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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 272, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/272\hfil Pullback attractor]
{Pullback attractor of a nonautonomous fourth-order parabolic equation
modeling epitaxial thin film growth}

\author[N. Duan, X. Zhao \hfil EJDE-2015/272\hfilneg]
{Ning Duan, Xiaopeng Zhao}

\address{Ning Duan \newline
School of Science, Jiangnan University, Wuxi 214122, China}
\email{123332453@qq.com}

\address{Xiaopeng Zhao \newline
Department of Mathematics, Southeast University, Nanjing 210096, China;
\newline
School of Science, Jiangnan University, Wuxi 214122, China}
\email{zhaoxiaopeng@sina.cn}

\thanks{Submitted July 30, 2015. Published October 21, 2015.}
\subjclass[2010]{35Q35, 35B40, 35B41}
\keywords{Pullback attractor;  asymptotic compactness;
\hfill\break\indent  nonautonomous fourth-order parabolic equation}

\begin{abstract}
 We study a nonautonomous fourth-order parabolic equation modeling epitaxial
 thin film growth. It is shown that a pullback attractor of the model exists
 when the external force has exponential growth.
\end{abstract}

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


\section{Introduction} \label{sect1}

The study of pullback attractors for nonautonomous infinite dimensional 
dynamical systems has attracted much attention and made fast progress 
in recent years; see for instance,  \cite{p6,p1,new2,new1,p4}. 
Recently, Caraballo et al \cite{p2} introduced the notion of the pullback 
$\mathcal{D}$-attractor for nonautonomous dynamical systems and gave a 
general method to prove the existence of pullback $\mathcal{D}$-attractor.  
In \cite{p3}, Li and Zhong proposed  the concept of norm-to-weak continuous 
process and proved the existence of pullback attractors for the nonautonomous 
reaction-diffusion equation. The authors in \cite{p5} considered the existence 
of pullback attractor for a nonautonomous modified Swift-Hohenberg equation 
when its external force has exponential growth.

Suppose that $\Omega$ is an open connected bounded domain in $\mathbb{R}^3$  
with a smooth boundary $\partial\Omega$, $p\in]2,\frac{10}3[$. 
We are concerned with the existence of pullback attractor for the following 
non-autonomous equation
\begin{equation}\label{1-1}
 u_t+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u-\nabla u\right)
 =g(x,t),\quad \text{in }\Omega\times[\tau,\infty),
\end{equation}
with the boundary value conditions
\begin{equation}\label{1-2}
u=\frac{\partial u}{\partial n}=0,\quad\text{on }
\partial\Omega\times[\tau,\infty),
\end{equation}
and the initial condition
\begin{equation}\label{1-3}
u(x,\tau)=u_{\tau}(x),\quad\text{in }\Omega.
\end{equation}

   Equation \eqref{1-1}  arises in epitaxial growth of nanoscale thin films
(see \cite{King,AZ}),  where $u(x,t)$ denotes the height from the
surface of the film in epitaxial growth, the term $\Delta^2u$ denotes the
capillarity-driven surface diffusion (see \cite{16}), 
$\nabla\cdot(|\nabla u|^{p-2}\nabla u)$ corresponding
to the upward hopping of atoms (see \cite{7}) and $\Delta u$ denotes the 
diffusion due to the evaporation-condensation (see \cite{9}).
Recently, Equation \eqref{1-1} was studied by several authors. 
By numerical simulations and heuristic arguments, Kohn and Yan \cite{RVK} 
proved that the standard deviation of $u$ in Equation \eqref{1-1} with $g\equiv0$
grows as $t^{-\frac 13}$, and the energy per unit area decays as
$t^{-\frac 13}$. Based on Schauder type estimates and Campanato spaces, 
Liu \cite{Liu1,Liu2} Studied the regularity of solutions for Equation 
\eqref{1-1} with nonlinear principal part and $g\equiv0$ in 1D and 2D case. 
The analysis of the long-time behavior of Equation \eqref{1-1} with
$g\equiv0$ has been  developed by \cite{Duan,Zhao1}.  There are also some 
papers which has done with the numerical analysis for the discrete scheme 
of  Equation \eqref{1-1}, such as \cite{n3,n2,n1} and so on. 
However, the existence of pullback attractor for the fourth-order evolution 
equation modeling epitaxial thin film growth has not been considered yet.

In this article, we are interested in the existence of pullback attractor 
for the  nonautonomous problem \eqref{1-1}-\eqref{1-3}.  
This article is organized as follows. In Section 2, we recall some 
abstract results on pullback attractors and give the main result. 
In Section 3, we prove the existence of pullback attractor for problem 
\eqref{1-1}-\eqref{1-3}.

Throughout this paper, we denote $(\cdot,\cdot)$ as the inner product 
of $L^2(\Omega)$ and $\|\cdot\|$ as the induced norm. $\|\cdot\|_X$ 
denotes the norm of a Banach space $X$. For simplicity, we denote 
$\|\cdot\|_{L^p(\Omega)}$ by $\|\cdot\|_p$, respectively. In the following, 
$c$ will represent generic positive constants that may change from line 
to line even if in the same inequality.

\section{Preliminaries}

In this section, we give some basic definitions and results on the existence 
of pullback attractor. Suppose that $X$ is a complete metric space and
 $\{U(t,\tau)\}=\{U(t,\tau):t\geq\tau,\tau\in\mathbb{R}\}$ is a two-parameter 
family of mappings act on $X$: 
$U(t,\tau):X\to X,t\geq\tau,\tau\in\mathbb{R}$.

\begin{definition}[\cite{add}] \label{def2.1} \rm
A two-parameter family of mappings $\{U(t,\tau)\}$ is said to be norm-to-weak 
continuous process in $X$ if
\begin{itemize}
\item $U(t,s)U(s,\tau)=U(t,\tau)$ for all $t\geq s\geq\tau$,
\item for all $\tau\in\mathbb{R}$, $U(t,\tau)=Id$ is the identity operator,
\item $U(t,\tau)x_n\rightharpoonup U(t,\tau)x$, if $x_n\rightharpoonup x$ in $X$.
\end{itemize}
\end{definition}

 Let $B$ be a bounded subset of $X$. The Kuratowski measure of noncompactness 
$\alpha(B)$ of $B$ is defined by
 $$
 \alpha(B)=\inf\{\delta>0: B \text{ has a finite open cover of sets of diameter}
\leq\delta\}.
 $$
 Suppose $\mathcal{D}$ is a nonempty class of parameterised sets 
$\hat{\mathcal{D}}=\{D(t):t\in\mathbb{R}\}\subset B(X)$.

 \begin{definition}[\cite{p3}] \label{def2.2} \rm
A process $\{U(t,\tau)\}$ is called pullback $\omega$-$\mathcal{D}$-limit 
compact if for any $\varepsilon>0$ and $\hat{D}\in\mathcal{D}$, 
there exists a $\tau_0(t,\hat{\mathcal{D}}\leq t$ such that
 $\alpha(U_{\tau\geq\tau_0}U(t,\tau)D(\tau))\leq\varepsilon$.
 \end{definition}

 \begin{definition}[\cite{p3}] \label{def2.3}
 The family $\hat{A}=\{A(t):t\in\mathbb{R}\}\subset B(X)$ is said to be 
a pullback $\mathcal{D}$-attractor for $U(t,\tau)$ if
 \begin{itemize}
 \item for all $t\in\mathbb{R}$, $A(t)$ is compact,
 \item $\hat{A}$ is invariant, i.e., $U(t,\tau)A(\tau)=A(\tau)$ for all $t\geq\tau$,
 \item $\hat{A}$ is pullback $\mathcal{D}$-attracting, i.e.,
 $$
 \lim_{\tau\to-\infty}\operatorname{dist}(U(t,\tau)D(\tau),A(t))=0,
\quad\forall\hat{\mathcal{D}}\in\mathcal{D},\; t\in\mathbb{R}.
 $$
 \item if $\{C(t)\}_{t\in\mathbb{R}}$ is another family of closed attracting sets, 
then $A(t)\subset C(t)$ for all $t\in\mathbb{R}$.
\end{itemize}
 \end{definition}

In the following, we give the result on the existence of pullback 
$\mathcal{D}$-attractor for nonautonomous systems which can be seen in \cite{p3}.

\begin{lemma} \label{lem-d}
Let $\{U(t,\tau)\}_{\tau\leq t}$ be a norm-to-weak continuous process such that 
$\{U(t,\tau)\}_{\tau\leq t}$ is pullback $\omega$-$\mathcal{D}$-limit compact. 
If there exists a family of pullback $\mathcal{D}$-absorbing sets 
$\{B(t):t\in\mathbb{R}\}\in\mathcal{D}$, i.e., for any $t\in\mathbb{R}$ and 
$\hat{D}\in\mathcal{D}$ there is a $\tau_0(t,\hat{D})\leq t$ such that 
$U(t,\tau)D(\tau)\subset B(t)$ for all $\tau\leq \tau_0$. 
Then, there is a pullback $\mathcal{D}$-attractor $\{A(t):t\in\mathbb{R}\}$ and
$$
A(t)=\cap_{s\leq t}\overline{\cup_{\tau\leq s}U(t,\tau)D(\tau)}.
$$
\end{lemma}


To study the existence of pullback attractor for problem \eqref{1-1}-\eqref{1-3}, 
we suppose that $g(x,t)$ is translation bounded in 
$L^2_{\rm loc}(\mathbb{R};L^2(\Omega))$; that is, 
$$
g(x,t)\in L^2_{\rm loc}(\mathbb{R};L^2(\Omega))\quad\text{and}\quad 
\sup_{t\in\mathbb{R}}\int_t^{t+1}|g(x,t)|^2ds<\infty.
$$
 We also suppose that, for any $t\in \mathbb{R}$, there exist $\beta\geq0$ and 
$0\leq\alpha<\big(\frac{20-6p}{6p-4}\big)^2\lambda$, such that
\begin{equation} \label{1-4}
\|g(t)\|^2\leq\beta e^{\alpha|t|},
\end{equation}
 where $\lambda$ is the first eigenvalue of $A=\Delta^2$.
By \eqref{1-4}, we have the following properties:
\begin{equation} \nonumber
 \begin{gathered}
G_1(t):=\int_{-\infty}^te^{\lambda s}\|g(s)\|^2ds<\infty,\quad\forall
  t\in\mathbb{R}, \\
G_2(t):=\int_{-\infty}^t\int_{-\infty}^se^{\lambda r}\|g(r)\|^2\,dr\,ds<\infty,
 \quad\forall t\in\mathbb{R},\\
\int_{-\infty}^te^{\frac{24-12p}{20-6p}\lambda s}\big[[G_1(s)]
^{\frac{6p-4}{20-6p}}+[G_2(s)]^{\frac{6p-4}{20-6p}}\big]ds<\infty,
 \quad\forall t\in\mathbb{R}.
\end{gathered} 
\end{equation}

Using a slight modification of the classical results in the autonomous framework, 
mainly of the Faedo-Galerkin method (see \cite{18}), we obtain the following 
result on the existence and uniqueness of solutions for problem
 \eqref{1-1}-\eqref{1-3} (see \cite{King,Zhao1}).

\begin{lemma} \label{lem1.1}
Suppose that $g\in L^2_{\rm loc}(\mathbb{R};L^2(\Omega))$. 
There is a unique solution $u(x,t)$ such that
\begin{itemize}
\item if $u_0=u_{\tau}\in L^2(\Omega)\Rightarrow 
 u(x,t)\in C^0([\tau,\infty);L^2(\Omega))$;
\item if $u_0=u(\tau)\in H_0^2(\Omega)\Rightarrow 
 u(x,t)\in C^0([\tau,\infty);H_0^2(\Omega))$.
\end{itemize}
\end{lemma}

Based on Lemma \ref{lem1.1}, we obtain that the solution $u(x,t)$ is 
continuous with respect to the initial value condition $u_{\tau}$ in the 
space $H_0^2(\Omega)$. In order to construct a process $\{U(t,\tau)\}$ 
for problem \eqref{1-1}-\eqref{1-3}, we define 
$U(t,\tau):H_0^2(\Omega)\to H_0^2(\Omega)$ by $U(t,\tau)u_{\tau}$. 
Thus, the process $\{U(t,\tau)\}$ is a norm-to-weak continuous process 
in the space $H_0^2(\Omega)$.

Now, we give the main result of this article, which will be proved in 
the next section.

\begin{theorem} \label{thm1.1}
The process corresponding to problem \eqref{1-1}-\eqref{1-3} possesses 
a unique pullback $\mathcal{D}$-attractor in the space $H_0^2(\Omega)$.
\end{theorem}

\section{Proof of main results}

In this section, we study the existence of pullback $\mathcal{D}$-attractors 
for non-autonomous problem \eqref{1-1}-\eqref{1-3}.
First of all, we derive uniform estimates of solutions for problem
 \eqref{1-1}-\eqref{1-3}, which are necessary for proving the existence 
of absorbing set of $\{u(t,\tau)\}$ associated with the problem.

\begin{lemma} \label{lem3.1}
Consider the problem \eqref{1-1}-\eqref{1-3}, for all $t\geq\tau$, we have
$$
\|u(t)\|^2\leq e^{-\lambda(t-\tau)}\|u_{\tau}\|^2
+\frac{2M}{\lambda}+\frac1{\lambda}e^{-\lambda t}G_1(t),
$$
and
\begin{equation}
\int_{\tau}^te^{\lambda s}\|\Delta u(s)\|^2ds
\leq [1+\lambda(t-\tau)]e^{\lambda\tau}\|u_{\tau}\|^2
 +\frac{4M}{\lambda}e^{\lambda t}+\frac1{\lambda}G_1(t)+G_2(t),
\label{xp}
\end{equation}
where $M=\frac p{p-2}(\frac p2)^{\frac{-2}{p-2}}$ is a positive constant. 
\end{lemma}

\begin{proof}
Multiplying \eqref{1-1} by $u$, integrating it over $\Omega$, using Young's 
inequality, we obtain
\begin{equation}
\begin{aligned}
&\frac12\frac d{dt}\|u(t)\|^2+\|\Delta u(t)\|^2+\|\nabla u(t)\|_p^p\\
&=\|\nabla u(t)\|^2+(g(t),u(t))
\\
&\leq \frac12\|\nabla u(t)\|^p_p+ M+\|g(t)\|\|u(t)\|\\
&\leq \frac12\|\nabla u(t)\|_p^p+ M+\frac1{2\lambda}\|g(t)\|^2
+\frac12\|\Delta u(t)\|^2,
\end{aligned} \label{2-0}
\end{equation}
where $M=\frac {2p}{p-2}(\frac p2)^{\frac{-2}{p-2}}$. 
It then follows from \eqref{2-0} that
\begin{gather} \label{2-1}
\frac d{dt}\|u(t)\|^2+\|\Delta u(t)\|^2+\|\nabla u(t)\|_p^p\leq 2M
+\frac{\|g(t)\|^2}{\lambda}, \\
\label{2-2}
\frac d{dt}\|u(t)\|^2+\lambda\| u(t)\|^2\leq 2M+\frac{\|g(t)\|^2}{\lambda},
\end{gather}
Multiplying \eqref{2-2} by $e^{\lambda(t-\tau)}$ and integrating it over
 $(\tau,t)$, we derive that
\begin{equation}\label{2-3}
\begin{aligned}
\|u(t)\|^2\leq e^{-\lambda(t-\tau)}\|u_{\tau}\|^2
+\frac{2M}{\lambda}+\frac1{\lambda}e^{-\lambda t}
\int_{-\infty}^te^{\delta s}\|g(s)\|^2ds.
\end{aligned}
\end{equation}
Multiplying \eqref{2-3} by $e^{\lambda t}$ and integrating it over 
$(\tau,t)$, we deduce that
\begin{equation} \label{2-4}
\int_{\tau}^te^{\lambda s}\|u(s)\|^2ds
\leq (t-\tau)e^{\lambda \tau}\|u_{\tau}\|^2+\frac{2M}{\lambda^2}e^{\lambda t}
+\frac1{\lambda}\int_{-\infty}^t\!\!\int_{-\infty}^se^{\lambda r}\|g(r)\|^2\,dr\,ds.
\end{equation}
Similarly, multiplying \eqref{2-1} by $e^{\lambda t}$ and integrating it over 
$(\tau,t)$, we obtain
\begin{equation}\label{2-5}
\begin{aligned}
&\int_{\tau}^te^{\lambda s}\|\Delta u(s)\|^2ds
 +\int_{\tau}^te^{\lambda s}\|\nabla u(s)\|_p^pds\\
&\leq e^{\lambda \tau}\|u_{\tau}\|^2+\frac{2M}{\lambda}e^{\lambda t}
 +\lambda\int_{\tau}^te^{\lambda s}\|u(s)\|^2ds
 +\frac1{\lambda}\int_{-\infty}^te^{\lambda s}\|g(s)\|^2ds
\\
&\leq \left[1+\lambda(t-\tau)\right]e^{\lambda \tau}\|u_{\tau}\|^2
 +\frac{4M}{\lambda}e^{\lambda t}
+\int_{-\infty}^t\!\!\int_{-\infty}^se^{\lambda r}\|g(r)\|^2\,dr\,ds
\\
&\quad +\frac1{\lambda}\int_{-\infty}^te^{\lambda s}\|g(s)\|^2ds.
\end{aligned}
\end{equation}
The proof is complete.
\end{proof}


\begin{lemma}\label{lem3.2}
In problem \eqref{1-1}-\eqref{1-3},  for all $t\geq\tau$, we have
\begin{equation}
\begin{aligned}
&\|\nabla u(t)\|^2\\
&\leq c\Big\{e^{-\lambda(t-\tau)}(\|u_{\tau}\|^2
 +\|\nabla u_{\tau}\|^2)+\frac{2M}{\lambda^2}
 +e^{-\lambda t}\big[\frac1{\lambda} G_2(t)+G_1(t)\big]\Big\}.
\end{aligned}\label{xp-0}
\end{equation}
\end{lemma}

\begin{proof}
Multiplying \eqref{1-1} by $\Delta u$, integrating it over $\Omega$, 
we derive that
\begin{equation}
\begin{aligned}
& \frac12\frac d{dt}\|\nabla u(t)\|^2+\|\nabla\Delta u(t)\|^2
 +(\nabla\cdot(|\nabla u(t)|^{p-2}\nabla u(t), \Delta u(t)) \\
&= \|\Delta u(t)\|^2-(g(t),\Delta u(t)).
\end{aligned} \label{yue-1}
\end{equation}
Note that
\begin{equation}
\begin{aligned}
& (\nabla\cdot(|\nabla u(t)|^{p-2}\nabla u(t),
\Delta u(t))\\
&= \int_{\Omega}|\nabla u|^{p-2}|\Delta u|^2dx+\int_{\Omega}\nabla
u(|\nabla u|^{p-2})\Delta udx
\\
&= \int_{\Omega}|\nabla u|^{p-2}|\Delta u|^2dx
+\frac{p-2}2\int_{\Omega}\nabla u(|\nabla u|^2)|\nabla u|^{p-4}\Delta u\,dx
\\
&= \int_{\Omega}|\nabla u|^{p-2}|\Delta
u|^2dx+(p-2)\int_{\Omega}(\sum_{i,j=1}^3u_{ij}u_iu_j)(\sum_{k=1}^3u_{kk})|\nabla
u|^{p-4}\,dx
\\
&= \int_{\Omega}|\nabla u|^{p-2}|\Delta
u|^2dx+(p-2)\int_{\Omega}(\sum_{i=1}^3u_{ii}u^2_i)(\sum_{k=1}^3u_{kk})|\nabla
u|^{p-4}dx\\
&\quad +(p-2)\int_{\Omega}(\sum_{i,j=1,i\neq
j}^3u_{ij}u_iu_j)(\sum_{k=1}^3u_{kk})|\nabla u|^{p-4}dx
\\
&= \int_{\Omega}|\nabla u|^{p-2}|\Delta u|^2dx
 +(p-2)\sum_{i=1}^3\int_{\Omega}|\nabla u|^{p-4}u_i^2u_{ii}^2dx
\\
&\quad +(p-2)\sum_{i,j=1,i\neq j}^3\int_{\Omega}|\nabla u|^{p-4}u_i^2u_{ii}u_{jj}dx\\
&\quad +(p-2)\int_{\Omega}|\nabla u|^{p-4}
 \Delta u(\sum_{i,j=1,i\neq j}^3u_{ij}u_iu_j)dx\,.
\end{aligned}\label{yue-2}
\end{equation}
Using H\"{o}lder's inequality, we have
\begin{equation}
\begin{aligned}\label{re-1}
&(p-2)\sum_{i,j=1,i\neq j}^3\int_{\Omega}|\nabla
u|^{p-4}u_i^2u_{ii}u_{jj}dx
\\
&\geq -\frac{p-2}2\sum_{i,j=1,i\neq j}^3\int_{\Omega}|\nabla
u|^{p-4}u_i^2(u_{ii}^2+u_{jj}^2)dx
\\
&= -\frac{p-2}2\sum_{i=1}^3\int_{\Omega}|\nabla u|^{p-4}u_i^2u_{ii}^2dx
 -\frac{p-2}2\sum_{i,j=1,i\neq j}^3\int_{\Omega}|\nabla u|^{p-4}u_i^2u_{jj}^2dx.
\end{aligned}
\end{equation}
Based on the regularity theorem of elliptic equation, we have
\begin{equation}
\begin{aligned}
&(p-2)\int_{\Omega}|\nabla
u|^{p-4}\Delta u\Big(\sum_{i,j=1,i\neq j}^3u_{ij}u_iu_j\Big)dx\\
&\geq-\frac {p-2}2\int_{\Omega}|\nabla u|^{p-2}|\Delta u|^2dx.
\end{aligned} \label{re-2}
\end{equation}
Combining \eqref{yue-2}, \eqref{re-1} and \eqref{re-2} together gives
\begin{equation}
\begin{aligned}
&(\nabla\cdot(|\nabla u(t)|^{p-2}\nabla u(t), \Delta u(t))\\
&\geq \int_{\Omega}|\nabla u|^{p-2}|\Delta
u|^2dx+\frac{p-2}2\sum_{i=1}^3\int_{\Omega}|\nabla
u|^{p-4}u_i^2u_{ii}^2dx \\
&\quad -\frac {p-2}2\int_{\Omega}|\nabla u|^{p-2}|\Delta u|^2dx
\geq 0.
\end{aligned}\label{yue-2-1}
\end{equation}
By Nirenberg's inequality, we obtain
\begin{equation}\label{re-3}
\|\Delta u(t)\|\leq c\|\nabla\Delta u(t)\|^{2/3}\|u(t)\|^{1/3}.
\end{equation}
It then follows from  \eqref{yue-1}, \eqref{yue-2-1} and \eqref{re-3}  that
\begin{equation} \label{yue-3}
\begin{aligned}
&\frac12\frac d{dt}\|\nabla u(t)\|^2+\|\nabla\Delta u(t)\|^2\\
&\leq \|\Delta u(t)\|^2+\|g(t)\|\|\Delta u(t)\|
\\
&\leq c\|\nabla\Delta u(t)\|^{\frac43}\|u(t)\|^{2/3}
 +c\|g(t)\|\|\nabla\Delta u(t)\|^{2/3}\|u(t)\|^{1/3} \\
&\leq \frac12\|\nabla \Delta u(t)\|^2+c(\|u(t)\|^2+\|g(t)\|^2);
\end{aligned}
\end{equation}
that is,
\begin{equation} \label{yue-4}
\frac d{dt}\|\nabla u(t)\|^2+\|\nabla\Delta u(t)\|^2\leq c(\|u(t)\|^2+\|g(t)\|^2),
\end{equation}
and
\begin{equation} \label{yue-5}
\frac d{dt}\|\nabla u(t)\|^2+\lambda\|\nabla u(t)\|^2\leq c(\|u(t)\|^2+\|g(t)\|^2).
\end{equation}
Multiplying \eqref{yue-5} by $e^{\lambda(t-\tau)}$ and integrating 
 over $(\tau,t)$, we derive that
\begin{equation}
\begin{aligned}
&\|\nabla u(t)\|^2\\
&\leq  e^{-\lambda(t-\tau)}\|\nabla u_{\tau}\|^2
 +ce^{-\lambda t}\int_{\tau}^te^{\lambda s}\|u(s)\|^2ds
 +ce^{-\lambda t}\int_{\tau}^te^{\lambda s}\|g(s)\|^2ds
\end{aligned}\label{yue-6}
\end{equation}
Combining \eqref{2-4} and \eqref{yue-6}  gives
\begin{equation}
\nonumber
\|\nabla u(t)\|^2
\leq c\Big\{e^{-\lambda(t-\tau)}(\|u_{\tau}\|^2
+\|\nabla u_{\tau}\|^2)+\frac{2M}{\lambda^2}+e^{-\lambda t}
\big[\frac1{\lambda} G_2(t)+G_1(t)\big]\Big\}.
\end{equation}
Hence, the proof is complete.
\end{proof}

\begin{lemma}\label{lem2.2}
Consider the problem \eqref{1-1}-\eqref{1-3}, for all $t\geq\tau$, we have
\begin{align*}
\|\Delta u(t)\|^2
&\leq  c\Big\{\big(1+(t-\tau)+\frac1{t-\tau}\big)e^{-\lambda(t-\tau)}
\big[\|u_{\tau}\|^2+\| u_{\tau}\|^{\frac{6p-4}{10-3p}}
+\|\nabla u_{\tau}\|^{\frac{6p-4}{10-3p}}\big]\\
&\quad +\big(1+\frac1{t-\tau}\big)\{1+e^{-\lambda t}[G_1(t)+G_2(t)]\}
\\
&\quad +e^{-\lambda t}\int_{-\infty}^te^{\frac{24-12p}{20-6p}\lambda s}
\left[[G_1(s)]^{\frac{6p-4}{20-6p}}+[G_2(s)]^{\frac{6p-4}{20-6p}}\right]ds\Big\}.
\end{align*}
\end{lemma}

\begin{proof}
Multiplying \eqref{1-1} by $\Delta^2u$, integrating it over $\Omega$, 
using Nirenberg's inequality and Young's inequality, we derive that
\begin{equation}
\begin{aligned}
&\frac12 \frac d{dt}\|\Delta u(t)\|^2+\|\Delta^2u(t)\|^2\\
&= \left(\nabla\cdot(|\nabla u(t)|^{p-2}\nabla u(t)),\Delta^2u(t)\right)
+(g(t),\Delta^2u(t))-(\Delta u(t),\Delta^2u(t))
\\
&\leq \frac14\|\Delta^2u(t)\|^2+c(\|\Delta u(t)\|^2+\|g(t)\|^2
 +\|\nabla\cdot(|\nabla u(t)|^{p-2}\nabla u(t))\|^2)\\
&\leq \frac14\|\Delta^2u(t)\|^2+c(\|\Delta u(t)\|^2+\|g(t)\|^2
 +\|\Delta u(t)\|_{\infty}^2\|\nabla u(t)\|_{2p-4}^{2p-4})
\\
&\leq \frac12\|\Delta^2u(t)\|^2+c(\|\Delta u(t)\|^2+\|g(t)\|^2\\
&\quad +\|\Delta^2u(t)\|^{\frac 53}\|\nabla u(t)\|^{1/3}
 \|\Delta^2u(t)\|^{p-3}\|\nabla u(t)\|^{p-1})
\\
&\leq \frac12\|\Delta u(t)\|^2+c(\|\Delta u(t)\|^2+\|g(t)\|^2
 +\|\nabla u(t)\|^{\frac{6p-4}{10-3p}});
\end{aligned} \label{revised}
\end{equation}
that is,
\begin{equation} \label{2-7}
\frac d{dt}\|\Delta u(t)\|^2+\|\Delta^2u(t)\|^2
\leq c(\|\Delta u(t)\|^2+\|g(t)\|^2+\|\nabla u(t)\|^{\frac{6p-4}{10-3p}}),
\end{equation}
and
\begin{equation} \label{2-8}
\frac d{dt}\|\Delta u(t)\|^2+\lambda\|\Delta u(t)\|^2\leq c(\|\Delta u(t)\|^2
+\|g(t)\|^2+\|\nabla u(t)\|^{\frac{6p-4}{10-3p}}),
\end{equation}
Multiplying this by $(t-\tau)e^{\lambda t}$ and integrating it over $(\tau,t)$,
 we obtain
\begin{align*}
&(t-\tau)e^{\lambda t}\|\Delta u(t)\|^2\\
&\leq  c\Big[\int_{\tau}^t[1+(s-\tau)]e^{\lambda s}\|\Delta u(s)\|^2ds
+\int_{\tau}^t(s-\tau)e^{\lambda s}\|g(s)\|^2ds \\
&+\int_{\tau}^t(s-\tau)e^{\lambda s}\|\nabla u(t)\|^{\frac{6p-4}{10-3p}}\Big].
\end{align*}
Therefore,
\begin{equation}\label{2-9}
\begin{aligned}
 \|\Delta u(t)\|^2
&\leq  c\Big(1+\frac1{t-\tau}\Big) e^{-\lambda t}\int_{\tau}^te^{\lambda s}
\|\Delta u(s)\|^2ds+ce^{-\lambda t}G_1(t)\\
&\quad +ce^{-\lambda t}\int_{\tau}^te^{\lambda s}\|\nabla u(t)
 \|^{\frac{6p-4}{10-3p}}ds
\\
&=:I_1+I_2+I_3.\end{aligned}
\end{equation}
It then follows from \eqref{xp} that
\begin{equation}
\begin{aligned}
I_1 &\leq c\Big(1+\frac1{t-\tau}\Big)e^{-\lambda t}
\Big[[1+\lambda(t-\tau)]e^{\lambda\tau}\|u_{\tau}\|^2
+\frac{4M}{\lambda}e^{\lambda t}+\frac1{\lambda}G_1(t)+G_2(t)\Big]
\\
&\leq c\Big(1+(t-\tau)+\frac1{t-\tau}\Big)
 e^{-\lambda(t-\tau)}\|u_{\tau}\|^2+c\Big(1+\frac1{t-\tau}\Big)\\
&\quad +c\Big(1+\frac1{t-\tau}\Big)e^{-\lambda t}[G_1(t)+G_2(t)].
\end{aligned}\label{xp-1}
\end{equation}
On the other hand, using H\"{o}lder's inequality and \eqref{xp-0} 
for sum, we obtain
\begin{equation}
\begin{aligned}
I_3&\leq ce^{-\lambda t}\int_{\tau}^te^{\lambda s}
 \Big\{e^{-\lambda(t-\tau)}(\|u_{\tau}\|^2+\|\nabla u_{\tau}\|^2)
 +\frac{2M}{\lambda^2}\\
&\quad +e^{-\lambda s}\big[\frac1{\lambda} G_2(s)+G_1(s)\big]
 \Big\}^{\frac{6p-4}{20-6p}}ds
\\
&\leq  ce^{-\lambda t}\int_{\tau}^te^{\lambda s}ds
 +ce^{-\lambda t}\int_{\tau}^te^{\lambda s}e^{-\frac{6p-4}{20-6p}
 \lambda (s-\tau)}(\|u_{\tau}\|^2
 +\|\nabla u_{\tau}\|^2)^{\frac{6p-4}{20-6p}}ds
\\
&\quad +ce^{-\lambda t}\int_{\tau}^te^{\lambda s}e^{-\frac{6p-4}{20-6p}\lambda s}
 \{[G_1(t)]^{\frac{6p-4}{20-6p}}+[G_2(t)]^{\frac{6p-4}{20-6p}}\}ds
\\
&\leq ce^{-\lambda t}[e^{\lambda t}-e^{\lambda \tau}]
 +ce^{-\lambda (t-\tau)}(\|u_{\tau}\|^{\frac{6p-4}{10-3p}} \\
&\quad +\|\nabla u_{\tau}\|^{\frac{6p-4}{10-3p}})\int_{\tau}^t
 e^{-\frac{6p-4}{20-6p}\lambda(s-\tau)}ds
\\
&\quad +ce^{-\lambda t}\int_{\tau}^te^{\frac{24-12p}{20-6p}
 \lambda s}\{[G_1(t)]^{\frac{6p-4}{20-6p}}+[G_2(t)]^{\frac{6p-4}{20-6p}}\}ds
\\
&\leq c+c(t-\tau)e^{-\lambda(t-\tau)}(\|u_{\tau}\|^{\frac{6p-4}{10-3p}}
 +\|\nabla u_{\tau}\|^{\frac{6p-4}{10-3p}})\\
&\quad +ce^{-\lambda t}\int_{\tau}^te^{\frac{24-12p}{20-6p}\lambda s}
 \{[G_1(t)]^{\frac{6p-4}{20-6p}}+[G_2(t)]^{\frac{6p-4}{20-6p}}\}ds.
\end{aligned}\label{xp-2}
\end{equation}
Combining  \eqref{xp-1} and \eqref{xp-2} with \eqref{2-9}, we complete the proof.
\end{proof}

\begin{remark} \label{rmk3.4} \rm
In this article, motivated by the ideas in \cite{p3,p5}, we establish the
 existence of pullback attractors for the nonautomous  fourth-order parabolic 
problem \eqref{1-1}-\eqref{1-3} in 3D case. To overcome the difficulty 
caused by the nonlinearity term $\nabla\cdot(|\nabla u|^{p-2}\nabla u)$, 
we  impose the exponential growth condition \eqref{1-4} on the external 
forcing term $g(x,t)$. On the other hand, in order to obtain the suitable 
a prior estimates, we have to restrict the parameter $p$ and take advantage 
of the Nirenberg's inequality more times. For example, in the six line of 
\eqref{revised}, the exponent of $\|\Delta^2u(t)\|$ is $\frac53+p-3$,
 which should be less than $2$, so we have to let the parameter
 $p\in]2,\frac{10}3[$. Furthermore, if the problem is studied in $n$D case, 
where $n\leq3$, we need only restrict the parameter $p\in]2,2+\frac4n[$. 
Using the same method as this article, we can obtain the result on 
the existence of pullback attractor for $n$D problem \eqref{1-1}-\eqref{1-3}
 when $p\in]2,2+\frac4n[$ and $n\leq 3$.
\end{remark}


Let $\mathfrak{R}$ be the set of all function $r:\mathbb{R}\to(0,\infty)$ such that
$$
\lim_{t\to-\infty}te^{\delta t}r^{\frac{6p-4}{10-3p}}(t)=0.
$$
Denote by $\mathcal{D}$ the class of all families 
$\hat{D}:=\{D(t):t\in\mathbb{R}\}\subset B(H_0^2(\Omega))$ such that 
$D(t)\subset\overline{B_0}(r(t))$ for some $r(t)\in\mathfrak{R}$, 
$\overline{B_0}(r(t))$ denote the closed ball in $H_0^2(\Omega)$ with radius 
$r(t)$. Let
\begin{align*}
r_0^2(t)
&= 2c\Big[1+e^{-\lambda t}G_1(t)+e^{-\lambda t}G_2(t)\\
&\quad +e^{-\lambda t}\int_{-\infty}^t
 e^{\frac{24-12p}{20-6p}\lambda s}
\big[[G_1(s)]^{\frac{6p-4}{20-6p}}+[G_2(s)]^{\frac{6p-4}{20-6p}}\big]ds\Big].
\end{align*}
Using the continuity of embedding $H_0^2(\Omega)\hookrightarrow L^2(\Omega)$, 
for any $\hat{D}\in\mathcal{D}$ and $t\in\mathbb{R}$ there exists 
$\tau_0(\hat{D},t)<t$ such that
\begin{equation}\label{2-10} 
\|\Delta u(t)\|\leq r_0(t),\quad \forall\tau<\tau_0.
\end{equation}
Furthermore, since $0\leq\alpha<\big(\frac{20-6p}{6p-4}\big)^2\lambda$, 
we obtain $\overline{B_0}(r_0(t))\in\mathcal{D}$. Therefore, 
$\overline{B_0}(r_0(t))$ is a family of bounded pullback $\mathcal{D}$-absorbing 
sets in $H_0^2(\Omega)$.

Now, we present the proof of the main result.

\begin{proof} [Proof of Theorem \ref{thm1.1}]
To prove the existence of pullback attractor for problem \eqref{1-1}-\eqref{1-3}, 
we need only prove that the process $\{u(t,\tau)\}$ is pullback 
$\omega-\mathcal{D}$-limit compact (PDC).
Thanks to $A^{-1}$ is a continuous compact operator in $L^2(\Omega)$, 
there is a sequence $\{\lambda_j\}_{j=1}^{\infty}$ satisfying
$$
\lambda_1\leq\lambda_2\leq\dots\leq\lambda_j\leq\dots,\quad
\lambda_j\to\infty,\quad \text{as }j\to\infty,
$$
and a family of elements $\{w_j\}_{j=1}^{\infty}$ of $H_0^2(\Omega)$ 
which are orthonormal in $L^2(\Omega)$ such that
$$
Aw_j=\lambda_jw_j,\quad\text{for}~j=1,2,\dots.
$$
Write $X_n=\text{span}\{w_1,w_2,\dots,w_n\}\subset H_0^2(\Omega)$ and 
$P_n:H_0^2(\Omega)\to X_n$ is an orthogonal projector. Hence 
$$
u=P_nu+(I-P_n)u:=u_1+u_2.
$$
Taking the scalar product of \eqref{1-1} with $\Delta^2u_2$, using Young's 
inequality, we obtain
\[
\frac12\frac d{dt}\|\Delta u_2(t)\|^2+\|\Delta^2u_2(t)\|^2
\leq \frac12\|\Delta^2u_2(t)\|^2+c(\|\Delta u(t)\|^2
 +\|g(t)\|^2+\|\nabla u(t)\|^{\frac{6p-4}{10-3p}}),
\]
which means
\begin{equation} \label{2-11}
\frac d{dt}\|\Delta u_2(t)\|^2+\lambda_n\|\Delta u_2(t)\|^2
\leq c(\|\Delta u(t)\|^2+\|g(t)\|^2+\|\nabla u(t)\|^{\frac{6p-4}{10-3p}}).
\end{equation}
Multiplying \eqref{2-11} by $(t-\tau)e^{\lambda_nt}$ and integrating it over 
$(\tau,t)$, we deduce that
\begin{equation}
\begin{aligned}
& (t-\tau)e^{\lambda_nt}\|\Delta u_2(t)\|^2\\
&\leq \int_{\tau}^te^{\lambda_ns}\|\Delta u_2(s)\|^2dx
+c\int_{\tau}^t(s-\tau)e^{\lambda_ns}\|\Delta u(s)\|^2ds\\
&\quad +c\int_{\tau}^t(s-\tau)e^{\lambda_ns}\|g(s)\|^2ds
 +c\int_{\tau}^t(s-\tau)e^{\lambda_ns}\|\nabla u(s)\|^{\frac{6p-4}{10-3p}}ds
\\
&\leq \int_{\tau}^te^{\lambda_ns}\|\Delta u(s)\|^2dx
 +c(t-\tau)\int_{\tau}^te^{\lambda_ns}\|\Delta u(s)\|^2ds\\
&\quad +c(t-\tau)\int_{\tau}^te^{\lambda_ns}\|g(s)\|^2ds
 +c\int_{\tau}^t(s-\tau)e^{\lambda_ns}\|\nabla u(s)\|^{\frac{6p-4}{10-3p}}ds.
\end{aligned}\label{2-12}
\end{equation}
It then follows from \eqref{2-10} and \eqref{2-12} that
\begin{equation}
\begin{aligned}
&\|\Delta u_2(t)\|^2\\
&\leq (t-\tau)^{-1}e^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}
 \|\Delta u(s)\|^2dx+ce^{-\lambda_nt}
 \int_{\tau}^te^{\lambda_ns}\|\Delta u(s)\|^2ds\\
&\quad +ce^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}\|g(s)\|^2ds
 +ce^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}\|\nabla u(s)\|^{\frac{6p-4}{10-3p}}ds
\\
&\leq c(t-\tau)^{-1}e^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}r_0^2(s)dx
 +ce^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}r_0^2(s)ds\\
&\quad +ce^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}\|g(s)\|^2ds
 +ce^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}r_0^{\frac{6p-4}{10-3p}}(s)ds,
\quad\forall\tau\leq \tau_0.
\end{aligned}\label{2-13}
\end{equation}
Note that
\begin{equation}
\begin{aligned}
e^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}r_0^2(s)dx
&= ce^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}[1+e^{-\lambda s}G_1(s)
+e^{-\lambda s}G_2(s)]ds
\\
&\leq c\lambda_n^{-1}+c(\lambda_n-\lambda)^{-1}e^{-\lambda t}[G_1(t)+G_2(t)].
\end{aligned} \label{2-14}
\end{equation}
and
\begin{equation}
\begin{aligned}
& e^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}r_0^{\frac{6p-4}{10-3p}}(s)ds
\\
&= ce^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}\Big[1+e^{-\lambda s}G_1(s)
+e^{-\lambda s}G_2(s)\\
&\quad +e^{-\lambda s}\int_{-\infty}^se^{\frac{24-12p}{20-6p}\lambda r}
\big[ [G_1(r)]^{\frac{6p-4}{20-6p}}+[G_2(r)]^{\frac{6p-4}{20-6p}}\big]dr
 \Big]^{\frac{6p-4}{20-6p}}ds
\\
&\leq c\lambda_n^{-1}+c\Big(\lambda_n-\frac{6p-4}{20-6p}\lambda\Big)^{-1}
 e^{-\frac{6p-4}{20-6p}\lambda t}
 \big\{[G_1(t)]^{\frac{6p-4}{20-6p}}+[G_2(t)]^{\frac{6p-4}{20-6p}}\big\}
\\
&\quad +c\Big(\lambda_n-\frac{6p-4}{20-6p}\lambda\Big)^{-1}
 e^{-\frac{6p-4}{20-6p}\lambda t}
\Big[\Big(\int_{-\infty}^te^{\frac{24-12p}{20-6p}\lambda s}[G_1(s)
 ]^{\frac{6p-4}{20-6p}}\Big)^{\frac{6p-4}{20-6p}}\\
&\quad +\Big(\int_{-\infty}^te^{\frac{24-12p}{20-6p}
 \lambda s}[G_2(s)]^{\frac{6p-4}{20-6p}}\Big)^{\frac{6p-4}{20-6p}}\Big].
\end{aligned} \label{zzz}
\end{equation}
On the other hand, simple calculations show that
\begin{equation} \label{2-15}
\begin{aligned}
e^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}\|g(s)\|^2ds
&\leq  \begin{cases}
  \beta e^{-\lambda_nt}\int_{\tau}^te^{\lambda_ns}e^{-\alpha s}ds,
& t\leq 0, \\[4pt]
  \beta e^{-\lambda_nt}\int_{\tau}^{t}e^{\lambda_ns}e^{\alpha|s|}ds,& t\geq 0,
\end{cases} 
\\
&\leq  \begin{cases}
 \frac{\beta e^{-\lambda_nt}}{\lambda_n-\alpha},& t\leq0, \\[4pt]
 \frac{\beta e^{-\lambda_nt}}{\lambda_n-\alpha}
 +\frac{\beta e^{\alpha t}}{\lambda_n+\alpha}, & t\geq 0.
\end{cases} 
\end{aligned}
\end{equation}
Adding \eqref{2-13}, \eqref{2-14}, \eqref{zzz} and \eqref{2-15}, we can obtain 
for any $\varepsilon>0$, there exist $\tau_0<t$ and $N\in\mathbb{N}$ such that
$$
\|\Delta u_2(t)\|\leq\varepsilon,\quad\forall \tau<\tau_0.
$$
It indicates that the process $\{U(t,\tau)\}$ is pullback 
$\omega-\mathcal{D}$-limit compact. Then, by Lemma \ref{lem-d}, 
the process corresponding to problem \eqref{1-1}-\eqref{1-3} possesses 
a unique pullback $\mathcal{D}$-attractor in $H_0^2(\Omega)$.
\end{proof}

\subsection*{Acknowledgements}
The authors would thank the referees for their
valuable suggestions for the revision and improvement of the original
manuscript. This research is supported by 
the Natural Science Foundation of China for Young Scholar (No. 11401258),
and by the Natural Science Foundation of Jiangsu Province for Young Scholar 
(No. BK20140130).

\begin{thebibliography}{00}

\bibitem{p6} T. Caraballo, R. Colucci;
\emph{Pullback attractor for a non-linear evolution equation in elasticity},
 Nonlinear Anal.: RWA, 15 (2014), 80-88.

\bibitem{p2} T. Caraballo, G. Lukaszewicz, J. Real;
\emph{Pullback attractors for asymptotically compact nonautonomous dynamical systems}, 
Nonlinear Anal., 64 (2006), 484-498.

\bibitem{p1} V. V. Chepyzhov, M. I. Vishik;
\emph{Attractors for Equations of Mathematics Physics}, 
American Mathematical Society, Providence, RI, 2002.

\bibitem{new2} A. Cheskidov, L. Kavlie;
\emph{Pullback attractors for generalized evolutionary systems}, 
Discrete and Continuous Dynamical Systems, Series B, 20(3)(2015), 749-779.

\bibitem{7} S. Das Sarma, S. V. Ghaisas;
\emph{Solid-on-solid rules and models for nonequilibrium growth in 2+1 dimensions},
Phys. Rev. Lett., 69 (1992), 3762-3765.

\bibitem{Duan} N. Duan, X. P. Zhao;
\emph{Golbal attractor for a fourth-order parabolic equation modeling epitaxial 
thin film growth}, Bull. Polish Acad. Sci. Math. 60 (2012), 259-268.

\bibitem{9} S. F. Edwards, D. R. Wilkinson;
\emph{The surface statistics of a granular aggregate}, Proc. Roy. Soc. London Ser.
A, 381(1982), 17-31.

\bibitem{King} B. B. King, O. Stein, M. Winkler;
\emph{A fourth order parabolic equation modeling epitaxial thin film growth}, 
J. Math. Anal. Appl, 286(2003), 459-490.

\bibitem{RVK} R. V. Kohn, X. Yan;
\emph{Upper bound on the coarsening rate for an epitaxial growth model},
Comm. Pure Appl. Math., 56(2003), 1549-1564.

\bibitem{new1} P. E. Kloeden, J. Simsen;
\emph{Pullback attractors for non-autonomous evolution with spatially variable 
exponents}, Comm. Pure Appl. Anal., 13(2014), 2543-2557.

\bibitem{add} Y. Li, S. Wang, H. Wu;
\emph{Pullback attractors for non-autonomous reaction-diffusion equations 
in $L^p$}, Appl. Math. Comput., 207 (2009), 373-379.

\bibitem{p3} Y. Li, C. K. Zhong;
\emph{Pullback attractors for the norm-to-weak continuous process and 
application to the nonautonomoous reaction-diffusion equations}, 
Appl. Math. Comput., 190(2007), 1020-1029.

\bibitem{Liu1} C. Liu;
\emph{Regularity of solutions for a fourth order parabolic equation},
 Bull. Belg. Math. Soc. Simon Stevin, 13 (3) (2006), 527-535.

\bibitem{Liu2} C. Liu;
\emph{A fourth order parabolic equation with nonlinear
principal part}, Nonlinear Anal., 68 (2008), 393-401.

\bibitem{n3} F. N. Liu, X. P. Zhao, B. Liu;
\emph{Finite element analysis of a nonlinear
parabolic equation modeling epitaxial thin-film growth}, 
Boundary Value Problems, 2014, 2014:46.

\bibitem{16} W. W. Mullins;
\emph{Theory of thermal grooving}, J. Appl. Phys., 28 (1957), 333-339.

\bibitem{p5}S. H. Park, J. Y. Park;
\emph{Pullback attractor for a non-autonomous modified Swift-Hohenberg equation}, 
Comput. Math. Appl., 67 (2014), 542-548.

\bibitem{n2} Z. H. Qiao, Z. Z. Sun, Z. R. Zhang;
\emph{The stability and convergence of some finite difference 
schemes for the nonlinear epitaxial growth model}, 
Numer. Methods Partial Differential Equations, (28) (6) 2012, 1893-1915.

\bibitem{p4} H. Song, H. Wu;
\emph{Pullback attractors of nonautonomous reaction-diffusion equations}, 
J. Math. Anal. Appl., 325 (2007), 1200-1215.

\bibitem{18} R. Temam;
\emph{Infinite Dimensional Dynamicl Systems in Mechanics and Physics}, 
second ed., Springer, 1997.

\bibitem{n1} C. J. Xu, T. Tang;
\emph{Stability analysis of large time-stepping methods for epitaxial growth 
models}, SIAM J. Numer. Anal., 44 (2006), 1759-1779.

\bibitem{AZ} A. Zangwill;
\emph{Some causes and a consequence of epitaxial roughening},
 J. Cryst. Growth, 163(1996), 8-21.

\bibitem{Zhao1} X. P. Zhao, C. C. Liu;
\emph{The existence of global attractor for a
fourth-order parabolic equation}, Appl. Anal., 92 (2013), 44-59.

\end{thebibliography}

\end{document}