\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 37, pp. 1--15.\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 - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/37\hfil Dynamics of the $p$-Laplacian equations]
{Dynamics of the $p$-Laplacian equations with nonlinear
dynamic boundary conditions}

\author[X. Cheng, L. Wei \hfil EJDE-2015/37\hfilneg]
{Xiyou Cheng, Lei Wei}

\address{Xiyou Cheng \newline
School of Mathematics and Statistics, Lanzhou University,
Lanzhou 730000, China.\newline
Key Laboratory of Applied Mathematics and Complex Systems,
Gansu Province, China}
\email{chengxy@lzu.edu.cn}

\address{Lei Wei (corresponding author)\newline
School of Mathematics and Statistics,
Jiangsu Normal University,
Xuzhou 221116, China}
\email{wlxznu@163.com}

\thanks{Submitted May 6, 2014. Published February 10, 2015.}
\subjclass[2000]{37L05, 35B40, 35B41}
\keywords{$p$-Laplacian equation; boundary condition;
asymptotic regularity; \hfill\break\indent attractor}

\begin{abstract}
 In this article, we study the long-time behavior of the $p$-Laplacian
 equation with nonlinear dynamic boundary conditions for both
 autonomous and non-autonomous cases. For the autonomous case, some
 asymptotic regularity of solutions is proved. For the non-autonomous
 case, we obtain the existence and structure of a compact uniform
 attractor in $L^{r_1}(\Omega)\times L^{r}(\Gamma)$
 ($r=\min(r_1,r_2)$).
\end{abstract}

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

\section{Introduction}

In this article, we consider the asymptotic behavior of solutions of
the following $p$-Laplacian equations with nonlinear dynamic boundary
conditions:
\begin{equation}\label{1.1.1}
 \begin{gathered}
 u_{t}-\Delta_p u+f(u)=h(x,t), \quad \text{in } \Omega, \\
 u_{t}+|\nabla u|^{p-2}\partial_n u+g(u)=0, \quad \text{on } \Gamma,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$
$(N\geqslant 3)$ with a smooth boundary $\Gamma$, 
$\Delta_p$ denotes the $p$-Laplacian operator, which is defined as
$\Delta_pu=\operatorname{div}(|\nabla u|^{p-2} \nabla u)$,
$p \geqslant 2$, and about the external forcing $h(x,t)$, we consider two cases:
the autonomous case $h(x,t)=h(x)\in L^{r_1'}(\Omega)$, where
$r_1'$ is conjugate to $r_1$, and the non-autonomous case $h(x,t)$,
which will be given later in Sections 3 and 4
respectively. The functions $f$ and
$g\in C^{1}(\mathbb{R},\mathbb{R})$, satisfy the following conditions:
\begin{gather}\label{1.1.2}
C_1|s|^{r_1}-k_1 \leq f(s)s\leq C_2|s|^{r_1}+k_2,\quad r_{1}\geq p,\\
\label{1.1.3}
 C_3|s|^{r_2}-k_3\leq g(s)s\leq C_4|s|^{r_2}+k_4,\quad r_{2}\geq2, \\
\label{1.1.4}
 f'(s)\geq -l, \quad g'(s)\geq -m,
\end{gather}
here $l,m>0$, $C_i,k_i>0$, $i=1,2,3,4$.

In the case $p=2$, the problem \eqref{1.1.1} is a general
reaction-diffusion equation, the dynamical behavior have been
studied in \cite{Ba,CV,CD2,Mar,Robin,Sun,Tem,ZYS} for the Dirichlet
boundary conditions and \cite{CS1,CS2,ES2,FZ,Y1,YY1} for the dynamic
boundary conditions.

The long-time behavior of the solutions of \eqref{1.1.1} has been
considered by many researchers, e.g., see \cite{Ba,CV,CD2,Tem} and the
references therein.

For the autonomous systems; i.e., $h(x,t) = h(x)$, in the Dirichlet
boundary case, the nonlinear eigenvalue problem for the
$p$-Laplacian operator was considered in \cite{A. Le} by using the
Ljusternik-Schnirelman principle. In \cite{Ba}, Babin \& Vishik
established the existence of a $(L^2(\Omega),\,
(W_0^{1,\,p}(\Omega)\cap L^q(\Omega))_w)$-global attractor. In \cite{Tem},
a special case of $f=ku$ was discussed by Temam. In \cite{CCD}, Carvalho,
Cholewa and Dlotko considered the existence
of global attractors for problems with monotone operators, and as an
application, they proved the existence of
$(L^2(\Omega),\,L^2(\Omega))$-global attractor for $p$-Laplacian
equation, see also Cholewa \& Dlotko \cite{CD2}. In \cite{CG2}, Carvalho \&
Gentile obtained that the corresponding
semigroup has a $(L^2(\Omega),W_0^{1,\,p}(\Omega))$-global attractor
under some additional conditions. In \cite{YSZ}, Yang, Sun and Zhong
obtained the existence of a $(L^2(\Omega),W_0^{1,\,p}(\Omega)\cap
L^{r_1}(\Omega))$-global attractor, which holds only under the
assumptions \eqref{1.1.2} and \eqref{1.1.4}. Some asymptotic
regularity of the solutions was proved by Liu, Yang and Zhong in \cite{LYZ}.
In the dynamic boundary case, recently, Gal $et$ $al$ \cite{G2,G1} presented
firstly the general result for the problem \eqref{1.1.1}, the
well-posedness and the asymptotic behavior of the solutions were
studied.

Inspired by the ideas of \cite{LYZ,Sun,YY1}, we obtain the
asymptotic regularity of the solutions of equation \eqref{1.1.1},
where we only assume the external forcing $h(x)\in
L^{r_1'}(\Omega)$, $r_1'$ is conjugate to $r_1$. As
a direct application of the asymptotic regularity results, we can
obtain the existence of a global attractor in $(W^{1,p}(\Omega)\cap
L^{r_1}(\Omega))\times (W^{1-1/p,p}(\Gamma)\cap L^{r_2}(\Gamma))$
immediately. Moreover, we also can show further that the global
attractor attracts every $L^2(\Omega)\times L^2(\Gamma)$-bounded
subset with $(W^{1,p}(\Omega)\cap L^{r_1+\delta}(\Omega))\times
(W^{1-1/p,p}(\Gamma)\cap L^{r_2+\gamma}(\Gamma))$-norm for any
$\delta,\gamma\in [0,\infty)$.

For the non-autonomous systems, in the Dirichlet boundary case, the
existence of the $(L^2(\Omega),W_0^{1,p}(\Omega)\cap
L^{r_1}(\Omega))$-uniform attractor was proved by Chen and Zhong in \cite{CZ}.
However, for the nonlinear dynamic boundary conditions, the
non-autonomous $p$-Laplacian equation is less considered. In this article, we obtain the existence and structure of a compactly uniform
attractor in $L^{r_1}(\Omega)\times L^{r}(\Gamma)$
($r=\min(r_1,r_2)$), which holds only under the assumptions
\eqref{1.1.2}--\eqref{1.1.4}, and no any restrictions on
$p,r_{1},r_{2}$ and $N$.

The main results of this article are Theorem \ref{t1.1} (asymptotic
regularity), Theorem \ref{c1.3} (global attractor) and Theorem
\ref{t3.9} (uniform attractor and its structure).

Hereafter, we  assume that
\[
2<p<N.
\]
For the case $p=2$, system \eqref{1.1.1} is a reaction-diffusion
equation and we refer the reader to \cite{FZ,Y1}; and if $p\geqslant
N$, then embeddings $W^{1,p}(\Omega)\hookrightarrow
L^{s_1}(\Omega)$ and $W^{1,p}(\Omega)\hookrightarrow
L^{s_2}(\Gamma)$ hold for any $s_1,s_2\in[1,\infty)$, which make the
nonlinear terms $f(\cdot)$ and $g(\cdot)$ to be trivial terms.

For convenience, hereafter $\|\cdot\|$ and $\|\cdot\|_{\Gamma}$
stand for the norm in $L^2(\Omega)$ and $L^2(\Gamma)$,
$\langle\cdot,\cdot\rangle$ and $\langle\cdot,\cdot\rangle_\Gamma$
stand for the inner product in $L^2(\Omega)$ and $L^2(\Gamma)$,
respectively. $C$, $C_i$ denote general positive constants,
$i=1,\dots$, which will be different in different estimates.

This article is organized as follows: in Section 2, we introduce some
preliminary results; in Section 3, for the autonomous
cases, i.e., $h(x,t) = h(x)$, we prove some asymptotic regularity of
the solution; in Section 4, for the non-autonomous cases, the
existence and structure of a uniform attractor in
$L^{r_1}(\Omega)\times L^{r}(\Gamma)$ ($r=\min(r_1,r_2)$) is
obtained.

\section{Preliminaries}

In this section, we give some auxiliary results which will be used
later.
We first introduce the spaces of time-dependent external forcing
$h(x,t)$ to be considered in this article (see\cite{CV}).

\begin{definition}[\cite{CV}] \rm
A function $\varphi$ is said to be
translation bounded in $L_{\rm loc}^2(\mathbb{R};X)$, if
$$
\|\varphi\|_b^2=\sup_{t\in\mathbb{
R}}\int_t^{t+1}\|\varphi\|_X^2ds<+\infty.
$$
Denote by $L_b^2(\mathbb{R};X)$ the set of all translation bounded
functions in $L_{\rm loc}^2(\mathbb{R};X)$.
\end{definition}

We now introduce a class of functions that was defined first in
\cite{LWZ}.


\begin{definition}[\cite{LWZ}] \label{d3.0} \rm
A function $\varphi\in L_{\rm loc}^2(\mathbb{R};X)$ is said to be normal
if for any $\varepsilon>0$, there exists $\eta>0$ such that
$$
\sup_{t\in\mathbb{
R}}\int_t^{t+\eta}\|\varphi\|_X^2ds\leq\varepsilon.
$$
Denote by $L_{n}^2(\mathbb{R};X)$ the set of all normal functions in
$L_{\rm loc}^2(\mathbb{R};X)$.
\end{definition}

\begin{lemma}[\cite{LWZ}] \label{l3.0}
If $\varphi_0\in L_{n}^2(\mathbb{R};X)$, then for any $\tau\in
\mathbb{R}$,
$$
\lim_{\gamma\to\infty}\sup_{t\geq\tau}
\int_{\tau}^te^{-\gamma(t-s)}\|\varphi(s)\|_X^2ds=0,
$$
uniformly (with respect to $\varphi\in H(\varphi_0)$),
 where $H(\varphi_0)= \{\varphi_0(t+h)\,|\,h\in\mathbb{R}\}$ .
\end{lemma}

The next result is an estimate of the $p$-Laplacian operator; see
\cite{Di} for the proof.

\begin{lemma}\label{l2.1}
Let $p\geqslant 2$. Then there exists constant $K >0$ such that for
any $a,b\in \mathbb{R}^N$,
\begin{equation}\label{0.1}
\langle |a|^{p-2}a-|b|^{p-2}b,\,a-b \rangle \geqslant K |a-b|^p,
\end{equation}
where $K$ depends only on $p$ and $N$; $\langle \cdot,\cdot\rangle$
denotes the inner product of $\mathbb{R}^N$.
\end{lemma}

\section{Autonomous cases: $h(x,t) = h(x)$}

In this section, we  consider the autonomous case of
\eqref{1.1.1}; that is,
\begin{equation}\label{1.3.1}
 \begin{gathered}
 u_{t}-\Delta_p u+f(u)=h(x), \quad \text{in } \Omega, \\
 u_{t}+|\nabla u|^{p-2}\partial_n u+g(u)=0, \quad \text{on } \Gamma, \\
 u(x,0)=u_0(x),
 \end{gathered}
\end{equation}
where $h(x)\in L^{r_1'}(\Omega)$, $r_1'$ is
conjugate to $r_1$.

\subsection{Mathematical setting}

At first, following \cite{G1}, it is more convenient to introduce
the unknown function $v(t):=u(t)_{\mid \Gamma}$, defined on the
boundary $\Gamma$, so we think of our problem as a coupled
system of two parabolic equations, one in the bulk $\Omega$ and the other
on the boundary $\Gamma$. Thus, the function $u ( t) $
tracks diffusion in the bulk, while $v ( t) $ records the
information on the boundary. Throughout the remainder of this
section, we formulate the problem as following:

\noindent \textbf{Problem (P).} Let $\Omega
\subset \mathbb{R}^{N}$ $(N\geq 3)$ be a bounded domain with a
smooth boundary $ \Gamma :=\partial \Omega $ (e.g., of class
$C^2$). The nonlinearities $f$ and $g$ satisfy
\eqref{1.1.2}--\eqref{1.1.4}. For any given pair of initial data
$(u_0,v _0)\in L^2( \Omega ) \times L^2( \Gamma
) $, find $(u ( t) ,v ( t) )$ with
\begin{equation}\label{1.1.2'}
\begin{gathered}
( u ,v )  \in C( [ 0,+\infty ) ;
L^2(\Omega)\times L^2(\Gamma)) \cap L^{\infty } ( (
0,+\infty ) ;W^{1,p}(\Omega)\times W^{1-1/p,p}(\Gamma)) , \\
( u ,v )  \in W_{\rm loc}^{1,2}((0,\infty );L^2(\Omega)\times L^2(\Gamma)), \\
u  \in L_{\rm loc}^{p}( [ 0,+\infty ) ;W^{1,p}(\Omega ) ) , \\
v  \in L_{\rm loc}^{p}( [ 0,+\infty ) ;W^{1-1/p,p}( \Gamma ) )
\end{gathered}
\end{equation}
such that $(u (0),v (0))=(u _0,v _0)$, and for almost all
$t\geq 0$, $( u ( t) ,v ( t) ) $
satisfies $u ( t) _{\mid \Gamma }=v ( t) $
a.e. for $t\in ( 0,\infty ) $, and the following partial
differential equations:
\begin{equation} \label{2.3}
\begin{gathered}
\partial _{t}u -\operatorname{div} (|\nabla u|^{p-2}\nabla u) +f( u )
=h(x),\quad \text{in }\Omega \times (0,+\infty ), \\
\partial _{t}v +|\nabla u|^{p-2}\partial_n u +g( v ) =0,
\quad \text{on }\Gamma \times (0,+\infty ).
\end{gathered}
\end{equation}

Secondly, we give the following existence and uniqueness results,
where we use the definition of weak solution as in
\cite[Definition 2.3]{G1}. For more details we refer the reader to \cite{G1}.

\begin{theorem}[\cite{G1}] \label{t3.0}
Let $\Omega$ be a bounded smooth
domain in $\mathbb{R}^N (N\geqslant 3)$, $f$ and $g$ satisfy
\eqref{1.1.2}--\eqref{1.1.4}, $h(x)\in L^{r_1'}(\Omega)$. Then for
any initial data $(u_0,v_0)\in L^2(\Omega)\times L^2(\Gamma)$
and any $T>0$, the problem {\rm (P)} has a unique weak solution
$(u(t),v(t))\in \mathcal{C}([0,T]; L^2(\Omega)\times
L^2(\Gamma))$. In addition to the regularity stated in
\eqref{1.1.2'}, we also have that
$$
u(t)\in L^{r_1}(0,T;L^{r_1}(\Omega)), \quad v(t)\in
L^{r_2}(0,T;L^{r_2}(\Gamma)).
$$
Furthermore, $(u_0,v_0)$ $\mapsto$
 $(u(t),v(t))$ is continuous on $L^2(\Omega)\times  L^2(\Gamma)$.
\end{theorem}

By Theorem \ref{l3.0}, we can define the operator semigroup
$\{S(t)\}_{t \geqslant 0}$ on the phase space $L^2(\Omega)\times
L^2(\Gamma)$ as follows:
\begin{equation} \label{3.1}
S(t):L^2(\Omega)\times L^2(\Gamma) \to L^2(\Omega)\times L^2(\Gamma), \quad
S(t)(u_0,v_0)= (u(t),v(t)),
\end{equation}
which is continuous in $L^2(\Omega)\times L^2(\Gamma)$.

Next, exactly as in \cite{G1}, we have the following dissipative
results.

\begin{lemma}[\cite{G1}] \label{l3.2}
Under the assumption of Theorem \ref{l3.0},
$\{S(t)\}_{t\geqslant 0}$ has a positively invariant
$(L^2(\Omega)\times L^2(\Gamma),W^{1,p}(\Omega)\cap L^{r_1}(\Omega)\times
W^{1-1/p,p}(\Gamma)\cap L^{r_2}(\Gamma))$-bounded absorbing
set; that is, there is a positive constant $M$, such that for any
bounded subset $B\subset L^2(\Omega)\times L^2(\Gamma)$, there
exists a positive constant $T$ which depends only on the
$L^2(\Omega)\times L^2(\Gamma)$-norm of $B$ such that
\[
\int_{\Omega}|\nabla u(t)|^p\,dx
+\int_{\Omega}|u(t)|^{r_1}\,dx+\int_{\Gamma}|v(t)|^{r_2}\,dS \leqslant M
\quad \text{for all $t \geqslant T$ and $(u_0,v_0)\in B$}.
\]
\end{lemma}

\begin{lemma}[\cite{G1}] \label{l4.1}
Under the assumption of Theorem \ref{l3.0},  for any bounded
subset $B\subset L^2(\Omega)\times L^2(\Gamma)$, there exists a
positive constant $T_1$ which depends only on the $L^2(\Omega)\times
L^2(\Gamma)$-norm of $B$ such that
\begin{equation} \label{2.00}
\int_\Omega |u_t(s)|^2\,dx+\int_\Gamma |v_t(s)|^2\,dS\leqslant M'\quad
\text{for all $s \geqslant T_1$ and $(u_0,v_0)\in B$},
\end{equation}
where $M'$ is a positive constant which depends on $M$.
\end{lemma}


Hereafter, from Lemma \ref{l3.2}, we denote one of the positively
invariant absorbing set by $B_0$ with
\[
B_0\subset\{(u(t),v(t)):\|u(t)\|_{W^{1,p}(\Omega)\cap
L^{r_1}(\Omega)}+ \|v(t)\|_{W^{1-1/p,p}(\Gamma)\cap L^{r_2}(\Gamma)}
\leqslant M\},
\]
note that here the positive invariance means $S(t)B_0\subset B_0$ for
any $t\geqslant 0$.


\subsection{Asymptotic regularity}
In this subsection, we  consider the asymptotic regularity
of solutions of systems \eqref{1.3.1}, which excel the regularity
allowed by the corresponding elliptic equation.

At first, we consider the  elliptic equation
\begin{equation}\label{3.5}
\begin{gathered}
-\operatorname{div} (|\nabla \phi|^{p-2}\nabla \phi)+f(\phi)=h(x) \quad
\text{in }\Omega,\\
|\nabla \phi|^{p-2}\partial_n \phi+g(\phi)=0 \quad \text{on } \Gamma.
\end{gathered}
\end{equation}
Due to the assumptions \eqref{1.1.2}--\eqref{1.1.4}, from the
classical results about elliptic equations, we know that \eqref{3.5}
at least has one solution $\phi(x)$ with
\begin{equation}\label{3.6}
\phi(x)\in W^{1,p}(\Omega)\cap L^{r_1}(\Omega).
\end{equation}

For the rest of this article,  we assume that $\phi(x)$  denotes a
fixed solution of  \eqref{3.5}. Then, for the
solution $(u(x,t),v(x,t))$ of \eqref{1.3.1}, we decompose
$(u(x,t),v(x,t))$ as follows
\begin{equation}
(u(x,t),v(x,t))=(\phi(x)+w(x,t),\phi(x)+\widetilde{w}(x,t))
\end{equation}
with $u_0(x)=\phi(x)+w(x,0),v_0(x)=\phi(x)+\widetilde{w}(x,0)$,
where $(w(x,t),\widetilde{w}(x,t))$ solves the  equation
\begin{equation}\label{3.7}
 \begin{gathered}
 w_{t}-\operatorname{div} (|\nabla u|^{p-2}\nabla u)+ \operatorname{div} (|\nabla
\phi|^{p-2}\nabla \phi)+f(u)-f(\phi)=0 \quad \text{in }  \Omega,\\
 \widetilde{w}_{t}+|\nabla u|^{p-2}\partial_n u
 -|\nabla \phi|^{p-2}\partial_n \phi+g(v)-g(\phi)=0,
 \quad \text{on }\ \Gamma, \\
 \widetilde{w}(x,t):=w(x,t)_{\mid \Gamma},\\
 w(x,0)=u_0(x)-\phi(x),\\
 \widetilde{w}(x,0)=v_0(x)-\phi(x).
 \end{gathered}
\end{equation}
It is easy to see that this equation is also globally well posed.
Moreover, thanks to Lemma \ref{l3.2}, without loss of generality,
hereafter we assume $(u_0,v_0)\in B_0$ and so
$(w(x,0),\widetilde{w}(x,0))\in (W^{1,p}(\Omega)\cap
L^{r_1}(\Omega))\times (W^{1-1/p,p}(\Gamma)\cap L^{r_2}(\Gamma))$.

At the same time, from the positive invariance of $B_0$ and
\eqref{3.6} we have that
\begin{equation}\label{a3.8}
\|w(x,t)\|_{W^{1,p}(\Omega)\cap L^{r_1}(\Omega)} +
\|\widetilde{w}(x,t)\|_{W^{1-1/p,p}(\Gamma)\cap L^{r_2}(\Gamma)}
\leqslant M_1 
\end{equation}
for all $t\geqslant 0$,
with some positive constant $M_1$.

The main result of this section reads as follows.

\begin{theorem}\label{t1.1}
Let $\Omega$ be a bounded smooth domain in 
$\mathbb{R}^N$ $(N\geqslant3)$, $f$ and $g$ satisfy \eqref{1.1.2}--\eqref{1.1.4},
 $h(x)\in L^{r_1'}(\Omega)$, and suppose that $\{S(t)\}_{t\geq 0}$ is the
semigroup generated by the solutions of equation \eqref{1.3.1} with
initial data $(u_0,v_0)\in L^2(\Omega)\times L^2(\Gamma)$. Then,
for any $\delta,\gamma\in[0,\infty)$, there exists a bounded subset
$B_{\delta,\gamma}$ satisfying the following properties:
\begin{align*}
B_{\delta,\gamma}=\Big\{&(w,\widetilde{w}):\|w\|_{W^{1,p}(\Omega)\cap
L^{r_1+\delta}(\Omega)} \\
&+ \|\widetilde{w}\|_{W^{1-1/p,p}(\Gamma)\cap
L^{r_2+\gamma}(\Gamma)} \leqslant
\Lambda_{p,r_1,r_2,N,\delta,\gamma}<\infty\Big\},
\end{align*}
and for any bounded subset $B\subset L^2(\Omega)\times L^2(\Gamma)$,
there exists a
\[
T=T(\|B\|_{L^2(\Omega)},\|B\|_{L^2(\Gamma)},\delta,\gamma)
\]
 such that
\begin{equation}\label{1.8}
S(t)B\subset \phi(x)+B_{\delta,\gamma}\quad \text{for all } t\geqslant T,
\end{equation}
where $\phi(x)$ is a fixed solution of \eqref{3.5},
$(w(x,t),\widetilde{w}(x,t))$ satisfies \eqref{3.7}; the constant
$\Lambda_{p,r_1,r_2,N,\delta,\gamma}$ depends only on $p,r_1,r_2,N,
\delta,\gamma$.
\end{theorem}



\begin{proof}
We  use the Moser-Alikakos iteration technique \cite{Al} to
prove the following induction estimates about the solution of
\eqref{3.7}.
For clarity, we separate our proof into two steps.
\smallskip

\emph{Step 1}: We first claim that

\emph{For each $k=0,1,2,\dots$, there exist two positive constants
$T_k$ and $M_k$, which depend only on $k,p,r_1,r_2,N$ and
$\|B_0\|_{W^{1,p}(\Omega)\cap L^{r_1}(\Omega)\times
W^{1-1/p,p}(\Gamma)\cap L^{r_2}(\Gamma)}$, such that for any
$(u_0,v_0)\in B_0$ and $t \geqslant T_k$, we have
\begin{equation*}\tag{$A_k$}\label{1}
\int_{\Omega}|w(t)|^{\sigma_k}\,dx
+\int_{\Gamma}|\widetilde{w}(t)|^{\sigma_k}\,dS\leqslant M_k,
\end{equation*}and
\begin{align*}\tag{$B_k$}\label{2}
\int_t^{t+1}\big(\int_{\Omega}|w(s)|^{\sigma_{k+1}}\,dx\big)^{\frac{N-p}{N-1}}ds
+\int_t^{t+1}\big(\int_{\Gamma}|\widetilde{w}(s)|^{\sigma_{k+1}}\,dS\big)^{\frac{N-p}{N-1}}ds\leqslant
M_k.
\end{align*}
where $(w(t),\widetilde{w}(t))$ is the solution of equation
\eqref{3.7}, and
\begin{equation}\label{a'}
\sigma_k={2(\frac{N-1}{N-p})^k+(p-2)\big[\sum_{i=0}^{k}(\frac{N-1}{N-p})^i-1\big]},\quad
k=0,1,2,\dots.
\end{equation}}

\noindent (i) Initialization of the induction ($k=0$).
From \eqref{a3.8}, we can deduce $(A_0)$ immediately. To prove
$(B_0)$, we multiply \eqref{3.7} by $w$ and $\widetilde{w}$,
and integrate over $\Omega$, then we obtain
\begin{equation} \label{3.01}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_{\Omega}
|w|^2\,dx+\frac{1}{2}\frac{d}{dt}\int_{\Gamma}
|\widetilde{w}|^2\,dS+\int_{\Omega} \langle |\nabla u|^{p-2}\nabla u-
|\nabla \phi|^{p-2}\nabla \phi, \nabla w \rangle \,dx \\
&+ \int_{\Omega} (f(u)-f(\phi))w \,dx +\int_{\Gamma}
(g(v)-g(\phi))\widetilde{w} \,dS=0.
\end{aligned}
\end{equation}
By  \eqref{1.1.4}, we have
\begin{gather}\label{3.02}
\int_{\Omega}(f(u)-f(\phi)) w\,dx \geqslant -l\int_{\Omega}|w|^2\,dx,\\
\label{3.03}
\int_{\Gamma}(g(v)-g(\phi)) \widetilde{w}\,dS \geqslant
-m\int_{\Gamma}|\widetilde{w}|^2\,dS.
\end{gather}
Then applying Lemma \ref{l2.1}, we have
\begin{equation} \label{3.9}
\int_{\Omega} \langle |\nabla u|^{p-2}\nabla u- |\nabla
\phi|^{p-2}\nabla \phi, \nabla w \rangle \,dx \geqslant K
\int_{\Omega} |\nabla w|^{p} \,dx.
\end{equation}
Inserting \eqref{3.02}--\eqref{3.9} into \eqref{3.01}, we obtain
\begin{equation} \label{3.04}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_{\Omega}
|w|^2\,dx+\frac{1}{2}\frac{d}{dt}\int_{\Gamma} |\widetilde{w}|^2\,dS + K
\int_{\Omega}  |\nabla w|^{p} \,dx\\
&\leqslant l \int_{\Omega}|w|^2\,dx+ m
\int_{\Gamma}|\widetilde{w}|^2\,dS\\&\leqslant
C\big(\int_{\Omega} |w|^2\,dx+\int_{\Gamma} |\widetilde{w}|^2\,dS\big).
\end{aligned}
\end{equation}
Then, for any $t\geqslant 0$, integrating the above inequality over
$[t,t+1]$ and using \eqref{a3.8}, we deduce that
\begin{equation}\label{3.05}
\int_t^{t+1}\int_{\Omega}|\nabla w(x,s)|^{p} \,dx\,ds\leqslant
C_{K,M,M_1} \quad \text{for all } t\geqslant 0.
\end{equation}
By the Sobolev embeddings (e.g., see Adams and Fourier \cite{Adams})
\[
W^{1,p}(\Omega) \hookrightarrow L^{\frac{p(N-1)}{N-p}}(\Omega),\quad
W^{1,p}(\Omega) \hookrightarrow L^{\frac{p(N-1)}{N-p}}(\Gamma),
\]
from \eqref{3.05}, for all $t\geqslant 0$,  we have
\begin{gather}\label{3.06}
\begin{aligned}
&\int_t^{t+1}\big(\int_{\Omega} |
w(x,s)|^{\frac{p(N-1)}{N-p}}\,dx\big)^{\frac{N-p}{N-1}}ds \\
&\leqslant C_1\int_t^{t+1}\int_{\Omega} |\nabla w(x,s)|^{p} \,dx\,ds \leqslant
C_{K,M,M_1,N},
\end{aligned} \\
\label{3.07}
\begin{aligned}
&\int_t^{t+1}\big(\int_{\Gamma} |
\widetilde{w}(x,s)|^{\frac{p(N-1)}{N-p}}\,dS\big)^{\frac{N-p}{N-1}}ds\\
&\leqslant C_2\int_t^{t+1}\int_{\Omega} |\nabla w(x,s)|^{p} \,dx\,ds
\leqslant C_{K,M,M_1,N},
\end{aligned}
\end{gather}
where $C_1,C_2$ are constants of embeddings $W^{1,p}(\Omega)
\hookrightarrow L^{\frac{p(N-1)}{N-p}}(\Omega)$ and $W^{1,p}(\Omega)
\hookrightarrow L^{\frac{p(N-1)}{N-p}}(\Gamma)$, note that here
$C_1,C_2$ depend only on $N$. This implies ($B_0$) holds.
\smallskip

\noindent (ii) The induction argument.
We now assume that $(A_k)$ and $(B_k)$ hold for $k\geqslant 1$, and
we need only to prove that $(A_{k+1})$ and $(B_{k+1})$ hold.
Multiplying \eqref{3.7} by $|w|^{\sigma_{k+1}-2} w$ and
$|\widetilde{w}|^{\sigma_{k+1}-2} \widetilde{w}$, and integrating over
$\Omega$, we obtain 
\begin{equation} \label{3.08}
\begin{aligned}
&\frac{1}{\sigma_{k+1}}\frac{d}{dt}\big(\int_{\Omega}|w|^{\sigma_{k+1}}\,dx+
\int_{\Gamma}|\widetilde{w}|^{\sigma_{k+1}}\,dS\big) \\
&+(\sigma_{k+1}-1)\int_{\Omega} \langle
|\nabla u|^{p-2}\nabla u- |\nabla \phi|^{p-2}\nabla \phi, \nabla w
\rangle|w|^{\sigma_{k+1}-2}\,dx \\
&+\int_{\Omega}\big(f(u)-f(\phi)\big)|w|^{\sigma_{k+1}-2} w\,dx+
\int_{\Gamma}\big(g(v)-g(\phi)\big)|\widetilde{w}|^{\sigma_{k+1}-2}
\widetilde{w}\,dS =0.
\end{aligned}
\end{equation}
Similar to \eqref{3.02}--\eqref{3.9}, we have
\begin{gather}\label{3.09}
\int_{\Omega}\big(f(u)-f(\phi)\big)|w|^{\sigma_{k+1}-2} w\,dx
\geqslant -l \int_{\Omega}|w|^{\sigma_{k+1}}\,dx, \\
\label{3.010}
\int_{\Gamma} (g(v)-g(\phi))|\widetilde{w}|^{\sigma_{k+1}-2}\widetilde{w}
\,dS\geqslant -m\int_{\Gamma}|\widetilde{w}|^{\sigma_{k+1}}\,dS, \\
\label{3.15}
\begin{aligned}
&(\sigma_{k+1}-1)\int_{\Omega} \langle |\nabla u|^{p-2}\nabla u-
|\nabla \phi|^{p-2}\nabla \phi, \nabla w \rangle|w|^{\sigma_{k+1}-2}\,dx  \\
& \geqslant K (\sigma_{k+1}-1) \int_{\Omega} |\nabla w|^{p}
|w|^{\sigma_{k+1}-2} \,dx,
\end{aligned}
\end{gather}
so we have
\begin{equation} \label{3.011}
\begin{aligned}
&\frac{1}{\sigma_{k+1}}\frac{d}{dt}\big(\int_{\Omega}|w|^{\sigma_{k+1}}\,dx
+\int_{\Gamma}|\widetilde{w}|^{\sigma_{k+1}}\,dS\big)+K
(\sigma_{k+1}-1) \int_{\Omega} |\nabla w|^{p} |w|^{\sigma_{k+1}-2}
\,dx \\
&\leqslant l \int_{\Omega}|w|^{\sigma_{k+1}}\,dx+ m
\int_{\Gamma}|\widetilde{w}|^{\sigma_{k+1}}\,dS\leqslant
C\big(\int_{\Omega} |w|^{\sigma_{k+1}}\,dx+\int_{\Gamma}
|\widetilde{w}|^{\sigma_{k+1}}\,dS\big).
\end{aligned}
\end{equation}
Then, combining with $(B_k)$ and application of the uniform Gronwall lemma
to \eqref{3.011} we can get ($A_{k+1}$) immediately. For
$(B_{k+1})$, we integrate the above inequality over $[t,t+1]$ and
use ($A_{k+1}$), we have
\begin{equation}\label{3.012}
\int_t^{t+1}\int_{\Omega}|\nabla w|^{p} |w|^{\sigma_{k+1}-2}
\,dx\,ds\leqslant M_{k+1} \quad \text{for all }t\geqslant 0,
\end{equation}
where $M_{k+1}$ depends on $k,p,r_1,r_2,N,M,M_1$. By the embeddings
$W^{1,p}(\Omega) \hookrightarrow L^{\frac{p(N-1)}{N-p}}(\Omega)$ and
$W^{1,p}(\Omega) \hookrightarrow L^{\frac{p(N-1)}{N-p}}(\Gamma)$
again, we have
\begin{gather}\label{3.013}
\begin{aligned}
&\Big( \int_{\Omega} |w|^{(\sigma_{k+1}-2+p)\frac{N-1}{N-p}}\,dx
\Big)^{\frac{N-p}{N-1}} \\
&\leqslant C_1\cdot \big(\frac{p}{\sigma_{k+1}-2+p}\big)^p \int_{\Omega}
|w|^{\sigma_{k+1}-2} |\nabla w|^{p} \,dx,
\end{aligned} \\
\label{3.014}
\begin{aligned}
&\Big( \int_{\Gamma}
|\widetilde{w}|^{(\sigma_{k+1}-2+p)\frac{N-1}{N-p}}\,dS \Big)^{\frac{N-p}{N-1}} \\
&\leqslant C_2\cdot \big(\frac{p}{\sigma_{k+1}-2+p}\big)^p \int_{\Omega}
|w|^{\sigma_{k+1}-2} |\nabla w|^{p} \,dx,
\end{aligned}
\end{gather}
and from the definition of $\sigma_k$, we have
\begin{equation} \label{3.18}
(\sigma_{k+1}-2+p)\frac{N-1}{N-p}=\sigma_{k+2}.
\end{equation}
Combining \eqref{3.012}--\eqref{3.18}, we deduce $(B_{k+1})$
immediately.
\smallskip

\emph{Step 2}: Based on Step 1, since $N\geqslant 3$, from
the definition of $\sigma_k$ given in \eqref{a'}, it is easy to
see that
$\sigma_k\to \infty$ as $k\to \infty$.

Hence, for any $\delta,\gamma\in [0,\infty)$, we can take $k$ so
large that $r_1+\delta \leqslant \sigma_k$, $r_2+\gamma \leqslant
\sigma_k$. Consequently, we can define
$\mathcal{B}_{\delta,\gamma}$ as 
\begin{align*}
\mathcal{B}_{\delta,\gamma}
:=\Big\{&(z,\tilde{z}):\,\|z+\phi\|^p_{W^{1,p}(\Omega)}
+ \|z\|^{r_1+\delta}_{L^{r_1+\delta}(\Omega)}\\
&+\|\tilde{z}+\phi\|^p_{W^{1-1/p,p}(\Gamma)}
+\|\tilde{z}\|^{r_2+\gamma}_{L^{r_2+\gamma}(\Gamma)} \leqslant M +
M_k\Big\},
\end{align*}
where $z(t) _{\mid\Gamma}=\tilde{z}(t)$, and recall that $\phi(x)$
is a fixed solution of \eqref{3.5}.
\end{proof}

Hence, from Theorem \ref{t1.1}, using the interpolation inequality, we
can obtain immediately the following results.

\begin{theorem}\label{c1.3}
Under the assumptions of Theorem \ref{t1.1}, the semigroup
$\{S(t)\}_{t\geqslant 0}$ has a $(L^2(\Omega)\times
L^2(\Gamma), W^{1,p}(\Omega)\cap L^{r_1}(\Omega)\times
W^{1-1/p,p}(\Gamma)\cap L^{r_2}(\Gamma))$-global attractor
$\mathscr{A}$. Moreover, $\mathscr{A}$ attracts every
$L^2(\Omega)\times L^2(\Gamma)$-bounded subset with
$(W^{1,p}(\Omega)\cap L^{r_1+\delta}(\Omega))\times
(W^{1-1/p,p}(\Gamma)\cap L^{r_2+\gamma}(\Gamma))$-norm for any
$\delta,\gamma\in [0,\infty)$; and $\mathscr{A}$ allows the
decomposition $\mathscr{A}=\phi(x)+\mathscr{A}_0$ with
$\mathscr{A}_0$ is bounded in $(W^{1,p}(\Omega)\cap
L^{r_1+\delta}(\Omega))\times (W^{1-1/p,p}(\Gamma)\cap
L^{r_2+\gamma}(\Gamma))$ for any $\delta,\gamma\in [0,\infty)$, and
$\phi(x)$ is a fixed solution of \eqref{3.5}.
\end{theorem}

\begin{proof}
From Theorem \ref{t1.1}, combining with the $(L^2(\Omega)\times
L^2(\Gamma),L^2(\Omega)\times L^2(\Gamma))$-asymptotic compactness
(obtained in \cite{G1}) and the interpolation inequality, it is easily
to verify that $\{S(t)\}_{t\geqslant 0}$ is asymptotically compact
in $L^{r_1}(\Omega)\times L^{r_2}(\Gamma)$, then it is sufficient to
verify that $\{S(t)\}_{t\geqslant 0}$ is asymptotically compact in
$W^{1,p}(\Omega)\times W^{1-1/p,p}(\Gamma)$.

Let $B_0$ be a $(W^{1,p}(\Omega)\cap L^{r_1}(\Omega))\times
(W^{1-1/p,p}(\Gamma)\cap L^{r_2}(\Gamma))$-bounded absorbing set
obtained in Lemma \ref{l3.2}, then we need only to show that
\begin{equation} \label{4.1'}
\parbox{11cm}{
for any $\{(u_{0n},v_{0n})\}\subset B_0$ and
$t_n\to \infty$,
$\{(u_{n}(t_n),v_{n}(t_n))\}_{n=1}^\infty$ is precompact in
$W^{1,p}(\Omega)\times W^{1-1/p,p}(\Gamma)$,
}
\end{equation}
where $u_{n}(t_n)=S(t_n)u_{0n},v_{n}(t_n)=S(t_n)v_{0n}$.

In fact, we know that $\{(u_{n}(t_n),v_{n}(t_n))\}_{n=1}^\infty$ is
precompact in $L^2(\Omega)\times L^2(\Gamma)$ and in
$L^{r_1}(\Omega)\times L^{r_2}(\Gamma)$. 

Without loss of generality, we assume that
$\{(u_{n_k}(t_{n_k}),v_{n_k}(t_{n_k}))\}_{n=1}^\infty$ is a Cauchy
sequence in $L^2(\Omega)\times L^2(\Gamma)$ and
$L^{r_1}(\Omega)\times L^{r_2}(\Gamma)$.

Now, we prove that
$\{(u_{n_k}(t_{n_k}),v_{n_k}(t_{n_k}))\}_{n=1}^\infty$ is a Cauchy
sequence in $W^{1,p}(\Omega)\times W^{1-1/p,p}(\Gamma)$. From
Lemma \ref{l2.1}, and after standard transformations, we know that
there exists a constant $K>0$, such that
\begin{align*}
&K\|\nabla(u_{n_k}(t_{n_k})-
u_{n_j}(t_{n_j}))\|_{L^p(\Omega)}^p  \\
\leq &\big\langle-\frac{d}{dt}u_{n_k}(t_{n_k}) - f(u_{n_k}(t_{n_k}))
+ \frac{d}{dt}u_{n_j}(t_{n_j}) + f(u_{n_j}(t_{n_j})),
u_{n_k}(t_{n_k})- u_{n_j}(t_{n_j})
\big\rangle \\
&+\big\langle-\frac{d}{dt}v_{n_k}(t_{n_k}) - g(v_{n_k}(t_{n_k})) +
\frac{d}{dt}v_{n_j}(t_{n_j}) + g(v_{n_j}(t_{n_j})),
v_{n_k}(t_{n_k})- v_{n_j}(t_{n_j})
\big\rangle_\Gamma \\
\leq &\int_\Omega\big|\frac{d}{dt}u_{n_k}(t_{n_k}) -
\frac{d}{dt}u_{n_j}(t_{n_j})\big||u_{n_k}(t_{n_k})-
u_{n_j}(t_{n_j})| \\
&+\int_\Omega|f(u_{n_k}(t_{n_k}))-f(u_{n_j}(t_{n_j}))|
|u_{n_k}(t_{n_k})- u_{n_j}(t_{n_j})| \nonumber\\
 &+\int_\Gamma\big|\frac{d}{dt}v_{n_k}(t_{n_k}) -
\frac{d}{dt}v_{n_j}(t_{n_j})\big||v_{n_k}(t_{n_k})-
v_{n_j}(t_{n_j})| \\
&+\int_\Gamma|g(v_{n_k}(t_{n_k}))-g(v_{n_j}(t_{n_j}))|
|v_{n_k}(t_{n_k})- v_{n_j}(t_{n_j})|,
\end{align*}
so we have
\begin{equation} \label{4.2}
\begin{aligned}
K&\|\nabla(u_{n_k}(t_{n_k})-
u_{n_j}(t_{n_j}))\|_{L^p(\Omega)}^p \\
\leq &\big\|\frac{d}{dt}u_{n_k}(t_{n_k}) -
\frac{d}{dt}u_{n_j}(t_{n_j})\big\|\text{ }\|u_{n_k}(t_{n_k})-
u_{n_j}(t_{n_j})\|\\&+\big\|\frac{d}{dt}v_{n_k}(t_{n_k}) -
\frac{d}{dt}v_{n_j}(t_{n_j})\big\|_\Gamma\text{ }\|v_{n_k}(t_{n_k})-
v_{n_j}(t_{n_j})\|_\Gamma\\
&+C\big(1+\|u_{n_k}(t_{n_k})\|_{L^{r_1}(\Omega)}^{r_1} +
\|u_{n_j}(t_{n_j})\|_{L^{r_1}(\Omega)}^{r_1}\big)\text{
}\big\|u_{n_k}(t_{n_k})- u_{n_j}(t_{n_j})\big\|_{L^{r_1}(\Omega)}
\\
&+\widetilde{C}\big(1+\|v_{n_k}(t_{n_k})\|_{L^{r_2}(\Gamma)}^{r_2} +
\|v_{n_j}(t_{n_j})\|_{L^{r_2}(\Gamma)}^{r_2}\big)\text{
}\big\|v_{n_k}(t_{n_k})- v_{n_j}(t_{n_j})\big\|_{L^{r_2}(\Gamma)}.
\end{aligned}
\end{equation}
Combining Lemma \ref{l3.2}, Lemma \ref{l4.1} and the compactness
of $L^{r_1}(\Omega)\times L^{r_2}(\Gamma)$, and since
$W^{1,p}(\Omega)\hookrightarrow W^{1-1/p,p}(\Gamma)$, we know that
the norms on $W^{1,p}(\Omega)\times W^{1-1/p,p}(\Gamma)$ and
$W^{1,p}(\Omega)$ are equivalent, \eqref{4.2} yields \eqref{4.1'}
immediately.
\end{proof}

\section{Non-autonomous case}

In this section, we discuss the non-autonomous case of
\eqref{1.1.1}; that is,
\begin{equation}\label{4.4.1}
 \begin{gathered}
 u_{t}-\Delta_p u+f(u)=h(x,t), \quad \text{in } \Omega, \\
 u_{t}+|\nabla u|^{p-2}\partial_n u+g(u)=0, \quad \text{on } \Gamma, \\
 u(x,\tau)=u_{\tau}(x),\quad \text{in } \bar{\Omega},
 \end{gathered}
\end{equation}
where $h(x,t)\in L_b^2(\mathbb{R}; L^2(\Omega))$.

\subsection{Mathematical setting}

Similar to the autonomous cases (e.g., Problem (p) and
Theorem \ref{l3.0}), for each $h\in\Sigma$, we can also easily
obtain the following well-posedness result and the time-dependent
terms make no essential complications.

\begin{theorem}[\cite{G1}] \label{l4.0}
Let $\Omega$ be a bounded smooth domain in 
$\mathbb{R}^N$ $(N\geqslant 3)$, $f$ and $g$ satisfy \eqref{1.1.2}--\eqref{1.1.4}, 
$h(x,t)\in L_b^2(\mathbb{R}; L^2(\Omega))$. Then for any initial data
$(u_{\tau},v_{\tau})\in L^2(\Omega)\times L^2(\Gamma)$, and any
$\tau,T\in \mathbb{R}$, $T>\tau$, the solution $(u(t),v(t))$ of
problem \eqref{4.4.1} is globally defined and satisfies
\begin{gather*}
u(t)\in \mathcal{C}([\tau,T];\,L^2(\Omega))\cap L_{\rm loc}^{p}(
\tau,T; W^{1,p}(\Omega))\cap L^{r_1}(\tau,T;L^{r_1}(\Omega)),\\
v(t)\in \mathcal{C}([\tau,T];\,L^2(\Gamma))
\cap L_{\rm loc}^{p}(\tau,T; W^{1-1/p,p}(\Gamma))\cap
L^{r_2}(\tau,T;L^{r_2}(\Gamma)),
\end{gather*}
where $v(t):=u(t)_{|\Gamma}$. Furthermore, 
$(u_{\tau},v_{\tau})\mapsto (u(t),v(t))$ is continuous on $L^2(\Omega)\times
 L^2(\Gamma)$.
\end{theorem}

We now define the symbol space $\Sigma$ for \eqref{4.4.1}. 
Taking a fixed symbol $\sigma_0(s)=h_0(s)$, 
$h_0(s) \in L_b^2(\mathbb{R}; L^2(\Omega))$. We denote by 
$L_{\rm loc}^{2, w}(\mathbb{R}; L^2(\Omega)) $ the space 
$L_{\rm loc}^2(\mathbb{R} ;L^2(\Omega)) $ endowed with local weak 
convergence topology. Set
\begin{equation}\label{4.4.5}
 \Sigma_0=\{h_0(s+h)| \:h\in\mathbb{R}\},
\end{equation}
and let
\begin{equation}\label{4.4.6}
 \Sigma \text{ be the closure of $\Sigma_0$  in }L_{\rm loc}^{2, w}(\mathbb{R}
; L^2(\Omega)).
\end{equation}

Systems \eqref{4.4.1} can be rewritten in the operator form
\begin{equation}\label{4.4.4}
 \partial_ty=A_{\sigma(t)}(y),\quad y|_{t=\tau}=y_{\tau},
\end{equation}
where $\sigma(t)=h(t)$ is the symbol of equation \eqref{4.4.4}.
Thus, from Theorem \ref{l4.0}, we know that  problem
\eqref{4.4.1} is well posed for all $\sigma(s)\in\Sigma$ and
generates a family of processes
$\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ given by the formula
$U_{\sigma}(t,\tau)y_{\tau}=y(t)$, and the $y(t)$ is the solution of
\eqref{4.4.1}.

\subsection{Existence of a bounded uniformly (w. r. t.
$\sigma\in\Sigma$) absorbing set in $(W^{1,p}(\Omega)\cap
L^{r_1}(\Omega))\times (W^{1-1/p,p}(\Gamma)\cap
L^{r_2}(\Gamma))$}

In this subsection, $(W^{1,p}(\Omega)\cap L^{r_1}(\Omega)\times
W^{1-1/p,p}(\Gamma)\cap L^{r_2}(\Gamma))$-bounded uniformly
(with respect to $\sigma\in\Sigma$) absorbing set is obtained. The proof is
similar to \cite{G1} (autonomous case).

\begin{theorem}\label{t3.5a}
Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$
$ (N\geqslant 3)$, $f$ and $g$ satisfy \eqref{1.1.2}--\eqref{1.1.4}, $h(x,t)\in
L_b^2(\mathbb{R}; L^2(\Omega))$. Then the family of processes
$\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ corresponding to
\eqref{4.4.1} has a bounded uniformly (with respect to $\sigma\in\Sigma$)
absorbing set $B_0$ in $(W^{1,p}(\Omega)\cap
L^{r_1}(\Omega))\times (W^{1-1/p,p}(\Gamma)\cap
L^{r_2}(\Gamma))$, that is, there is a positive constant $M$,
such that for any $\tau\in \mathbb{R}$ and any bounded subset $B$,
there exists a positive constant $T=T(B,\tau)\geq\tau$ such that
\[
\int_{\Omega}|\nabla u(t)|^p\,dx
+\int_{\Omega}|u(t)|^{r_1}\,dx+\int_{\Gamma}|v(t)|^{r_2}\,dS \leqslant M
\]
for all $t \geqslant T$, $(u_{\tau},v_{\tau})\in B$,
$\sigma\in\Sigma$.
\end{theorem}

\begin{proof} 
Multiplying \eqref{4.4.1} by $u$ and $v$, and integrating by
parts, we obtain
\begin{equation} \label{4.4.7}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\int_\Omega
|u|^2\,dx+\frac{1}{2}\frac{d}{dt}\int_{\Gamma} |v|^2\,dS + \int_{\Omega}
 |\nabla u|^{p} \,dx+\int_{\Omega}f(u)u\,dx+
\int_{\Gamma}g(v)v\,dS\\
&=\int_{\Omega}h_0(t)u\,dx,
\end{aligned}
\end{equation}
combining with assumptions \eqref{1.1.2}--\eqref{1.1.4}, Young's
inequality and Poincar\'{e} inequality, we obtain
\begin{equation} \label{4.4.8}
\begin{aligned}
&\frac{d}{dt}\int_\Omega |u|^2\,dx+\frac{d}{dt}\int_{\Gamma}
|v|^2\,dS+\mathcal{C}(\int_\Omega |u|^2\,dx+\int_{\Gamma}
|v|^2\,dS)\\
&\leq \mathcal{C}_{|\Omega|,S(\Gamma)}+\mathcal{C}\|h_0\|^2.
\end{aligned}
\end{equation}
Applying the suitable version of Gronwall's inequality to
\eqref{4.4.8}, we can find $T_0>0$ and $\rho_0>0$, such that
\begin{align}\label{4.4.9}
\|u(t)\|^2+\|v(t)\|_{\Gamma}^2\leq \rho_0^2, \quad\text{for any }t\geq T_0.
\end{align}
Let $F(s)=\int_0^sf(\tau)d\tau$, $G(s)=\int_0^sg(\tau)d\tau$, by
 assumptions \eqref{1.1.2}--\eqref{1.1.3} again, from
\eqref{4.4.7}, we obtain
\begin{align*}
&\frac{d}{dt}\int_\Omega |u|^2\,dx+\frac{d}{dt}\int_{\Gamma}
|v|^2\,dS+\int_{\Omega} |\nabla
u|^p\,dx+\mathcal{C}_1\int_{\Omega}F(u)\,dx+
\mathcal{C}_2\int_{\Gamma}G(v)\,dS\\&\leq
\mathcal{C}_{|\Omega|,S(\Gamma)} +\mathcal{C}\|h_0\|^2.
\end{align*}
Integrating this inequality above from $t$ to $t+1$, and combining
\eqref{4.4.9}, it follows that for any $t\geq T_0$,
\begin{equation} \label{4.4.15}
\begin{aligned}
&\int_{t}^{t+1}(\int_{\Omega} |\nabla
u|^p\,dx+\mathcal{C}_1\int_{\Omega}F(u)\,dx+
\mathcal{C}_2\int_{\Gamma}G(v)\,dS)ds\\
&\leq \mathcal{C}_{|\Omega|,S(\Gamma),\rho_0}
 +\mathcal{C}\int_{t}^{t+1}\|h_0\|^2ds\\
&\leq \mathcal{C}_{|\Omega|,S(\Gamma),\rho_0,\|h_0\|_b^2}.
\end{aligned}
\end{equation}
On the other hand, multiplying \eqref{1.1.1} by $u_t$ and $v_t$, we
have
\begin{equation} \label{4.4.15'}
\begin{aligned}
&\int_\Omega|u_t|^2\,dx+\int_\Gamma|v_t|^2\,dS+\frac{1}{p}\frac{d}{dt}\int_\Omega
|\nabla u|^p\,dx+\frac{d}{dt}\big(\int_{\Omega}F(u)\,dx+
\int_{\Gamma}G(v)\,dS\big)\\
&\leq\frac{1}{2}\int_\Omega|h_0|^2\,dx+\frac{1}{2}\int_\Omega|u_t|^2\,dx,
\end{aligned}
\end{equation}
so we obtain
\begin{equation} \label{4.4.16}
\frac{d}{dt}(\int_\Omega |\nabla u|^p\,dx+p\int_{\Omega}F(u)\,dx+
p\int_{\Gamma}G(v)\,dS)\leq \mathcal{C}\|h_0\|^2.
\end{equation}
Combining \eqref{4.4.15} and \eqref{4.4.16}, by the uniformly
Gronwall lemma, we have that for any $t\geq T_0+1$, $\sigma\in\Sigma$,
\begin{equation} \label{4.4.17}
\int_\Omega |\nabla u|^p\,dx+\int_{\Omega}F(u)\,dx+ \int_{\Gamma}G(v)\,dS
\leq \mathcal{C}_{|\Omega|,S(\Gamma),\rho_0,\|h\|_b^2},
\end{equation}
which implies that for any $t\geq T_0+1$, $\sigma\in\Sigma$,
\begin{equation} \label{4.4.18}
\int_\Omega |\nabla u|^p\,dx+\int_{\Omega}|u|^{r_1}\,dx+
\int_{\Gamma}|v|^{r_2}\,dS\leq M,
\end{equation}
where $M$ depends on $|\Omega|,S(\Gamma),\rho_0,\|h\|_b^2$.
\end{proof}

As a direct result of Theorem \ref{t3.5a}, we have the existence
of a uniform attractor in $L^2(\Omega)\times L^2(\Gamma)$:

\begin{corollary} \label{t3.3}
Under the assumptions of Theorem \ref{t3.5a},  the family of
processes $\{U_{\sigma}(t,\tau)\},\sigma\in\Sigma$ corresponding to
\eqref{4.4.1} has a uniform attractor $\mathcal{A}_{\Sigma}$ in
$L^2(\Omega)\times L^2(\Gamma)$ , which is compact in
$L^2(\Omega)\times L^2(\Gamma)$ and attracts every
$L^2(\Omega)\times L^2(\Gamma)$-bounded subset with
$L^2(\Omega)\times L^2(\Gamma)$-norm. Moreover,
\[
\mathcal{A}_{\Sigma}=\omega_{0,\Sigma}(B_0)
=\cup_{\sigma\in
\Sigma}\mathcal{K}_{\sigma}(s),\quad \forall\; s\in \mathbb{R},
\]
where $\mathcal{K}_{\sigma}(s)$ is the section at $t=s$ of the
kernel $\mathcal{K}_{\sigma}$ of the process
$\{U_{\sigma}(t,\tau)\}$ with symbol $\sigma$.
\end{corollary}

\begin{proof}
 Theorem \ref{t3.5a} and the Sobolev compactness imbedding
theorem imply the existence of a uniform attractor
$\mathcal{A}_{\Sigma}$ in $L^2(\Omega)\times L^2(\Gamma)$
immediately.
\end{proof}

\subsection{Existence of a uniform attractor in
$L^{r_1}(\Omega)\times L^{r}(\Gamma)$ ($r=\min(r_1,r_2)$)}

First, we give some a priori estimates for the solution of
\eqref{4.4.1} to verify the uniformly asymptotic compactness in
$L^{r_1}(\Omega)\times L^{r_1}(\Gamma)$. The idea of the proof comes
from \cite{ZYS}.

\begin{theorem} \label{t3.4}
Assume that $h(t)$ is normal in $L_{\rm loc}^2(\mathbb{R};L^2(\Omega))$,
$f$ and $g$ satisfy \eqref{1.1.2}--\eqref{1.1.3}. Then for any
$\varepsilon > 0$, $\tau\in\mathbb{R}$ and any bounded subset
$B\subset L^2(\Omega)\times L^2(\Gamma)$, there exist two positive
constants $T=T(B,\varepsilon,\tau)$ and $M=M(\varepsilon)$, such
that
\[
\int_{\Omega(|U_\sigma(t,\tau)u_\tau|\geq
M)}{|U_\sigma(t,\tau)u_\tau|^{r_1}}+\int_{\Gamma(|U_\sigma(t,\tau)v_\tau|\geq
M)}{|U_\sigma(t,\tau)v_\tau|^{r_1}}\leq \varepsilon,
\]
for all $t\geq T$, $(u_{\tau},v_{\tau})\in B$,
$\sigma\in\Sigma$.
\end{theorem}

\begin{proof} 
We multiply \eqref{4.4.1} by $(u-M)^{r_1-1}_{+}$ and
$(v-M)^{r_1-1}_{+}$, and integrate over $\Omega$, then we have
\begin{equation} \label{4.4.27}
\begin{aligned}
&\frac{1}{r_1}\frac{d}{dt}\int_{\Omega(u\geq M)}|u-M|^{r_1}\,dx
 +\frac{1}{r_1}\frac{d}{dt}
\int_{\Gamma(v\geq M)}|v-M|^{r_1}\,dS\\
&+(r_1-1)\int_{\Omega(u\geq M)}(u-M)^{r_1-2}|\nabla
u|^p\,dx+\int_{\Omega(u\geq M)}f(u)(u-M)^{r_1-1}\,dx\\
&+\int_{\Gamma(v\geq M)}g(v)(v-M)^{r_1-1}\,dS\\
&=\int_{\Omega(u\geq M)}h_0(t)(u-M)^{r_1-1}\,dx,
\end{aligned}
\end{equation}
where $(u-M)_{+}$ denotes the positive part of $(u-M)$; that is,
\[
(u-M)_{+}= \begin{cases}
 u-M, & u\geq M, \\
 0, & u\leq M.
\end{cases}
\]
From conditions \eqref{1.1.2}--\eqref{1.1.3}, we can take $M$ large
enough such that
\begin{gather*}
\mathcal{C}_3|v|^{r_2-1}\leq g(v),\quad \text{in }\Gamma(v(t)\geq M),\\
\mathcal{C}_4|u|^{r_1-1}\leq f(u),\quad \text{in }\Omega(u(t)\geq M).
\end{gather*}
Let $\Omega_1=\Omega(u(t)\geq M)$, $\Gamma_1=\Gamma(v(t)\geq M)$,
using Young's inequality and the inequalities above, we obtain
\begin{equation} \label{4.4.32}
\begin{aligned}
&\frac{1}{r_1}\frac{d}{dt}\int_{\Omega_1}|u-M|^{r_1}\,dx
+\frac{1}{r_1}\frac{d}{dt}\int_{\Gamma_1}|v-M|^{r_1}\,dS\\
&+(r_1-1)\int_{\Omega_1}(u-M)^{r_1-2}|\nabla u|^p\,dx\\
&+\mathcal{C}_4\int_{\Omega_1}|u|^{r_1-1}(u-M)^{r_1-1}\,dx
+\mathcal{C}_3\int_{\Gamma_1}|v|^{r_2-1}(v-M)^{r_1-1}\,dS\\
&\leq \frac{\mathcal{C}_4}{2}\int_{\Omega_1}|u-M|^{2r_1-2}\,dx
+\frac{1}{2\mathcal{C}_4}\int_{\Omega_1}|h_0(t)|^2\,dx,
\end{aligned}
\end{equation}
so we have
\begin{align*}
&\frac{1}{r_1}\frac{d}{dt}\int_{\Omega_1}|u-M|^{r_1}\,dx
+\frac{1}{r_1}\frac{d}{dt}\int_{\Gamma_1}|v-M|^{r_1}\,dS\\
&+(r_1-1)\int_{\Omega_1}(u-M)^{r_1-2}|\nabla u|^p\,dx\\
&+\frac{\mathcal{C}_4M^{r_1-2}}{2}\int_{\Omega_1}|u-M|^{r_1}\,dx
+\mathcal{C}_3M^{r_2-2}\int_{\Gamma_1}|v-M|^{r_1}\,dS
\\
&\leq\frac{1}{2\mathcal{C}_4}\int_{\Omega_1}|h_0(t)|^2\,dx.
\end{align*}
By using the Gronwall lemma and together with the Lemma
\ref{l3.0}, we can choose $M$ large enough, such that
\begin{align}\label{4.4.33}
\int_{\Omega_1}|u-M|^{r_1}\,dx
+\int_{\Gamma_1}|v-M|^{r_1}\,dS\leq\varepsilon.
\end{align}
Noting that
\begin{gather}\label{4.4.34}
\frac{1}{2^{r_1}}\int_{\Omega(u\geq 2M)}|u|^{r_1}\,dx\leq
\int_{\Omega(u\geq M)}|u-M|^{r_1}\,dx,\\
\frac{1}{2^{r_1}}\int_{\Gamma(v\geq 2M)}|v|^{r_1}\,dS\leq
\int_{\Gamma(v\geq M)}|v-M|^{r_1}\,dS,\label{4.4.34'}
\end{gather}
combining \eqref{4.4.33}--\eqref{4.4.34'}, we obtain
\begin{equation} \label{4.4.35}
\int_{\Omega(u\geq 2M)}|u(t)|^{r_1}\,dx+\int_{\Gamma(v\geq
2M)}|v(t)|^{r_1}\,dS\leq 2^{r_1}\varepsilon.
\end{equation}
Repeating the same steps above, just taking $(u+M)_{-}^{r_1-1}$
instead of $(u-M)_{+}^{r_1-1}$, $(v+M)_{-}^{r_1-1}$ instead of
$(v-M)_{+}^{r_1-1}$, we deduce that
\begin{equation} \label{4.4.36}
\int_{\Omega(u\leq -2M)}|u(t)|^{r_1}\,dx
+\int_{\Gamma(v\leq -2M)}|v(t)|^{r_1}\,dS\leq 2^{r_1}\varepsilon.
\end{equation}
Combining \eqref{4.4.35}--\eqref{4.4.36}, we  obtain
\begin{equation} \label{4.4.37}
\int_{\Omega(|u(t)|\geq 2M)}|u(t)|^{r_1}\,dx+\int_{\Gamma(|v(t)|\geq
2M)}|v(t)|^{r_1}\,dS\leq 2^{r_1}\varepsilon.
\end{equation}
\end{proof}


Now we state the existence and structure of a uniform attractor in
$L^{r_1}(\Omega)\times L^{r}(\Gamma)$ ($r=\min(r_1,r_2)$).

\begin{theorem} \label{t3.9}
Assume that $h(t)$ is normal in $L_{\rm loc}^2(\mathbb{R};L^2(\Omega))$,
$f$ and $g$ satisfy \eqref{1.1.2}--\eqref{1.1.4}. Then the family of
processes $\{U_{\sigma}(t,\tau)\}, \sigma\in\Sigma$ corresponding to
\eqref{4.4.1} has a compact uniform (with respect to $\sigma\in\Sigma$)
attractor $\mathscr{A}_{\Sigma}$ in $L^{r_1}(\Omega)\times
L^{r}(\Gamma)$ ($r=\min(r_1,r_2)$) and $\mathscr{A}_{\Sigma}$
satisfies
\[
\mathscr{A}_{\Sigma}=\omega_{0,\Sigma}(B_0)=\cup_{\sigma\in
\Sigma}\mathcal{K}_{\sigma}(s),\quad \forall s\in \mathbb{R},
\]
where $\mathcal{K}_{\sigma}(s)$ is the section at $t=s$ of the
kernel $\mathcal{K}_{\sigma}$ of the process
$\{U_{\sigma}(t,\tau)\}$ with symbol $\sigma$.
\end{theorem}

\begin{proof}
From Corollary \ref{t3.3} and Theorem \ref{t3.4}, it is easy
to verify that $\{U_{\sigma}(t,\tau)\}, \sigma\in\Sigma$ has
uniformly asymptotic compactness in $L^{r_1}(\Omega)\times
L^{r_1}(\Gamma)$, which combining with Theorem \ref{t3.5a}, we can
 obtain the existence of a compactly uniform attractor in
$L^{r_1}(\Omega)\times L^{r}(\Gamma)$ ($r=\min(r_1,r_2)$). Then,
similar to \cite{PZ,Y1}, we can obtain the structure of
$\mathscr{A}_{\Sigma}$, see more details in \cite{PZ,Y1}.
\end{proof}

\subsection*{Acknowledgments}
This work is partly supported by the NNSF of China \\
(11101404, 11201204, 11471148)
 and by the State Scholarship Fund of China Scholarship Council (201308620021).


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\end{document}