\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 09, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/09\hfil Exponential attractors]
{Exponential attractors for a nonclassical diffusion equation}

\author[Y.-F. Liu, Q. Ma\hfil EJDE-2009/09\hfilneg]
{Yong-Feng Liu, Qiaozhen Ma} % in alphabetical order

\address{Yong-Feng Liu \newline
College of Mathematics and Information Science,
Northwest Normal University, Lanzhou, Gansu 730070, China}
\email{liuyongfeng1982@126.com}

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

\thanks{Submitted October 22, 2008. Published January 9, 2009.}
\thanks{Supported  by grants: 3ZS061-A25-016 from the Natural Sciences
Foundation of Gansu \hfill\break\indent Province, 0801-02 from the
Education Department Foundation of Gansu Province,
\hfill\break\indent and  NWNU-KJCXGC-03-40}
\subjclass[2000]{35B41, 35Q35} 
\keywords{Nonclassical diffusion equation; exponential attractors}

\begin{abstract}
 In this article, we prove the existence of exponential attractors
 for a nonclassical diffusion equation in
 ${H^{2}(\Omega)}\cap{H}^{1}_{0}(\Omega)$
 when the space dimension is less than 4.
\end{abstract}

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

\section{Introduction}

 Let $\Omega$ be an open bounded set of
$\mathbb{R}^3$ with smooth boundary $\partial\Omega$. We consider
the  equation
\begin{gather}
u_{t}-\Delta {u_t}-\Delta {u}+f(u)=g(x),  \quad \text{in }
\Omega\times \mathbb{R}_{+},\label{e1.1}\\
u=0,         \quad  \text{on } \partial\Omega,\label{e1.2}\\
u(x,0)=u_{0},\quad x\in\Omega. \label{e1.3} \end{gather}
    This equation is a special form of the nonclassical diffusion
equation used in fluid mechanics, solid mechanics and heat
conduction theory \cite{a1,l1}. Existence of the global attractors for
problem \eqref{e1.1}-\eqref{e1.3} was studied originally by
Kalantarov in \cite{k1} in the Hilbert space ${H}^{1}_{0}(\Omega)$. In
recent years, many authors have proved the existence of  global
attractors under different assumptions, \cite{k1,s1,t1,x1} in the Hilbert
space ${H}^{1}_{0}(\Omega)$, and \cite{m1,w1} in the Hilbert space
${H^{2}(\Omega)}\cap{H}^{1}_{0}(\Omega)$. In this paper, we
study the existence of exponential attractors in the Hilbert space
${H^{2}(\Omega)}\cap{H}^{1}_{0}(\Omega)$.

In this article the nonlinear function satisfies the following conditions:
\begin{itemize}
\item[(G1)]  There exists $l>0$ such that
$ f'(s)\geq-l$ for all $s\in \mathbb{R}$;

\item[(G2)]  there exists $\kappa_1>0$ such that
$f'(s)\leq \kappa_1(1+|s|^2)$ for all $s\in \mathbb{R}$;

\item[(G3)] $\liminf_{|s| \to\infty} F(s)/s^2\geq0$,
where $$ F(s)=\int_0^sf(r)\,dr;$$

\item[(G4)] there exists $\kappa_2>0$ such that
$$
\liminf_{|s| \to\infty}{\frac{sf(s)-\kappa_2F(s)}{s^2}}\geq0.
$$

\end{itemize}
The main results of this paper will be stated as Theorem \ref{thm3.10}
below.

\section{Preliminaries}

 Let $H=L^{2}(\Omega)$, $V_1=H^{1}_0(\Omega)$ ,
$V_2=H^{2}(\Omega)\cap H^{1}_0(\Omega)$. We denote by $(\cdot,\cdot)$
denote the scalar product, and $\|\cdot\|$ the norm of $H$.
The scalar product in $V_1$ and $V_2$ are denoted by
\begin{gather*}
((u,v))=\int_\Omega \nabla u \nabla v  \,dx,\quad\forall u,v\in V_1 ,\\
[u,v]=\int_\Omega \Delta  u \Delta  v \,dx,\quad\forall u,v\in V_2 .
\end{gather*}
The corresponding norms  are denoted by  ${\|\cdot\|}_1$, ${\|\cdot\|}_2$.
 It is well known that the norm
${\|\cdot\|}_s$ is equivalent to the usual norm of $V_s$ for $s=1,2$.
Let $X$ be a separable Hilbert space and $\mathscr{B}$ be a
  compact subset of $X$, $ \{S(t)\}_{t\geq0}$ be a nonlinear
  continuous semigroup that leaves the set $\mathscr{B}$ invariant
  and $\mathscr{A}=\cap_{t>0}S(t)\mathscr{B}$, that is,
  $\mathscr{A}$ is a global attractor for $ \{S(t)\}_{t\geq0}$ on
  $\mathscr{B}$.

\begin{definition}[\cite{e1}]  \label{def2.1}\rm
 A compact set
$\mathscr{A}\subseteq\mathscr{M}\subseteq\mathscr{B}$ is called an
exponential attractor for $(S(t),\mathscr{B})$ if:
\begin{enumerate}
\item $\mathscr{M}$ has finite fractal dimension;

\item $\mathscr{M}$ is a positive invariant set of
$S(t):S(t)\mathscr{M}\subseteq\mathscr{M}$, for all $t>0$;

\item $\mathscr{M}$ is an exponentially attracting set for the
semigroup  $ \{S(t)\}_{t\geq0}$; i.e. there exist
universal constants $\alpha$,$\beta>0$ such that
$$
\mathop{\rm dist}{}_X(S(t)u,\mathscr{M})\leq\alpha e^{-\beta
t},\quad\forall u\in\mathscr{B},\; t>0,
$$
where dist denotes the nonsymmetric Hausdorff distance between sets.
\end{enumerate}
\end{definition}

    A sufficient condition for the existence of an exponential
attractor depends on a dichotomy principle called the squeezing
property; we recall this property as follows.

\begin{definition}[\cite{e1}] \label{def2.2} \rm
 A continuous semigroup of operators $\{S(t)\}_{t\geq0}$
is said to satisfy the squeezing property on
$\mathscr{B}$ if there exists $t_\ast>0$ such that
$S_\ast=S(t_\ast)$ satisfies that there exists an orthogonal
projection operator $P$ of rank $N_0$ such that, for every $u$ and
$v$ in $\mathscr{B}$, either
\begin{gather*}
\|(I-P)(S(t_\ast)u_1-S(t_\ast)u_2)\|_X\leq \|
P(S(t_\ast)u_1-S(t_\ast)u_2)\|_X,\quad\text{or}\\
\| S(t_\ast)u_1-S(t_\ast)u_2\|_X\leq\frac{1}{8}\| u_1-u_2\|_X.
\end{gather*}
\end{definition}

\begin{definition}[\cite{e1}] \label{def.2.3} \rm
 For every $u$, $v$ in the compact set
$\mathscr{B}$, if there exists a local bounded function $l(t)$ such
that
$$
\|S(t)u-S(t)v\|_X\leq l(t)|u-v\|_X,
$$
then $S(t)$ is Lipschitz continuous in $\mathscr{B}$. Here $l(t)$
does not depend on $u$ or $v$.
\end{definition}

\section{Exponential Attractor in $V_2$}

\begin{lemma}[\cite{w1}] \label{lem3.1}
 Assume that $g\in V_s^{'}$ $(s=1,2)$. Then for each $u_0\in V_s $
the problem \eqref{e1.1}-\eqref{e1.3}
has  a unique solution $u=u(t)=u(t;u_0)$ with
$u\in C^1([0,\tau),V_s)$ on some interval $[0,\tau)$. Also for each $t$
fixed, $u$ is continuous in $u_0$.
\end{lemma}

\begin{lemma}[\cite{k1}] \label{lem3.2}
Assume that $g\in H$, then for any $R>0$, there exist
positive constants $E_1(R)$, $\rho_1$ and $t_1(R)$ such that for every
solution $u$ of problem \eqref{e1.1}-\eqref{e1.3},
\begin{gather*}
\|u\|_1\leq E_1(R),\quad t\geq 0,\\
\|u\|_1\leq \rho_1,\quad t\geq t_1(R),
\end{gather*}
provided $\| u_0\| _1 \leq R$.
\end{lemma}

\begin{lemma}[\cite{w1}] \label{lem3.3}
 Assume $g\in V_1$, then for any $R>0$, there exist
positive constants $E_2(R)$, $\rho_2$ and $t_2(R)$ such that for every
solution $u$ of problem \eqref{e1.1}-\eqref{e1.3},
\begin{gather*}
\|u\|_2\leq E_2(R),\quad t\geq 0,\\
\|u\|_2\leq \rho_2,\quad t\geq t_2(R),
\end{gather*}
provided $\| u_0\| _2 \leq R$.
\end{lemma}

\begin{remark} \label{rmk3.4} \rm
From the proof of Lemma \ref{lem3.3} \cite[Theorem 3.2]{w1},
 we obtain
$$
\int_t^{t+1}(\|u_t\|_1^2+\| u_t\|_2^2)\leq m,
$$
where $m$ is a positive constant.
\end{remark}

   According to Lemmas \ref{lem3.2} and \ref{lem3.3}, we have
\begin{equation}
\mathscr{B}_0=\{u\in V_2: \|\nabla u \|\leq \rho_1,
\|\Delta  u \|\leq \rho_2\}\label{e3.1}
\end{equation}
is a compact absorbing set of a semigroup of operators
$ \{S(t)\}_{t\geq0}$ generated by \eqref{e1.1}-\eqref{e1.3}.
Namely, for any given $u_0\in V_2$,
there exists $T=T(u_0)>0$ such that\ $\| S(t)u_0\|
\leq\rho,$ for all $t\geq T$.
Hence
$$
\mathscr{B}=\overline{\cup_{0\leq t\leq T}S(t)\mathscr{B}_0}
$$
is a compact positive invariant set in $V_2$ under $S(t)$.


\begin{lemma}[\cite{w1}] \label{lem3.5}
 Assume that $f\in C^2(\mathbb{R};\mathbb{R})$ and satisfies
{\rm (G1)--(G4)} with $f(0)=0$, $g\in V_1$. Then the semigroup $S(t)$
generated by \eqref{e1.1}--\eqref{e1.3} possesses  a
global attractor $\mathscr{A}$ in $V_2$.
\end{lemma}


\begin{lemma} \label{lem3.6}
Assume that $f$ satisfies {\rm (G1)--(G4)}, $u(t),v(t)$ are two solutions
of \eqref{e1.1}--\eqref{e1.3} with initial values $u_{0},v_{0}\in
\mathscr{B}$, then
\begin{equation}
\|u(t)-v(t)\|_2\leq e^{c_1t}\|u(0)-v(0)\|_2\label{e3.2}
\end{equation}
\end{lemma}

\begin{proof}
 Setting $w(t)=u(t)-v(t)$, we see that $w(t)$ satisfies
\begin{equation}
w_t-\Delta  w_t-\Delta  w+f(u)-f(v)=0.\label{e3.3}
\end{equation}
Taking the inner product with $-\Delta  w$ of \eqref{e3.3}, we obtain
\begin{equation}
\frac{1}{2}\frac{d}{dt}(\|\Delta
w\|^{2}+\|\nabla w\|^2)+\|\Delta
w\|^{2}+(f(u)-f(v),-\Delta  w)=0.\label{e3.4}
\end{equation}
Using $H_0^1(\Omega)\subset L^6(\Omega)$ and (G2), it follows that
\begin{equation} \label{e3.5}
\begin{aligned}
& \big|\int_\Omega(f(u)-f(v)) \Delta  w \,dx\big|\\
& \leq \int_\Omega |f'(\theta u+(1-\theta)v)||w||\Delta  w|\,dx \quad
  (0<\theta<1) \\
& \leq  c\int_\Omega (1+|u|^2+|v|^2)|w||\Delta  w|\,dx  \\
& \leq  c\int_\Omega |w||\Delta  w|dx+c\int_\Omega |u|^2|w||\Delta
  w|dx+c\int_\Omega |v|^2|w||\Delta  w|\,dx \\
& \leq  c\|w\|\|\Delta w\|+c\|u\|_6^2\|w\|_6\|\Delta  w\|
  +c\|v\|_6^2\|w\|_6\|\Delta w\|.
\end{aligned}
\end{equation}
Since $\mathscr{B}$ is a bounded absorbing set given by \eqref{e3.1},
$u_{0},v_{0}\in \mathscr{B}$, from \eqref{e3.5} we get
\begin{equation}
|\int_\Omega (f(u)-f(v))\Delta  w\,dx|\leq c\|\nabla
w\|\|\Delta  w\|\leq\frac{\|\Delta w\|^2}{2}+\frac{c_1}{2}\|\nabla
w\|^2,\label{e3.6}
\end{equation}
where $ c_1 $ is dependent on $\rho_1$ and
$\rho_2$.
Combining \eqref{e3.4} with \eqref{e3.6}, we deduce that
\begin{equation}
\frac{d}{dt}(\|\Delta w\|^{2}+\|\nabla w\|^2)+\|\Delta  w\|^{2}
\leq c_1\|\nabla w\|^2. \label{e3.7}
\end{equation}
This yields
\begin{equation}
\frac{d}{dt}(\|\Delta w\|^{2}+\|\nabla w\|^2)\leq
c_1(\|\nabla w\|^2+\|\Delta  w \|^2).\label{e3.8}
\end{equation}
By the Gronwall Lemma, we get
$$
\|\Delta w(t)\|^{2}+\|\nabla w(t)\|^2\leq e^{c_1t}(\|\Delta
w(0)\|^{2}+\|\nabla w(0)\|^2).
$$
\end{proof}

\begin{lemma} \label{lem3.7}
Under the assumptions of Lemma \ref{lem3.5}, there exists $L>0$ such
that
$$
\sup_{u_0\in \mathscr{B}}\| u_t(t)\|_2\leq L,\quad \forall t\geq 0.
$$
\end{lemma}

\begin{proof}
Differentiating \eqref{e1.1} with respect to time
and denoting $v=u_t$, we have
\begin{equation}
v_t-\Delta v_t-\Delta v=-f'(u)v\label{e3.9}
\end{equation}
Multiplying the above equality by $-\Delta  v$ and using (G1),
\begin{equation}
\frac{1}{2}\frac{d}{dt}(\|
\nabla v\|^{2}+\|\Delta  v\|^2)+\|\Delta  v\|^2\leq
l\|\nabla v\|^2.\label{e3.10}
\end{equation}
 This inequality and Remark \ref{rmk3.4}, by the
uniform Gronwall lemma,  complete the proof.
\end{proof}

\begin{lemma} \label{lem3.8}
Under the assumptions of lemma \ref{lem3.5}, for every $T>0$,
the mapping $(t,u)\mapsto S(t)u $ is Lipschitz continuous
on $[0,T]\times\mathscr{B}$.
\end{lemma}

\begin{proof} For $u_1$, $u_2\in\mathscr{B}$ and $t_1$,
$t_2\in[0,T]$ we have
\begin{equation}
\| S(t_1)u_1-S(t_2)u_2\|_2\leq\| S(t_1)u_1-S(t_1)u_2\|_2
+\| S(t_1)u_2-S(t_2)u_2\|_2 \label{e3.11}
\end{equation}
The fist term of the above inequality is handled by estimate
\eqref{e3.2}. For the second term, we have
\begin{equation}
\|u(t_1)-u(t_2)\|_2\leq|\int_{t_1}^{t_2}\|
u_t(y)\|_2dy|\leq L| t_1-t_2|. \label{e3.12}
\end{equation}
Hence
\begin{equation}
\| S(t_1)u_1-S(t_2)u_2\|_2\leq L[|t_1-t_2|+\| u_1-u_2\|_2].\label{e3.13}
\end{equation}
for some $L=L(T)\geq0$.
\end{proof}

\begin{lemma} \label{lem3.9}
Assume that $f$ satisfies {\rm (G1)--(G4)}, $u(t),v(t)$ are two
solutions of problem
\eqref{e1.1}--\eqref{e1.3} with initial values
$u_{0},v_{0}\in \mathscr{B}$, then the semigroup S(t) generated from
\eqref{e1.1}--\eqref{e1.3} satisfies the squeezing property;
i.e., there exist $t_*$ and $N=N_0=N(t_*)$ such that
$$
\|(I-P)(S(t_\ast)u_0-S(t_\ast)v_0)\|_2> \|
P(S(t_\ast)u_0-S(t_\ast)v_0)\|_2
$$
then
$$
\| S(t_\ast)u_0-S(t_\ast)v_0\|_2\leq\frac{1}{8}\| u_0-v_0\|_2.
$$
\end{lemma}

\begin{proof}
  We consider the operator $A=-\Delta $. Since $A$
is self-adjoint, positive operator and has a compact inverse,
there exists a complete set of
eigenvectors $\{\omega_{i}\}^{\infty}_{i=1}$ in $H$, the
corresponding eigenvalues  $\{\lambda_{i}\}^{\infty}_{i=1}$
satisfy
$$
A\omega_{i}=\lambda_{i}\omega_{i},\quad
0<\lambda_{1}\leq\lambda_{2}\leq\dots\leq\lambda_{i}\leq\dots
\to+\infty,\quad i\to+\infty.
$$
We set $H_N=\mathop{\rm span}\{\omega_{1},\omega_{2} ,\dots , \omega_{N}\}$.
$P_N$ is the orthogonal projection onto $H_N$, and $Q_{N}=I-P_{N} $ is
 the orthogonal projection onto the orthogonal complement of $H_N$,
$w=P_Nw+Q_{N}w = p+q$. Assume that $\|P_Nw(t)\|\leq\|Q_Nw(t)\|$,
taking the inner product of \eqref{e3.3} with  $-\Delta  q$, we
obtain
\begin{equation}
\frac{1}{2}\frac{d}{dt}(\|\Delta q\|^{2}+\|\nabla q\|^2)+\|\Delta
q\|^{2}+(f(u)-f(v),-\Delta  q)=0. \label{e3.14}
\end{equation}
Similar to \eqref{e3.5}, it leads to
\begin{equation} \label{e3.15}
\begin{aligned}
\big|\int_\Omega (f(u)-f(v))\Delta  q\,dx\big|
&\leq  c\int_\Omega |w||\Delta  q |\,dx+c\int_\Omega |u|^2|w||\Delta
 q|\,dx \\
&\quad +c\int_\Omega |v|^2|w||\Delta  q|\,dx.
\end{aligned}
\end{equation}
Since
$$
\int_\Omega |u|^2|w||\Delta  q|\,d x\leq\|u\|^2_\infty\|w\|\|\Delta  q\|
$$
and by the Agmon inequality:
$\|u\|_\infty\leq c\|\nabla u\|^{1/2}\|\Delta
u\|^{1/2}$, and \eqref{e3.1},  from \eqref{e3.15} we obtain
\begin{equation}
|\int_\Omega (f(u)-f(v))\Delta  q\,dx|\leq c\|w\|\|\Delta  q\|
\leq\frac{\|\Delta q\|^2}{2}+\frac{c_2}{2}\|w\|^2, \label{e3.16}
\end{equation}
where $ c_2 $  depends on $\rho_1$ and $\rho_2$.
Combining \eqref{e3.14} and  \eqref{e3.16}, we deduce that
\begin{equation}
\frac{d}{dt}(\|\Delta q\|^{2}+\|\nabla q\|^2)+\|\Delta  q\|^{2}
\leq c_2\|w\|^2.\label{e3.17}
\end{equation}
Furthermore, by lemma \ref{lem3.6} and the
Poincar\'e inequality, we have
\begin{equation}
\begin{aligned}
\frac{d}{dt}(\|\Delta q\|^{2}+\|\nabla q\|^2)+\frac{\|\Delta
q\|^{2}}{2}+\frac{\lambda _{N+1}}{2}\|\nabla q\|^2
&\leq c_2\|w\|^2\leq c_2\|p+q\|^2\\
&\leq 2c_2\|q\|^2 \leq 2c_2\lambda _{N+1}^{-1} \|\nabla q\|^2\\
&\leq c_3\lambda _{N+1}^{-2} \|\Delta w\|^2\\
&\leq c_3\lambda _{N+1}^{-2}e^{c_1t} \|\Delta w(0)\|^2.
\end{aligned} \label{e3.18}
\end{equation}
Since $\lambda_1\leq\lambda_{N+1}$,
\begin{equation}
\frac{d}{dt}(\|\Delta
q\|^{2}+\|\nabla q\|^2)+\frac{\|\Delta
q\|^{2}}{2}+\frac{\lambda _{1}}{2}\|\nabla q\|^2\leq
c_3\lambda _{N+1}^{-2}e^{c_1t} \|\Delta w(0)\|^2. \label{e3.19}
\end{equation}
Let $c_4=\min\{\frac{1}{2},\frac{\lambda _{1}}{2}\}$. Then
\begin{equation}
\frac{d}{dt}(\|\Delta q\|^{2}+\|\nabla q\|^2)+c_4(\|\Delta q\|^{2}
  +\|\nabla q\|^2)
\leq c_3\lambda _{N+1}^{-2}e^{c_1t} \|\Delta  w(0)\|^2.\label{e3.20}
\end{equation}
By the Gronwall Lemma, we
conclude that
\begin{align*}
\|\Delta q(t)\|^{2}+\|\nabla q(t)\|^2
&\leq e^{-c_4t}(\|\Delta q(0)\|^{2}+\|\nabla q(0)\|^2)
  +c_5\lambda _{N+1}^{-2}e^{c_1t} \|\Delta  w(0)\|^2\\
&\leq c_6(e^{-c_4t}+c_7\lambda _{N+1}^{-2}e^{c_1t}) \|\Delta w(0)\|^2.
\end{align*} %\label{e3.21}
 Hence
\begin{equation}
\|\Delta  w(t)\|^2\leq 2\|\Delta  q(t)\|^2\leq
c_8(e^{-c_4t}+c_9\lambda _{N+1}^{-2}e^{c_1t}) \|\Delta
w(0)\|^2.\label{e3.21}
\end{equation}
Choose $t_*>0$, such that
 $c_8e^{-c_4t_*}\leq 1/128$, and then let $t_*$ be fixed, and
 $N$ large enough , such that
$c_8c_9\lambda_{N+1}^{-2}e^{c_1t_*}\leq 1/128$.
 We obtain
$$
\|\Delta w(t_*)\|\leq\frac{1}{8}\|\Delta  w(0)\|.
$$
\end{proof}

\begin{theorem} \label{thm3.10}
 Assume that $f\in C^2(\mathbb{R};\mathbb{R})$ and satisfies
{\rm (G1)--(G4)} with $f(0)=0$, $g\in V_1$. Then there exists an
 exponential attractor $\mathscr{M} \subset V_{2}$ for the semigroup
of operators $ \{S(t)\}_{t\geq0}$
 generated by \eqref{e1.1}--\eqref{e1.3}.
\end{theorem}

\begin{proof}
 From Lemma \ref{lem3.9}, $S(t_*)$ satisfies the squeezing
property for some $t_*>0$. According to \cite[Theorem 2.1]{e1}, there
exists an exponential attractor $\mathscr{M_*}$ for
$(S(t_*),\mathscr{B})$ and we set
$$
\mathscr{M}={\bigcup_{0\leq t\leq t_*}S(t)\mathscr{M}_*}.
$$
By Lemma \ref{lem3.8}, $(t,u)\mapsto S(t)u $ is Lipschitz continuous
from $[0,T]\times\mathscr{B}$ to $\mathscr{B}$.
Then as in the proof of \cite[Theorem 3.1]{e1},
 $\mathscr{M}$ is an exponential attractor for
$(\{S(t)\}_{t\geq0},\mathscr{B})$.
\end{proof}

\subsection*{Acknowledgements}
The authors would like to thank the anonymous referee for
his or her many vital comments and suggestions.

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\end{document}