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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 61, 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 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/61\hfil Existence and continuity of
global attractors]
{Existence and continuity of global attractors for a degenerate
semilinear parabolic equation}

\author[C. T. Anh, T. D. Ke\hfil EJDE-2009/??\hfilneg]
{Cung The Anh, Tran Dinh Ke}  % in alphabetical order

\address{Department of Mathematics,
Hanoi National University of Education,
 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam}
\email[Cung The Anh]{anhctmath@hnue.edu.vn}
\email[Tran Dinh Ke]{ketd@hn.vnn.vn}

\thanks{Submitted December 20, 2008. Published May 4, 2009.}
\subjclass[2000]{35B41, 35K65, 35D05}
\keywords{Semilinear degenerate parabolic equation;
Grushin operator; \hfill\break\indent
global solution; global attractor;
upper semicontinuity; nonlinearity; shape of domain}

\begin{abstract}
 In this article, we study the existence and the upper semicontinuity
 with respect to the nonlinearity and the shape of the domain of global
 attractors  for a semilinear degenerate parabolic equation involving
 the Grushin operator.
\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}

 Understanding  the asymptotic behavior of dynamical systems
is one of the most important problems of modern mathematical
physics. One way to attack this problem for dissipative dynamical
systems is to consider its global attractors. A first question is
to study the existence of a global attractor. Once a global
attractor is obtained, a next natural question is to study the
most important properties of the global attractor, such as
dimension, dependence on parameters, regularity of the attractor,
determining modes, etc. In the previous decades, many authors have
paid attention to these problems and obtained results for a large
class of PDEs; see \cite{c1,h1,r1,t1}
 and references therein. However,
to the best of our knowledge, little seems to be known for the
asymptotic behavior of solutions of degenerate equations.

One of the classes of degenerate equations that has been studied
widely, in recent years, is the class of equations involving an
operator of Grushin type
$$
G_su=\Delta_{x_1}u+|x_1|^{2s}\Delta_{x_2} u,\quad (x_1,x_2)\in
\Omega\subset \mathbb{R}^{N_1}\times \mathbb{R}^{N_2},\; s\geqslant
0.
$$
This operator was first introduced in \cite{g1}. Noting that
$G_0=\Delta$ and  $G_s$, when $s>0$, is not elliptic in domains in
$\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}$ intersecting with the
hyperplane $\{x_1=0\}$. The local properties of $G_s$ were
investigated in \cite{b1,g1}. The existence and nonexistence results for
the elliptic equation
\begin{gather*}
-G_su+f(u)=0, \quad x\in \Omega\\
u=0,\quad x\in \partial \Omega
\end{gather*}
were proved in \cite{t2}. Furthermore, the semilinear elliptic systems
with the Grushin type operator, which are in the Hamilton form or
in the potential form, were also studied in \cite{c2,c3,k1}.

To study  boundary value problems for equations involving Grushin
operators, we have usually used the natural energy space
$S_{0}^1(\Omega)$ defined as the completion of
$C_0^1(\bar{\Omega})$ in the  norm
$$
\|u\|_{S_0^1(\Omega)}=\Big(\int_\Omega
\big(|\nabla_{ x_1}u|^2+|x_1|^{2s}|\nabla_{x_2}u|^2\big)dx\Big)^{1/2}.
$$
We have the continuous embedding $S^1_0(\Omega)\hookrightarrow
L^p(\Omega)$, for $2\leqslant p \leqslant
2^*_s=\frac{2N(s)}{N(s)-2}$, where $N(s)=N_1+(s+1)N_2$. Moreover,
this embedding is compact if $2\leqslant p<2^*_s$ (for more
details, see \cite{t2}).

In a recent paper \cite{a1}, we considered the  initial boundary value
problem
\begin{equation} \label{e1.1}
\begin{gathered}
u_t-G_su+ f(u)+g(x)=0,\quad x\in \Omega, t>0\\
u(x,t)=0,\quad  x\in \partial \Omega, t>0  \\
u(x, 0)=u_0(x),\quad x\in \Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in
$\mathbb{R}^N=\mathbb{R}^{N_1}\times\mathbb{R}^{N_2}$
$(N_1, N_2\geq 1)$ with smooth boundary $\partial\Omega$,
$u_0\in S^1_0(\Omega)$ is given, $g\in L^2(\Omega)$, and
$f: \mathbb{R}\to \mathbb{R}$ satisfies
\begin{gather*}
|f(u)-f(v)|\leqslant C_0|u-v|(1+|u|^{\rho}+|v|^{\rho}), \quad
0\leqslant \rho< \frac{4}{N(s)-2}, \\
F(u)\geqslant-\frac{\mu}{2} u^2-C_1, \\
f(u)u\geqslant -\mu u^2-C_2,
\end{gather*}
where $C_0, C_1, C_2\geqslant 0$, $F$ is the primitive
$F(y)=\int_0^y f(s)ds$ of $f$, $\mu<\lambda_1$, $\lambda_1$
is the first eigenvalue of the operator $-G_s$ in $\Omega$
with homogeneous Dirichlet condition. Under the above assumptions
of $f$, we proved that problem \eqref{e1.1} defines a semigroup
$S(t): S_{0}^{1}(\Omega)\to S_{0}^{1}(\Omega)$, which possesses
a compact connected global attractor $\mathcal{A}=W^u(E)$ in the
space $S_0^1(\Omega)$. Furthermore, for each $u_0\in S_0^1(\Omega)$,
the corresponding solution $u(t)$ tends to the set $E$ of equilibrium
points in $ S_0^1(\Omega)$ as $t\to+\infty$. The basic tool for the
approach in this case is the following Lyapunov functional
$$
\Phi(u)=\frac12\|u\|^2_{S_{0}^{1}(\Omega)}+\int_{\Omega}(F(u)+gu)dx.
$$
Noting that the critical exponent of the embedding
$S^1_0(\Omega)\hookrightarrow L^p(\Omega)$ is $2^*_s=\frac{2N(s)}{N(s)-2}$,
so the condition $0\leqslant \rho< \frac{4}{N(s)-2}$ is necessary to
prove the existence of a mild solution by the fixed point method and
to ensure the existence of the Lyapunov functional $\Phi$.

In this article, we continue studying the long-time behavior of solutions
to problem \eqref{e1.1} by removing the restrictions on the growth
 of the nonlinearity $f$. More precisely, we assume that the
initial data $u_0\in L^2(\Omega)$ and the nonlinearity
$f: \mathbb{R}\to \mathbb{R}$ is a $C^1$ function that satisfies
the following conditions:
\begin{gather}
C_1|u|^p-C_0 \leqslant f(u)u\leqslant C_2|u|^p+C_0,\quad  p>2, \label{e1.2}\\
f'(u) \geqslant -C_3, \quad\text{for all } u\in \mathbb{R}, \label{e1.3}
\end{gather}
where $C_0,C_1, C_2$ and $C_3$ are positive constants.
A typical example of the nonlinearity $f$ satisfying
\eqref{e1.2}-\eqref{e1.3} is the following
$$
f(u)=\sum_{j=0}^{2p+1}b_j u^j, \;\;\text{where } b_j\in \mathbb{R}, b_{2p+1}>0.
$$
It is clear that the fixed point method for proving the existence of
solutions is not valid here, and the system is no longer  a gradient system.
However, thanks to the structure of the nonlinearity, we may use the
compactness method \cite{l1} to prove the global existence of a weak solution
and use {\it a priori} estimates to show the existence of an absorbing
set $B_0$ in the space $S^1_0(\Omega)$ for the semigroup $S(t)$
generated by the solutions of the problem \eqref{e1.1}.
By the compactness of the embedding
$S^1_0(\Omega)\hookrightarrow L^2(\Omega)$, the semigroup $S(t)$
is asymptotically compact in $L^2(\Omega)$. This implies the existence
of a compact global attractor $\mathcal{A}=\omega(B_0)$ for $S(t)$
in $L^2(\Omega)$.

Besides the problem of existence of the global attractor, the
dependence of the global attractor on the parameters is also an
important object of study (see \cite{r1} for an excellent review of the
subject). In particular, the problem of continuity of the global
attractor with respect to variations of the domain where the
problem is posed has been studied recently for the
reaction-diffusion equation with various boundary conditions. In
\cite{o1,p1}, the authors assume that $\Omega$ is a small regular
perturbation of a fixed smooth domain $\Omega_0$ and use the
approach suggested by Henry \cite{h2}. This approach is simple, but
quite limited since it requires that $\Omega_0$ is $C^2$ and
$\Omega$ is only a $C^k$ ($k\geqslant 2$) small perturbation of
$\Omega$, i.e. there exists a $C^k$-diffeomorphic $h:\Omega_0 \to
\mathbb{R}^N$ such that $\Omega=h(\Omega_0)$ and
$\|h-id_{\Omega_0}\|_{C^k}$ is small. In \cite{a2}, the authors used a
different method based on the spectral convergence which allows
more irregular perturbations. However, as indicated in \cite{o1}, this
approach is quite technical and gives less detailed results for
the regular case.

In this paper, we use another approach to study the upper
semicontinuity  of the global attractor with respect to the shape
of the domain, which allows us to consider the more general
situations and requires less smoothness of the domain than one
used in \cite{o1,p1}, and it is simpler than one used in \cite{a2}.
We can also use this method to study the upper semicontinuity of the
global attractor with respect to the nonlinear term when taking
the nonlinearity as a parameter.  However, the more delicate
question of the lower semicontinuity of global attractor is not
treated in the present paper.

The rest of the paper is organized as follows. In Section 2, we
prove first the existence and uniqueness of a weak solution of the
problem by using the compactness method, and then the existence of
a compact global attractor $\mathcal{A}$ in $L^2(\Omega)$ for the
semigroup $S(t)$ generated by \eqref{e1.1}. In Section 3, we
study the upper semicontinuity of the global attractor with
respect to the nonlinearity. In the last section, the
upper-continuous dependence of the global attractors on the shape
of the domain is investigated.

\subsection*{Notation}
The $L^2(\Omega)$-norm will be denoted as $\|\cdot \|$,  and the
$S^1_0(\Omega)$-norm will be denoted by $\|\cdot\|_{S^1_0(\Omega)}$.
By $S^{-1}(\Omega)$ we denote the dual space
of $S^1_0(\Omega)$. Let $(X,d)$ be a metric space, we usually use
the semi-distance $\delta_X(.,.)$ defined on the subsets of $X$ by
$$
\delta_X(A,B)=\sup_{a\in A}\inf_{b\in B}d(a,b), \quad
\forall  A, B \subset X.
$$
Denote by $Q_T=\Omega\times(0,T)$ the cylinder with the base $\Omega$.

Let $X_1,X_2$ be two Banach spaces and $Z$ be a topological vector
space such that $X_1\hookrightarrow Z, X_2\hookrightarrow Z$.
Then $X_1\cap X_2$ and $X_1+X_2$ are two Banach spaces equipped
with the norms
\begin{gather*}
\|u\|_{X_1\cap X_2}=\|u\|_{X_1}+\|u\|_{X_2},\\
\|u\|_{X_1+X_2}=\inf \{\|u_1\|_{X_1}+\|u_2\|_{X_2}: u=u_1+u_2\}.
\end{gather*}
It is known that if $X_1\cap X_2$ is dense both in $X_1$ and $X_2$
then $(X_1\cap X_2)^*=X_1^*+X_2^*$.

\section{Existence of the Global Attractor}

In this section, we prove the global existence of a weak solution and
of a global attractor of problem \eqref{e1.1}, under the assumptions
$u_0\in L^2(\Omega)$, $g\in L^2(\Omega)$ given,  and $f$ satisfying
the conditions \eqref{e1.2}-\eqref{e1.3}.
First, we give the definition of the weak solution to the
problem \eqref{e1.1}.

\begin{definition} \label{def1} \rm
Let $T>0$ and $u_0\in L^2(\Omega)$ be given. A function $u$ is
called a weak solution of the problem \eqref{e1.1} on $(0,T)$
 if $u\in W_{0,T}=L^p(Q_T)\cap L^2(0,T;S_0^1(\Omega))
 \cap C([0,T];L^2(\Omega)),\frac{\partial u}{\partial t}
 \in L^2(0,T;S^{-1}(\Omega))+L^q(Q_T), u(0)=u_0$, and
\begin{align*}
&\int_0^T\langle u_t,\varphi \rangle dt
+\int_0^T\int_{\Omega}(\nabla_{x_1} u\nabla_{x_1}
\varphi+|x_1|^{2s}\nabla_{x_2}u\nabla_{x_2}\varphi ) \,dx\,dt\\
&+\int_0^T\int_{\Omega} f(u)\varphi \,dx\,dt
+\int_0^T\int_{\Omega}g(x)\varphi \,dx\,dt=0
\end{align*}
for all test functions $\varphi\in L^p(Q_T)\cap L^2(0,T;S_0^1(\Omega))$,
where $q$ is the conjugate of $p$ (i.e. $\frac{1}{p}+\frac{1}{q}=1$).
\end{definition}

We remark that under condition \eqref{e1.2}, one can prove
that $f(u)\in L^q(Q_T)$ if $u\in W_{0,T}$
(see the proof of Theorem \ref{thm2.1} below).
Thus, the integral $\int_0^T\int_\Omega f(u)\varphi \,dx\,dt$
is well-defined.


To prove the existence of solutions by the compactness method,
we need the following Compactness Lemma (see e.g.  \cite[p. 58]{l1}).

\begin{lemma} \label{lem2.1}
Let $X_0, X$, and $X_1$ be three Banach spaces such that
$X_0\hookrightarrow X\hookrightarrow X_1$,
the injection of $X$ into $X_1$ is continuous, the injection of $X_0$
into $X$ is compact, and $X_0, X_1$ are reflexive.
Let $1< \alpha_0, \alpha_1<\infty$, we set
$$
E=\Big\{u\in L^{\alpha_0}(0,T;X_0), \frac{du}{dt}\in L^{\alpha_1}(0,T; X_1)
\Big\}
$$
equipped with the norm
$$
\|u\|_E=\|u\|_{L^{\alpha_0}(0,T;X_0)}
+\|\frac{du}{dt}\|_{L^{\alpha_1}(0,T;X_1)}.
$$
Then the inclusion $E\hookrightarrow L^{\alpha_0}(0,T;X)$ is compact.
\end{lemma}

The following lemma shows the continuity of solutions.

\begin{lemma} \label{lem2.2}
If $u\in L^2(0,T;S_0^1(\Omega))\cap L^p(Q_T)$ and
$\frac{\partial u}{\partial t}\in L^2(0,T;S^{-1}(\Omega))+L^q(Q_T)$
then $u\in C([0,T];L^2(\Omega))$.
\end{lemma}

\begin{proof} We select a sequence $u_n\in C^1([0,T];S^1_0(\Omega))$ such
that
\begin{gather*}
u_n \to u \quad \text{in } L^2(0,T;S^1_0(\Omega))\cap L^p(Q_T)\\
\frac{\partial u_n}{\partial t}\to \frac{\partial u}{\partial t}\quad
\text{in } L^2(0,T;S^{-1}(\Omega))+L^q(Q_T).
\end{gather*}
Then, for all $t,t_0\in [0,T]$, we have
\begin{align*}
\|u_n(t)-u_m(t)\|^2=\|u_n(t_0)-u_m(t_0)\|^2
+2\int_{t_0}^t\langle u_n'(s)-u_m'(s),u_n(s)-u_m(s)\rangle ds.
\end{align*}
We choose $t_0$ so that
$$
\|u_n(t_0)-u_m(t_0)\|^2=\frac1T\int_0^T\|u_n(t)-u_m(t)\|^2dt.
$$
Setting $X(t_1,t_2)=L^2(t_1,t_2;S^1_0(\Omega))\cap L^p(Q_T)$ and
$X^*(t_1,t_2)=L^2(t_1,t_2;S^{-1}(\Omega))+L^q(Q_T)$, we have
\begin{align*}
&\int_{\Omega}|u_n(t)-u_m(t)|^2dx\\
&=\frac1T\int_{\Omega}\int_{0}^T|u_n(t)-u_m(t)|^2dtdx
 +2\int_{\Omega}\int_{t_0}^t(u_n'(s)-u_m'(s))(u_n(s)-u_m(s))dsdx\\
&\leqslant \frac1T\int_{\Omega}\int_{0}^T|u_n(t)-u_m(t)|^2dtdx
 + 2\|u_n'-u_m'\|_{X^*(t_0,t)}\|u_n-u_m\|_{X(t_0,t)}\\
&\leqslant \frac1T\int_{\Omega}\int_{0}^T|u_n(t)-u_m(t)|^2dtdx
 +2\|u_n'-u_m'\|_{X^*(0,T)}\|u_n-u_m\|_{X(0,T)}.
\end{align*}
Hence, $\{u_n\}$ is a Cauchy sequence in $C([0,T];L^2(\Omega))$ thanks
to the choosing of the sequence $u_n$.  Thus the sequence $\{u_n\}$
converges in $C([0,T];L^2(\Omega))$ to a function
$v\in C([0,T];L^2(\Omega))$. Since
$u_n(t)\longrightarrow u(t)\in L^2(\Omega)$ for a.e. $t\in [0,T]$,
we deduce that $u=v$ a.e. It implies that $u\in C([0,T];L^2(\Omega))$
(after possibly being redefined on a set of measure zero).
\end{proof}

\begin{theorem} \label{thm2.1}
Under the conditions \eqref{e1.2}-\eqref{e1.3}, problem \eqref{e1.1}
 has a unique weak solution $u(t)$ satisfying
\begin{gather*}
u\in C([0,\infty); L^2(\Omega))\cap L^2_{loc}(0,\infty;S_0^1(\Omega))
\cap L^p_{loc}(0, \infty; L^p(\Omega)),
\\
\frac{\partial u}{\partial t}\in L^2_{loc}(0,\infty;S^{-1}(\Omega))
+L^q_{loc}(0, \infty; L^q(\Omega)),
\end{gather*}
where $q$ is the conjugate of $p$. Moreover, the mapping
$u_0\mapsto u(t)$ is continuous on $L^2(\Omega)$.
\end{theorem}

\begin{proof}  \textbf{(i) Existence.}
We will use the compactness method for showing the existence of a
weak solution to the problem \eqref{e1.1}.

We look for an approximate solution $u_n(t)$ that belongs to the
finite-dimensional space spanned by the first $n$ eigenfunctions of
$-G_s$ such that
$$
u_n(t)=\sum_{j=1}^n u_{nj}(t)e_j,
$$
and solves the problem
\begin{equation} \label{e2.1}
\begin{gathered}
\langle \frac{\partial u_n}{\partial t},e_j\rangle-\langle G_su_n,e_j
\rangle+\langle\ f(u_n),e_j \rangle+(g,e_j)=0, \quad
 1\leqslant j\leqslant n, \\
(u_n(0),e_j)=(u_0,e_j).
\end{gathered}
\end{equation}
Hence we have a system of first-order ordinary differential equations
for the functions $u_{n1},u_{n2},\ldots,u_{nn}$,
\begin{gather*}
 u_{nj}'+\lambda_ju_{nj}+\langle f(u_n),e_j\rangle+( g,e_j)=0,\quad
  j=\overline{1,n}\\
 u_{nj}(0)=(u_0,e_j).
\end{gather*}
According to theory of ODEs, we obtain the existence of approximate
solutions $u_n(t)$.
We now establish some {\it a priori} estimates for $u_n$. Since
\[
\frac12 \frac{d}{dt}\|u_n\|^2+\|u_n\|^2_{S_0^1(\Omega)}
+\int_{\Omega}f(u_n)u_ndx+\int_\Omega g u_ndx=0,
\]
it follows from  \eqref{e1.2} that
\begin{equation} \label{e2.2}
\begin{aligned}
\quad &\frac12\frac{d}{dt}\|u_n(t)\|^2+\|u_n(t)\|_{S_0^1(\Omega)}^2
+C_1\int_{\Omega}|u_n(t)|^pdx \\
\quad & -C_0|\Omega|-\frac{1}{2\lambda_1}\|g\|^2
-\frac{\lambda_1}{2}\|u_n(t)\|^2\leqslant 0,
\end{aligned}
\end{equation}
where $\lambda_1 >0$ is the first eigenvalue of $-G_s$ in $\Omega$
with the homogeneous Dirichlet condition (noting that
$\|u\|^2_{S_0^1(\Omega)}\geq \lambda_1 \|u\|^2$ for all
$u\in S^1_0(\Omega)$).
Hence
\[
\frac{d}{dt}\|u_n(t)\|^2\leqslant - \lambda_1\|u_n(t)\|^2+C_4,
\]
where $C_4=\frac{1}{\lambda_1}\|g\|^2+2C_0|\Omega|$.
Using the Gronwall inequality, we obtain
\begin{equation}
\|u_n(t)\|^2\leqslant e^{-\lambda_1 t}\|u_n(0)\|^2
+\frac{C_4}{\lambda_1}(1-e^{-\lambda_1 t}).\label{e2.3}
\end{equation}
This estimate implies that the solution $u_n(t)$ of   \eqref{e2.1}
can be extended to $+\infty$.

 From \eqref{e2.2}, we have
\begin{align*}
\frac{d}{dt}\|u_n(t)\|^2+\|u_n(t)\|_{S_0^1(\Omega)}^2
+2C_1\int_{\Omega}|u_n(t)|^pdx\leqslant C_4.
\end{align*}
Let $T$ be an arbitrary positive number, integrating both sides of
the above inequality from $0$ to $T$, we obtain
\begin{align*}
\|u_n(T)\|^2+ \int_0^T\|u_n(t)\|^2_{S_0^1(\Omega)}dt
+2C_1\int_{0}^T\int_{\Omega}|u_n|^p\,dx\,dt\leqslant \|u_n(0)\|^2+C_4T.
\end{align*}
This inequality yields $\{u_n\}$ is bounded in
$L^{\infty}(0,T;L^2(\Omega))$,
 in $L^{2}(0,T;S_0^1(\Omega))$,
 and  in $L^{p}(Q_T)$.

We first use the boundedness of $\{u_n\}$ in $L^{p}(Q_T)$ to prove
the boundedness of $\{f(u_n)\}$ in $L^{q}(Q_T)$, where $q$ is conjugate
of $p$. Indeed, the condition \eqref{e1.2} implies
\begin{align*}
|f(u)|\leqslant C_5\Big(1+|u|^{p-1}\Big).
\end{align*}
Therefore,
\begin{align*}
\|f(u_n)\|^q_{L^q(Q_T)}
&=\int_0^T\int_{\Omega}|f(u_n)|^q\,dx\,dt\\
&\leqslant C \int_0^T\int_{\Omega}\Big(1+|u_n|^{q(p-1)}\Big)\,dx\,dt\\
&\leqslant C\int_0^T\int_{\Omega}\Big(1+|u_n|^{p}\Big)\,dx\,dt.
\end{align*}
Hence $\{f(u_n)\}$ is bounded in $L^q(Q_T)$.

Next, we show that $\{\frac{\partial u_n}{\partial t}\}$ is bounded
in the space $L^q(0,T;S^{-1}(\Omega))$. Indeed, since
$$
\frac{\partial u_n}{\partial t}=G_s u_n-f(u_n)-g
$$
we have $\frac{\partial u_n}{\partial t}\in L^2(0,T;S^{-1}(\Omega))
+L^q(Q_T)$. Combining this with the fact that $L^2(0,T;S^{-1}(\Omega))$
and $L^q(Q_T)$ are continuously embedded into $L^q(0,T;S^{-1}(\Omega))$,
we obtain the boundedness of $\{\frac{\partial u_n}{\partial t}\}$
in $L^q(0,T;S^{-1}(\Omega))$. Hence, by choosing a subsequence,
we can assume that
$\frac{\partial u_n}{\partial t} \rightharpoonup\frac{\partial u}{\partial t}$
in $L^q(0,T;S^{-1}(\Omega))$.

From the above results, we can assume that
\begin{gather*}
u_n\rightharpoonup u \quad\text{in } L^2(0,T;S_0^1(\Omega)),\\
u_n\rightharpoonup u \quad \text{in } L^p(Q_T), \\
f(u_n)\rightharpoonup \eta \quad \text{in } L^q(Q_T).
\end{gather*}
From the fact that $u\in L^2(0,T;S_0^1(\Omega))\cap L^p(Q_T)$
and $u_t\in L^2(0,T;S^{-1}(\Omega))+L^q(Q_T)$, by Lemma \ref{lem2.2}, we
infer that  $u\in C([0,T];L^2(\Omega))$ and thus $u\in W_{0,T}$.

It remains to be shown that $\eta=f(u)$ and $u(0)=u_0$.
Since $\{u_n\}$ is bounded in $L^2(0,T;S_0^1(\Omega))$ and
$\{\frac{\partial u_n}{\partial t}\}$ is bounded in
$L^q(0,T;S^{-1}(\Omega))$, it follows from the Compactness Lemma that
$$
u_n\to u \text{ in } L^2(0,T;L^2(\Omega)).
$$
Hence we can choose a subsequence $\{u_{n_k}\}$ such that
$$
u_{n_k}(t,x)\to u(t,x) \quad\text{for a.e. } (t,x)\in Q_T.
$$
It follows from the continuity of the function $f$ that
$$
f(u_{n_k}(t,x))\to f(u(t,x))\quad\text{for a.e. } (t,x)\in Q_T.
$$
 In view of the boundedness of $\{f(u_{n_k})\}$ in $L^q(Q_T)$,
by \cite[Lemma 1.3]{l1}, we conclude that
$$
f(u_{n_k})\rightharpoonup f(u)\quad \text{in } L^q(Q_T).
$$
Taking into account the uniqueness of a weak limit, we get $\eta=f(u)$.

We are in a position to show that $u(0)=u_0$. Choosing a test function
$\varphi\in C^1([0,T]; S^1_0(\Omega)\cap L^p(\Omega))$ with
$\varphi(T)=0$, we  see that
$\varphi\in L^p(Q_T)\cap L^2(0,T;S_0^1(\Omega))$. Taking integration by
parts in the $t$ variable, we have
\begin{align*}
&\int_0^T-(u,\varphi')+\int_0^T\int_\Omega (\nabla_{x_1}
u\nabla_{x_1}\varphi +|x_1|^{2s}\nabla_{x_2}u\nabla_{x_2}\varphi)
+\int_0^T\int_\Omega(f(u)+g)\varphi\\
&=(u(0),\varphi(0)).
\end{align*}
Doing the same in the Galerkin approximations yields
\begin{align*}
&\int_0^T-(u_n,\varphi')
+\int_0^T\int_\Omega (\nabla_{x_1}u_n\nabla_{x_1}\varphi
+|x_1|^{2s}\nabla_{x_2}u_n\nabla_{x_2}\varphi)
+ \int_0^T\int_\Omega(f(u_n)+g)\varphi\\
&=(u_n(0),\varphi(0)).
\end{align*}
Taking limits as $n\to \infty$ we conclude that
\begin{align*}
&\int_0^T-(u,\varphi)'+\int_0^T\int_\Omega (\nabla_{x_1}u\nabla_{x_1}\varphi
+|x_1|^{2s}\nabla_{x_2}u\nabla_{x_2}\varphi)+\int_0^T\int_\Omega(f(u)+g)
\varphi\\
&=(u_0,\varphi(0))
\end{align*}
since $u_n(0)\to u_0$. Thus, $u(0)=u_0$.

We now prove existence of a global solution $u$.
Analogously to \eqref{e2.3} we have
\begin{equation}
\|u(t)\|^2\leqslant e^{-\lambda_1 t}\|u(0)\|^2
+\frac{C_4}{\lambda_1}(1- e^{-\lambda_1 t}). \label{e2.4}
\end{equation}
This implies that the solution $u$ exists globally in time.


\textbf{(ii) Uniqueness and continuous dependence.}
 Let $u_0, v_0 \in L^2(\Omega)$. Denote by $u, v$ two corresponding
solutions of the problem \eqref{e1.1} with initial data $u_0, v_0$.
Then $w=u-v$ satisfies
\begin{gather*}
w_t-G_sw+ f(u)-f(v)=0,\\
w_{|_{\partial \Omega }}=0\\
w(0)=u_0-v_0.
\end{gather*}
Hence
$$
\frac12\frac{d}{dt}\|w\|^2+\|w\|^2_{S_0^1(\Omega)}
+\int_{\Omega}(u-v)(f(u)-f(v))dx=0,\quad \text{for a.e. } t\in[0,T].
$$
Using  \eqref{e1.3}, we have
\[
\frac{d}{dt}\|w\|^2+2\|w\|^2_{S_0^1(\Omega)}
\leqslant   2C_3\|w\|^2,\quad \text{ for a.e. }t\in[0,T].
\]
Applying the Gronwall inequality, we obtain
$\|w(t)\|\leqslant \|w(0)\|e^{2C_3t}$.
This implies the uniqueness (if $u_0=v_0$) and the continuous
dependence of solutions.
\end{proof}

Note that Theorem \ref{thm2.1} allows us to define a continuous semigroup
$$
S(t): u_0\in L^2(\Omega) \mapsto u(t)\in L^2(\Omega)
$$
associated with problem \eqref{e1.1}.
We now prove that the semigroup $S(t)$  possesses a compact connected
global attractor  $\mathcal{A}$ in $L^2(\Omega)$.

First, from \eqref{e2.4} we deduce the existence of an absorbing
set in $L^2(\Omega)$: There is a constant $R$ and a time $t_0(\|u_0\|)$
such that, for the solution $u(t)=S(t)u_0$,
$$
\|u(t)\|\leqslant R \quad \text{for all }t\geqslant t_0(\|u_0\|).
$$
Multiplying  \eqref{e1.1} by $u$ and using \eqref{e1.2}, we obtain
\[
\frac12\frac{d}{dt}\|u(t)\|^2+\|u(t)\|_{S_0^1(\Omega)}^2
+C_1\int_{\Omega}|u(t)|^pdx-C_0|\Omega|+\int_{\Omega}gu\,dx\leqslant 0.
\]
Integrating between $t$ and $t+1$, we obtain
$$
\int_t^{t+1}\Big[\frac 1 2\|u(s)\|^2_{S^1_0}
+C_1\int_{\Omega}|u(s)|^pdx+\int_{\Omega}gu\,dx\Big]ds\leqslant
 C_0|\Omega|+\frac{1}{2}\|u(t)\|^2.
$$
This shows that
$$
\int_t^{t+1}\Big[\frac 1 2\|u(s)\|^2_{S^1_0}
+C_1\int_{\Omega}|u(s)|^pdx+\int_{\Omega}gudx\Big]ds\leqslant
C_0|\Omega|+\frac 1 2 R^2, \quad \forall  t\geqslant t_0(\|u_0\|).
$$
Noting that
\begin{equation}
C_5(|u|^p-1)\leqslant F(u)\leqslant C_6(|u|^p+1), \label{e2.5}
\end{equation}
where $F(u)=\int_0^u f(\sigma)d\sigma$, we obtain
\begin{equation}
\int_t^{t+1}\Big[\frac 1 2\|u(s)\|^2_{S^1_0}
+\int_{\Omega}(F(u) +gu)dx\Big]ds\leqslant C_7, \quad
\text{for all } t\geqslant t_0(\|u_0\|). \label{e2.6}
\end{equation}
In what follows, we shall formally derive an {\it a priori}
estimate in $S^1_0(\Omega)\cap L^p(\Omega)$ on the solutions
which holds for smooth functions and will become rigorous by
using a Galerkin truncation and a limiting process.
Taking the inner product of \eqref{e1.1} with $u_t$, we obtain
\begin{equation}
\frac{d}{dt}\Big[\frac 1 2\|u\|^2_{S_0^1}
+\int_{\Omega}( F(u)+gu)dx\Big] = - 2\|u_t\|^2\leqslant 0. \label{e2.7}
\end{equation}
To deduce the existence of an absorbing set in $S^1_0(\Omega)$,
we need the uniform Gronwall inequality that we recall
(see e.g.  \cite[p. 91]{t1}).

\begin{lemma} \label{lem2.3}
 Let $g, h, y$ be three locally integrable functions on $(t_0,+\infty)$
which satisfy
\begin{gather*}
\frac{dy}{dt} \in L^1_{loc}(t_0,+\infty) \quad\text{and}\quad
 \frac{dy}{dt}\leqslant gy+h,\quad \text{for } t\geqslant t_0, \\
\int_t^{t+r}g(s)ds\leqslant a_1, \quad
\int_{t}^{t+r}h(s)ds\leqslant a_2, \quad
\int_t^{t+r}y(s)ds\leqslant a_3,\quad \text{for  } t\geqslant t_0,
\end{gather*}
where $r, a_1, a_2, a_3$ are positive constants. Then
$$
y(t)\leqslant (\frac{a_3}{r}+a_2)e^{a_1}, \quad\text{for all }
t\geqslant t_0+r.
$$
\end{lemma}

Combining \eqref{e2.6}, \eqref{e2.7} and using the above lemma, we obtain
$$
\frac 1 2\|u\|^2_{S_0^1}+\int_{\Omega}( F(u)+gu)dx\leqslant C_7, \quad
\text{for all } t\geqslant t_0(\|u_0\|)+1.
$$
Using \eqref{e2.5}, the Cauchy inequality and the fact that
$\|u\|_{S_0^1(\Omega)}^2\geqslant \lambda_1\|u\|_{L^2(\Omega)}^2$,
we deduce from the last inequality that
$$
\|u(t)\|^2_{S^1_0}+\int_\Omega |u|^pdx\leqslant C_8
$$
provided that $t\geqslant t_0(\|u_0\|)+1$. It follows from here
that the ball $B_0$ centered at $0$ with radius $C_8$ is an absorbing
set for $S(t)$ in $S^1_0(\Omega)\cap L^p(\Omega)$.

Using the absorbing set $B_0$ in $S^1_0(\Omega)$, and noting that
the embedding $S^1_0(\Omega)\hookrightarrow L^2(\Omega)$ is compact,
and that $L^2(\Omega)$ is connected, we obtain the following theorem.

\begin{theorem} \label{thm2.2}
Under conditions \eqref{e1.2}-\eqref{e1.3}, the semigroup $S(t)$
generated by the problem \eqref{e1.1} possesses a compact connected
global attractor $\mathcal{A}=\omega(B_0)$ in $L^2(\Omega)$.
\end{theorem}

\begin{remark} \label{rmk2.3} \rm
In fact, if we are only concerned with the existence of the global
attractor for the semigroup $S(t)$ in $L^2(\Omega)$, then the
assumption \eqref{e1.3} can be replaced by the weaker assumption
$$
\big(f(u)-f(v)\big)(u-v)\geqslant -C|u-v|^2 \quad
\text{for any }u,v\in \mathbb{R}.
$$
However, we need to use the stronger assumptions, namely
$f\in C^1(\mathbb{R})$ and \eqref{e1.3}, in the next section
(for proving \eqref{e3.2}).
\end{remark}

\section{Continuous dependence of attractors on the nonlinearity}

In this section we consider a family of $C^1$ functions
$f_\lambda, \lambda\in \Lambda $, such that for each
$\lambda\in {\Lambda}$,  $f_\lambda$ satisfies conditions
\eqref{e1.2}-\eqref{e1.3} with the constants independent
of $\lambda$. The family $\Lambda$ is considered with a topology
$\mathcal{T}$ such that the convergence ${\lambda_j}\to \lambda$
with respect to $\mathcal{T}$ implies that
$$
f_{\lambda_j}(u)\to f_\lambda(u) \quad \text{for any } u.
$$
Let $S_t(\lambda,u_0)$ be the semigroup generated by the  problem
\begin{gather*}
u_t-G_su+ f_\lambda(u)+g(x)=0,\quad x\in \Omega, t>0\\
u(x,t)=0, \quad x\in \partial \Omega, \; t>0  \\
u(x, 0)=u_0(x),\quad x\in \Omega.
\end{gather*}
 From the results in Section 2, this semigroup has a compact
absorbing set
$$
B_\lambda=\{u\in L^2(\Omega):\|u\|_{S^1_0(\Omega)}\leqslant R_\lambda\}
$$
and a compact global attractor
${\mathcal{A}}_\lambda=\omega(B_\lambda)$ in $X=L^2(\Omega)$.

\begin{lemma} \label{lem3.1}
$S_t(.,.)$ is continuous in ${\Lambda}\times X$ for any fixed $t>0$.
\end{lemma}
\begin{proof}
Let $(\lambda_0,u_0)\in {\Lambda}\times X$ and
$(\lambda_j, u_{j0})\in {\Lambda}\times X$ such that
$\lambda_j\to \lambda_0$ and $u_{j0}\to u_0$.
Let $u_j(t)=S_t(\lambda_j,u_{j0})$ be the solution of \eqref{e1.1}
with the nonlinearity $f_{\lambda_j}$ and the initial data $u_{j0}$.
Since $f_{\lambda_j}$ satisfies \eqref{e1.2}-\eqref{e1.3}
with the same constants and $\{u_{j0}\}$ is bounded,
by using arguments as in the proof of Theorem \ref{thm2.1}, we have
\begin{gather*}
\{u_j\} \text{ is bounded  in } L^\infty(0,T; L^2(\Omega))\\
\{u_j\} \text{ is bounded  in } L^2(0,T; S_{0}^{1}(\Omega))\\
\{f_{\lambda_j}(u_j)\} \text{  is bounded  in } L^q(0,T; L^q(\Omega))\\
\{\partial_tu_j\} \quad\text{is bounded  in }
 L^2(0,T,S^{-1}(\Omega))+{L^{q}(0,T; L^{q}(\Omega))}.
\end{gather*}
We may apply the Compactness Lemma to conclude that $\{u_j\}$
is relatively compact in $L^2(0,T; L^2(\Omega))$.
Hence, there exists a subsequence (still denoted by) $u_j$ such that
\begin{equation} \label{e3.1}
\begin{gathered}
u_j\overset{\ast}\rightharpoonup u \quad
 \text{in } L^\infty(0,T; L^2(\Omega))\\
u_j\rightharpoonup u \quad
 \text{in } L^2(0,T; S_{0}^{1}(\Omega))\\
u_j\to u \quad\text{almost everywhere in } \Omega\times(0,T)\\
f_{\lambda_j}(u_j)\rightharpoonup \omega \quad
 \text{in } L^q(0,T; L^q(\Omega))\\
\partial_tu_j \rightharpoonup  \partial_tu \quad
  \text{in }L^2(0,T; S^{-1}(\Omega))+{L^{q}(0,T; L^{q}(\Omega))}.
\end{gathered}
\end{equation}
Combining \eqref{e3.1} with the hypotheses imposed on
$f_\lambda$ and the fact that $f_{\lambda_j}$  converges almost
everywhere to $f_{\lambda_0}$ we have
\begin{equation}
f_{\lambda_j}(u_j)\to f_{\lambda_0}(u) \quad \text{almost everywhere in }
 \Omega\times(0,T). \label{e3.2}
\end{equation}
 From \cite[Lemma 1.3]{l1}, we have $\omega=f_{\lambda_0}(u)$.
By passing to limit in the weak form, we obtain that $u$ is the
solution of   the problem \eqref{e1.1}.

Now, let $t\in (0,T)$.  Since $u_j(t)$ is bounded in $S^1_0(\Omega)$,
there is a subsequence, still denoted by $u_j$, such that
$u_j(t) \to v(t)$ strongly in $L^2(\Omega)$. Therefore,
$$
S_t(\lambda_j,u_{j0}) \to S_t(\lambda_0,u_0).
$$

We have proved that for any $(\lambda_j,u_{j0}) \to (\lambda_0,u_0)$,
there exists a subsequence of $S_t(\lambda_j,u_{j0})$ which converges
to $S_t(\lambda_0,u_0)$ and the limit is independent of the subsequence,
so the whole sequence $S_t(\lambda_j,u_{j0})$ converges to
$S_t(\lambda_0,u_0)$. This completes the proof.
\end{proof}

\begin{theorem} \label{thm3.1}
 The family $\{{\mathcal{A}}_\lambda: \lambda\in {\Lambda}\}$ depends
upper semi-continuously on the parameter $\lambda$, i.e.
$$
\limsup_{\lambda\to \lambda_0}\delta_X({\mathcal{A}}_\lambda,
{\mathcal{A}}_{\lambda_0})=0.
$$
\end{theorem}

\begin{proof}
For any $\lambda_j\in {\Lambda}$ the semigroup $S_t(\lambda_j,u)$
has a compact absorbing set
$$
B_{\lambda_j}=\{u\in L^2(\Omega):\|u\|_{{S}^1_0(\Omega)}\leqslant R\},
$$
where $R$ is sufficiently large constant depending only on the
constants in \eqref{e1.2}-\eqref{e1.3}. Hence,  we can choose
$R$ independent of $\lambda_j$.
Hence, there exists
$$
B_0=\{u\in L^2(\Omega) :\|u\|_{{S}^1_0(\Omega)}\leqslant R\}
$$
such that for any bounded set $B\subset L^2(\Omega)$ and for
any $\lambda$, there is $\tau=\tau(\lambda,B)$ with the property
$$
S_t(\lambda,B)\subset B_0 \text{ for } t\geq \tau.
$$
 Let $\varepsilon >0$, there exists $T=T(\varepsilon)>0$ such that
$$
\delta_X(S_T(\lambda_0,B_0),{\mathcal{A}}_{\lambda_0})<\varepsilon.
$$
 By Lemma \ref{lem3.1},  for any $x\in B_0$, there are open neighborhoods $V(x)$
and $W(\lambda_0)$ in $X$ and ${\Lambda}$ such that
$$
\delta_X(S_T(\lambda,V(x)),{\mathcal{A}}_{\lambda_0})
 <\varepsilon \quad \text{for any } \lambda\in W(\lambda_0) .
$$
 Since $B_0$ is compact in $X$, there exists a neighborhood $W$
of $\lambda_0$ such that
$$
\delta_X(S_T(\lambda,B_0),{\mathcal{A}}(\lambda_0))<\varepsilon \quad
\text{for any } \lambda \in W .
$$
 Therefore,
$$
\delta_X({\mathcal{A}}(\lambda),{\mathcal{A}}(\lambda_0))
<\varepsilon \quad \text{for any } \lambda\in W .
$$
The proof is complete.
\end{proof}

\section{Continuous dependence of attractors on the shape of domain}

Let $\Omega_0$ be a bounded domain in ${\mathbb R}^N$ with boundary
$\partial \Omega_0$. We consider a family $\mathcal G$ of diffeomorphism
$G$ such that:
\begin{itemize}
\item Any $G\in {\mathcal G}$ is a diffeomorphism of class $C^1$ of a neighborhood of $\overline{\Omega}_0$.\\
We denote $\Omega_G=G(\Omega_0)$ and let
\begin{gather*}
\|G\|_{C^0(\overline{\Omega}_0)}=\max_{x\in\overline{\Omega}_0}|G(x)|\\
\|G\|_{C^1(\overline{\Omega}_0)}=\|G\|_{C^0(\overline{\Omega}_0)}
 +\max_{x\in\overline{\Omega}_0}|\frac{\partial G}{\partial x}(x)|.
\end{gather*}

\item Assume that
\begin{equation}
\sup_{G\in {\mathcal G}}\|G\|<+\infty,\quad
\sup_{G\in {\mathcal G}}\|G^{-1}\|<+\infty.\label{e4.1}
\end{equation}

\item The family $\mathcal G$ is equipped with the topology $\mathcal T$
such that $G_j\to G$ with respect to $\mathcal T$ if and only if
$$
\|G_j-G\|_{C^0(\overline{\Omega}_0)}\to 0.
$$

\end{itemize}
 Let $X=L^2(\Omega_0)$ and $X_G=L^2(\Omega_G)$, we define
$G^*: X_G\to X$ as follows:
$$
 G^*u(x)=u(G(x))\quad \text{for } u\in X_G.
$$
We consider  \eqref{e1.1} on $\Omega_G\times[0,+\infty)$ and assume
that  \eqref{e1.2}-\eqref{e1.3} are satisfied.
Denote by $\Sigma_t(G,u_0)$ the semigroup in $X_G$ generated by
this problem. From the results in Section 2, this semigroup
has a compact absorbing set
$$
B_G=\{u\in X_G: \|u\|_{S^1_0(\Omega_G)}\leqslant R_G\}
$$
and has a  global attractor $\mathcal{A}_G=\omega(B_G)$.

Denote $\mathcal{A}(G)=G^*(\mathcal{A}_G)$ and we define the semigroup
of operators
$S_t(G,.): X\longrightarrow X$, $G\in {\mathcal G}$,
 by
$$
S_t(G,u_0)=G^*\Sigma_t(G,(G^*)^{-1}u_0).
$$

\begin{lemma} \label{lem4.1}
  $S_t$ is continuous in ${\mathcal G}\times X$ for any fixed $t>0$.
\end{lemma}

\begin{proof}
Let $(G_0,u_0)\in {\mathcal G}\times X$ and assume that $G_j\to G_0$
and $u_{j0}\to u_0$. Putting $v_{j0}=(G^*_j)^{-1}(u_{j 0})$ and
$u_j(t)=\Sigma_t(G_j,v_{j0})$, then $u_j(t)$  is the solution of
 \eqref{e1.1} in $\Omega_{G_j}\times (0,T)$,
$u_j(0)=v_{j0}=(G^*_j)^{-1}(u_{j 0})$. By \eqref{e4.1},
there exists $R>0$ such that
\begin{equation}
\|u_j\|_{L^2(0,T; S^1_0(\Omega_{G_j}))}
+\|u_j\|_{L^\infty(0,T;L^2(\Omega_{G_j}))}\leqslant R \quad
\text{for all } j.\label{e4.2}
\end{equation}
Putting $v_j(t)=G^*_j(u_j(t))$ then $v(0)=u_{j0}$.
It follows from \eqref{e4.1} and \eqref{e4.2} that $v_j$
is uniformly bounded in
$L^2(0,T; S^1_0(\Omega_{G_0}))\cap L^\infty(0,T;L^2(\Omega_{G_0})) $.
There exists a subsequence, still denoted by $v_j$, such that
\begin{gather*}
v_j\overset{\ast}\rightharpoonup v \quad\text{in } L^\infty(0,T; L^2(\Omega))\\
v_j\rightharpoonup v \quad\text{in } L^2(0,T; S_{0}^{1}(\Omega)).
\end{gather*}

Putting $u(t)=(G^*_0)^{-1}(v(t))$. Since $u_j$ is the solution of
\eqref{e1.1} in $\Omega_{G_j}\times(0,T)$, $u$ is the solution
in the sense of distributions of \eqref{e1.1} in $\Omega_{G_0}\times(0,T)$.
Moreover,
$$
u(0)=(G^*_0)^{-1}(v(0))=\lim_{j} (G^*_0)^{-1}(u_{j0})=(G^*_0)^{-1}(u_0).
$$
On the other hand, putting $\tilde{u}(t)=\Sigma_t(G_0,v_0)$ where
$v_0=(G^*_0)^{-1}(u_0)$ then  by the uniqueness of solution we
have $\tilde{u}=u$.

Now, let $t\in (0,T)$. Since $v_j(t)$ is bounded in
$S^1_0(\Omega_0)$, there is a subsequence (still denoted by) $v_j$
such that $v_j(t) \to v(t)$ strongly in $L^2(\Omega_0)$. Therefore,
$$
S_t(G_j,u_{j0}) \to S_t(G_0,u_0).
$$
For any $(G_j,u_{j0}) \to (G_0,u_0)$, there exists a subsequence
of $S_t(G_j,u_{j0})$ which converges to $S_t(G_0,u_0)$, the limit
is independent on the subsequence, so the whole sequence
$S_t(G_j,u_{j0})$ converges to $S_t(G_0,u_0)$.
The proof is complete.
\end{proof}

\begin{theorem} \label{thm4.1}
The family $\{\mathcal{A}(G):G\in {\mathcal G}\}$ depends upper
semi-continuously on the parameter $G$, i.e.
$$
\limsup_{G\to G_0}\delta_X(\mathcal{A}(G), \mathcal{A}(G_0))=0.
$$
\end{theorem}

\begin{proof}
For any $G\in {\mathcal G}$, the semigroup $\Sigma_t(G,u)$ has a compact
absorbing set
$$
B_{G}=\{u\in X_{G}:\|u\|_{S^1_0(\Omega_{G})}\leqslant R\},
$$
where $R$ is a sufficiently large constant depending on the constants
in \eqref{e1.2}-\eqref{e1.3} and on the volume of $\Omega_{G}$.
Hence,  we can choose $R$ independent on $G$.
It follows from \eqref{e4.1} that  there exists
$$
B_0=\{u\in X:\|u\|_{S^1_0(\Omega_0)}\leqslant R\}
$$
such that for any bounded set $B\subset X$ and for any $G\in {\mathcal G}$,
there is $\tau=\tau(G,B)$ with the property
$$
S_t(G,B)\subset B_0 \text{ for } t\geq \tau.
$$
 Let $\varepsilon >0$, there exists $T=T(\varepsilon)>0$ such that
$$
\delta_X(S_T(G_0,B_0),\mathcal{A}(G_0))<\varepsilon.
$$
 By Lemma \ref{lem4.1},  for any $x\in B_0$, there are open neighborhoods $V(x)$
and $W(G_0)$ in $X$ and ${\mathcal G}$ such that
$$
\delta_X(S_T(G,V(x)),\mathcal{A}(G_0))<\varepsilon
\quad\text{for any } G\in W(G_0) .
$$
 Since $B_0$ is compact in $X$, there exists a neighborhood
$W$ of $G_0$ such that
$$
\delta_X(S_T(G,B_0),\mathcal{A}(G_0))<\varepsilon \quad\text{for any }
G\in W .
$$
 Therefore,
$$
\delta_X(\mathcal{A}(G),\mathcal{A}(G_0))<\varepsilon \quad\text{for any }
G\in W .
$$
The proof is complete.
\end{proof}

\noindent{\bf Acknowledgements.} The authors would like to thank the referee, who gave useful comments and suggestions to improve the manuscript. This work was supported by National Foundation for Science and Technology Development (NAFOSTED).

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