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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 76, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/76\hfil Blowup and asymptotic stability]
{Blowup and asymptotic stability of weak solutions to wave
equations with nonlinear degenerate damping and source terms}

\author[Q. Y. Hu, H. W. Zhang\hfil EJDE-2007/76\hfilneg]
{Qingying Hu, Hongwei Zhang}  % in alphabetical order


\thanks{Submitted February 27, 2007. Published May 22, 2007.}
\subjclass[2000]{35B40}
\keywords{Wave equation; degenerate damping and source terms;
\hfill\break\indent
asymptotic stability; blow up of solutions}

\begin{abstract}
 This article  concerns the blow-up and asymptotic
 stability of weak solutions to the wave equation 
 $$
 u_{tt}-\Delta u  +|u|^kj'(u_t)=|u|^{p-1}u
 \quad \text{in }\Omega \times (0,T),
 $$
 where $p>1$ and $j'$  denotes the  derivative of a $C^1$ convex 
 and real value function $j$.
 We prove that every weak solution  is asymptotically stability,
 for every $m$ such that $0<m<1$, $p<k+m$ and the the initial energy is
 small; the solutions blows up in finite time, whenever $p>k+m$ and
 the initial data is positive, but  appropriately bounded.
\end{abstract}

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

\section{Introduction}

In this article we study the  initial boundary value
problem
\begin{gather}
u_{tt}-\Delta u+|u|^kj'(u_t)=|u|^{p-1}u, \quad\text{in }\Omega\times (0,T), \label{1.1}\\
u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad\text{in }\Omega,\label{1.2}\\
u(x,t)=0, \quad\text{on }\Gamma\times (0,T),
\label{1.3}
\end{gather}
where $\Omega$ is a bounded domain in $R^n$ with smooth boundary
$\Gamma $ and $j(s)$ is a $C^1$ convex real function defined on
$R$, and $j'$ denotes the derivative of $j$ \cite{BLR1}.
Furthermore, the following assumptions on the convex function $j$
and the parameters $k,m,p$ are imposed throughout the paper.


\subsection*{Assumptions} \quad
\begin{itemize}
\item[(A1)] $k,m,p>0$, and $k<\frac {n}{n-2}$, $p+1<\frac {2n}{n-2}$
if $n \ge 3$;

\item[(A2)] There exist positive constants $C,C_0,C_1$ such that for all
$s,v \in R$, $j(s) \ge C|s|^{m+1}$, $|j'(s)|\le C_0|s|^m$,
$(j'(s)-j'(v))(s-v)\ge C_1|s-v|^{m+1}$.
\end{itemize}

The partial differential  equation \eqref{1.1} is a special case of the
prototype evolution equation
\begin{equation}\label{1.4}
 u_{tt}-\Delta u+Q(x,t,u,u_t)=f(x,u),
 \end{equation}
where  the nonlinearities satisfy the structural
conditions
$vQ(x,t,u,v)\geq0$,
$$
Q(x,t,u,0)=f(x,0)=0
$$ and
$f(x,u)\sim |u|^{p-1}u$ for large $|u|$. Various special cases of \eqref{1.4}
 arise in many contexts, for instance, in classical mechanics, fluid
dynamics, quantum field theory, see \cite{Jo} and \cite{Se}.


A special case of \eqref{1.1}, is the following well known
polynomially damped wave equation studied extensively in the
literature(see for instance \cite{PR,RS}),
\begin{equation}\label{1.5}
 u_{tt}-\Delta u+|u|^k|u_t|^{m-1}u_t=|u|^{p-1}u.
\end{equation}
Indeed, by taking $j(s)=\frac {1}{m+1}|s|^{m+1}$ we easily verify
that Assumption (A1) and (A2) is satisfied. It is easy to see in
this case that equation \eqref{1.1} is equivalent to \eqref{1.5}.
It is worth noting that there has been an extensive body of work
on the global existence and nonexistence for the equation
\eqref{1.1} with $k=0$, see, for example \cite{BV}-\cite{LRS},
\cite{PS1}-\cite{PuSe3},\cite{Se,BLR2} and the references therein.
One of the pioneering papers in this area was by Lions and Strauss
\cite{LS}. We also note here the work of Georgiev and Todorova
\cite{GT} and Levine and Serrin \cite{LeSe}.


The situation, however, is different when the damping is
degenerate. From the applications point of view degenerate problem
of this type arise quite often in specific physical contexts: for
example when the friction is modulated by the strain. However,
from the mathematical point of view this leads to that some
standard arguments to establish the existence of solutions to
problem \eqref{1.1}-\eqref{1.3} is not applicable. These difficulties makes
the problem interesting and the analysis more subtle. The problem
with degenerate damping has been first addressed in Levine and
Serrin \cite{LeSe}, where the global nonexistence of solutions was
shown for the case $k+m<p$ under several other restrictions
imposed on the parameters $n, k,m,p$ and the negative initial
energy. However Levine and Serrin \cite{LeSe} provide only
negative results (blow up of solutions in finite time if initial
energy is negative) without any assurance that a relevant local
solutions does indeed exist. Pitts and Rammaha \cite{PR}
established local and global (when $m+k \ge p$) existence and
uniqueness for the case of sub-linear damping, i.e., $m<1$. In
addition, the blow up of solutions (when $m+k<p$ and the negative
initial energy) is also proved in \cite{PR} for the relevant class
of solutions. Barbu, Lasiecka and Rammaha \cite{BLR1,BLR2}
introduced the suitable concept of solution, provided results on
the existence and uniqueness of various types of solutions such as
generalized solutions, weak solutions, and strong solutions to
\eqref{1.1}-\eqref{1.3}. In \cite{BLR2,BLR3,PR,RS} blow up of the weak or
generalized solution was shown if $p>m+k$ and the initial energy
is negative. The negativity of the initial energy was used to
prove blow up in the above paper \cite{BLR2,BLR3,PR,RS}. However,
the blow up of the solutions for \eqref{1.1}-\eqref{1.3} in case of positive
initial energy has not been discussed, and the asymptotic behavior
of the solutions for \eqref{1.1}-\eqref{1.3} is much less understood. In this
paper, following the ideas of  ``potential well" theory introduced
by Payne and Sattinger \cite{PS1}, we extend the results about
asymptotic stability and blowup of the solution to \eqref{1.1}-\eqref{1.3}
with $k=0$ (see, for example, \cite{BV,PS2,Vi1,Vi2,ZC} to the
problem \eqref{1.1}-\eqref{1.3} with $k>0$.


It is worth mentioning here that Levine, Park and Serrin [9]
studied the existence and nonexistence of the solution to the
quasilinear evolution equation of formally parabolic type, namely
\begin{equation} \label{1.6}
 Q(t,u,u_t)+A(t,u)=f(t,u).
\end{equation}

The purpose of the paper is, first, to show that the weak solution
of the problem \eqref{1.1}-\eqref{1.3} blow up in the case of positive initial
energy $E(0)>0$ and $p>k+m$, which we do in section 3. The another
purpose of this paper is to give an asymptotic stability results
of the problem \eqref{1.1}-\eqref{1.3} with $0<m<1$, $p<k+m$, which do in
section 4.

The following notation will be used in the sequel:
\begin{gather*}
|u|_{s,\Omega}\equiv
\|u\|_{H^s(\Omega)}, \quad \|u\|_p\equiv \|u\|_{L^p(\Omega)},\\
\|u\|\equiv \|u\|_{L^2(\Omega)},(u,v)=\int_{\Omega}u(x)v(x)dx,
p^*=\frac{2n}{n-2},
\end{gather*}
where $H^s(\Omega)$ and $L^p(\Omega)$ stands for the classical
Sobolev spaces and the Lebesgue spaces, respectively.

\section{Preliminaries}\label{Pre}

In this section we introduce some notations, definitions and some
known results which are necessary for the remaining sections of
the paper.


\begin{definition}[\cite{BLR1,BLR2,BLR3}] \label{def2.1} \rm
We say that $u$ is a
weak solution to the problem \eqref{1.1}-\eqref{1.3} on $[0,T]$ if $u\in
C_w([0,T];H^1_0(\Omega))\cap C^1_w([0,T],L^2(\Omega)),\Delta
u-u_{tt}\in L^2(\Omega \times (0,T)),|u|^kj(u_t)\in L^2(\Omega
\times (0,T))$ which satisfies $u(0)=u_0$, $u_t(0)=u_1$ and for all
$0<t\le T$ the following variational equality holds
\begin{align*}
& \int_0^t\int_\Omega(-u_t(s)v_t(s)+\nabla
u\nabla v)\,dx\,ds- \int_\Omega u_1v(0)dx\\
&+\int_0^t\int_\Omega |u(s)|^k j'(u_t)(s)v(s)\,dx\,ds\\
&=\int_0^t\int_\Omega|u(s)|^{p-1}u(s)v(s)\,dx\,ds
\end{align*}
for all test functions $v$ satisfying  $v\in H^1(0,T;L^2(\Omega))\cap
L^2(0,T;H_0^1(\Omega)),v(T)=0$.
\end{definition}

\begin{theorem}[\cite{BLR2}] \label{thm2.2}
In addition to Assumption (A1)  and (A2) and $p\le
\max\{\frac{p^*}{2},\frac {p^*m+k}{m+1}\}$; $m<1$ if $n=1,2$;
$\frac{k}{p^*}+\frac{m}{2}\le\frac{1}{2}$ if $n\ge 3$. Let $u_0\in
H_0^1(\Omega),u_1\in L^2(\Omega)$, then there exists a constant
$T>0$ such that the initial boundary problem
\eqref{1.1}-\eqref{1.3} has a unique weak solution on $[0,T]$ if
$p\le k+m$.
\end{theorem}

Now, we define the energy associated with problem \eqref{1.1}-\eqref{1.3} by
$$
E(t)=\frac{1}{2}\|u_t(t)\|^2+\frac{1}{2}\|\nabla
u(t)\|^2-\frac{1}{p+1}\|u(t)\|_{p+1}^{p+1}.
$$
We see that the
energy has the so-called energy identity
$$
E(t)+\int_0^t\int_{\Omega}|u(s)|^kj(u_t)(s)ds=E(0),
$$
where
$$
E(0)=\frac{1}{2}\|u_1\|^2+\frac{1}{2}\|\nabla
u_0\|^2-\frac{1}{p+1}\|u_0\|_{p+1}^{p+1}.
$$
It is clear that
\begin{equation}
E'(t)=-\int_{\Omega}|u(s)|^kj(u_t)(s)ds\leq 0 \label{2.1}
\end{equation}
and
$E(t)$ is a non-increasing function in time, then
\begin{equation}
E(t)\geq E(0).\label{2.2}
\end{equation}
Finally, we set
\begin{gather*}
\lambda_1=B_1^{-\frac{2}{p-1}},\quad
E_1=(\frac{1}{2}-\frac{1}{p+1})\lambda_1^{p+1},
\\
\lambda_2=(\frac{1}{(p+1)B_1^2})^{\frac{1}{p-1}},\quad
E_2=\frac{p+1}{2}(\frac{1}{2}-\frac{1}{p+1})\lambda_2^{p+1},
\\
{\sum }_{1}=\{(\lambda,E)\in R^2,\lambda>\lambda_1,0<E<E_1\}, \\
{\sum} _{2}=\{(\lambda,E)\in R^2,0\leq\lambda<\lambda_2,0<E<E_2\},
\end{gather*}
where $B_1$ is the embedding constant (where $H_0^1(\Omega)$ is
embedded into $L^{p+1}(\Omega)$). We call $\sum_1$  the unstable
set, $\sum_2$  the stable set.

\section{Blow-up of the solutions}

In this section, we assume that $p>k+m$ and $u$ be a weak solution
to \eqref{1.1}-\eqref{1.3} on the interval $[0,T]$ in the sense of
Definition \ref{def2.1}.


\begin{lemma} \label{lem3.1}
Let $(\|u_0\|_{p+1},E(0))\in \sum_1$, then
 $E(t)\leq E_0$ for all $t\in [0,T]$, and there exist
 $\lambda_0>\lambda_1$ such that
 $\|u(t)\|_{p+1}\geq\lambda_0>\lambda_1$ for all $t\in [0,T]$.
\end{lemma}

The proof is similar to that of \cite[Lemma 1]{Vi1}, so we omit
it.


\begin{theorem} \label{thm3.2}
Let $(\|u_0\|_{p+1},E(0))\in \sum_1,~p>k+m$, and
$u$ be a weak solution to \eqref{1.1}-\eqref{1.3} on the interval $[0,T]$ in
the sense of Definition \ref{def2.1}, then $T$ is necessarily finite, i.e.
$u$
can not be continued for all $t>0$.
 \end{theorem}

\begin{proof} We argue by contradiction. Let
$F(t)=\|u(t)\|^2$, $H(t)=E_1-E(t)$.
 From \eqref{2.1}, we have
\begin{equation}
H'(t)=-E'(t)=\int_{\Omega}|u(t)|^kj(u_t)(t)dx\geq 0.\label{3.1}
\end{equation}
 Therefore, $H(t)$ is an increasing function, then
\begin{equation}\label{3.2}
H(t)\geq H(0)=E_1-E(0)>0,  \quad t\geq 0.
\end{equation}
Next, by the definition of $E(t)$ and Lemma \ref{lem3.1},
\begin{equation}
\begin{aligned}
H(t)&\leq E_1-\frac{1}{2}\|\nabla
u(t)\|^2+\frac{1}{p+1}\|u(t)\|_{p+1}^{p+1}\\
&\leq
E_1-\frac{1}{2}B^{-2}_1\lambda_1^2+\frac{1}{p+1}\|u(t)\|_{p+1}^{p+1},
\quad t\geq 0.
\end{aligned}\label{3.3}
\end{equation}
Hence, since $E_1-\frac{1}{2}B^{-2}_1\lambda_1^2=
-\frac{1}{p+1}\lambda_1^{p+1}<0$, we have
\begin{equation}\label{3.4}
0<H(0)\leq H(t)\leq \frac{1}{p+1}\|u(t)\|_{p+1}^{p+1}, \quad t\geq 0.
\end{equation}
 For simplicity, we denote
$$
I(t)=\int_{\Omega}|u(t)|^ku(t)j'(u_t)(t)dx.
$$
By the definition of the solution and the definition of $H(t)$,
\begin{align*}
\frac{1}{2}F''(t)
&=\frac{d}{dt}\int_{\Omega}u(t)u_t(t)dx=\|u_t(t)\|^2-\|\nabla
u(t)\|^2+\|u(t)\|_{p+1}^{p+1}-I(t)\\
&=2\|u_t(t)\|^2+(1-\frac{2}{p+1})\|u(t)\|_{p+1}^{p+1}-2E(t)-I(t)\\
&= 2\|u_t(t)\|^2+(1-\frac{2}{p+1})\|u(t)\|_{p+1}^{p+1}+2H(t)-2E_1-I(t).
\end{align*}
By Lemma \ref{lem3.1} again (i.e
$\|u(t)\|_{p+1}^{p+1}\lambda_0^{-(p+1)}>1$, or
$E_1\|u(t)\|_{p+1}^{p+1}\lambda_0^{-(p+1)}>E_1$),
\begin{equation}
\begin{aligned}
\frac{1}{2}F''(t)&\geq
2\|u_t(t)\|^2+(1-\frac{2}{p+1}-2E_1\lambda_0^{-(p+1)})\|u(t)\|_{p+1}^{p+1}+2H(t)-I(t)\\
&=2\|u_t(t)\|^2+C_2\|u(t)\|_{p+1}^{p+1}+2H(t)-I(t),
\end{aligned}\label{3.5}
\end{equation}
where $C_2=1-\frac{2}{p+1}-2E_1\lambda_0^{-(p+1)}>0$, because
$\lambda_0>\lambda_1$ by Lemma \ref{lem3.1}.

 Now, to estimate the last term $I(t)$ in
\eqref{3.5}, since $p>k+m$ and Assumption (A1)  and (A2) and by
applying Holder's inequality and Young's inequality, we obtain
\begin{equation}
\begin{aligned}
|I(t)|&\leq
C_0\int_{\Omega}|u(t)|^{k+1-\frac{k+m+1}{m+1}}|u(t)|^\frac{k+m+1}{m+1}|u_t(t)|^mdx\\
&\leq C_0
(\int_{\Omega}|u(t)|^k|u_t(t)|^{m+1}dx)^\frac{m}{m+1}(\int_{\Omega}|u(t)|^{k+m+1}dx)^\frac{1}{m+1}\\
&\leq
C_0B_0(H'(t))^\frac{m}{m+1}\|u(t)\|^\frac{k+m+1}{m+1}_{p+1}\\
&\leq
C_0B_0(\frac{1}{\delta}H'(t)+\delta^m\|u(t)\|^{k+m+1}_{p+1}),
\end{aligned}\label{3.6}
\end{equation}
where $\delta$ is a constant to be chosen later, $B_0$ is the
embedding constants from $L^{k+m+1}(\Omega)$ to
$L^{p+1}(\Omega)$(since $k+m<p$).

Now, we introduce the  auxiliary function
$$
y(t)=H^{1-\alpha}(t)+\epsilon F'(t),
$$
where $\epsilon$ is a small positive constant to be fixed later,
and $\alpha=\min\{\frac{p-(k+m)}{m(p+1)},\frac{p-1}{2(p+1)}\}$.
Clearly, $0<\alpha<\frac{1}{2}$. Therefore, \eqref{3.5}, \eqref{3.6} yield
\begin{align*}
 y'(t)&=(1-\alpha)H^{-\alpha}(t)H'(t)+\epsilon F''(t)\\
&\geq
(1-\alpha)H^{-\alpha}(t)H'(t)+4\epsilon\|u_t(t)\|^2+4\epsilon
H(t)+2\epsilon C_2\|u(t)\|^{p+1}_{p+1}-2\epsilon I(t)\\
&\geq [(1-\alpha)H^{-\alpha}(t)-\frac{2\epsilon
C_0B_0}{\delta}]H'(t)+4\epsilon\|u_t(t)\|^2+4\epsilon H(t)\\
&\quad +2\epsilon
C_2\|u(t)\|^{p+1}_{p+1}-2C_0B_0\epsilon\delta^m\|u(t)\|^{k+m+1}_{p+1}.
\end{align*}
Choosing
$\delta=(\frac{C_2}{2C_0B_0}\|u(t)\|_{p+1}^{p-k-m})^\frac{1}{m}$,
then
$$
C_2\epsilon\|u(t)\|^{p+1}_{p+1}-2C_0B_0\epsilon\delta^m\|u(t)\|^{k+m+1}_{p+1}=0.
$$
Therefore,
\begin{equation}
y'(t)\geq
[(1-\alpha)H^{-\alpha}(t)-\frac{2\epsilon
C_0B_0}{\delta}]H'(t)+4\epsilon\|u_t(t)\|^2+4\epsilon
H(t)+\epsilon
 C_2\|u(t)\|^{p+1}_{p+1}.\label{3.7}
 \end{equation}
  By \eqref{3.4} and the choice
$\delta$, then
\begin{equation}
\begin{aligned}
&(1-\alpha)H^{-\alpha}(t)-\frac{2\epsilon
C_0B_0}{\delta}=H^{-\alpha}(t)[1-\alpha-\frac{2\epsilon
C_0B_0}{\delta}H^{\alpha}(t)]\\
&\geq H^{-\alpha}(t)[1-\alpha-2^{1+\frac{1}{m}}\epsilon
(C_0B_0)^{1+\frac{1}{m}}C_2^{-\frac{1}{m}}(\frac{1}{p+1})^\alpha\|u(t)\|_{p+1}^\frac{k+m-p+\alpha
m(p+1)}{m}].
\end{aligned} \label{3.8}
\end{equation}
 Furthermore, since
$\|u(t)\|_{p+1}\geq[(p+1)H(0)]^\frac{1}{p+1}$ by \eqref{3.4} and
$\alpha$ was chosen so that $k+m-p+\alpha m(p+1)\leq 0$, it
follows from \eqref{3.8} that
\begin{equation}
\begin{aligned}
&(1-\alpha)H^{-\alpha}(t)-\frac{2\epsilon
C_0B_0}{\delta}\\
\geq&H^{-\alpha}(t)[1-\alpha-2^{1+\frac{1}{m}}\epsilon
(C_0B_0)^{1+\frac{1}{m}}C_2^{-\frac{1}{m}}(\frac{1}{p+1})^
\frac{p-k+m}{m(p+1)}(H(0))^{\alpha+\frac{k+m-p}{m(p+1)}}].
 \end{aligned} \label{3.9}
\end{equation}
We choose $\epsilon$ sufficiently small such that
\begin{equation}
1-\alpha-2^{1+\frac{1}{m}}\epsilon
(C_0B_0)^{1+\frac{1}{m}}C_2^{-\frac{1}{m}}(\frac{1}{p+1})^\frac{p-k+m}{m(p+1)}(H(0))^{\alpha+\frac{k+m-p}{m(p+1}}\geq
0.\label{3.10}
\end{equation}
 Therefore, \eqref{3.8}-\eqref{3.10} yield
\begin{equation}
(1-\alpha)H^{-\alpha}(t)-\frac{2\epsilon C_0B_0}{\delta}\geq
0. \label{3.11}
\end{equation}
Thus, by \eqref{3.11} and \eqref{3.7}, we
obtain
\begin{equation}
y'(t)\geq \epsilon
C_3[H(t)+\|u_t(t)\|^2+\|u(t)\|_{p+1}^{p+1}],\label{3.12}
\end{equation}
where $C_3>0$
is a constant which does not depended on $\epsilon$. In particular,
\eqref{3.12} shows that $y(t)$ is increasing on $(0,T)$, with
\begin{equation*}
y(t)=H^{1-\alpha}(t)+\epsilon F'(t)\geq
H^{1-\alpha}(0)+\epsilon F'(0).
\end{equation*}
We further choose
$\epsilon$ sufficiently small such that $y(0)>0$, so  $y(t)\geq
y(0)>0$ for $t\geq 0$.

 Now, let $r=\frac{1}{1-\alpha}$. Since
$0<\alpha<\frac{1}{2}$, it is evident that $r>1$. Using Young's
inequality again
\begin{equation}
\begin{aligned}
y^r(t)&\leq 2^{r-1}(H(t)+\epsilon
\|u(t)\|^r\|u_t(t)\|^r)\\
&\leq
C_4(H(t)+\|u_t(t)\|^2+\|u(t)\|^{\frac{1}{\frac{1}{2}-\alpha}}).\label{3.13}
\end{aligned}
\end{equation}
By the choice of $\alpha$, we have
$\frac{1}{2}-\alpha>\frac{1}{p+1}$. Now apply the inequality
\begin{equation*}
x^\sigma \leq (1+\frac{1}{a})(a+x),\quad x\geq 0,\quad 0\leq
\sigma \leq 1,\quad a>0,
\end{equation*}
 and take
$x=\|u(t)\|^{p+1},\sigma=\frac{1}{(\frac{1}{2}-\alpha)(p+1)}<1,a=H(0)$,
and $d=1+\frac{1}{H(0)}$, we obtain
\begin{equation}
\|u(t)\|^\frac{1}{\frac{1}{2}-\alpha}\leq
d(H(0)+\|u(t)\|^{p+1})\leq
C_5(H(t)+\|u(t)\|_{p+1}^{p+1}).
\label{3.14}
\end{equation}
Hence, from \eqref{3.13} and \eqref{3.14} there results
\begin{equation}
y^r(t)\le C(H(t)+\|u_t(t)\|^2+\|u(t)\|_{p+1}^{p+1}).\label{3.15}
\end{equation}
Thus, \eqref{3.12} and \eqref{3.15} show that
\[
y'(t)\geq C_6y^r(t),\quad t\in [0,T].
\]
 Finally,
 from this inequality and $r=\frac{1}{1-\alpha}>1$, we see that
$y(t)=H^{1-\alpha}(t)+\epsilon F'(t)$ blow up in finite time. This
completes the proof.
\end{proof}

\section{Asymptotic stability  of the solutions}

To obtain the asymptotic stability of the solution, we start
with a series of lemmas. The assumption of Theorem \ref{thm2.2} will
be valid throughout this section.

\begin{lemma} \label{lem4.1}
If $(\|u_0\|_{p+1},E(0))\in \sum_2$, then
\begin{equation}
(\|u(t)\|_{p+1},E(t))\in {\sum}_2,\quad  t\geq 0. \label{4.1}
\end{equation}
Moreover
\begin{equation}
E(t)\geq\frac{1}{2}\|u_t(t)\|^2+\frac{1}{4}\|\nabla
u(t)\|^2, \quad\quad t\geq 0. \label{4.2}
\end{equation}
\end{lemma}

\begin{proof}  By \eqref{2.2} and
the embedding theorem, for all $t\geq 0$, there holds
\begin{equation}
\begin{aligned}
E_2>E(0) & \geq E(t)\geq
\frac{1}{2}\|u_t(t)\|^2+\frac{1}{4}\|\nabla
u(t)\|^2+\frac{1}{4}B_1^{-2}\|u(t)\|^2_{p+1}-\frac{1}{2}\|u(t)\|_{p+1}^{p+1}\\
&\geq \frac{1}{2}\|u_t(t)\|^2+\frac{1}{4}\|\nabla
u(t)\|^2+g(\|u(t)\|_{p+1}),
\end{aligned} \label{4.3}
\end{equation}
 where
$g(\lambda)=\frac{1}{4}B_1^{-2}\lambda^2-\frac{1}{2}\lambda^{p+1}$,
for $\lambda\geq 0$. It is easy to see that $g(\lambda)$ attains
its maximum $E_2$ for $\lambda=\lambda_2,\quad g(\lambda)$ is
strictly decreasing for $\lambda\geq \lambda_2$ and
$g(\lambda)\to {-\infty}$ as $\lambda\to\infty$.
By the continuity of $\|u(t)\|_{p+1}$ and
$\lambda(0)=\|u_0\|_{p+1}<\lambda_2$, so that
$\lambda(t)<\lambda_2$ for all $t\geq 0$. Also, of course,
$E(t)<E_2$ by \eqref{4.3}. Then, \eqref{4.1} holds. To obtain \eqref{4.2}, it
remains to note that $g(\lambda)\geq 0$ whenever
$0\leq\lambda<\lambda_2$.
Then \eqref{4.2} follows at once.
\end{proof}


\begin{lemma} \label{lem4.2}
If $(\|u_0\|_{p+1},E(0))\in\sum_2$, then
$\|\nabla u(t)\|^2\geq 2\|u(t)\|^{p+1}_{p+1}$,
or
\begin{equation}
\|\nabla u(t)\|^2-\|u(t)\|^{p+1}_{p+1}\geq\frac{1}{2}\|\nabla
u(t)\|^2. \label{4.4}
\end{equation}
\end{lemma}

\begin{proof}  By the
embedding theorem
 \begin{align*}
 \frac{1}{2}\|\nabla
u(t)\|^2-\frac{1}{2}\|u(t)\|^{p+1}_{p+1}&\geq\frac{1}{4}\|\nabla
u(t)\|^2+\frac{1}{4}B_1^{-2}\|u(t)\|^2_{p+1}-\frac{1}{2}\|u(t)\|^{p+1}_{p+1}\\
&=\frac{1}{4}\|\nabla
u(t)\|^2+g(\|u(t)\|_{p+1}).
\end{align*}
Hence \eqref{4.4} is true, since $g(\lambda)\geq 0$, if
$0\leq\lambda<\lambda_2$ and $0\leq \|u(t)\|_{p+1}<\lambda_2$ by
Lemma \ref{lem4.1}.
\end{proof}

\begin{lemma} \label{lem4.3} If  $(\|u_0\|_{p+1},E(0))\in\sum_2$, then
\begin{enumerate}
\item  $ \|u_t(t)\|\in L^2(0,\infty),H'(t)\in L^1(0,\infty)$
\item $\|\nabla u(t)\|,\|u(t)\|_{p+1},\|u_t(t)\|\leq C $.
\end{enumerate}
\end{lemma}

\begin{proof} The first result in (1) follows the
definition of weak solution.  The second result in (1) follows by
$H'(t)=-E'(t)$, since $E(t)\geq 0$ for $t\geq 0$ and $H(t)\in
AC(0,\infty)$, while (2)
follows \eqref{4.2} (or\eqref{4.3}) and \eqref{4.4}.
\end{proof}

\begin{lemma} \label{lem4.4}
Let $(\|u_0\|_{p+1},E(0))\in\sum_2$ and
$E(t)\geq\beta$, where $\beta>0$ is a constant, then there exists
$\alpha=\alpha(\beta)>0$ such that
\begin{equation}
\|u_t(t)\|^2+\|\nabla
u(t)\|^2-\|u(t)\|^{p+1}_{p+1}\geq\alpha,  \quad t\geq 0.\label{4.5}
\end{equation}
\end{lemma}

\begin{proof}
By the definition of $E(t)$ and $E(t)\geq\beta$,
we have
\begin{equation}
\|u_t(t)\|^2+\|\nabla u(t)\|^2\geq 2\beta,\quad t\geq 0. \label{4.6}
\end{equation}
Now suppose that \eqref{4.5} does not hold. From \eqref{4.4}, there is a
sequences ${t_n}\subset R^+$ such that
\begin{equation*}
\|u_t(t_n)\|^2+\|\nabla
u(t_n)\|^2-\|u(t_n)\|_{p+1}^{p+1}\geq
\|u_t(t_n)\|^2+\frac{1}{2}\|\nabla u(t_n)\|^2\to 0,(n\to\infty).
\end{equation*}
Then, we get
\begin{equation*}
\|u_t(t_n)\|^2\to 0, \quad \|\nabla
u(t_n)\|^2\to 0,\quad \text{as }n\to\infty.
\end{equation*}
This is contradiction with \eqref{4.6}. The
lemma is proved.
\end{proof}

\begin{theorem} \label{thm4.5}
Assume the conditions of Theorem \ref{thm2.2}, that  $(\|u_0\|_{p+1},E(0))\in\sum_2$,
and that $u$ is a weak solution to \eqref{1.1}-\eqref{1.3}. Then
\begin{equation}
\lim_{t\to \infty}E(t)=0 , \lim_{t\to\infty}\|\nabla
u(t)\|=0.\label{4.7}
\end{equation}
\end{theorem}

\begin{proof}
  Suppose that \eqref{4.7} fails, then there exists
$\beta>0$ such that $E(t)\geq\beta$ for all $t\geq 0$ since \eqref{2.2}
and $E(t)\geq 0$.
 Multiplying both sides of \eqref{1.1} by $u$, integrating over
$[T,t]\times \Omega$
 $(0<T\leq t<\infty)$ and integrating by parts with respect to
 $t$, we obtain
\begin{equation}
\begin{aligned}
(u_t(s),u(s))|_{s=T}^{t}
&=\int_T^t[2\|u_t(s)\|^2-(\|u_t(s)\|^2+\|\nabla
u(s)\|^2-\|u(s)\|_{p+1}^{p+1})\\
&\quad -\int_\Omega |u(s)|^k u(s)j'(u_t)(s)dx]ds \\
&=\int_T^t(I_1+I_2+I_3)ds.
\end{aligned} \label{4.8}
\end{equation}
 By \eqref{4.2}, \eqref{2.2}
and Lemma \ref{lem4.3} (1), we have
\begin{equation}
\int_T^tI_1ds=\int_T^t2\|u_t(s)\|^2ds
\leq 4E^\frac{1}{2}(0)(\int_T^t\|u_t(s)\|^2ds)^\frac{1}{2}
(\int_T^tds)^\frac{1}{2}
\leq C_7(\int_T^tds)^\frac{1}{2}.
 \end{equation} \label{4.9}
 Here and in the following, $C_i$ denotes a positive constant which do not
depend on $t$ and $T$. By Lemma \ref{lem4.4}
\begin{equation}
\int_T^tI_2ds=-\int_T^t{(\|u_t(s)\|^2+\|\nabla
u(s)\|^2-\|u(s)\|_{p+1}^{p+1})}ds\leq
-\alpha\int_T^tds.\label{4.10}
\end{equation}
By Holder inequality, Lemma \ref{lem4.3} (1), Lemma \ref{lem4.3} (2) and embedding
theorem, we have
\begin{equation}
\begin{aligned}
\int_T^tI_3dt
&=-\int_T^t\int_\Omega |u(s)|^k u(s)j'(u_t)(s)\,dx\,ds\\
&\leq
\int_T^t\int_{\Omega}|u(s)|^{k+1-\frac{k+m+1}{m+1}}|u(s)|^\frac{k+m+1}{m+1}|u_t(s)|^m\,dx\,ds\\
&\leq
(\int_T^t\int_{\Omega}|u(s)|^kj(u_t)(s)\,dx\,ds)^\frac{m}{m+1}(\int_T^t\int_{\Omega}|u(s)|^{k+m+1}\,dx\,ds)^\frac{1}{m+1} \\
&\leq
C_8(\int_T^tH'(s)ds)^\frac{m}{m+1}(\int_T^t\|u(s)\|^{k+m+1}_{k+m+1}ds)^\frac{1}{m+1}\\
&\leq C_9(\int_T^t\|\nabla
u(s)\|^{k+m+1}ds)^\frac{1}{m+1}\leq
C_{10}(\int_T^tds)^\frac{1}{m+1},
\end{aligned} \label{4.11}
\end{equation}
 here we have
used the embedding theorem from $H_0^1(\Omega)$ to
$L^{k+m}(\Omega)$ since $k+m+1<\frac{1-m}{2}p^*+m+1<p^*$. Then
from \eqref{4.9}-\eqref{4.11}, since $\frac{1}{m+1}>\frac{1}{2}$,
we know
\begin{equation}
(u_t(s),u(s))|_{s=T}^{t}\leq
C_{11}(\int_T^tds)^\frac{1}{m+1}-\alpha\int_T^tds. \label{4.12}
\end{equation}
 On the other hand, from Holder inequality and Lemma \ref{lem4.3} (2),
\begin{equation*}
|(u_t(t),u(t))|\leq C_{12}(\|u_t(t)\|^2+\|\nabla u(t)\|^2)<\infty.
\end{equation*}
 In turn, we reach a contradiction
with \eqref{4.12} for fixing $T$ when $t\to\infty$. Hence, we
derive $ \lim_{t\to\infty}E(t)=0$ and
$\lim_{t\to\infty}\|\nabla u(t)\|^2=0$ by
\eqref{4.2}. This completes the proof.
\end{proof}

\begin{remark} \label{rmk4.1}\rm
 The set $\sum_2$ is called stable set. It is
smaller than the potential well introduced by Payne and
Sattinger\cite{PS1}. Moreover the value $\lambda_2$ in this paper
can be chosen larger than now but $\lambda_2<\lambda_1$.
\end{remark}

\begin{remark} \label{rmk4.2} \rm
 The method seems general enough to apply to the
 generate equation \eqref{1.4} with $f(x,u)$ being source
term and also let $Q$ and $F$ depending on time but this will be
discuss in a future paper.
\end{remark}

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Qingying Hu \newline
Department of Mathematics, Henan University of  Technology\\
Zhengzhou 450052, China\\
email address: slxhqy@yahoo.com.cn \smallskip

Hongwei Zhang \newline
Department of Mathematics, Henan University of  Technology\\
Zhengzhou 450052, China\\
email addres: wei661@yahoo.com.cn


\section*{Editors note: September 10, 2007}

 A reader informed us that that parts of the introduction were
 copied from reference [2], without giving the proper credit.
 Also that the first statement in Lemma 4.3 maybe false;
 so that Theorem 4.5 has not been proved. The
 authors agreed to post a new proof, if they succeed in proving 
 the lemma.
  
 Errata: Assumption (A1) should include $p>1$. 
 Inequality (2.2) should read $E(t)\leq E(0)$.


\end{document}