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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 163, pp. 1--13.\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-2013/163\hfil Solutions to fourth-order evolution equations]
{Properties of solutions of fourth-order nonlinear evolution equations}

\author[N. Duan, X. Zhao, B. Liu \hfil EJDE-2013/163\hfilneg]
{Ning Duan, Xiaopeng Zhao, Bo Liu}  % in alphabetical order

\address{Ning Duan \newline
College of Mathematics, Jilin University, Changchun 130012, China}
\email{123332453@qq.com}

\address{Xiaopeng Zhao \newline
School of Science, Jiangnan University, Wuxi 214122, China}
\email{zxp032@126.com}

\address{Bo Liu \newline
College of Mathematics, Jilin University, Changchun 130012, China}
\email{liubom@jlu.edu.cn}

\thanks{Submitted August 24, 2012. Published July 19, 2013.}
\subjclass[2000]{35B41, 35K55}
\keywords{Fourth-order nonlinear evolution equation;  existence;
\hfill\break\indent uniqueness; global attractor}

\begin{abstract}
 In this article, we consider the existence and uniqueness of global
 solutions for a fourth-order nonlinear evolution equation which
 models the formation of facets and corners in the course of kinetically
 controlled crystal growth. Moreover, the existence of global attractor
 in $H^2$ and $H^k$ $(k\geq0)$ space is also considered.
\end{abstract}

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


\section{Introduction} \label{sect1}

In the study of the formation of facets and corners in the course 
of kinetically controlled crystal growth \cite{GDN}, 
there arises the fourth-order nonlinear evolution equation  
$$
h_t+m\nabla^2h+\nu\nabla^4 h=\frac12(\nabla h)^2+h_{xx}[ah_x^2+bh_y^2]
+h_{yy}[bh_x^2+ah_y^2]+ch_{xy}h_xh_y.
$$
Such equation is derived for the faceting of crystal surfaces with 
unstable orientations when there is no surface growth. The linear damping
coefficient $\nu>0$ characterizes the stabilizing effect of the additional
energy of edges and determines their widths. The coefficient $m>0$ 
characterize the linear faceting instability of the thermodynamically unstable 
surface, and the coefficients of the nonlinear terms
determine the stable orientations of the appearing facets and
the symmetry of the faceted structure. The coefficients $a$, $b$ and $c$ 
characterizing the stable orientation
of facets are also taken to be positive.

In this article, we consider the 1D case of the above equation
\begin{equation}
u_t+\nu u_{xxxx}+mu_{xx}-\frac12(u_x)^2-a(u_x)^2u_{xx}=0,\quad\text{in }Q_T,
\label{1-1}
\end{equation}
with the Newmann boundary conditions
\begin{equation} \label{1-2}
u_x(0,t)=u_x(1,t)=u_{xxx}(0,t)=u_{xxx}(1,t)=0,
\end{equation}
 and the initial condition
\begin{equation}
u(x,0)=u_0(x),\quad\text{in }(0,1),\label{1-3}
\end{equation}
where $Q_T=(0,1)\times(0,T)$, $\nu$, $m$ and $a$ are also positive constants.

This article is organized as follows. 
In the next section, we establish the existence and
uniqueness of global weak solution in the space $H^{4,1}(Q_T)$;
In Section 3, by uniform a priori estimates methods, we show the existence 
of the global attractor in the space $H^2(0,1)$; 
In the last section, based on the iteration technique and regularity 
estimates for the semigroups, we study the existence of global attractor 
for problem \eqref{1-1}-\eqref{1-3}
in a more generalized space $H^k(0,1)$, where $0\leq k<\infty$.

For notational convenience, we denote by $\|\cdot\|$ the norm of $L^2(0,1)$ 
with the usual inner product $(\cdot,\cdot)$, $\|\cdot\|_p$ denotes the 
norm of $L^p(0,1)$ for $1\leq p\leq+\infty$ $(\|\cdot\|_2=\|\cdot\|)$, 
$\|\cdot\|_Y$ denotes the norm of any Banach space $Y$. 
In the following, $C$, $C_i$, $C'_i,~(i=1,2\cdots)$ will represent generic 
positive constants that may change from line to line even if in the same
inequality.

\section{Existence and uniqueness of global solutions}

In this section, we consider the existence and uniqueness of global
weak solutions of the problem \eqref{1-1}-\eqref{1-3}. First of all, we
define
\begin{gather*}
L^{\infty}(0,1)=\{v;\|v\|_{L^{\infty}}
=\operatorname{ess,sup}_{x\in(0,1)}|v|<+\infty\},\\
H^2_E(0,1)=\{v\in H^2(0,1); v_x(0,t)=v_x(1,t)=0\}, \\
 H^4_E(0,1)=\{v\in H^4(0,1); v_x(x,t)=v_{xxx}(x,t)=0,x=0,1\}, \\
H^{4,1}(Q_T)=\{v;v_t\in L^2(Q_T),v,v_x,v_{xx},v_{xxx},v_{xxxx}\in L^2(Q_T)\}.
\end{gather*}

\begin{definition} \label{def2.1} \rm
A function $u(x, t)$ is called a weak solution to  problem
\eqref{1-1}-\eqref{1-3}, if $u\in H^{4,1}(Q_T)$, and it satisfies
$$
\iint_{Q_T}u_tv\,dx\,dt+\iint_{Q_T}(\nu u_{xxxx}+mu_{xx}
-\frac12u_x^2-au_x^2u_{xx})v\,dx\,dt=0,\quad \forall v\in L^2(Q_T).
$$
\end{definition}

From the classical approach, it is not difficult to conclude that 
\eqref{1-1}-\eqref{1-3} admits a unique solution local in time. 
So, to obtain the result on the global solution, it is sufficient to make a
priori estimates.

\begin{theorem} \label{thm2.1}
Assume that $u_0\in H^2_E(0,1)$ and $T>0$, then 
 problem \eqref{1-1}-\eqref{1-3} admits one and only one 
 solution $u\in H^{4,1}(Q_T)$.
\end{theorem}

\begin{proof}
Multiplying both sides of \eqref{1-1} by $u$, then integrating resulting
relation with respect to $x$ over $(0, 1)$, we obtain
$$
\frac12\frac d{dt}\|u\|^2+\nu\|u_{xx}\|^2-m\|u_x\|^2
-\frac12\int_0^1u_x^2u\,dx-a\int_0^1u_x^2u_{xx}u\,dx=0.
$$
Note that
\begin{gather*}
a\int_0^1u_x^2u_{xx}u\,dx=-\frac a3\|u_x\|_4^4, \\
\frac12\int_0^1u_x^2u\,dx\leq\frac a3\|u_x\|_4^4+\frac3{16a}\|u\|^2,\\
m\|u_x\|^2=-m(u_{xx},u)\leq\frac{\nu}2\|u_{xx}\|^2+\frac{m^2}{2\nu}\|u\|^2.
\end{gather*}
Summing up, we derive that
\begin{equation}
\label{2-1}
\frac d{dt}\|u\|^2+\nu\|u_{xx}\|^2\leq(\frac{3}{8a}+\frac{m^2}{\nu})\|u\|^2.
\end{equation}
By Gronwall's inequality, we obtain 
\begin{equation} \label{2-2}
\|u\|^2\leq e^{(\frac{3}{8a}+\frac{m^2}{\nu})t}\|u_0\|^2\leq C,
\quad \forall t\in(0,T).
\end{equation}
Integrating \eqref{2-1} over $(0,T)$, using
\eqref{2-2}, we deduce that
\begin{equation} \label{2-3}
\int_0^T\|u_{xx}\|^2dt\leq \frac1{\nu}
\Big((\frac{3}{8a}+\frac{m^2}{\nu})\int_0^T\|u\|^2dt+\|u_0\|^2\Big)\leq C.
\end{equation}
Multiplying both sides of equation \eqref{1-1} by $-u_{xx}$, then integrating 
with respect to $x$ over $(0, 1)$, we obtain
$$
\frac12\frac d{dt}\|u_x\|^2+\nu\|u_{xxx}\|^2-m\|u_{xx}\|^2
+\frac12\int_0^1u_x^2u_{xx}dx+a\int_0^1u_x^2u_{xx}^2dx=0.
$$
Note that
\begin{gather*}
\int_0^1u_x^2u_{xx}dx=-2\int_0^1u_x^2u_{xx}dx=0,\\
m\|u_{xx}\|^2=-m(u_{xxx},u_x)\leq\frac{\nu}2\|u_{xxx}\|^2
+\frac{m^2}{2\nu}\|u_x\|^2.
\end{gather*}
Then, summing up, we derive that
\begin{equation}
\label{2-4}
\frac d{dt}\|u_x\|^2+\nu\|u_{xxx}\|^2\leq\frac{m^2}{\nu}\|u_x\|^2.
\end{equation}
By Gronwall's inequality, we obtain
\begin{equation}
\label{2-5}
\|u_x\|^2\leq e^{\frac{m^2}{\nu}t}\|u_{x0}\|^2\leq C,\quad \forall t\in(0,T).
\end{equation}
Integrating \eqref{2-4} over $(0,T)$, using
\eqref{2-5}, we deduce that
\begin{equation}
\label{2-6a}
\int_0^T\|u_{xxx}\|^2dt\leq \frac1{\nu}
\Big(\frac{m^2}{\nu}\int_0^T\|u_x\|^2dt+\|u_{x0}\|^2\Big)\leq C.
\end{equation}
Here, using Sobolev's embedding theorem, by \eqref{2-2} and \eqref{2-5}, 
we have
\begin{equation}
\|u\|_{\infty}=\sup_{x\in[0,1]}|u(x,t)|\leq C,\quad \forall t\in(0,T).\label{2-11}
\end{equation}
Multiplying both sides of \eqref{1-1} by $u_{xxxx}$, then integrating 
with respect to $x$ over $(0, 1)$, we obtain
$$
\frac12\frac d{dt}\|u_{xx}\|^2+\nu\|u_{xxxx}\|^2-m\|u_{xxx}\|^2
-\frac12\int_0^1u_x^2u_{xxxx}dx-a\int_0^1u_x^2u_{xx}u_{xxxx}dx=0.
$$
Using Nirenberg's inequality, we have
\begin{gather*}
\|u_x\|^4\leq C'_1\|u_{xxxx}\|^{1/12}\|u_x\|^{11/12}+C'_2\|u_x\|,\\
\|u_x\|_8\leq C'_1\|u_{xxxx}\|^{1/8}\|u_x\|^{7/8}+C'_2\|u_x\|, \\
\|u_{xx}\|_4\leq C'_1\|u_{xxxx}\|^{5/12}\|u_x\|^{7/12}+C'_2\|u_x\|.
\end{gather*}
Then
\begin{gather*}
\frac12\int_0^1u_x^2u_{xxxx}dx\leq\frac{\nu}{12}\|u_{xxxx}\|^2
+\frac3{2\nu}\|u_x\|_4^4\leq\frac{\nu}{6}\|u_{xxxx}\|^2+C_1,\\
a\int_0^1u_x^2u_{xx}u_{xxxx}dx\leq\frac{\nu}{12}\|u_{xxxx}\|^2
+\frac{3a^2}{2\nu}\|u_x\|_8^8+\frac{3a^2}{2\nu}\|u_{xx}\|_4^4
\leq\frac{\nu}{6}\|u_{xxxx}\|^2+C_2, \\
m\|u_{xxx}\|^2=-m(u_{xxxx},u_{xx})\leq \frac{\nu}6\|u_{xxxx}\|^2
+\frac{3m^2}{2\nu}\|u_{xx}\|^2.
\end{gather*}
Summing up, we derive that
\begin{equation}
\label{2-7}
\frac d{dt}\|u_{xx}\|^2+\nu\|u_{xxxx}\|^2\leq\frac{3m^2}{\nu}\|u_{xx}\|^2
+2C_1+2C_2.
\end{equation}
By Gronwall's inequality, we deduce that
\begin{equation}
\label{2-8}
\|u_{xx}\|^2\leq e^{3m^2t/\nu}\|u_{xx0}\|^2+\frac2{\nu}(C_1+C_2)\leq C,
\quad \forall t\in(0,T).
\end{equation}
Integrating \eqref{2-7} over $(0,T)$, using
\eqref{2-8}, we deduce that
\begin{equation}\label{2-9}
\int_0^T\|u_{xxxx}\|^2dt
\leq \frac1{\nu}\Big(\frac{3m^2}{\nu}\int_0^T\|u_{xx}\|^2dt
+2(C_1+C_2)T+\|u_{xx0}\|^2\Big)\leq C.
\end{equation}
By Sobolev's imbedding theorem,  from 
\eqref{2-2}, \eqref{2-5}, \eqref{2-8}  it follows that
\begin{equation}
\label{2-10}
\|u_x\|_{\infty}=\sup_{x\in[0,1]}|u_x(x,t)|\leq C,\quad \forall t\in(0,T).
\end{equation}
The a priori estimates \eqref{2-11}, \eqref{2-6a} and \eqref{2-9} 
complete the proof of global existence. 
Because the proof of the uniqueness of the solution is easy, we omit it here.
The proof is complete.
\end{proof}


\section{Global attractor in $H^2(0,1)$}

The dynamic properties of diffusion equations such as the global attractors 
and global asymptotic behavior of solutions are important
for the study of diffusion model. There are many studies on the existence 
of global attractors for diffusion equations, such as 
\cite{Dlotko,GW,Hale,Temam} and so on. In this section, we are interested 
in the existence of global attractors in the space $ H^2(0, 1)$ for
problem \eqref{1-1}-\eqref{1-3}.

By Theorem \ref{thm2.1}, we can also obtain $u(x,t)\in L^{\infty}(0,T;H^2(0,1))$. 
Define the operator semigroup
$\{S(t)\}_{t\geq 0}$ in $H^2(0,1)$ space as
\begin{equation}
S(t)u_0=u(t),\quad\forall u_0\in H^2_E(0,1),\; t\geq 0, \label{4-1}
\end{equation}
where $u(t)$ is the solution of \eqref{1-1}-\eqref{1-3} corresponding to
initial value $u_0$.

 Notice that the total mass is
conserved for all time; we let
\begin{equation}
\mathcal{U}=\big\{u\in
H^2_E(0,1): \int_0^1u\,dx=0\big\}. \label{4-2}
\end{equation}
 It is sufficient to see that the
restriction of $\{S(t)\}$ on  the affined space
$\mathcal{U}$ is a well defined semigroup.

Now, we give the result on the existence of global attractor
for problem \eqref{1-1}-\eqref{1-3} in $H^2(0,1)$.

\begin{theorem} \label{thm4.1}
Assume that $\nu$ is sufficiently large, then  the semiflow associated 
with the solution $u$ of \eqref{1-1}-\eqref{1-3} possesses in $\mathcal{U}$ a
global attractor $\mathcal{A}$ which attracts all the
bounded sets in $\mathcal{U}$.
\end{theorem}

To prove Theorem \ref{thm4.1}, we establish some a priori
estimates for the solution $u$ of  \eqref{1-1}-\eqref{1-3}. In
this section we always assume that $\{S(t)\}_{t\geq 0}$ is the
semigroup generated by the weak solutions of equation \eqref{1-1} with
initial data $u_0\in H^2_E(0,1)$. We have the following lemmas.

\begin{lemma} \label{lem4.1} 
For initial data $u_0$ varying in a bounded set
$B\subset \mathcal{U}$, there exists a $t_0(B)>0$ such that
$$
 \|u(t)\|_{H^2(0,1)}\leq C,\quad t\geq t_0(B).
 $$
which implies that  $\{S(t)\}_{t\geq 0}$ has a bounded absorbing set
in $\mathcal{U}$.
\end{lemma}

\begin{proof}
We prove this lemma in the following three steps.

\noindent\textbf{Step 1.}
 Based on Poincar\'{e}'s inequality, we have
\begin{equation}
\|u\|^2\leq \frac12\|u_x\|^2,
\label{4-3}
\end{equation}
H\"{o}lder's inequality gives
\begin{equation}\label{4-4}
\|u_x\|^2\leq\frac12\|u\|^2+\frac12\|u_{xx}\|^2.
\end{equation}
Adding \eqref{4-3} and \eqref{4-4} together gives
\begin{equation}
\|u\|^2\leq\frac13\|u_{xx}\|^2.
\label{4-5}
\end{equation}
 Using \eqref{2-1} and \eqref{4-5}, we immediately obtain the following inequality
 $$
 \frac d{dt}\|u\|^2+3\nu\|u\|^2\leq(\frac3{8a}+\frac{m^2}{\nu})\|u\|^2;
 $$
 that is,
$$
\frac d{dt}\|u\|^2+(3\nu-\frac3{8a}-\frac{m^2}{\nu})\|u\|^2\leq0,
$$
where $\nu$ is sufficiently large, which satisfies 
$3\nu-\frac3{8a}-\frac{m^2}{\nu}>0$.
By Gronwall's inequality, we get
\begin{equation}
\label{4-6}
\|u\|^2\leq e^{-(3\nu-\frac3{8a}-\frac{m^2}{\nu})t}\|u_0\|^2.
\end{equation}
Thus, for initial data in any bounded set
$B\subset\mathcal{U}$, there is a uniform time $t_1(B)$
depending on $B$ such that for $t\geq t_1(B)$, %%
\begin{equation}
\label{4-7} \|u\|^2\leq C.
\end{equation}

\noindent\textbf{Step 2.}
 By \eqref{4-3} and \eqref{4-4}, we can also obtain
\begin{equation}
\|u_x\|^2\leq\frac23\|u_{xx}\|^2.
\label{4-8-}
\end{equation}
Adding \eqref{2-4} and \eqref{4-8-}  gives
$$
\frac d{dt}\|u_x\|^2+\frac32\nu\|u_x\|^2\leq\frac{m^2}{\nu}\|u_x\|^2;
$$
that is,
$$
\frac d{dt}\|u_x\|^2+(\frac{3\nu}2-\frac{m^2}{\nu})\|u_x\|^2\leq 0,
$$
where $\nu$ is sufficiently large, which satisfies 
$\frac{3\nu}2-\frac{m^2}{\nu}>0$.
By Gronwall's inequality, we deduce that
\begin{equation}
\label{4-8}
\|u_x\|^2\leq e^{-(\frac{3\nu}2-\frac{m^2}{\nu})t}\|u_{x0}\|^2.
\end{equation}
Thus, for initial data in any bounded set
$B\subset\mathcal{U}$, there is a uniform time $t_2(B)$
depending on $B$ such that for $t\geq t_2(B)$, 
\begin{equation}
\label{4-9} \|u_x\|^2\leq C.
\end{equation}


\noindent\textbf{Step 3.}
 For \eqref{2-7}, applying the regularity theorem of elliptic operator, we have
$$
\frac d{dt}\|u_{xx}\|^2+\nu C'(\|u_{xx}\|^2+\|u_{xxx}\|^2)
\leq\frac{3m^2}{\nu}\|u_{xx}\|^2+2(C_1+C_2),
$$
which means
\begin{equation} \label{4-9-}
\frac d{dt}\|u_{xx}\|^2+(\nu C'-\frac{3m^2}{\nu})\|u_{xx}\|^2\leq 2(C_1+C_2),
\end{equation}
where $\nu$ is sufficiently large, which satisfies $\nu C'-\frac{3m^2}{\nu}>0$. 
Then, using Gronall's inequality, we derive that
\begin{equation}
\label{4-10}
\|u_{xx}\|^2\leq e^{(\nu C'-\frac{3m^2}{\nu})t}\|u_{xx0}\|^2
+\frac{2\nu(C_1+C_2)}{\nu^2C'-3m^2}.
\end{equation}
Thus, for initial data in any bounded set
$B\subset\mathcal{U}$, there is a uniform time $t_3(B)$
depending on $B$ such that for $t\geq t_3(B)$, %%
\begin{equation}
\label{4-11} \|u_{xx}\|^2\leq\frac{4\nu(C_1+C_2)}{\nu^2C'-3m^2}.
\end{equation}
By Sobolev's embedding theorem, we have
$$
\|u_x(x,t)\|_{\infty}\leq C.
$$
Combining \eqref{4-7}, \eqref{4-9} and \eqref{4-11} , we complete the proof.
\end{proof}

In the following, we prove the precompactness of the orbit in
$\mathcal{U}$.

\begin{lemma} \label{lem4.2}
For initial data $u_0$ varying in a bounded set $B\subset
\mathcal{U}$, there exists a $T(B)>0$ such that
$$
\|u(t)\|_{H^3(0,1)}\leq C,\quad\forall t\geq T >0,
$$
which implies that $\bigcup_{t\geq T}u(t)$ is relatively compact in
$\mathcal{U}$.
\end{lemma}

\begin{proof} 
The uniform bound of $H^2$-norm of $u(x, t)$ has been obtained in 
Lemma \ref{lem4.1}. In what follows we derive the estimate on $H^3$-norm.

Differentiating \eqref{1-1} with respect to $x$, then multiplying 
by $u_{xxxxx}$ and integrating on $(0, 1)$, using the boundary conditions,
we have
\begin{align*}
&\frac12\frac d{dt}\|u_{xxx}\|^2+\nu\|u_{xxxxx}\|^2-m\|u_{xxxx}\|^2
\\
&-\frac12\int_0^1(u_x^2)_xu_{xxxxx}dx-a\int_0^1(u_x^2u_{xx})_xu_{xxxxx}dx=0.
\end{align*}
Note that
$$
m\|u_{xxxx}\|^2=-m\int_0^1u_{xxx}u_{xxxxx}dx\leq\frac{\nu}3\|u_{xxxxx}\|^2
+\frac{3m^2}{4\nu}\|u_{xxx}\|^2,
$$
and
\begin{align*}
\frac12\int_0^1(u_x^2)_xu_{xxxxx}dx
&=\int_0^1u_xu_{xx}u_{xxxxx}dx\leq\|u_x\|_{\infty}\|u_{xx}\|\|u_{xxxxx}\|
\\
&\leq \frac{\nu}3\|u_{xxxxx}\|^2+\frac{3}{4\nu}\|u_x\|_{\infty}^2
\|u_{xx}\|^2\leq\frac{\nu}3\|u_{xxxxx}\|^2+C_3.
\end{align*}
On the other hand,
Nirenberg's inequality gives
$$
\|u_{xx}\|_4\leq C'_1\|u_{xxxxx}\|^{\frac1{12}}\|u_{xx}\|^{\frac{{11}}{12}}
+C'_2\|u_{xx}\|.
$$
Hence
\begin{align*}
a\int_0^1(u_x^2u_{xx})_xu_{xxxxx}dx
&=2a\int_0^1u_xu_{xx}^2u_{xxxxx}dx+a\int_0^1u_x^2u_{xxx}u_{xxxxx}dx
\\
&\leq 2a\|u_x\|_{\infty}\|u_{xx}\|_4^2\|u_{xxxxx}\|
+a\|u_x\|_{\infty}^2\|u_{xxx}\|\|u_{xxxxx}\|
\\
&\leq \frac{\nu}6\|u_{xxxxx}\|^2+ C\|u_{xx}\|_4^4+C_4\|u_{xxx}\|^2+C
\\
&\leq \frac{\nu}3\|u_{xxxxx}\|^2+C_4\|u_{xxx}\|^2+C_5.
\end{align*}
Summing up, we have
\begin{equation}
\frac d{dt}\|u_{xxx}\|^2\leq (\frac{3m^2}{2\nu}+2C_4)\|u_{xxx}\|^2+2C_3+2C_5.
\label{4-12}
\end{equation}
Integrating \eqref{4-9-} between $t$ and $t+ 1$, using \eqref{4-11}, we have
\begin{equation}
\label{4-13}
\int_t^{t+1}\|u_{xxx}\|^2dt\leq C.
\end{equation}
Due to \eqref{4-12}, \eqref{4-13}, and the uniform Gronwall inequality 
in \cite{Temam}, we obtain that
$$
\|u_{xxx}\|^2\leq C,\quad \forall t\geq 1.
$$
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.1}]
By Lemmas \ref{lem4.1}-\ref{lem4.2} and \cite[Theorem I.1.1]{Temam}, 
we  conclude that
$\mathcal{A}=\omega (\mathcal{B})$, the $\omega$-limit set
of absorbing set $\mathcal{B}$ is a global attractor in
$\mathcal{U}$. By lemma \ref{lem4.2}, this global attractor
is a bounded set in $H^3(0,1)$. The proof is complete.
\end{proof}

\section{Attractor in $H^k(0,1)$}

We introduce the following spaces: 
\begin{equation}\label{5-1} 
\begin{gathered}
H =\{u\in L^2(0,1):\int_{0}^1u(x,t)dx=0\}, \\
H_{1/2}  = H^2_{E}(0,1)\cap H=\mathcal{U},\\
H_1= H^4_{E}(0,1)\cap H.
\end{gathered}
\end{equation}
Define the linear operator $L$ and the nonlinear
operator $G$ by
\begin{equation}\label{5-2} 
\begin{gathered} 
Lu =-\nu u_{xxxx}, \\
Gu =g(u)=-mu_{xx}+\frac12(u_x)^2+a(u_x)^2u_{xx}.
\end{gathered}
\end{equation}
It is easy to check that $L$ given by \eqref{5-2} is a sectorial
operator and the tractional power operator $(-L)^{1/2}$ is given
by $ (-L)^{1/2}=\nu^{1/2}\frac{\partial^2}{\partial x^2}$.
The space $H_{1/2}$ is the same as
\eqref{5-1}, $H_{\frac14}$ is given by $H_{\frac14}=$ closure of
$H_{1/2}$ in $H^1(\Omega)$ and $H_k=H^{4k}\cap H_1$ for
$k\geq 1$.

Based on \cite{Ma}, the solution $u(t,u_0 )$ of the problem
\eqref{1-1} can be written as
\begin{equation}
u(t,u_0 )=e^{tL}u_0 +\int_0^te^{(t-\tau )L}Gud\tau= e^{tL}u_0
+\int_0^te^{(t-\tau )L}g(u)d\tau . \label{5-3}
\end{equation}
We introduce a result on the sectorial operator $L$ in \eqref{5-2},
which is important in this section and can be found in
\cite{Luo,Ma,Song1,Song2,Song3,ZL1}.

\begin{lemma} 
Assume that $L: H\to H$ is a
sectorial operator which generates an analytic semigroup
$T(t)=e^{tL}$. If all eigenvalues $\lambda$ of $L$ satisfy
$\hbox{Re}\lambda<-\lambda_0$ for some real number $\lambda_0>0$,
then for $\mathcal{L}^{\kappa}(\mathcal{L}=-L)$ we have
\begin{itemize}
\item[(C1)] $T(t):H\to H$ is bounded for all $\kappa\in
R^1$ and $t>0$;

\item[(C2)] $T(t)\mathcal{L}^{\kappa} x=\mathcal{L}^{\kappa}T(t)x,\forall
x\in H$;

\item[(C3)] For each $t>0$, $\mathcal{L}^{\kappa}T(t):H\to H$ is
bounded, and
$$
 \|\mathcal{L}^{\kappa}T(t)\|_H\leq
Ct^{-\kappa}e^{-\delta t};
$$
where some $\delta >0$ and $C>0$ is a constant
depending only on $\kappa$;

\item[(C4)] The $H-$norm can be defined by $
\|x\|_{H_{\kappa}}=\|\mathcal{L}^{\kappa}x\|_H$.
\end{itemize}
\end{lemma}

 Now, we give the  main result of this section.

\begin{theorem}\label{thm5.1} 
Let $u_0\in H_{\kappa}(0,1)$ and $\nu$ is sufficiently large. 
Then, for any $\kappa\geq 0$, the semigroup associated with the problem
\eqref{1-1}-\eqref{1-3} possesses a global attractor in $H_{\kappa}(0,1)$,
which attracts all the bounded sets in the $H_{\kappa}$-norm.
\end{theorem}

To prove Theorem \ref{thm5.1}, we should prove the following two lemmas.

\begin{lemma} \label{lem5.1}
Let $u_0\in H_{\kappa}(0,1)$ and $\nu$ is sufficiently large. 
Then for any $\kappa \geq 0$, the semigroup $S(t)$
generated by the problem \eqref{1-1}-\eqref{1-3} is uniformly
compact in $H_{\kappa}$.
\end{lemma}

\begin{proof}
It suffices to prove that for any bounded set $U\subset H_{\kappa}$,
there exists $C>0$ such that 
\begin{equation}
\|u(t,u_0 )\|_{H_{\kappa}}\leq C,\quad\forall t\geq 0,\; 
u_0\in U\subset H_{\kappa},\; \kappa\geq 0. \label{2-2-}
\end{equation}
For $\kappa =1/2$, this follows form Theorem \ref{thm4.1}; i.e., for
any bounded set $U\subset H_{1/2}$, there is a constant $C>0$
such that 
\begin{equation}
\|u(t,u_0 )\|_{H_{1/2}}\leq C,\quad\forall t\geq 0,\;
u_0\in U\subset H_{1/2}. \label{2-3-}
\end{equation}
Then, we  shall prove \eqref{2-2-} for any $\kappa>\frac{1}{2}$,
which will be proved in the following steps.

\noindent\textbf{Step 1.} 
We prove that for any bounded set $U\subset H_{\kappa}$
$(\frac{1}{2}<\kappa <1)$, there is a constant $C>0$ such that
%%
\begin{equation}
\|u(t,u_0 )\|_{H_{\kappa}}\leq C,\quad\forall t\geq 0,~u_0\in
U,~\frac12<\kappa <1.
 \label{2-4-}
\end{equation}
By the embedding theorem, we have
$$
H_{1/2}(0,1)\hookrightarrow W^{1,4}(0,1),\quad
H_{1/2}(0,1)\hookrightarrow W^{1,\infty}(0,1).
$$
Hence
\begin{align*}
\|g(u)\|_H
&=\int_0^1(-mu_{xx}+\frac12u_x^2+au_x^2u_{xx})^2dx\\
&\leq C\int_0^1(u_{xx}^2+u_{x}^4+u_x^4u_{xx}^2)dx\\
&\leq C(\|u\|_{H_{1/2}}^2+\|u\|_{W^{1,4}}^4
 +\|u\|_{W^{1,\infty}}^4\|u\|_{H_{1/2}}^2)\\
&\leq C(\|u\|_{H_{1/2}}^2+\|u\|_{H_{1/2}}^4
 +\|u\|_{H_{1/2}}^6)\leq C.
\end{align*}
which implies that $g:H_{1/2}\to H$ is bounded. Hence,
\begin{equation}
\begin{aligned}
\|u(t,u_0 )\|_{H_{\kappa}}
&=\|e^{tL}u_0 +\int_0^te^{(t-\tau)L}g(u)d\tau \|_{H_{\kappa}}
\\
&\leq  C\|u_0\|_{H_{\kappa}}+\int_0^t\|(-L)^{\kappa}e^{(t-\tau
)L}g(u)\|_Hd\tau
\\
&\leq  C\|u_0\|_{H_{\kappa}}+\int_0^t\|(-L)^{\kappa}e^{(t-\tau
)L}\|\cdot \|g(u)\|_Hd\tau
\\
&\leq C\|u_0
\|_{H_{\kappa}}+C\int_0^t\tau^{-\kappa}e^{-\delta\tau}d\tau
\\
&\leq C,\quad\forall t\geq 0,\; u_0\in U\subset H,
\end{aligned}\label{3.9}
\end{equation}
where $0<\kappa <1$. Then, \eqref{2-4} holds.

\noindent\textbf{Step 2.} We prove that for any bounded set 
$U\subset H_{\kappa}$ $(1\leq\kappa <\frac54)$, there is a constant 
$C>0$ such that
\begin{equation}
\|u(t,u_0 )\|_{H_{\kappa}}\leq C,\quad\forall t\geq 0,\; 
u_0\in U,\; 1\leq\kappa <\frac54.  \label{2-5-}
\end{equation}
In fact, by the embedding theorems, we derive that
$$
H_{\kappa}(0,1)\hookrightarrow H^3(0,1),\quad
H^3(0,1)\hookrightarrow W^{1,\infty}(0,1),\quad 
H^3(0,1)\hookrightarrow W^{2,4}(0,1),
$$
where $\frac{3}{4}\leq\kappa <1$. Then, using \eqref{1-1}, we obtain
\begin{equation}
\begin{aligned}
\|g(u)\|_{\frac14}^2 
&=\int_0^1(g(u)_x)^2dx \\
&=\int_0^1(-mu_{xxx}+u_xu_{xx}+2au_xu_{xx}^2+au_x^2u_{xxx})^2dx \\
&\leq C\int_0^1(u_{xxx}^2+u_x^2u_{xxx}^2+u_x^2u_{xx}^4+u_{x}^4u_{xxx}^2)dx\\
&\leq C(\|u\|_{H^3}^2+\|u\|_{W^{1,\infty}}^2\|u\|_{H^3}^2
+\|u\|_{W^{1,\infty}}^2\|u\|_{W^{2,4}}^4+\|u\|^4_{W^{1,\infty}}\|u\|_{H^3}^2) \\
&\leq C(\|u\|_{H_{\kappa}}^2+\|u\|_{H_{\kappa}}^4+\|u\|_{H_{\kappa}}^6
+\|u\|_{H_{\kappa}}^6)\leq C,
\end{aligned}\label{aa3}
\end{equation}
which implies that $g:H_{\kappa}\to H_{\frac14}$ is bounded for
$\frac{3}{4}\leq\kappa <1$. Hence,
\begin{align*}
\|u(t,u_0 )\|_{H_{\kappa}}
&= \|e^{tL}u_0+\int_0^te^{(t-\tau )L}g(u)d\tau\|_{H_{\kappa}}
\\
&\leq C\|u_0\|_{H_{\kappa}}+\int_0^t\|(-L)^{\kappa}e^{(t-\tau)L}g(u)\|_Hd\tau
\\
&\leq C\|u_0\|_{H_{\kappa}}+\int_0^t\|(-L)^{\kappa
-\frac{1}{4}}e^{(t-\tau )L}\|\cdot\|g(u)\|_{H_{\frac14}}d\tau
\\
&\leq C\|u_0\|_{H_{\kappa}}+C\int_0^t\tau^{-\epsilon}e^{-\delta\tau}d\tau
\\
&\leq C,\quad \forall t\geq 0,u_0\in U\subset H,
\end{align*}
where $\epsilon =\kappa -\frac{1}{4}$ $(0<\epsilon <1)$. 
Then, \eqref{2-5} holds.

\noindent\textbf{Step 3.}
We prove that for any bounded set $U\subset H_{\kappa}$
$(5/4 \leq\kappa <3/2)$, there is a constant $C>0$ such that 
\begin{equation}
\|u(t,u_0 )\|_{H_{\kappa}}\leq C,\quad\forall t\geq 0, \quad 
u_0\in U,\; \frac54\leq\kappa <\frac32.
 \label{2-6}
\end{equation}
In fact, by the embedding theorems,
 we have
\begin{gather*}
H_{\kappa}(0,1)\hookrightarrow H^4(0,1),\quad
H^4(0,1)\hookrightarrow W^{1,\infty}(0,1),\quad
H^4(0,1)\hookrightarrow W^{2,4}(0,1),\\
H^4(0,1)\hookrightarrow W^{3,4}(0,1),\quad
H^4(0,1)\hookrightarrow W^{2,6}(0,1),
\end{gather*}
where $1\leq\kappa<5/4$.
 Then,
\begin{equation}
\begin{aligned}
&\|g(u)\|^2_{H_{1/2}}\\
&=\int_0^1(g(u)_{xx})^2dx 
\\
&=\int_0^1(-mu_{xxxx}+u_{x}u_{xxxx}+u_{xx}u_{xxx}+2au_{xx}^3
\\
&\quad +6au_xu_{xx}u_{xxx}+au_x^2u_{xxxx})^2dx
\\
&\leq C\int_0^1(u_{xxxx}^2+u_x^2u_{xxxx}^2+u_{xx}^4+u_{xxx}^4+u_{xx}^6+u_x^4u_{xx}^4+u_x^4u_{xxxx}^2)dx
\\
&\leq C(\|u\|_{H^4}^2+\|u\|_{W^{1,\infty}}^2\|u\|_{H^4}^2+\|u\|_{W^{2,4}}^4+\|u\|_{W^{3,4}}^4+\|u\|_{W^{2,6}}^6
\\
&\quad +\|u\|_{W^{1,\infty}}^4\|u\|_{W^{2,4}}^4+\|u\|_{W^{1,\infty}}
^4\|u\|_{H^4}^2)
\\
&\leq C(\|u\|_{H_{\kappa}}^2+\|u\|_{H_{\kappa}}^4+\|u\|_{H_{\kappa}}^6
+\|u\|_{H_{\kappa}}^8)\leq C.
\end{aligned}\label{aa4}
\end{equation}
which implies that $g:H_{\kappa}\to H_{1/2}$ is bounded for
$\kappa\geq\frac34$. Hence,
\begin{equation}
\begin{aligned}
\|u(t,u_0 )\|_{H_{\kappa}}
&=\|e^{tL}u_0+\int_0^te^{(t-\tau )L}g(u)d\tau\|_{H_{\kappa}}
\\
&\leq C\|u_0\|_{H_{\kappa}}+\int_0^t\|(-L)^{{\kappa}-\frac{1}{2}}e^{(t-\tau)L}\|\cdot\|g(u)\|_{H_{1/2}}d\tau
\\
&\leq C\|u_0\|_{H_{\kappa}}+C\int_0^t\tau^{-\epsilon}e^{-\delta\tau}d\tau
\\
&\leq  C,\quad \forall t\geq 0,\; u_0\in U\subset H,
\end{aligned} \label{3.17}
\end{equation}
where $\epsilon =\kappa -\frac{1}{2}~(0<\epsilon <1)$. Then, \eqref{2-6}
holds.

\noindent\textbf{Step 4.} We prove that for any bounded set $U\subset
H_{\kappa}$ $(3/2\leq\alpha <7/4)$, there exists a
constant $C>0$ such that
\begin{equation}
\|u(t,u_0)\|_{H_{\kappa}}\leq C,\quad\forall t\geq 0,\; 
u_0\in U\subset H,\; \frac{3}{2}\leq\kappa <\frac{7}{4}.
 \label{3.18}
\end{equation}
Based on the following embedding theorems, we deduce that
\begin{gather*}
H_{\kappa}(0,1)\hookrightarrow H^5(0,1),\quad
H^5(0,1)\hookrightarrow W^{1,\infty}(0,1),\quad
H^5(0,1)\hookrightarrow W^{2,4}(0,1), \\
H^5(0,1)\hookrightarrow W^{4,4}(0,1),\quad 
H^5(0,1)\hookrightarrow W^{2,8}(0,1),\quad
H^5(0,1)\hookrightarrow W^{3,2}(0,1), \\
H^5(0,1)\hookrightarrow W^{3,4}(0,1),\quad 
H^5(0,1)\hookrightarrow W^{1,8}(0,1),
\end{gather*}
where $5/4\leq\alpha <3/2$.
Then
\begin{align*}
&\|g(u)\|_{H_{3/4}}^2\\
&=\int_0^1(g(u))_{xxx}^2dx
\\
&=\int_0^1(-mu_{xxxxx}+u_xu_{xxxxx}+2u_{xx}u_{xxxx}+u_{xxx}^2+12au_{xx}^2u_{xxx}
\\
&\quad +6au_xu_{xxx}^2+8au_xu_{xx}u_{xxxx}+au_x^2u_{xxxxx})^2dx
\\
&\leq C\int_0^1(u_{xxxxx}^2+u_x^2u_{xxxxx}^2+u_{xx}^4+u_{xxxx}^4
+u_{xxx}^2+u_{xx}^8+u_{xxx}^4\\
&\quad +u_x^2u_{xxx}^4+u_{xxxx}^4+u_x^8+u_{xx}^8
+u_x^4u_{xxxxx}^2)dx
\\
&\leq C(\|u\|_{H^5}^2+\|u\|_{W^{1,\infty}}^2\|u\|_{H^5}^2+\|u\|_{W^{2,4}}^4
 +\|u\|_{W^{4,4}}^4+\|u\|_{W^{2,8}}^8+\|u\|_{W^{3,2}}^2\\
&\quad + \|u\|_{W^{3,4}}^4+\|u\|_{W^{1,\infty}}^2\|u\|_{W^{3,4}}^4
 +\|u\|_{W^{1,8}}^8+\|u\|_{W^{1,\infty}}^4\|u\|_{H^5}^2)
\\
&\leq C(\|u\|_{H_{\kappa}}^2+\|u\|_{H_{\kappa}}^4+\|u\|_{H_{\kappa}}^8
 +\|u\|_{H_{\kappa}}^6)\leq C.
\end{align*}
which implies that $g:H_{\kappa}\to H_{3/4}$ is bounded for
$\kappa\geq1$. Hence,
\begin{equation}
\begin{aligned}
\|u(t,u_0 )\|_{H_{\kappa}}
&=\|e^{tL}u_0+\int_0^te^{(t-\tau )L}g(u)d\tau\|_{H_{\kappa}}
\\
&\leq C\|u_0\|_{H_{\kappa}}+\int_0^t\|(-L)^{{\kappa}-\frac{3}{4}}e^{(t-\tau)L}\|\cdot\|g(u)\|_{H_{3/4}}d\tau
\\
&\leq C\|u_0\|_{H_{\kappa}}+C\int_0^t\tau^{-\epsilon}e^{-\delta\tau}d\tau
\\
&\leq C,\quad \forall t\geq 0,~u_0\in U\subset H,
\end{aligned}\label{3.21}
\end{equation}
where $\epsilon =\kappa -\frac{3}{4}$ $(0<\epsilon <1)$. Then, \eqref{2-7}
holds.

In the same way as in the proof of \eqref{3.18}, by iteration we
can prove that for any bounded set $U\subset H_{\kappa}$
$(\kappa \geq0)$ there exists a constant $C>0$ such that \eqref{2-2-} holds;
i.e., for all $\kappa\geq 0$ the semigroup $S(t)$ generated by
problem \eqref{1-1} is uniformly compact in $H_{\kappa}$.
The proof is complete.
\end{proof}

\begin{lemma}\label{lem5.2} 
Let $u_0\in H_{\kappa}(0,1)$ and $\nu$ is sufficiently large. 
Then for any $\kappa\geq 0$, the problem
\eqref{1-1}-\eqref{1-3} has a bounded absorbing set in $H_{\kappa}$.
\end{lemma}

\begin{proof}
It suffices to prove that for any bounded set
 $U\subset H_{\kappa}$ $(\kappa \geq 0)$, there exist $T>0$ and a 
constant $C>0$ independent of $u_0$, such that
\begin{equation}
\|u(t,u_0 )\|_{H_{\kappa}}\leq C,\quad\forall t\geq T,~u_0\in
U\subset H_{\kappa}.  \label{2-8-}
\end{equation}
For $\kappa =1/2$, this follows from Theorem \ref{thm4.1}. So
we shall prove \eqref{2-8-} for any $\kappa >1/2$. We prove
it in the following steps:

\noindent\textbf{Step 1.} We prove that for any $\frac{1}{2}< \kappa <1$, 
problem \eqref{1-1}-\eqref{1-3} has a bounded absorbing set in $H_{\kappa}$. By
\eqref{5-3}, we deduce that 
\begin{equation}
u(t,u_0)=e^{(t-T)L}u(T,u_0 )+\int_T^te^{(t-\tau)L}g(u)d\tau .
\label{5-4}
\end{equation}
Suppose that $B$ is a bounded absorbing set of problem \eqref{1-1}-\eqref{1-3}, 
which satisfies $B\subset H_{1/2}$, we also assume $T_0>0$ such that
\begin{equation}
\label{2-10-} u(t,u_0 )\in B,\quad\forall t>T_0,\; u_0\in U\subset
H_{\kappa},\; \kappa>\frac{1}{2}.
\end{equation}
It is easy to check that
$$
\|e^{tL}\|\leq Ce^{-\lambda_1^2t},
$$
where $\lambda_1>0$ is the first eigenvalue of the equation 
\begin{equation}
\begin{gathered}
-\nu^{1/2} u_{xx} = \lambda u,\\
u_x(0,t)=u_x(1,t)  =0.
\end{gathered}
\end{equation}
Thus, for any given $T>0$ and $u_0\in U\subset
H_{\kappa}$ $(\kappa>1/2)$, we deduce that 
\begin{equation}
\lim_{t\to\infty}\|e^{(t-T)L}u(T,u_0)\|_{H_{\kappa}}= 0.
\label{2-11-}
\end{equation}
Using \eqref{5-4}, \eqref{2-10-} and \eqref{2-11-}, we have
\begin{equation}
\begin{aligned}
\|u(t,u_0)\|_{H_{\kappa}}
&\leq \|e^{(t-T_0)L}u(T_0,u_0)\|_{H_{\kappa}}
+\int_{T_0}^t\|(-L)^{\kappa}e^{(t-T)L}\|\cdot\|g(u)\|_Hd\tau
\\
&\leq \|e^{(t-T_0)L}u(T_0,u_0
)\|_{H_{\kappa}}+C\int_{T_0}^t\|(-L)^{\kappa}e^{(t-T)L}\|
\\
&\leq \|e^{(t-T_0)L}u(T_0,u_0
)\|_{H_{\kappa}}+C\int_0^{T-T_0}\tau^{-\kappa}e^{-\delta\tau}d\tau
\\
&\leq \|e^{(t-T_0)L}u(T_0,u_0 )\|_{H_{\kappa}}+C,
\end{aligned}\label{3.26}
\end{equation}
where $C>0$ is a constant independent of $u_0$.
Then \eqref{2-8-} holds for all $1/2<\kappa <1$.

\noindent\textbf{Step 2.}
We shall show that for any $1\leq\kappa<5/4$,  problem
\eqref{1-1}-\eqref{1-3} has a bounded absorbing set in $H_{\kappa}$.
Using \eqref{5-4} and \eqref{aa3}, we deduce that
\begin{align*}
\|u(t,u_0)\|_{H_{\kappa}}
&\leq \|e^{(t-T_0)L}u(T_0,u_0)\|_{H_{\kappa}}
 +\int_{T_0}^t\|(-L)^{\kappa-\frac14}e^{(t-\tau)L}\|
 \cdot\|g(u)\|_{H_{\frac14}}d\tau
\\
&\leq \|e^{(t-T_0)L}u(T_0,u_0
)\|_{H_{\kappa}}+C\int_{T_0}^t\|(-L)^{\kappa-\frac14}e^{(t-\tau)L}\|dx
\\
&\leq \|e^{(t-T_0)L}u(T_0,u_0
)\|_{H_{\kappa}}+C\int_0^{T-T_0}\tau^{-(\kappa-\frac14)}e^{-\delta\tau}d\tau
\\
&\leq \|e^{(t-T_0)L}u(T_0,u_0 )\|_{H_{\kappa}}+C,
\end{align*}
where $C>0$ is a constant independent of $u_0$. Then, \eqref{2-8}
holds for all $1\leq\kappa<5/4$.

\noindent\textbf{Step 3.} 
We shall show that for any $5/4\leq\kappa<3/2$, problem
\eqref{1-1}-\eqref{1-3} has a bounded absorbing set in $H_{\kappa}$.
Using \eqref{5-4} and \eqref{aa4}, we deduce that
\begin{align*}
\|u(t,u_0)\|_{H_{\kappa}}
&\leq \|e^{(t-T_0)L}u(T_0,u_0)\|_{H_{\kappa}}
 +\int_{T_0}^t\|(-L)^{\kappa-\frac12}e^{(t-\tau)L}\|\cdot
\|g(u)\|_{H_{1/2}}d\tau
\\
&\leq \|e^{(t-T_0)L}u(T_0,u_0)\|_{H_{\kappa}}
 +C\int_{T_0}^t\|(-L)^{\kappa-\frac12}e^{(t-\tau)L}\|dx
\\
&\leq \|e^{(t-T_0)L}u(T_0,u_0)\|_{H_{\kappa}}
+C\int_0^{T-T_0}\tau^{-(\kappa-\frac12)}e^{-\delta\tau}d\tau
\\
&\leq \|e^{(t-T_0)L}u(T_0,u_0 )\|_{H_{\kappa}}+C,
\end{align*}
where $C>0$ is a constant independent of $u_0$. Then \eqref{2-8}
holds for all $5/4\leq\kappa<3/2$.

By the iteration method, we have that \eqref{2-8-} holds 
for all $\kappa>1/4 $.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm5.1}]
By Lemma \ref{lem5.1} and Lemma \ref{lem5.2}, we
immediately conclude that the statement of the theorem.
\end{proof}


\begin{remark} \rm 
Since the tools used work for the periodic boundary values, 
the results of this article are also valid for  equation \eqref{1-1} 
with the periodic boundary conditions in the sense \cite{Temam},
That is, for any $u_0\in H_{\rm per}^k(0,1)$, there exists a global 
unique weak solution $u(x,t)$, a global attractor in 
$H^k$ $(0\leq k<\infty)$ space for equation \eqref{1-1} under the 
initial value condition \eqref{1-3} and the periodic boundary conditions
$$
 \varphi|_{x=0}=\varphi|_{x=1},
$$
for $u$ and the derivatives of $u$ at least of order $\leq 3$.
\end{remark}

\subsection*{Acknowledgements}
The author would like to thank the anonymous referees for their
valuable comments and suggestions about this paper.

\begin{thebibliography}{99}

\bibitem{Dlotko} T. Dlotko;
 \textit{Global attractor for the Cahn-Hilliard equation in $H^{2}$ and $H^{3}$},
 J. Differential Equations, 113 (1994), 381-393.

\bibitem{GDN} A. A. Golovin, S. H. Davis, A. A. Nepomnyashchy;
\textit{Model for faceting in a kinetically controlled crystal growth}, 
Physical Review E, 59(1) (1999), 803-825.

\bibitem{GW}B. Guo, B. Wang;
\textit{Long-time behaviour of the solutions for the multidimensional 
Kolmogorov-Spieqel-Sivashinsky equation}, Acta Math. Sin. (Engl. Ser.), 
18 (2002), 579-596.

\bibitem{Hale} J. K. Hale;
 \textit{Asymptotic Behaviour of Dissipative Systems},
 Amer. Math. Soc., Providence, RI, 1988.

\bibitem{Luo} H. Luo, Q. Zhang;
\textit{Regularity of global attractor for the fourth-order
reaction-diffusion equation}, Commun. Nonlinear Sci. Numer. Simul.,
3824-3831.

\bibitem{Ma} T. Ma, S. H. Wang;
\textit{Stability and Bifurcation of
Nonlinear Evolution Equations}, Science Press, Beijing, 2006, (in
Chinese).

\bibitem{Song1} L. Song, Y. Zhang, T. Ma;
\textit{Global attractor of the Cahn-Hilliard euqation in $H^k$ spaces},
 J. Math. Anal. Appl., 355(2009), 53--62.

\bibitem{Song2} L. Song, Y. He, Y. Zhang;
\textit{The existence of global attractors for semilinear parabolic equation
in $H^k$ space}, Nonlinear Anal., 68(2008), 3541--3549.

\bibitem{Song3} L. Song, Y. Zhang, T. Ma;
\textit{Global attractor of a modified Swift-Hohenberg equation in $H^k$ space}, 
Nonlinear Anal., 72(2010), 183--191.

\bibitem{Temam} R. Temam;
\textit{Infinite-Dimensional Dynamical Systems in Mechanics and Physics},
Springer-Verlag, New York, 1988.

\bibitem{ZL1} X. Zhao, B. Liu;
\textit{The existence of global attractor for convective Cahn-Hilliard equation}, 
J. Korean Math. Soc., 49(2)(2012), 359-378.

\end{thebibliography}

\end{document}