H^2(\Omega)\cap H_0^1(\Omega))\times L^2(\Omega)$ with finite
energy norm
$$
\|y\|_{E_0}=\|\Delta u\|^2+\|u_t\|^2.
$$
Then system \eqref{1.1.1} is equivalent to:
\begin{equation}\label{4.4.5}
 \begin{gathered}
 \partial_tu_t=-\Delta^2 u-\gamma g(u_t)
-(\Gamma-\int_{\Omega}|\nabla u|^2dx)\Delta u+p(x,t), \quad \text{for }
  t\ge \tau, \\
 u|_{\partial\Omega}=\frac{\partial}{\partial \nu}u|_{\partial\Omega}=0,\quad
 u(x,\tau)=u_{\tau}^{0}(x),\quad
 u_t(x,\tau)=u_{\tau}^{1}(x).
 \end{gathered}
\end{equation}
We can also rewrite \eqref{4.4.5} in the operator form:
\begin{equation}\label{4.4.6}
 \partial_ty=A_{\sigma(t)}(y),\quad y|_{t=\tau}=y_{\tau},
\end{equation}
where $\sigma(t)=p(t)$ is symbol of equation \eqref{4.4.6}. We now
define the symbol space for \eqref{4.4.6}, take a fixed symbol
$\sigma_0(s)=p_0(s)$, $p_0\in L^{\infty}(\mathbb{R};L^2(\Omega))\cap
W_b^{1,r}(\mathbb{R};L^r(\Omega))$ for some $r>\frac{2n}{n+4}$, and
set
\begin{gather}\label{4.4.7}
 \Sigma_0=\{p_0(x,t+h)\quad h\in\mathbb{R}\}, \\
\label{4.4.8}
 \Sigma\text{is the }*\text{-weakly closure of
 }\Sigma_0\text{ in }L^{\infty}(\mathbb{R};L^2(\Omega))\cap
W_b^{1,r}(\mathbb{R};L^r(\Omega)).
\end{gather}
Then we have the following properties.

\begin{proposition}  
$\Sigma$ is bounded in 
$L^{\infty}(\mathbb{R};L^2(\Omega))\cap
W_b^{1,r}(\mathbb{R};L^r(\Omega))$, and for any $\sigma\in\Sigma$,
the following estimate holds
$$
\|\sigma\|_{L^{\infty}(\mathbb{R};L^2(\Omega))\cap
W_b^{1,r}(\mathbb{R};L^r(\Omega))}
\le\|p_0\|_{L^{\infty}(\mathbb{R};L^2(\Omega))\cap
W_b^{1,r}(\mathbb{R};L^r(\Omega))}.
$$
\end{proposition}

Thus, from Theorem \ref{dingli0.1}, we know that \eqref{1.1.1} is
well posed for all $\sigma(s)\in\Sigma$ and generates a family of
processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ given by the
formula $U_{\sigma}(t,\tau)y_{\tau}=y(t)$. The $y(t)$ is the
solution of \eqref{1.1.1}-\eqref{1.1.5} and
$\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ satisfies
\eqref{4.4.1}-\eqref{4.4.2}. At the same time, due to the unique
solvability, we know $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$
satisfies the translation identity \eqref{4.4.3}-\eqref{4.4.4}.

In what follows, we denote by
$\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ the family of processes
which is generated by \eqref{4.4.6}-\eqref{4.4.8}.
Next we recall some criterion developed in \cite{SCD}.

\begin{definition}[\cite{SCD}] \rm
Let $X$ be a Banach space, $B$ a
bounded subset of $X$ and $\Sigma$ a symbol (or parameter) space. We
call a function $\phi(\cdot,\cdot;\cdot,\cdot)$, defined on $(X
\times X)\times (\Sigma\times\Sigma)$, to be a contractive function
on $B\times B$ if for any sequence $\{x_n\}_{n=1}^{\infty}\subset B$
and any $\{\sigma_n\}\subset\Sigma$, there is a subsequence
$\{x_{n_k}\}_{k=1}^{\infty}\subset\{x_n\}_{n=1}^{\infty}$ and
$\{\sigma_{n_k}\}_{k=1}^{\infty}\subset\{\sigma_n\}_{n=1}^{\infty}$
such that
$$
\lim_{k\to\infty}\lim_{l\to\infty}\phi(x_{n_k},x_{n_l};\sigma_{n_k},\sigma_{n_l})=0.
$$
 We
denote the set of all contractive functions on $B \times B$ by
$\operatorname{contr}(B, \Sigma)$.
\end{definition}


\begin{theorem}[\cite{SCD}]\label{contr}
 Let
$\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ be a family of processes
which satisfies the translation identity \eqref{4.4.3}-\eqref{4.4.4}
on Banach space $X$ and has a bounded uniformly (w.r.t.
$\sigma\in\Sigma$) absorbing set $B_0\subset X$. Moreover, assume
that for any $\varepsilon>0$ there exist $T=T(B_0,\varepsilon)$ and
$\phi_T\in \operatorname{contr}(B_0,\Sigma)$ such that
$$
\|U_{\sigma_1}(T,0)x-U_{\sigma_2}(T,0)y\|\le
\varepsilon+\phi_T(x,y;\sigma_1,\sigma_2),\quad \forall x,y\in
B_0,\quad \forall \sigma_1,\sigma_2\in\Sigma.
$$
Then $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ is uniformly (w.r.t.
$\sigma\in\Sigma$) asymptotically compact in $X$.
\end{theorem}

Applying \cite[Proposition 7.1]{Robinson}, we
obtain the following results.

\begin{proposition} \label{proposition1}
Let $p\in L^{\infty}(\mathbb{R};L^2(\Omega))\cap
W^{1,r}_{b}(\mathbb{R};L^r(\Omega))$ $(r>\frac{2n}{n+4})$. Then there
is an $M>0$ such that
$$
\sup_{t\in\mathbb{R}}\|p(x,t+s)\|_{L^2(\Omega)}\le M \quad \text{for all } 
s\in \mathbb{R}.
$$
\end{proposition}

\begin{proposition}\label{proposition2}
Let $s_i\in \mathbb{R}$ $(i=1,2,\dots)$, 
$p\in L^{\infty}(\mathbb{R};L^2(\Omega))\cap
W^{1,r}_{b}(\mathbb{R};L^r(\Omega))$ $(r>\frac{2n}{n+4})$,
$\{u_n(t): t\ge0,n=1,2,\dots\}$ be bounded in 
$H^2(\Omega)\cap H_0^1(\Omega)$, and for any
$T_1>0$, $\{u_{n_t}(t) \big|\; n=1,2,\dots\}$ bounded in $
L^{\infty}(0,T_1;L^2(\Omega))$. Then for any $T>0$, there exist
subsequences $\{u_{n_k}\}_{k=1}^{\infty}$ of
$\{u_{n}\}_{n=1}^{\infty}$ and $\{s_{n_k}\}_{k=1}^{\infty}$ of
$\{s_{n}\}_{n=1}^{\infty}$ such that
$$
\lim_{k\to\infty}\lim_{l\to\infty}\int_{0}^{T}
\int_{s}^{T}\int_{\Omega}(p(x,\tau+s_{n_k})-p(x,\tau+s_{n_l}))(u_{n_k}-u_{n_l})_t(\tau)\,dx\,d\tau\,ds=0.
$$
\end{proposition}

\subsection {Uniformly  absorbing set in
$(H^2(\Omega)\cap H_0^1(\Omega))\times L^2(\Omega)$}

In this subsection, we start with the following result on the
existence of uniformly (w.r.t. $\sigma\in\Sigma$) absorbing set in
$(H^2(\Omega)\cap H_0^1(\Omega))\times L^2(\Omega)$. The proof is
similar to the autonomous case \cite{CHL1}, so we omit it here.

\begin{theorem}\label{dingli0.2}
Assume that $g$ satisfies \eqref{1.1.2}-\eqref{1.1.3}. If 
\[
p_0\in L^{\infty}(\mathbb{R};L^2(\Omega))\cap
W_b^{1,r}(\mathbb{R};L^r(\Omega))\quad\text{for some }r>\frac{2n}{n+4}
\]
and $\Sigma$ is defined by \eqref{4.4.8}, then the family of processes
$\{U_{\sigma}(t,\tau)\}$, $\sigma\in\Sigma$ corresponding to problem
\eqref{1.1.1} has a bounded uniformly (w.r.t. $\sigma\in\Sigma$)
absorbing set in $(H^2(\Omega)\cap H_0^1(\Omega))\times
L^2(\Omega)$.
\end{theorem}

\section{Uniform (w.r.t. $\sigma\in\Sigma$) asymptotic compactness in $(H^2(\Omega)\cap
H_0^1(\Omega))\times L^2(\Omega)$}

In this section, we first prove some a priori estimates about the
energy inequalities based on the idea presented in
\cite{CHL1,Khanmamedov,SCD}. Then, we establish the uniform (w.r.t.
$\sigma\in\Sigma$) asymptotic compactness in $E_0$.

For convenience, we   denote by $B_0$ the bounded uniformly
(w.r.t. $\sigma\in\Sigma$) absorbing set obtained in Theorem
\ref{dingli0.2}, and without loss of generality, we assume that
$\gamma\equiv1$ from now on.
Hereafter, we   use the notation
$$
E_w(t)=\frac{1}{2}\int_{\Omega}|w_t(t)|^2+\frac{1}{2}\int_{\Omega}|\triangle
w(t)|^2.
$$

\subsection{A priori estimates}

The main purpose of this part is to establish
\eqref{4.013}-\eqref{4.015}, which will be used to obtain the
uniform (w.r.t. $\sigma\in\Sigma$) asymptotic compactness. Based on
the technique in \cite{CHL1,Khanmamedov,SCD}, we have the following
subsequent procedure.

For any $(u_0^i,v_0^i)\in B_0$, let $(u_i(t),u_{i_t}(t))$ be the
corresponding solution to $\sigma_i$ with respect to initial data
$(u_0^i,v_0^i), i=1,2,$ that is, $(u_i(t),u_{i_t}(t))$ is the
solution of the following equation
\begin{equation}\label{4.00}
 \begin{gathered}
 u_{tt}+ g(u_t)+\Delta^2 u+(\Gamma-\int_{\Omega}|\nabla u|^2dx)\Delta u
=\sigma_i(x,t), \\
 u|_{\partial\Omega}=\frac{\partial}{\partial \nu}u|_{\partial\Omega}=0 \\
 (u(0),u_t(0))= (u_0^i,v_0^i).
 \end{gathered}
\end{equation}

\begin{lemma}\label{lemma1}
Assume that $g$ satisfies \eqref{1.1.2}--\eqref{1.1.3}. Then for any
fixed $T>0$, there exist a constant $C_{M,T}$ and a function
$\phi_T=\phi_T((u_0^1,\upsilon_0^1),(u_0^2,\upsilon_0^2);\sigma_1,\sigma_2)$
such that
$$
\|u_1(T)-u_2(T)\|_{E_0}\leq
C_{M,T}+\phi_T((u_0^1,\upsilon_0^1),(u_0^2,\upsilon_0^2);\sigma_1,\sigma_2),
$$
where $C_{M,T}$ and $\phi_T$ depend on $T$.
\end{lemma}

\begin{proof}
For convenience, we denote
\begin{gather*}
p_i(t)=\sigma_i(x,t), \quad t\ge 0,\; i=1,2, \\
w(t)=u_1(t)-u_2(t).
\end{gather*}
Then $w(t)$ satisfies
\begin{equation}\label{4.01}
 \begin{gathered}
\begin{aligned}
&w_{tt}+ g(u_{1_t})- g(u_{2_t})+\Delta^2 w
-\Big(\int_{\Omega}|\nabla u_1|^2dx\Delta u_1-\int_{\Omega}|\nabla u_2|^2dx\Delta
 u_2\Big)\\
&+\Gamma \Delta w
 =p_1(t)-p_2(t),
\end{aligned} \\
 w|_{\partial\Omega}=\frac{\partial}{\partial \nu}w|_{\partial\Omega}=0, \\
 (w(0),w_t(0))= (u_0^1,v_0^1)-(u_0^2,v_0^2).
 \end{gathered}
\end{equation}
Multiplying \eqref{4.01} by $w_t$ and integrating over
$[s,T]\times\Omega$, we obtain
\begin{equation} \label{4.02}
\begin{aligned}
 & E_{w}(T)-E_{w}(s)+\int_{s}^{T}\int_{\Omega}(g(u_{1_t}(\tau))-g(u_{2_t}(\tau)))
 w_t(\tau)\,dx\,d\tau \\
 &=\int_{s}^{T}\int_{\Omega}(\|\nabla u_1(\tau)\|^2\Delta u_1(\tau)
 -\|\nabla u_2(\tau)\|^2\Delta u_2(\tau))w_t(\tau)\,dx\,d\tau \\
 &\quad +\frac{1}{2} \Gamma\ \int_{\Omega}|\nabla w(T)|^2dx
-\frac{1}{2}\Gamma \int_{\Omega}|\nabla  w(s)|^2dx
+\int_{s}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,d\tau,
\end{aligned}
\end{equation}
where $0\leq s\leq T$. Then we have
\begin{equation} \label{4.03}
\begin{aligned}
&\int_{0}^{T}\int_{\Omega}(g(u_{1_t}(\tau))-g(u_{2_t}(\tau)))w_t(\tau)\,dx\,d\tau \\
&\leq E_{w}(0)+\frac{1}{2}\Gamma\ \int_{\Omega}|\nabla
w(T)|^2dx-\frac{1}{2}\Gamma \int_{\Omega}|\nabla w(0)|^2dx \\
&\quad +\int_{0}^{T}\int_{\Omega}(\|\nabla u_1(\tau)\|^2\Delta
u_1(\tau)-\|\nabla u_2(\tau)\|^2\Delta u_2(\tau))w_t(\tau)\,dx\,d\tau \\
&\quad +\int_{0}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,d\tau.
\end{aligned}
\end{equation}
Combining this with Lemma \ref{yingli1}, we obtain that for any
$\delta>0$,
\begin{equation} \label{4.04}
\begin{aligned}
&\int_{0}^{T}\int_{\Omega}|w_t(\tau)|^2\,dx\,d\tau\\
& \leq \delta T\operatorname{meas}(\Omega)+C_{\delta}E_{w}(0)
 +\frac{1}{2}C_{\delta}\Gamma
\int_{\Omega}|\nabla w(T)|^2dx \\
&\quad -\frac{1}{2}C_{\delta}\Gamma \int_{\Omega}|\nabla w(0)|^2dx
 +C_{\delta}\int_{0}^{T}\int_{\Omega}\Big(\|\nabla u_1(\tau)\|^2\Delta
u_1(\tau) \\
&\quad -\|\nabla u_2(\tau)\|^2\Delta u_2(\tau)\Big)w_t(\tau)\,dx\,d\tau
+C_{\delta}\int_{0}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,d\tau.
\end{aligned}
\end{equation}
Secondly, multiplying \eqref{4.01} by $w$ and integrating over
$[0,T]\times\Omega$, we obtain
\begin{equation} \label{4.05}
\begin{aligned}
&\int_{0}^{T}\int_{\Omega}|\triangle
w(s)|^2\,dx\,ds+\int_{\Omega}w_t(T)w(T)dx-\Gamma
\int_{0}^{T}\int_{\Omega}|\nabla
w(s)|^2\,dx\,ds \\
&=\int_{0}^{T}\int_{\Omega}|w_t(s)|^2\,dx\,ds
 -\int_{0}^{T}\int_{\Omega}(g(u_{1_t}(s))-g(u_{2_t}(s)))w(s)\,dx\,ds \\
&\quad +\int_{\Omega}w_t(0)w(0)dx
 +\int_{0}^{T}\int_{\Omega}\Big(\|\nabla u_1(s)\|^2\Delta u_1(s) \\
&\quad -\|\nabla u_2(s)\|^2\Delta
u_2(s)\Big)w(s)\,dx\,ds+\int_{0}^{T}\int_{\Omega}(p_1-p_2)w\,dx\,ds.
\end{aligned}
\end{equation}
So from \eqref{4.04}-\eqref{4.05}, we have
\begin{equation} \label{4.06}
\begin{aligned}
&\int_{0}^{T}E_{w}(s)ds \\
&\leq \delta T
\operatorname{meas}(\Omega)+C_{\delta}E_{w}(0)+\frac{1}{2}C_{\delta}\Gamma
\int_{\Omega}|\nabla w(T)|^2dx-\frac{1}{2}C_{\delta}\Gamma
\int_{\Omega}|\nabla
w(0)|^2dx \\
&\quad +C_{\delta}\int_{0}^{T}\int_{\Omega}(\|\nabla u_1(s)\|^2\Delta
u_1(s)-\|\nabla u_2(s)\|^2\Delta u_2(s))w_t(s)\,dx\,ds \\
&-\frac{1}{2}\int_{\Omega}w_t(T)w(T)dx+\frac{1}{2}\Gamma\int_{0}^{T}
 \int_{\Omega}|\nabla w(s)|^2\,dx\,ds \\
&\quad -\frac{1}{2}\int_{0}^{T}\int_{\Omega}(g(u_{1_t}(s))-g(u_{2_t}(s)))w(s)\,dx\,ds
 +\frac{1}{2}\int_{\Omega}w_t(0)w(0)dx \\
&\quad +\frac{1}{2}\int_{0}^{T}\int_{\Omega}(\|\nabla u_1(s)\|^2\Delta
u_1(s)-\|\nabla u_2(s)\|^2\Delta u_2(s))w(s)\,dx\,ds \\
&\quad +C_{\delta}\int_{0}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,ds
+\frac{1}{2}\int_{0}^{T}\int_{\Omega}(p_1-p_2)w\,dx\,ds.
\end{aligned}
\end{equation}
Integrating \eqref{4.02} over [0,T] with respect to $s$, we obtain
\begin{equation} \label{4.07}
\begin{aligned}
& TE_{w}(T) \\
&\leq \int_{0}^{T}\int_{s}^{T}\int_{\Omega}(\|\nabla u_1(\tau)\|^2\Delta
u_1(\tau)-\|\nabla u_2(\tau)\|^2\Delta u_2(\tau))w_t(\tau)\,dx\,d\tau\, ds\\
&\quad +\int_{0}^{T}E_{w}(s) ds
 +\frac{1}{2}T\Gamma \int_{\Omega}|\nabla w(T)|^2dx\\
&\quad -\frac{1}{2}\Gamma\ \int_{0}^{T} \int_{\Omega}|\nabla
 w(s)|^2\,dx\,ds
+\int_{0}^{T}\int_{s}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,d\tau\,ds.
\end{aligned}
\end{equation}
Therefore, from \eqref{4.06} and \eqref{4.07}, we have
\begin{equation} \label{4.08}
\begin{aligned}
& TE_{w}(T) \\
&\le\delta T
\operatorname{meas}(\Omega)+C_{\delta}E_{w}(0)+\frac{1}{2}C_{\delta}\Gamma
\int_{\Omega}|\nabla w(T)|^2dx-\frac{1}{2}C_{\delta}\Gamma
\int_{\Omega}|\nabla w(0)|^2dx \\
&\quad +C_{\delta}\int_{0}^{T}\int_{\Omega}(\|\nabla u_1(s)\|^2\Delta
u_1(s)-\|\nabla u_2(s)\|^2\Delta u_2(s))w_t(s)\,dx\,ds \\
&\quad +\frac{1}{2}T\Gamma \int_{\Omega}|\nabla
w(T)|^2dx-\frac{1}{2}\int_{\Omega}w_t(T)w(T)dx
+\frac{1}{2}\int_{0}^{T}\int_{\Omega}(p_1-p_2)w\,dx\,ds \\
&\quad -\frac{1}{2}\int_{0}^{T}\int_{\Omega}(g(u_{1_t}(s))-g(u_{2_t}(s)))w(s)\,dx\,ds
+\frac{1}{2}\int_{\Omega}w_t(0)w(0)dx \\
&\quad +\frac{1}{2}\int_{0}^{T}\int_{\Omega}(\|\nabla u_1(s)\|^2\Delta
u_1(s)-\|\nabla u_2(s)\|^2\Delta u_2(s))w(s)\,dx\,ds \\
&\quad +\int_{0}^{T}\int_{s}^{T}\int_{\Omega}(\|\nabla
u_1(\tau)\|^2\Delta
u_1(\tau)-\|\nabla u_2(\tau)\|^2\Delta u_2(\tau))w_t(\tau)\,dx\,d\tau\,ds
 \\
&\quad +\int_{0}^{T}\int_{s}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,d\tau\,ds
+C_{\delta}\int_{0}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,ds.
\end{aligned}
\end{equation}

Next, we need to study
$\int_{0}^{T}\int_{\Omega}(g(u_{1_t})-g(u_{2_t}))w\,dx\,ds$.
The following estimate can be derived by using similar arguments as
in \cite[Chap. 5]{CHL1}. However, for the sake of completeness we
give the proof.
From  condition \eqref{1.1.3}, we have
\[
|g(s)|^\frac{q+1}{q}=|g(s)|^{1/q}\cdot|g(s)|\leq
C(1+|s|)|g(s)|,
\]
combining this with \eqref{1.1.2}, we obtain
\begin{equation}\label{4.09}
 |g(s)|^\frac{q+1}{q}
\leq \begin{cases}
 C, & |s|\leq 1, \\
 2Cg(s)s, & |s|\geq 1,
 \end{cases}
\end{equation}
where $C$ is a constant which is independent of $s$. Multiplying
\eqref{4.00} by $u_{i_t}(t)$, we obtain
\[
 \frac{1}{2}\frac{d}{dt}\int_{\Omega}\left(|u_{i_t}|^2+|\Delta
 u_i|^2\right)+\int_{\Omega}g(u_{i_t})u_{i_t}+\int_{\Omega}(\Gamma-\|\nabla u_i\|^2)\Delta u_iu_{i_t}=\int_{\Omega}p_iu_{i_t},
\]
which, combined with the existence of bounded uniformly absorbing
set, implies 
\begin{equation} \label{4.010}
\int_{0}^{T}\int_{\Omega}g(u_{i_t})u_{i_t}\leq C_{\rho,T},
\end{equation}
where $C_{\rho,T}$ is a constant which depends on the size of $B_0$
in $(H^2(\Omega)\cap H_0^1(\Omega))\times L^2(\Omega)$ and $T$.
Therefore, from \eqref{4.09} and \eqref{4.010}, we have
\begin{equation} \label{4.011}
\begin{aligned}
&\big|\int_{0}^{T}\int_{\Omega}g(u_{i_t})w\big| \\
&\leq \int_{0}^{T}\int_{\Omega(|u_{i_t}|\leq1)}|g(u_{i_t})w|+\int_{0}^{T}
\int_{\Omega(|u_{i_t}|\geq1)}|g(u_{i_t})w| \\
&\leq C\int_{0}^{T}\int_{\Omega(|u_{i_t}|\leq1)}|w|
 +\int_{0}^{T}\int_{\Omega(|u_{i_t}|\geq1)}|g(u_{i_t})||w|  \\
&\leq C\int_{0}^{T}\int_{\Omega(|u_{i_t}|\leq1)}|w|
 +\Big(\int_{0}^{T}\int_{\Omega(|u_{i_t}|\geq1)}|g(u_{i_t})|^\frac{q+1}{q}
\Big)^\frac{q}{q+1} \\
&\quad\times \Big(\int_{0}^{T}\int_{\Omega(|u_{i_t}|\geq1)}|w|^{q+1}
 \Big)^\frac{1}{q+1} \\
&\leq C\int_{0}^{T}\int_{\Omega(|u_{i_t}|\leq1)}|w|
 +2C\Big(\int_{0}^{T}\int_{\Omega(|u_{i_t}|\geq1)}g(u_{i_t})u_{i_t}
 \Big)^\frac{q}{q+1} \\
&\quad\times \Big(\int_{0}^{T}\int_{\Omega(|u_{i_t}|\geq1)}|w|^{q+1}
 \Big)^\frac{1}{q+1} \\
&\leq C\int_{0}^{T}\int_{\Omega(|u_{i_t}|\leq1)}|w|
 +C_{\rho,T}\Big(\int_{0}^{T}\int_{\Omega(|u_{i_t}|\geq1)}|w|^{q+1}
 \Big)^\frac{1}{q+1}.
\end{aligned}
\end{equation}
Combining \eqref{4.08} and \eqref{4.011}, we obtain
\begin{equation} \label{4.012}
\begin{aligned}
&TE_{w}(T) \\
&\le\delta T \operatorname{meas}(\Omega)+C_{\delta}E_{w}(0)
 +\frac{1}{2}C_{\delta}\Gamma \int_{\Omega}|\nabla w(T)|^2dx
 -\frac{1}{2}C_{\delta}\Gamma \int_{\Omega}|\nabla w(0)|^2dx \\
&\quad +C_{\delta}\int_{0}^{T}\int_{\Omega}(\|\nabla u_1(s)\|^2\Delta
u_1(s)-\|\nabla u_2(s)\|^2\Delta u_2(s))w_t(s)\,dx\,ds \\
&\quad +\frac{1}{2}T\Gamma \int_{\Omega}|\nabla
w(T)|^2dx-\frac{1}{2}\int_{\Omega}w_t(T)w(T)dx
+\frac{1}{2}\int_{0}^{T}\int_{\Omega}(p_1-p_2)w\,dx\,ds \\
&\quad +C\int_{0}^{T}\int_{\Omega}|w|\,dx\,ds
+C_{\rho,T}\Big(\int_{0}^{T}\int_{\Omega}|w|^{q+1}\,dx\,ds\Big)^\frac{1}{q+1}
+\frac{1}{2}\int_{\Omega}w_t(0)w(0)dx \\
&\quad +\frac{1}{2}\int_{0}^{T}\int_{\Omega}(\|\nabla u_1(s)\|^2\Delta
u_1(s)-\|\nabla u_2(s)\|^2\Delta u_2(s))w(s)\,dx\,ds \\
&\quad +\int_{0}^{T}\int_{s}^{T}\int_{\Omega}(\|\nabla u_1(\tau)\|^2\Delta
u_1(\tau)-\|\nabla u_2(\tau)\|^2\Delta u_2(\tau))w_t(\tau)\,dx\,d\tau\,ds \\
&\quad +\int_{0}^{T}\int_{s}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,d\tau\,ds
+C_{\delta}\int_{0}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,ds.
\end{aligned}
\end{equation}
Set
\begin{gather} \label{4.013}
\begin{aligned}
C_{M,T}&=\delta T
\operatorname{meas}(\Omega)+C_{\delta}E_{w}(0)+\frac{1}{2}C_{\delta}\Gamma
\int_{\Omega}|\nabla w(T)|^2dx \\
&\quad -\frac{1}{2}C_{\delta}\Gamma \int_{\Omega}|\nabla
w(0)|^2dx-\frac{1}{2}\int_{\Omega}w_t(T)w(T)dx 
 +\frac{1}{2}\int_{\Omega}w_t(0)w(0)dx,
\end{aligned}\\
\label{4.014}
\begin{aligned}
&\phi_{\delta,T}((u_0^1,\upsilon_0^1),(u_0^2,\upsilon_0^2);\sigma_1,\sigma_2) \\
&=C_{\delta}\int_{0}^{T}\int_{\Omega}(\|\nabla u_1(s)\|^2\Delta
u_1(s)-\|\nabla u_2(s)\|^2\Delta u_2(s))w_t(s)\,dx\,ds \\
&\quad +\frac{1}{2}\int_{0}^{T}\int_{\Omega}(\|\nabla u_1(s)\|^2\Delta
u_1(s)-\|\nabla u_2(s)\|^2\Delta u_2(s))w(s)\,dx\,ds \\
&\quad +\int_{0}^{T}\int_{s}^{T}\int_{\Omega}(\|\nabla
u_1(\tau)\|^2\Delta
u_1(\tau)-\|\nabla u_2(\tau)\|^2\Delta u_2(\tau))w_t(\tau)\,dx\,d\tau\,ds \\
&\quad +C\int_{0}^{T}\int_{\Omega}|w|\,dx\,ds
+C_{\rho,T}\Big(\int_{0}^{T}\int_{\Omega}|w|^{q+1}\,dx\,ds\Big)^\frac{1}{q+1}
+\frac{1}{2}T\Gamma \int_{\Omega}|\nabla w(T)|^2dx \\
&\quad +\int_{0}^{T}\int_{s}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,d\tau\,ds
+C_{\delta}\int_{0}^{T}\int_{\Omega}(p_1-p_2)w_t\,dx\,ds
 \\
&\quad +\frac{1}{2}\int_{0}^{T}\int_{\Omega}(p_1-p_2)w\,dx\,ds.
\end{aligned}
\end{gather}
Then we have
\begin{equation} \label{4.015}
E_w(T)\le
\frac{C_{M,T}}{T}+\frac{1}{T}\phi_{\delta,T}((u_0^1,\upsilon_0^1),
(u_0^2,\upsilon_0^2);\sigma_1,\sigma_2).
\end{equation}
\end{proof}

\subsection{Uniform asymptotic compactness}

In this subsection, we prove the uniform (w.r.t.
$\sigma\in\Sigma$) asymptotic compactness in $(H^2(\Omega)\cap
H_0^1(\Omega))\times L^2(\Omega)$, which is given in the following
theorem.

\begin{theorem}\label{dingli1}
Assume that $g$ satisfies \eqref{1.1.2}-\eqref{1.1.3}. If 
\[
p_0\in L^{\infty}(\mathbb{R};L^2(\Omega))\cap
W_b^{1,r}(\mathbb{R};L^r(\Omega))\quad\text{for some }r>\frac{2n}{n+4}\]
 and $\Sigma$ is defined by \eqref{4.4.8}, then the family of processes
$\{U_{\sigma}(t,\tau)\}$, $\sigma\in\Sigma$ corresponding to problem
\eqref{1.1.1}, is uniformly (w.r.t. $\sigma\in\Sigma$)
asymptotically compact in $(H^2(\Omega)\cap H_0^1(\Omega))\times
L^2(\Omega)$.
\end{theorem}

\begin{proof}
Since the family of processes $\{U_{\sigma}(t,\tau)\}$
$\sigma\in\Sigma$ has a bounded uniformly absorbing set and from the
Lemma \ref{lemma1}, for any fixed $\varepsilon>0$, we can choose
first $\delta\le\frac{\varepsilon}{2\operatorname{meas}(\Omega)}$, and let $T$ so
large that
$$
\frac{C_{M,T}}{T}\le\varepsilon.
$$
Hence, thanks to Theorem \ref{contr}, it is sufficient to prove
that $\phi_{\delta,T}(\cdot,\cdot;\cdot,\cdot)$ defined in
\eqref{4.014} belongs to $\operatorname{contr}(B_0,\Sigma)$ for each fixed $T$.

From Theorem \ref{dingli0.2}, we can deduce that for any fixed
$T$,
\begin{align}\label{4.015'}
 \cup_{\sigma\in\Sigma}\cup_{t\in[0,T]}U_{\sigma}(t,0)B_0\text{ is bounded in
 } E_0,
\end{align}
and the bound depends on $T$.

Let $(u_n,u_{n_t})$ be the solutions corresponding to initial data
$(u_0^n,v_0^n)\in B_0$ with respect to symbol $\sigma_n\in\Sigma,
 n=1,2,\dots$. From \eqref{4.015'}, without loss of generality (at
most by passing subsequence), we assume that
\begin{gather}\label{4.016}
 u_n\to u\quad \text{weakly star in }
 L^{\infty}(0,T;H^2(\Omega)\cap H_0^1(\Omega)),\\
\label{4.017}
 u_{n_t}\to u_t\quad \text{weakly star in }  L^{\infty}(0,T;L^2(\Omega)),\\
\label{4.018}
 u_n\to u\quad \text{in }  L^{2}(0,T; L^{2}(\Omega)), \\
\label{4.019}
u_n\to u\quad \text{in }\quad  L^{q+1}(0,T;L^{q+1}(\Omega)), \\
\label{4.020}
 u_n(T)\to u(T)\quad \text{strongly in } H_0^1(\Omega),
\end{gather}
for $q<\frac{n+4}{n-4}$, where we use the compact embeddings
$H^2\hookrightarrow H_0^1$ and $H^2\hookrightarrow L^{q+1}$.

Now we deal with each term corresponding to that in
\eqref{4.014}.
First, from Proposition \ref{proposition1} and \eqref{4.019},
we can obtain
\begin{align}\label{4.021}
\lim_{n\to\infty}\lim_{m\to\infty}\int_0^T\int_{\Omega}(p_n(x,s)-p_m(x,s))(u_{n}(s)-u_{m}(s))\,dx\,ds=0,
\end{align}\label{4.022}
and from Proposition \ref{proposition2} we can get
\begin{gather}\label{4.023}
\lim_{n\to\infty}\lim_{m\to\infty}
\int_0^T\int_{\Omega}(p_n(x,s)-p_m(x,s))(u_{n_t}(s)-u_{m_t}(s))\,dx\,ds=0,\\
\lim_{n\to\infty}\lim_{m\to\infty}
\int_0^T\int_s^T\int_{\Omega}(p_n(x,\tau)-p_m(x,\tau))(u_{n_t}(\tau)
-u_{m_t}(\tau))\,dx\,d\tau\,ds=0.
\end{gather}
Secondly, from \eqref{4.016} and \eqref{4.019}, we can get that
\begin{gather}\label{4.024}
 \lim_{n\to\infty}\lim_{m\to\infty}\|\nabla u_n(T)-\nabla
 u_m(T)\|^2=0, \\
\label{4.025}
\lim_{n\to\infty}\lim_{m\to\infty}\int_0^T\int_{\Omega}|u_{n}(s)-u_{m}(s)|\,dx\,ds=0,\\
\label{4.026}
\lim_{n\to\infty}\lim_{m\to\infty}
\Big(\int_{0}^{T}\int_{\Omega}|u_{n}(s)-u_{m}(s)|^{q+1}\,dx\,ds\Big)
^\frac{1}{q+1}=0.
\end{gather}
Since $\{(u_n,u_{n_t})\}_{n=1}^{\infty}$ is bounded in $C(0,T;
(H^2(\Omega)\cap H_0^1(\Omega))\times L^2(\Omega))$ and the
embedding $H^2\hookrightarrow C(\bar{\Omega})$ is compact, by the Arzela
theorem $\{u_n\}_{n=1}^{\infty}$ is compact in
$C(0,T;C(\bar{\Omega}))$.

 On the other hand,
$\{u_n\}_{n=1}^{\infty}$ converges weakly star in $L^{\infty}(0,
T;(H^2(\Omega)\cap H_0^1(\Omega)))$. Thus $\{u_n\}_{n=1}^{\infty}$
strongly converges in $C(0,T;C(\bar{\Omega}))$ and then we find that
\begin{equation} \label{4.027}
\begin{aligned}
&\big|\int_{0}^{T}\int_{\Omega}(\|\nabla u_n\|^2\Delta u_n-\|\nabla
u_m\|^2\Delta u_m)(u_n-u_m)\,dx\,ds\big| \\
&\le C_{R,T} \|u_n-u_m\|_{C(0,T;C(\bar{\Omega}))}.
\end{aligned}
\end{equation}
From \eqref{4.027}, we obtain
\begin{equation} \label{4.028}
\lim_{n\to\infty}\lim_{m\to\infty}
\int_{0}^{T}\int_{\Omega}(\|\nabla u_n\|^2\Delta u_n-\|\nabla
u_m\|^2\Delta u_m)(u_n-u_m)\,dx\,ds=0.
\end{equation}
Finally, Since (for smooth solutions) we have
\[
\int_{\Omega}\|\nabla u\|^2\Delta uu_t\,dx
=-\frac{1}{4}\frac{\partial}{\partial t}\|\nabla u\|^4,
\]
from the above equality, we obtain
\begin{align*}
&\int_{0}^{T}\int_{\Omega}(\|\nabla u_n(s)\|^2\Delta u_n(s)-\|\nabla
u_m(s)\|^2\Delta u_m(s))(u_{n_t}(s)-u_{m_t}(s))\,dx\,ds \\
&=\int_{0}^{T}\int_{\Omega}\|\nabla u_n(s)\|^2\Delta
u_n(s)u_{n_t}(s)\,dx\,ds+\int_{0}^{T}\int_{\Omega}\|\nabla
u_m(s)\|^2\Delta u_m(s)u_{m_t}(s)\,dx\,ds \\
&\quad -\int_{0}^{T}\int_{\Omega}\|\nabla u_n(s)\|^2\Delta
u_n(s)u_{m_t}(s)\,dx\,ds \\
&\quad -\int_{0}^{T}\int_{\Omega}\|\nabla
u_m(s)\|^2\Delta u_m(s)u_{n_t}(s)\,dx\,ds \\
&=\frac{1}{4}\;\big[\;\|\nabla u_n(0)\|^4 -\|\nabla u_n(T)\|^4 +
\|\nabla
u_m(0)\|^4 -\|\nabla u_m(T)\|^4\;\big] \\
&\quad -\int_{0}^{T}\int_{\Omega}\|\nabla u_n(s)\|^2\Delta
u_n(s)u_{m_t}(s)\,dx\,ds \\
&\quad -\int_{0}^{T}\int_{\Omega}\|\nabla
u_m(s)\|^2\Delta u_m(s)u_{n_t}(s)\,dx\,ds,
\end{align*}
using \eqref{4.016}, \eqref{4.017} and \eqref{4.020}, taking first
$m\to\infty$, then $n\to\infty$, we obtain
\begin{equation} \label{4.029}
\begin{aligned}
&\lim_{n\to\infty}\lim_{m\to\infty}\int_{0}^{T}\int_{\Omega}\Big(\|\nabla
u_n(s)\|^2\Delta u_n(s) \\
&\quad -\|\nabla u_m(s)\|^2\Delta u_m(s)\Big)(u_{n_t}(s)-u_{m_t}(s))\,dx\,ds \\
&=\frac{1}{2}[\|\nabla u(0)\|^4 -\|\nabla
u(T)\|^4]-2\int_{0}^{T}\int_{\Omega}\|\nabla u(s)\|^2\Delta
u(s)u_t(s)\,dx\,ds
= 0.
\end{aligned}
\end{equation}
Similarly, we have
\begin{align*}
&\int_{s}^{T}\int_{\Omega}(\|\nabla u_n(\tau)\|^2\Delta
u_n(\tau)-\|\nabla
u_m(\tau)\|^2\Delta u_m(\tau))(u_{n_t}(\tau)-u_{m_t}(\tau))\,dx\,d\tau  \\
&= \frac{1}{4}\;\big[\;\|\nabla u_n(s)\|^4 -\|\nabla u_n(T)\|^4 +
\|\nabla
u_m(s)\|^4 -\|\nabla u_m(T)\|^4\;\big] \\
&\quad -\int_{s}^{T}\int_{\Omega}\|\nabla u_n(\tau)\|^2\Delta
u_n(\tau)u_{m_t}(\tau)\,dx\,d\tau \\
&\quad -\int_{s}^{T}\int_{\Omega}\|\nabla
u_m(\tau)\|^2\Delta u_m(\tau)u_{n_t}(\tau)\,dx\,d\tau.
\end{align*}

At the same time, $|\int_{s}^{T}\int_{\Omega}(\|\nabla
u_n(\tau)\|^2\Delta u_n(\tau)-\|\nabla u_m(\tau)\|^2\Delta
u_m(\tau))(u_{n_t}(\tau)-u_{m_t}(\tau))\,dx\,d\tau|$ is bounded for each
fixed $T$, by the Lebesgue dominated convergence theorem we have
\begin{equation} \label{4.030}
\begin{aligned}
&\lim_{n\to\infty}\lim_{m\to\infty}\int_{0}^{T}\int_{s}^{T}\int_{\Omega}\Big(\|\nabla
u_n(\tau)\|^2\Delta u_n(\tau) \\
&\quad -\|\nabla u_m(\tau)\|^2\Delta
u_m(\tau)\Big)(u_{n_t}-u_{m_t})\,dx\,d\tau\,ds \\
&=\int_{0}^{T}\Big(\lim_{n\to\infty}\lim_{m\to\infty}\int_{s}^{T}\int_{\Omega}
\Big(\|\nabla u_n(\tau)\|^2\Delta u_n(\tau) \\
&\quad -\|\nabla u_m(\tau)\|^2\Delta
u_m(\tau)\Big)(u_{n_t}-u_{m_t})\,dx\,d\tau
\Big)ds \\
&= \int_{0}^{T}0ds=0.
\end{aligned}
\end{equation}
Hence, combining \eqref{4.021}-\eqref{4.030}, we obtain that
$\phi_{\delta,T}(\cdot,\cdot;\cdot,\cdot)\in \operatorname{contr}(B_0,\Sigma)$
immediately.
\end{proof}


\subsection{Existence of a compact uniform attractor}

\begin{theorem}\label{dingli2}
Assume that $g$ satisfies \eqref{1.1.2}-\eqref{1.1.3}. If 
\[
p_0\in L^{\infty}(\mathbb{R};L^2(\Omega))\cap
W_b^{1,r}(\mathbb{R};L^r(\Omega))\quad\text{for some }r>\frac{2n}{n+4}
\]
and $\Sigma$ is defined by \eqref{4.4.8}, then the family of processes
$\{U_{\sigma}(t,\tau)\}$, $\sigma\in\Sigma$ corresponding to problem
\eqref{1.1.1} has a compact uniform (w.r.t. $\sigma\in\Sigma$)
attractor $\mathcal{A}_{\Sigma}$ in $(H^2(\Omega)\cap
H_0^1(\Omega))\times L^2(\Omega)$.
\end{theorem}

\begin{proof}
Theorem \ref{dingli0.2} and Theorem \ref{dingli1} imply the
existence of a compact uniform attractor immediately.
\end{proof}

\begin{remark} \rm
For the autonomous case of \eqref{1.1.1}, that is $p(x,t) = p(x)$,
the growth order of nonlinear damping $g$ is equal to
$\frac{n+4}{n-4}$ if $n>4$. As for the non-autonomous system, the
constant $C_{\rho,T}$ in \eqref{4.010} depends on $T$, which is
different from the autonomous case, and to some extent,
\eqref{4.010} requires that the growth order of $g$ is strictly less
than $\frac{n+4}{n-4}$ with $n>4$.
\end{remark}

\begin{remark} \rm
The technique (scheme) used in this paper is also applicable to
another non-autonomous plate models, e.g., the model of
non-autonomous extensible beam with nonlinear damping and source
terms.
\end{remark}

\subsection*{Acknowledgments}
This work was supported by the NSFC Grants (11361053, 11471148),
by the Fundamental Research Funds for the
Central Universities Grant (lzujbky-2016-98), and by
the Young Teachers Scientific Research Ability Promotion Plan of
 Northwest Normal University
(NWNU-LKQN-11-5).


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\bibitem{ball2} J. M. Ball; 
\emph{Initial and boundary value problems
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\end{thebibliography}

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