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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 114, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/114\hfil Existence of exponential attractors]
{Existence of exponential attractors for the plate equations with strong damping}

\author[Q. Ma, Y. Yang, X. Zhang \hfil EJDE-2013/114\hfilneg]
{Qiaozhen Ma, Yun Yang, Xiaoliang Zhang}  % in alphabetical order

\address{Qiaozhen Ma \newline
College of Mathematics and Statistics, Northwest
Normal University, Lanzhou 730070, China}
\email{maqzh@nwnu.edu.cn}

\address{Yun Yang \newline
College of Mathematics and Statistics, Northwest
Normal University, Lanzhou 730070, China}
\email{yangyun880@163.com}

\address{Xiaoliang Zhang \newline
College of Mathematics and Statistics, Northwest
Normal University, Lanzhou 730070, China}
\email{zhangxl258@163.com}

\thanks{Submitted November 29, 2012. Published May 6, 2013.}
\subjclass[2000]{35Q35, 35B40, 35B41}
\keywords{Plate equation; critical exponent; exponential attractor}

\begin{abstract}
  We show the existence of
 $(H_0^2(\Omega)\times L^2(\Omega), H_0^2(\Omega)\times  H_0^2(\Omega))$-global
 attractors for plate equations with critical nonlinearity when
 $g\in H^{-2}(\Omega)$. Furthermore we prove that for each fixed
 $T > 0$, there is an ($H_0^2(\Omega)\times L^2(\Omega),
 H_0^2(\Omega)\times  H_0^2(\Omega))_{T}$-exponential attractor
 for all $g\in L^2(\Omega)$, which  attracts 
 any $H_0^2(\Omega)\times L^2(\Omega)$-bounded set under the stronger
 $H^2(\Omega)\times H^2(\Omega)$-norm for all $t\geq T$.
\end{abstract}

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

\section{Introduction}

 We consider the long-time behavior of the solutions for the
following equation on a bounded domain $\Omega \subset\mathbb{R}^{5}$
with smooth boundary $\partial\Omega$:
\begin{equation}
\begin{gathered}
u_{tt}+\Delta^2u_t+\Delta^2u+f(u)=g(x), \quad x\in\Omega,\\
u\big|_{\partial\Omega}=\frac{\partial
u}{\partial\nu}\big|_{\partial\Omega}=0,\\
u(x,0)=u_0(x),\quad u_t(x,0)=u_{1_{}}(x),\quad x\in\Omega,
\end{gathered} \label{e1.1}
\end{equation}
where $g\in H^{-2}(\Omega)$, $f\in \mathcal {C}^{1}(\mathbb{R}),f(0)=0$
and satisfies the following conditions:
\begin{gather}
|f'(s)|\leq C(1+|s|^{8}),\quad\forall s\in \mathbb{R},\label{e1.2}\\
 \liminf_{|s|\to \infty}\frac{f(s)}{s}>-\lambda_1^2,\label{e1.3}
\end{gather}
where $\lambda_1$ is the first eigenvalue of $\Delta^2$ on
$H_0^2(\Omega)$.

Problem \eqref{e1.1} stems from the elastic equation established
by Woinowsky-Krieger \cite{w1}. The asymptotic behavior and
the existence of global solutions of the linear plate equations
were studied by Ball \cite{b1,b2} in 1973.
The asymptotic behavior of the plate equations with linear damping
and nonlinear damping have been extensively studied,
see for example \cite{k1,k2,x1,y1,y2}. The existence of the global attractors
of the autonomous plate equations with critical exponent on the unbounded
domain was investigated by several authors in \cite{k2,k3,x1}.
In \cite{y1,y2}, the authors discussed the existence of compact attractors
for the autonomous and non-autonomous plate equations in a
 bounded domain, respectively. For the best of our knowledge,
the existence of bi-space global attractor and exponential attractor
of \eqref{e1.1} has not been published.
Therefore, it is necessary to continue researching.
As we know, existence and regularity of global attractors of the wave
equations with strong damping have been studied in \cite{p1,p2,y3,y4,y5}.
The authors in \cite{y3} proved  the existence of global attractors for
the wave equation when the nonlinearity is critical and
$g\in L^2(\Omega)$. Then in \cite{y5}, they showed
the existence of a global attractor when nonlinearity is critical
 and $g\in H^{-1}(\Omega)$; moreover, they  showed the
existence of exponential attractor for $g\in L^2(\Omega)$.
In this article, we borrow the ideas and methods  in \cite{y3,y5}
 to prove existence of bi-space global attractor for
$g\in H^{-2}(\Omega)$ and bi-space T-exponential attractor
for $g\in L^2(\Omega)$. For other results of attractors about
the dynamical systems, please refer the reader to \cite{s1,t1,y4}
and the references therein.

\section{Preliminaries}

Let $A=\Delta^2$ with domain $D(A)=H_0^2(\Omega)\cap H^{4}(\Omega)$.
Consider the family of Hilbert spaces
$D(A^{s/2})$, $s\in\mathbb{R}$ with inner products and norms
$$
(\cdot,\cdot)_{D(A^{s/2})}=(A^{s/2}\cdot,A^{s/2}\cdot), \quad
\|\cdot\|_{D(A^{s/2})}=\|A^{s/2}\cdot\|,
$$
where $(\cdot,\cdot)$ and $\|\cdot\|$ mean the $L^2(\Omega)$ inner
product and norm respectively.
 For convenience, we denote
$\mathcal {H}_s=D(A^{(1+s)/2})\times D(A^{s/2}),\quad \forall
s\in\mathbb{R}$, whose norm is $\|\cdot\|_s$.
In particular,
$\mathcal {H}_0=H_0^2(\Omega)\times L^2(\Omega)$ and
$\mathcal {V}=H_0^2(\Omega)\times H_0^2(\Omega)$. Note that
\begin{equation}
\begin{gathered}
D(A^{s/2})\hookrightarrow D(A^{r/2}),\quad\text{for } s>r;\\
D(A^{s/2})\hookrightarrow L^{10/(5-4s)}(\Omega),\quad\text{for }
s\in[0,\frac{5}{4}).
\end{gathered}\label{e2.1}
\end{equation}
Given $s>r>q$, for any $\epsilon>0$, there exists
$C_{\epsilon}=C_{\epsilon}(s,r,q)$ such that
\begin{equation}
\|A^{r/2}u\|\leq\epsilon\|A^{s/2}u\|+C_{\epsilon}\|A^{\frac{q}{2}}u\|,\quad
\text{for any } u\in D(A^{s/2}).\label{e2.2}
\end{equation}
For the nonlinear function $f$, we know that $f$ allows the
decomposition
\begin{equation}
f=f_0+f_1,\label{e2.3}
\end{equation}
where $f_0,f_1\in\mathcal {C}(\mathbb{R})$ and satisfy
\begin{gather}
|f_0(u)|\leq C(|u|+|u|^{9})\quad \text{for all }u\in\mathbb{R},\label{e2.4}\\
f_0(u)u\geq 0 \quad \text{for all }u\in\mathbb{R},\label{e2.5}\\
|f_1(u)|\leq C(1+|u|^{\gamma})\quad \text{for all }u\in\mathbb{R},\;
\gamma<9,\label{e2.6}\\
\liminf_{|u|\to \infty}\frac{f_1(u)}{u}>-\lambda_1^2,\label{e2.7}
\end{gather}
where $C$ is a positive constant.
Denote
\begin{equation}
\sigma=\min\{\frac{1}{8}, \frac{9-\gamma}{4}\}.\label{e2.8}
\end{equation}
Under the above assumptions, equation \eqref{e1.1} has an unique weak
solution satisfying
$$
u\in C([0,T],H_0^2(\Omega)),\quad
u_t\in C([0,T],L^2(\Omega))\cap L^2([0,T],H_0^2(\Omega)).
$$
We also need the following properties.

\begin{lemma}[\cite{y5}] \label{lem2.1}
 Let $\mathscr{T}$ be a H\"older mapping from $(\mathscr{X},\|\cdot\|_1)$
to $(\mathscr{X},\|\cdot\|_2)$ with constant $\mathscr{L}$ and
H\"older exponent $\gamma\in(0,1]$; that is,
$$
\|\mathscr{T}x_1-\mathscr{T}x_2\|_2
\leq\mathscr{L}\|x_1-x_2\|_1^{\gamma},\quad\forall x_1,x_2\in\mathscr{X},
$$
Then for any $\mathcal {E}\subset \mathscr{X}$, the following estimates hold:
\begin{itemize}
\item[(i)] $\dim_{F}(\mathscr{T}\mathcal{E},\|\cdot\|_2)\leq\frac{1}{\gamma}
\dim_{F}(\mathcal{E},\|\cdot\|_1)$;

\item[(ii)] if, further, $\{S(t)\}_{t\geq0}$ is a semigroup on $\mathscr{X}$,
satisfies $S(t)\mathscr{X}\subset\mathscr{X}$ for all $t\geq0$, then
\begin{equation}
\operatorname{dist}_{\|\cdot\|_2}(\mathscr{T}S(t)\mathscr{X},
\mathscr{T}\mathcal {E})\leq2\mathscr{L}
\operatorname{dist}_{\|\cdot\|_1}^{\gamma}(S(t)\mathscr{X},
\mathcal {E}),\quad\forall t\geq0,\label{e2.9}
\end{equation}
where $\operatorname{dist}_{\|\cdot\|_{i}}(\cdot,\cdot)$ is the
Hausdorff semidistance of two sets with respect to $\|\cdot\|_{i}$,
$i=1,2$.
\end{itemize}
\end{lemma}

\section{Global attractors and regularity for $g$ in $H^{-2}(\Omega)$}

Since the injection $i:L^2(\Omega)\hookrightarrow H^{-2}(\Omega)$ is dense,
we know that for every $g\in H ^{-2}(\Omega)$ and any $\eta>0$, there is a
$g_{\eta}\in L^2(\Omega)$ which depends on $g$ and $\eta$ such that
\begin{equation}
\|g-g_{\eta}\|_{H^{-2}}<\eta.\label{e3.1}
\end{equation}
We decompose the solution $u(t)$ of \eqref{e1.1} corresponding to
initial data $(u_0,u_1)$ as $u(t)=v^{\eta}(t)+w^{\eta}(t)$,
where $v^{\eta}(t) and  w^{\eta}(t)$
satisfy the following two equations
\begin{equation}
\begin{gathered}
v^{\eta}_{tt}+\Delta^2v^{\eta}_t+\Delta^2v^{\eta}+f_0(v^{\eta})=g-g_{\eta}, \\
(v^{\eta}(0),v^{\eta}_t(0))=(u_0,u_1),\quad v^{\eta}|_{\partial\Omega}=0
\end{gathered} \label{e3.2}
\end{equation}
and
\begin{equation}
\begin{gathered}
w^{\eta}_{tt}+\Delta^2w^{\eta}_t+\Delta^2w^{\eta}+f(u)-f_0(v^{\eta})=g_{\eta}, \\
(w^{\eta}(0),w^{\eta}_t(0))=(0,0),\quad w^{\eta}|_{\partial\Omega}=0.
\end{gathered}\label{e3.3}
\end{equation}
We first recall some results for the bounded dissipative case.

\begin{lemma} \label{lem3.1}
Let $f$ satisfy \eqref{e1.2} and \eqref{e1.3}, $g\in H^{-2}(\Omega)$
and $\{S(t)\}_{t\geq0}$ be the semigroup generated by the weak
solution of \eqref{e1.1} in the natural energy space $\mathcal{H}_0$.
Then $\{S(t)\}_{t\geq0}$ has a bounded absorbing set $B_0$ in $\mathcal{H}_0$;
that is, for any bounded subset $B\subset \mathcal{H}_0$,
there exists $T=T(B_0)$ such that
\begin{equation}
S(t)B\subset B_0,\quad\forall t\geq T.\label{e3.4}
\end{equation}
\end{lemma}

 The proof of the above lemma and the following corollary are similar
 to those in \cite{y3,y5}, so we omit them.


\begin{corollary} \label{coro3.2}
Under the assumptions of Lemma \ref{lem3.1}, for a given $R>0$, there exists
 $K_0=K_0(R)$ and $\Lambda_0=\Lambda_0(R)$, for $\|z_0\|_0\leq R$,
the corresponding solution $S(t)z_0=(u(t),u_t(t))$ satisfy
\begin{gather*}
\|S(t)z_0\|_0\leq K_0,\quad\forall t\in\mathbb{R}^{+};\\
\int^{+\infty}_0\|\Delta u_t(y)\|^2dy\leq \Lambda_0.
\end{gather*}
\end{corollary}

Next, we obtain the existence of the global attractors,
so we need the following asymptotic compactness result.

\begin{lemma} \label{lem3.3}
For any $\epsilon>0$, there is a $\eta=\eta(\epsilon,g)$ such
that the solutions of \eqref{e3.2} satisfy
\begin{equation}
\|v^{\eta}_t\|^2+\|\Delta v^{\eta}\|^2\leq Q_0(\|z_0\|_0)e^{-Ct}
+\epsilon,\quad\forall t\geq 0,\label{e3.5}
\end{equation}
where the constant $C$ only depends on $\|z_0\|_0$ and
$\|g-g_{\eta}\|_{H^{-2}}$, $Q_0(\cdot)$ is a nondecreasing function
on $[0,\infty)$.
\end{lemma}

\begin{proof} Multiplying \eqref{e3.2} by ($v^{\eta}_t+\delta v^{\eta}$)
and integrating over $\Omega$, we have
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\Big(\|v^{\eta}_t+\delta v^{\eta}\|^2+(1+\delta)\|\Delta v^{\eta}\|^2
+2\int_{\Omega}F(v^{\eta})\Big)+\frac{\delta}{2}\|\Delta
v^{\eta}\|^2\\
&+\frac{1}{2}\|\Delta v^{\eta}_t\|^2
+\Big(\frac{\lambda_1}{2}-\delta-\frac{\delta^2}{2}\Big)\|v^{\eta}_t\|^2
+\frac{\delta(\lambda_1-\delta)}{2}\|v^{\eta}\|^2\\
&\leq 4\|g-g_{\eta}\|_{H^{-2}}^2+\frac{1}{4}\|\Delta
v^{\eta}_t\|^2+\frac{\delta^2}{4}\|\Delta v^{\eta}\|^2,
\end{aligned}\label{e3.6}
\end{equation}
 where $F(v^{\eta})=\int^{v^{\eta}}_0f_0(s)ds$.

Let $\delta$ be small enough, then from \eqref{e3.6} we have the estimate
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\Big(\|v^{\eta}_t+\delta v^{\eta}\|^2+(1+\delta)\|\Delta v^{\eta}|^2
+2\int_{\Omega}F(v^{\eta})\Big)\\
&+C_{\delta}(\|\Delta v^{\eta}_t\|^2+\|\Delta v^{\eta}\|^2)
\leq 4\|g-g_{\eta}\|_{H^{-2}}^2.
\end{aligned}
\label{e3.7}
\end{equation}
Multiplying \eqref{e3.2} by $v_t^{\eta}$ we can deduce that
(similar to Lemma \ref{lem3.1})
\begin{equation}
\|v^{\eta}_t\|^2+\|\Delta v^{\eta}\|^2
\leq Q'(\|z_0\|_0, \|g-g_{\eta}\|_{H^{-2}}):=M_0,\quad\forall t\geq0.
\label{e3.8}
\end{equation}
On the other hand, this inequality and \eqref{e2.4} yield
\begin{equation}
\int_{\Omega}F(v^{\eta})dx
\leq C\int_{\Omega}(|v^{\eta}(t)|^2+|v^{\eta}(t)|^{10})dx\label{e3.9}
\end{equation}
which combining with \eqref{e3.8}  imply
\begin{equation}
\int_{\Omega}F(v^{\eta})dx\leq C_{M_0}\int_{\Omega}|\Delta v^{\eta}|^2dx.
\label{e3.10}
\end{equation}
Hence, from \eqref{e3.7} and \eqref{e3.10}, taking $C_{\delta,M_0}$
small enough, we have
\begin{equation}
\begin{aligned}
&\frac{d}{dt}\Big(\|v^{\eta}_t+\delta
v^{\eta}\|^2+(1+\delta)\|\Delta
v^{\eta}\|^2+2\int_{\Omega}F(v^{\eta})dx\Big)\\
&+C_{\delta,M}(\|v^{\eta}_t+\delta
v^{\eta}\|^2+(1+\delta)\|\Delta
v^{\eta}\|^2+2\int_{\Omega}F(v^{\eta})dx)\\
&\leq 4 \|g-g_{\eta}\|_{H^{-2}}^2.
\end{aligned}\label{e3.11}
\end{equation}
Applying Gronwall lemma, we  obtain
$$
\|v^{\eta}_t+\delta v^{\eta}\|^2+(1+\delta)\|\Delta
v^{\eta}\|^2+2\int_{\Omega}F(v^{\eta})dx\leq
Q_0(\|z_0\|_0)e^{-C_{\delta,M}t}
+\frac{\|g-g_{\eta}\|_{H^{-2}}^2}{4C_{\delta,M_0}}.
$$
Therefore, we can complete our proof by taking
$\eta^2\leq4C_{\delta,M_0}\epsilon$ in \eqref{e3.1}.
\end{proof}

\begin{lemma} \label{lem3.4}
For any $T>0$ and $\eta>0$, there is a positive constant $M_1=M_1(T,\eta)$
which depends on $(T,\eta)$, such that the solutions of \eqref{e3.3} satisfy
\begin{equation}
\|w^{\eta}(T)\|^2_{1+\sigma}+\|w^{\eta}_t(T)\|^2_{\sigma}\leq M_1,\label{e3.12}
\end{equation}
where $\sigma=\min\{\frac{1}{8}, \frac{9-\gamma}{4}\}$.
\end{lemma}

\begin{proof}
 According to Corollary \ref{coro3.2} and Lemma \ref{lem3.3}, 
\begin{equation}
\|\Delta u\|+\|\Delta v^{\eta}\|\leq M_2,\quad t \geq 0.\label{e3.13}
\end{equation}
Multiplying \eqref{e3.3} by $A^{\sigma}w^{\eta}_t$, we have
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}(\|A^{\frac{\sigma}{2}}w_t^{\eta}\|^2
 +\|A^{\frac{\sigma+1}{2}}w^{\eta}\|^2)
+\|A^{\frac{\sigma+1}{2}}w^{\eta}_t\|^2\\
&=-(f(u)-f_0(v^{\eta}),A^{\sigma}w^{\eta}_t)
+(g_{\eta},A^{\sigma}w^{\eta}_t).
\end{aligned}\label{e3.14}
\end{equation}
Recall that the nonlinear term $f(u)$ satisfies
$$
|(f(u)-f_0(v^{\eta}),A^{\sigma}w^{\eta}_t)|
\leq|(f(u)-f(v^{\eta}),A^{\sigma}w^{\eta}
_t)|+|(f_1(v^{\eta}),A^{\sigma}w^{\eta}_t)|.
$$
From \eqref{e1.2}, \eqref{e3.13} and using the H\"older inequality, we have
\begin{align*}
|(f(u)-f(v^{\eta}),A^{\sigma}w^{\eta}_t)|
&\leq C\int_{\Omega}(1+|u|^{8}+|v^{\eta}|^{8})
|w^{\eta}||A^{\sigma}w^{\eta}_t|\\
& \leq C(1+\|u\|^{8}_{L^{10}}+\|v^{\eta}\|^{8}_{L^{10}})
\|w^{\eta}\|_{L^{\frac{10}{1-4\sigma}}}
\|A^{\sigma}w^{\eta}_t\|_{L^{\frac{10}{1+4\sigma}}}\\
&\leq C(1+\|\Delta u\|^{8}+\|\Delta v^{\eta}\|^{8})
\|A^{\frac{\sigma+1}{2}}w^{\eta}\|\|A^{\frac{\sigma+1}{2}}w^{\eta}_t\|\\
&\leq C_{M_2}\|A^{\frac{\sigma+1}{2}}w^{\eta}\|^2
 +\frac{1}{3}\|A^{\frac{\sigma+1}{2}}w^{\eta}_t\|^2;%\eqref{e3.15}
\end{align*}
In addition, noticing that $\frac{\gamma}{9-4\sigma}\leq 1$, we obtain
\begin{align*}
|(f_1(v^{\eta}),A^{\sigma}w^{\eta}_t|
& \leq C(1+\|v^{\eta}\|^{\gamma}_{L^{\frac{10\gamma}{9-4\sigma}}})
\|A^{\sigma}w^{\eta}_t\|_{L^{\frac{10}{1+4\sigma}}}\\
&\leq C(1+ \|\Delta v^{\eta}\|^{\gamma})\|A^{\frac{\sigma+1}{2}}
w^{\eta}_t\|\\
&\leq C_{M_2}+\frac{1}{3}\|A^{\frac{\sigma+1}{2}}w^{\eta}_t\|^2;
%\eqref{e3.16}
\end{align*}
Finally, for $\sigma<1$, we obtain
\begin{equation}
|(g_{\eta},A^{\sigma}w^{\eta}_t)|\leq
C\|g_{\eta}\|^2+\frac{1}{3}\|A^{\frac{\sigma+1}{2}}w^{\eta}_t\|^2.\label{e3.17}
\end{equation}
Combining \eqref{e3.14} and \eqref{e3.17}, it follows that
 $$
\frac{d}{dt}(\|A^{\frac{\sigma}{2}}w_t^{\eta}\|^2+\|A^{\frac{\sigma+1}{2}}
 w^{\eta}\|^2)\leq C_{M_2}(\|A^{\frac{\sigma}{2}}w_t^{\eta}\|^2
+\|A^{\frac{\sigma+1}{2}}  w^{\eta}\|^2)+C'_{M_2}.
$$
Thus, we can complete our proof by applying Gronwall lemma.
\end{proof}

Using Lemmas \ref{lem3.3} and \ref{lem3.4}, we have the following lemma.

 \begin{lemma} \label{lem3.5}
Let $f$ satisfy \eqref{e1.2} and \eqref{e1.3}, $g\in H^{-2}(\Omega)$
and $\{S(t)\}_{t\geq0}$ be the semigroup generated by the weak solution
of \eqref{e1.1} in the natural energy space $\mathcal{H}_0$.
Then $\{S(t)\}_{t\geq0}$ is asymptotically smooth in $\mathcal{H}_0$.
\end{lemma}

To prove that the global attractors $\mathscr{A}_{\mathcal{H}_0}$ in
$\mathcal{H}_0$ are bounded in $\mathcal{V}$, we need the following lemma.

\begin{lemma} \label{lem3.6}
Under conditions of Lemma \ref{lem3.5}, and \eqref{e1.2}, \eqref{e1.3},
 for every $t>0$, the following estimate holds:
 $$
\min\{1,t\}\|\Delta u_t\|^2+\min\{1,t^2\}\|u_{tt}\|^2
\leq Q_1(\|z_0\|_0+\|g\|_{H^{-2}}),
$$
where $Q_1(\cdot)$ is a nondecreasing function on $[0,\infty)$,
and $(u(t),u_t(t))$ is the solution corresponding to the initial
data $z_0 \in \mathcal{H}_0$.
\end{lemma}

The results in the above lemma, are obtained suing the
 same derivation process as in \cite{y3,y5}.
Combining Lemmas \ref{lem3.1},  \ref{lem3.5} and \ref{lem3.6}, according to the
abstract conclusion in \cite{t1,y3,y5}, we have the following theorem.

\begin{theorem} \label{thm3.7}
Under the assumptions of Lemma \ref{lem3.5}, $\{S(t)\}_{t\geq0}$ has a global
attractor $\mathscr{A}_{\mathcal{H}_0}$ in $\mathcal{H}_0$,
and $\mathscr{A}_{\mathcal{H}_0}$ is bounded in $\mathcal{V}$.
\end{theorem}

Next, we  prove  that $\mathscr{A}_{\mathcal{H}_0}$ is a
 $(\mathcal{H}_0,\mathcal{V})$-global attractor.
First, By  Theorem \ref{thm3.7} and Lemma \ref{lem3.6}, we have the following statement.

\begin{lemma} \label{lem3.8}
Let $f$ satisfy \eqref{e1.2} and \eqref{e1.3}, $g\in H^{-2}(\Omega)$,
then the semigroup $\{S(t)\}_{t\geq0}$ possesses
$(\mathcal {H}_0 ,\mathcal {V})$-bounded absorbing set, that is, there
exists $B_{\mathcal {V}}\subset \mathcal {V}$ such that,
for any bounded  set $B\subset \mathcal {H}_0$,  there exists
$T_1=T_1(B)$, there holds
$$
S(t)B\subset B_{\mathcal {V}},\quad\forall t\geq T_1.
$$
\end{lemma}

Therefore, to obtain the existence of $(\mathcal {H}_0,\mathcal {V})$-global
attractor, we only need prove $\{S(t)\}_{t\geq0}$ is
$(\mathcal {H}_0,\mathcal {V})$-asymptotic compactness and continuity.

Let $\bar{B}_1=\cup_{t\geq T_{B_{\mathcal{V}}}}S(t)B_{\mathcal {V}}$,
 where $T_{B_{\mathcal{V}}}=\max\{T_1,1\}$, $T_1$ is from Lemma 
\ref{lem3.8}.
Then $\bar{B}_1$ is bounded absorbing set, and positive invariant.
At the same time, due to Lemma \ref{lem3.6} and uniqueness of the solution,
for any initial value $(u_0,u_1)\in\bar{B}_1$, we have the
estimate
$$
\|u_{tt}\|^2\leq C_{\|B_{\mathcal{V}}\|,\|g\|_{H^{-2}}},\quad\forall t\geq 0.
$$

\begin{lemma} \label{lem3.9}
Suppose that $z^{n}_0=(u^{n}_0,u^{n}_1)\in
\bar{B}_1,n=1,2,\dots$ is convergent sequence about $\mathcal {H}$-norm,
then for any $t\geq0$, $S(t)z^{n}_0$ is convergent sequence about
$\mathcal {V}$-norm in $\bar{B}_1$.
\end{lemma}

\begin{proof}
Suppose that $(u^{i}(t),u^{i}_t(t))(i=1,2)$ is the solution for the
initial value $(u^{i}_0,u^{i}_1)\in \bar{B}_1$, let $z(t)=u^{1}(t)-u^2(t)$.
 Then $z$ satisfy
\begin{equation}
z_{tt}+\Delta^2z_t+\Delta^2z+f(u^{1})-f(u^2)=0,\label{e3.18}
\end{equation}
the corresponding initial condition
 $(z(0),z_t(0))=(u^{1}_0,u^{1}_1)-(u^2_0,u^2_1)$£¬boundary value
conditions $z|_{\partial\Omega}=0$.

Multiplying \eqref{e3.18} by $z_t$, we have
$$
\|\Delta z_t\|^2=-(z_{tt},z_t)-(\Delta^2z,z_t)-(f(u^{1})-f(u^2),z_t).
$$
Due to
$$
|-(z_{tt},z_t)-(\Delta^2z,z_t)|\leq\|z_{tt}\|\|z_t\|+\|\Delta z\|^2+
\frac{1}{4}\|\Delta z_t\|^2,
$$
and
$$
|-(f(u^{1})-f(u^2),z_t)|\leq C\int_{\Omega}|f'(u^{1}+\theta(u^{1}-u^2))||z||z_t|\leq C_{M}\|\Delta z\|^2
+\frac{1}{4}\|\Delta z_t\|^2,
$$
we get
$$
\|\Delta z_t\|^2\leq C_{M}(\|z_t\|+\|\Delta z\|^2),
$$
where $ C_{M}$ only depends on $\|\bar{B}_1\|_0$.
By means of the continuity of semigroup $S(t)$ about $\mathcal{H}_0$-norm
and the arbitrariness of $(u^{i}_0,u^{i}_1)$, we can easily obtain
the results of Lemma \ref{lem3.9} hold.
\end{proof}

So, according to Theorem \ref{thm3.7} and Lemma \ref{lem3.9}, we have
$(\mathcal{H}_0,\mathcal{V})$-asymptotic compactness.

 \begin{lemma} \label{lem3.10}
Under the assumptions of Lemma \ref{lem3.5}, $\{S(t)\}_{t\geq0}$ is
$(\mathcal{H}_0, \mathcal{V})$-asymptotic compact.
\end{lemma}

Now we have the existence of ($\mathcal{H}_0,\mathcal{V}$)-Global Attractors:

\begin{theorem} \label{thm3.11}
Let $f$ satisfy \eqref{e1.2}, \eqref{e1.3},
$g\in H^{-2}(\Omega)$ and $\{S(t)\}_{t\geq0}$ be the semigroup generated
 by the weak solution of \eqref{e1.1} in the natural energy
space $\mathcal{H}_0$.
Then $\{S(t)\}_{t\geq0}$ has a $(\mathcal{H}_0, \mathcal{V})$-global
attractor $\mathscr{A}$; that is, $\mathscr{A}$ is compact,
invariant in $\mathcal {V}$, and attracts every bounded
(in $\mathcal {H}_0$) subset of $\mathcal {H}_0$ under the $\mathcal {V}$-norm.
\end{theorem}

\section{Exponential attractor for $g$ in $L^2(\Omega)$}

In this section, we consider a slightly stronger 
$(\mathcal{H}_0,\mathcal{V})$-exponential attraction for $\{S(t)\}_{t\geq0}$. 
Borrowing the main idea and methods in \cite{y3,y5} we prove 
the following main results.

 \begin{theorem} \label{thm4.1}
Let $g\in L^2(\Omega)$ and $f$ satisfy \eqref{e1.2}, \eqref{e1.3}.
Then there exists a set $\mathcal{E}$ which is compact in $\mathcal {V}$ 
and bounded in $D(A)\times H_0^2(\Omega)$, satisfying the following conditions:
\begin{itemize}
\item[(i)] $\mathcal {E}$ is positive invariant; i.e., 
$S(t)\mathcal {E}\subset\mathcal {E}$, for all $t\geq0$;

\item[(ii)] $\dim_{F}(\mathcal {E},\mathcal{V})<\infty$; i.e., $\mathcal {E}$
has finite fractal dimension in $\mathcal{V}$;

\item[(iii)] there exists an increasing function 
$\tilde{Q}:\mathbb{R}^{+}\to \mathbb{R}^{+}$ and $\alpha>0$ such that 
for any subset $B\subset\mathcal {H}_0$ with 
$\sup_{z_0\in B}\|z_0\|_{\mathcal {H}_0}\leq R$ there holds 
$$
\operatorname{dist}_{\mathcal{V}}(S(t)B,\mathcal {E})
\leq \tilde{Q}(R)\frac{1}{\sqrt{t}}e^{-\alpha t},\quad\text{for all }t>0.
$$
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk4.2} \rm
From the proof of Theorem \ref{thm4.1} given below, we can require in 
Theorem \ref{thm4.1} that $\mathcal {E}$ be bounded in $D(A)\times D(A)$.
\end{remark}

We first state a crucial result about the asymptotic regularity 
of the solutions of \eqref{e1.1} with $g\in L^2(\Omega)$,
which can be found in \cite{y5}.

\begin{theorem}[\cite{y3,y5}] \label{thm4.3}
Let $f$ satisfy \eqref{e1.2} and \eqref{e1.3}, $g\in L^2(\Omega)$,
 $B_0$ be a bounded absorbing set of $\{S(t)\}_{t\geq 0}$ in the 
natural energy space $H_0^2(\Omega)\times L^2(\Omega)$. 
Then the global attractor $\mathcal{A}_{\mathcal{H}_0}$ is bounded in 
$D(A)\times D(A)$. Moreover, there exists positive constants $M$
(which depends only on the $H_0^2\times L^2$-bounds of $B_0$) and 
$v$ (which is independent of $B_0$ but may depend on the coefficients in 
\eqref{e1.1}), and a set $\mathcal{B}_1$, closed and bounded in 
$D(A)\times D(A)$, such that 
\begin{equation}
\operatorname{dist}_{\mathcal {H}}(S(t)B_0,\mathcal{B}_1)\leq
Me^{-\nu t}, \quad\forall t \geq 0,\label{e4.1}
\end{equation}
where $\operatorname{dist}_{\mathcal {H}}$ denotes the usual
 Hausdorff semidistance in $\mathcal{H}_0$.
\end{theorem}

As a results, based on the regularity and exponential attraction results, 
Theorem \ref{thm4.3}, we can repeat  the process in \cite{p1,y5} 
to prove the existence of the exponential attractor in $\mathcal{H}_0$ 
for the critical case. That is,

\begin{proposition} \label{prop4.4} 
Let $g\in L^2(\Omega)$ and $f$ satisfy \eqref{e1.2} and \eqref{e1.3}. 
Then the semigroup  $\{S(t)\}_{t\geq 0}$ has an exponential attractor
 $\mathcal {E}_0$ in $\mathcal{H}_0$; that is,
\begin{itemize}
\item[(i)] $\mathcal {E}_0$ is positive invariant; i.e., 
$S(t)\mathcal{E}_0\subset\mathcal {E}_0$, for all $t\geq0$;

\item[(ii)] $\dim_{F}(\mathcal {E}_0,\mathcal{H}_0)<\infty$;
 i.e., $\mathcal {E}_0$ has finite fractal dimension in $\mathcal {H}_0$;

\item[(iii)] There exists an increasing function 
$\mathscr{J}:\mathbb{R}^{+}\to \mathbb{R}^{+}$ and $\mu_0$ such 
that for any subset $B\subset\mathcal{H}_0$ with
$\sup_{z_0\in B}\|z_0\|_{\mathcal {H}_0}\leq R$ there holds
 $$
\operatorname{dist}_{\mathcal {H}_0}(S(t)B,\mathcal {E}_0)\leq
\mathscr{J}(R)e^{-\mu_0t}, \quad\forall t>0.
$$
\end{itemize}
\end{proposition}

As in \cite{p1,y5}, we have the following Lipschitz continuity in $\mathcal{H}_0$.

\begin{lemma} \label{lem4.5} 
For any bounded subset $B\subset \mathcal{H}_0$ and each fixed $T>0$, 
there exists a positive constant $M_{T,B}$ which depends only on $T$
and $\|B\|_{\mathcal{H}_0}$ such tat
\begin{equation}
\|S(T)z_0-S(T)z_1\|_{\mathcal{H}_0}\leq M_{T,B}\|z_0-z_1\|_{\mathcal{H}_0},
\quad\forall z_0,z_1\in B.\label{e4.2}
\end{equation}
and, $S(t)$ maps the bounded set of $\mathcal {H}_0$ into a bounded set
of $\mathcal {H}_0$, that is, there exists an increasing function
$Q_1: \mathbb{R}^{+} \to  \mathbb{R}^{+}$ such that, for any subset
$B\subset \mathcal{H}_0$,
\begin{equation}
\|S(t)B\|_{\mathcal{H}_0}\leq Q_1(\|B\|_{\mathcal{H}_0}),
\quad\forall t\geq 0.\label{e4.3}
\end{equation}
\end{lemma}

Thanks to Lemma \ref{lem3.6}, we can deduce the following  H\"older continuity.

\begin{lemma} \label{lem4.6}
 For any bounded subset $B\subset\mathcal {H}_0$ and each fixed $T>0$,
 the mapping $S(T):(\cup_{t\geq0}S(t)B, \|\cdot\|_{\mathcal{H}_0})
\to (\cup_{t\geq T}S(t)B,\|\cdot\|_{\mathcal{V}})$ is 
$\frac{1}{2}$-H\"older continuous; that is, 
there exists an increasing function $Q_{T}(\cdot):[0,\infty)\to [0,\infty)$, 
which depends only on $T$, such that
\begin{equation}
\|S(T)z_0-S(T)z_1\|_{\mathcal {V}}\leq Q_{T}(\|B\|_{\mathcal {H}_0})
\|z_0-z_1\|^{1/2}_{\mathcal {H}_0},
\quad\text{for all }z_0,z_1\in\cup_{t\geq0}S(t)B.\label{e4.4}
\end{equation}
\end{lemma}

\begin{proof}
 From Lemma \ref{lem3.6} we know that $\cup_{t\geq T}S(t)B$ is bounded in 
$\mathcal{V}$ for every $T>0$.
 For any $z^{i}=(u^{i}_0,u_1^{i})\in\mathcal {H}_0(i=1,2)$,
let $(u_{i}(t),u_{i_t}(u))=S(t)z^{i}$ be the corresponding solution of 
\eqref{e1.1}, and denote
$z(t)=u_1(t)-u_2(t)$, then $z$ satisfies
\begin{equation} 
\begin{gathered}
z_{tt}+\Delta^2z_t+\Delta^2z+f(u^{1})-f(u^2)=0,\\
(z(0),z_t(0))=z_1-z_2,\quad z|_{\partial\Omega}=0.
\end{gathered} \label{e4.5}
\end{equation}
Multiplying \eqref{e4.5} by $z_t$ and integrating over $\Omega$, we have
$$
\|\Delta z_t\|^2\leq\|z_{tt}\|\|z_t\|+\|\Delta z_t\|\|\Delta
z\|+\int_{\Omega}|f(u_1)-f(u_2)||z_t|.
$$
From \eqref{e1.2} and using the H\"older inequality, we have
\begin{align*}
\int_{\Omega}|f(u_1)-f(u_2)||z_t|
&\leq C\int_{\Omega}(1+|u_1|^{8}+|u_2|^{8})|z||z_t| \\
&\leq C_{M}\|z\|_{L^{10}}\|z_t\|_{L^{10}} \\
&\leq C_{M}\|\Delta z\|\|\Delta z_t\|,
\end{align*}
where the constant $C_{M}$ depends only on the $\mathcal{H}_0$-bounds of $B$.
The above inequality  with Lemma \ref{lem4.5} and Lemma \ref{lem3.6} imply
$$
\|\Delta z_t\|^2\leq \bar{M}_1(\|z_0-z_1\|_{\mathcal{H}_0}
+\|z_0-z_1\|^2_{\mathcal{H}_0})\leq
\bar{M}_2\|z_0-z_1\|_{\mathcal{H}_0},
$$
where $\bar{M}_1,\bar{M}_2$ depend only on $T$ and $\|B\|_{\mathcal{H}_0}$;
 Which, noticing \eqref{e4.2} again, implies \eqref{e4.4}.
\end{proof}

For convenience, we first iterate the following so-called 
T-exponential attractor.

\begin{definition}[\cite{y5}] \label{def4.7} \rm
 Let $X,Y$ be two Banach spaces, $Y \hookrightarrow X$ and 
$\{S(t)\}_{t\geq0}$ be a semigroup on $X$. 
A set $\mathcal {E}_{T}\subset Y$ is called a $(X,Y)_{T}$-exponential 
attractor for $\{S(t)\}_{t\geq0}$ if the following conditions hold:
\begin{itemize}
\item[(i)] $\mathcal {E}_{T}$ is compact in $Y$ and positive invariant;
 that is, $S(t)\mathcal{E}_{T}\subset\mathcal {E}_{T}$, for every $t\geq0$;

\item[(ii)] $\dim_{F}(\mathcal {E}_{T},Y)<\infty$;
 that is $\mathcal {E}_{T}$ has finite fractal dimension in $Y$;

\item[(iii)] There exists an increasing function 
$J_{T}:\mathbb{R}^{+}\to \mathbb{R}^{+}$ and $k>0$ such that, for any 
set $B\subset X with \sup_{z_0\in B}\|z_0\|_{X}\leq R$ there holds
$$
\operatorname{dist}_{Y}(S(t)B,\mathcal {E}_{T})\leq J_{T}(R)e^{-k t},
\quad\text{for all }t\geq T.
$$
\end{itemize}
\end{definition}

Then, we have the  existence of an
$(\mathcal {H}_0,\mathcal{V})_{T}$-exponential attractor.

\begin{lemma} \label{lem4.8}
Let $f$ satisfy \eqref{e1.2} and \eqref{e1.3}, $g\in L^2(\Omega)$. 
Then for each fixed $T>0$,
$\{S(t)\}_{t\geq0}$ has an $(\mathcal {H}_0,\mathcal{V})_{T}$-exponential 
attractor.
\end{lemma}

\begin{proof}
 For each fixed $T>0$, we will verify $S(T)\mathcal {E}_0$ is an 
$(\mathcal {H}_0,\mathcal{V})_{T}$-exponential attractor, where 
$\mathcal{E}_0$ is the exponential attractor given in 
Proposition \ref{prop4.4}.

We verify that $S(T)\mathcal{E}_0$ satisfies all the conditions of 
Definition \ref{def4.7} corresponding to spaces $\mathcal {H}_0$ and 
$\mathcal {V}$ as follows

(1) The positive invariance of $S(T)\mathcal{E}_0$ is obvious since 
$\mathcal {E}_0$ is positive invariant; The compactness of 
$S(T)\mathcal {E}_0$ in $\mathcal {V}$ follows from the compactness 
of $\mathcal{E}_0$ in $\mathcal {H}_0$ and continuity 
(Lemma \ref{lem4.6}) of $S(T)$.

(2) Applying property (i) of Lemma \ref{lem2.1}, the finiteness of  
$\dim_{F}(S(T)\mathcal {E}_0,\mathcal{V})$ follows from Lemma \ref{lem4.6} and 
the finiteness of  $\dim_{F}(\mathcal {E}_0,\mathcal{H}_0)$.

(3) For any bounded subset $B\subset\mathcal {H}_0$, denote 
$\hat{B}=B\cup\mathcal{E}_0$. Then from Lemma \ref{lem4.6} we have 
$S(T):(\cup_{t\geq0}S(t)B,\|\cdot\|_{\mathcal
{H}_0})\to (\cup_{t\geq T}S(t)B,\|\cdot\|_{\mathcal{V}})$ is 
$\frac{1}{2}-$H\"older continuous. Hence, applying property (ii) 
of Lemma \ref{lem2.1}, 
the exponential attraction of
$S(T)\mathcal{E}_0$ with respect to $\mathcal{V}$-norm follows 
from the exponential attraction of $\mathcal{E}_0$ with respect to 
$\mathcal{H}_0$-norm immediately.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.1}]
For any fixed $T_0\geq1$, let $\mathcal {E}_{T_0}$ be the 
$(\mathcal {H}_0,\mathcal{V})_{T_0}$-exponential attractor obtained 
in Lemma \ref{lem4.8}. Then we claim that $\mathcal{E}_{T_0}$ satisfies 
conditions (i)-(iii) of Definition \ref{def4.7}.

 We  need to verify only (iii). 
Let $J_{T_0}(\cdot)$ and $k_0$ be the mapping and exponent given in 
Definition \ref{def4.7} and Lemma \ref{lem4.8} corresponding to $T_0$.
 Note that there is a $t_0 > 0$ such that
  $$
e^{-\frac{k_0}{2}t}\leq\frac{1}{\sqrt{t}},\quad\text{for all }t\geq t_0.
$$
  Then, to complete the proof, we can set $\alpha=\frac{k_0}{2}$ and
$$
\tilde{Q}(\cdot)=(J_{T_0}(\cdot)+Q_0(\cdot
+\|\mathcal {E}_{T_0}\|_{\mathcal{H}_0})+Q_1(\cdot+
 \|\mathcal {E}_{T_0}\|_{\mathcal{H}_0}+\|g\|_{H^{-2}}))e^{(t_0+T_0)\alpha},
$$
where $Q(\cdot)$ is given in Lemma \ref{lem3.6} and $Q_1(\cdot)$ is given 
in \eqref{e4.3}.
\end{proof}

\subsection*{Acknowledgments}
This work was partly supported by grant 11101334 from
the NSFC, grant 1107RJZA223 from the NSF of
Gansu Province, and by the Fundamental Research Funds of Gansu
Universities.


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\end{document}