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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 139, 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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/139\hfil Hyperbolic hemivariational inequality]
{On asymptotic behavior of global solutions for hyperbolic
hemivariational inequalities}

\author[J. Y. Park, S. H. Park\hfil EJDE-2006/139\hfilneg]
{Jong Yeoul Park, Sun Hye Park}  % in alphabetical order

\address{Jong Yeoul Park \newline
Department of Mathematics,
 Pusan National University,
30 Changjeon-dong,
Keumjeong-ku,  Busan, 609-735, South Korea}
\email{jyepark@pusan.ac.kr}

\address{Sun Hye Park \newline
Department of Mathematics,
 Pusan National University,
30 Changjeon-dong,
Keumjeong-ku,  Busan, 609-735, South Korea}
\email{sh-park@pusan.ac.kr}

\date{}
\thanks{Submitted March 31, 2006. Published October 31, 2006.}
\thanks{Supported  by a research grant from Pusan National University}
\subjclass[2000]{35L85, 35B40}
\keywords{Weak solutions; asymptotic behavior; hemivariational ineuqlity}

\begin{abstract}
 In this paper we study the existence of global weak solutions
 for a  hyperbolic differential inclusion with a discontinuous
 and nonlinear multi-valued term. Also we investigate the
 asymptotic behavior of solutions.
\end{abstract}

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

\section{Introduction}

 The main purpose of this paper is to
investigate the initial boundary value problem for the hyperbolic
differential inclusion
\begin{gather}
u''+A^2u+M(\|A^{1/2}u\|^2)Au+\varphi(u')\ni 0
\quad  \text{in }   (0,\infty)\times \Omega,  \label{e1.1}\\
u(0)=u_0, u'(0)=u_1  \quad \text{in } \Omega, \label{e1.2}
\end{gather}
where $\varphi$ is a discontinuous and nonlinear set-valued mapping
by filling in jumps a function $b\in L^{\infty}_{\rm loc}(\mathbb{R})$. The
precise hypothesis on the above system will be given in the next
section.

Recently, a class of nonlinear Cauchy problems are
studied by many authors \cite{m1,m2,n1,p1,r2}
 Medeiros \cite{m2} studied the equation
\begin{equation*}
u''+A^2u+M(\|A^{1/2}u\|^2)Au=0,
\end{equation*}
where $A$ is a linear operator in a Hilbert space $H$ and $M$ is a
real function. Rivera \cite{r2} investigated the equation
\begin{equation} \label{e1.3}
u''+A^2u+M(\|A^{1/2}u\|^2)Au+g(u')=0,
\end{equation}
when the damping term is linear, i.e., $g(x)=\delta x$ and Patcheu
\cite{p1} studied the existence and asymptotic behavior of the solutions
of \eqref{e1.3} when $g$ is a nonlinear and nondecreasing continuous
functions. Motivated by works of Patcheu \cite{p1}, we consider more
generalized problem \eqref{e1.1} with a discontinuous and nonlinear
multi-valued term $\varphi$. Thus, in this paper we shall deal with
the existence and asymptotic behavior of the global weak solution of
the hemivariational inequality \eqref{e1.1}-\eqref{e1.2}. The background of these
variational problems are in physics, especially in continuum
mechanics, where nonmonotone, multi-valued constitutive laws lead to
the above-cited hemivariational inequalities.

 At this point it
is important to mention that such hemivariational inequalities were
studied by some authors \cite{m3,m4,r1}, but, in their works no decay rates
were obtained as in this present paper. The plan of this paper as
follows. In section 2, the assumptions and the main results are
given. In section 3, the existence of a solution to the
 \eqref{e1.1}-\eqref{e1.2}
is proved by using the Faedo-Galerkin method and finally in section
4, the decay of solutions is investigated.


\section{Assumptions and main results }

 First we explain the notation used
throughout this paper. Let $\Omega$ be a bounded domain in
$\mathbb{R}^n$ with a smooth boundary $\partial \Omega$ and
$Q=[0,T]\times\Omega$, where $T$ be any positive real number.
 Let $H=L^2(\Omega)$
with inner product and norm respectively denoted by $(\cdot, \cdot)$
and $\|\cdot\|$. Let $A$ be a linear operator in $H$, with domain
$D(A)=V$ dense in $H$ and the graph norm denoted by $\|\cdot\|_V$.
Let $V'$ be the dual of $V$ and let $\langle\cdot, \rangle$ denote the
duality pairing between $V'$ and $V$. We assume that the imbedding
$V\subset H$ is compact. Identifying $H$ and $H'$, it follows that
$V\subset H\subset V'$ and the imbedding $H\subset V'$ is also
compact.

The following hypothesis will be used throughout
this paper.

\begin{itemize}
\item[(H1)] $A$ is self-adjoint and positive, i.e., there is a
constant $\mu_0>0$ such that
\begin{equation} \label{e2.1}
(Av,v)\geq \mu_0\|v\|^2,  \quad \forall v \in V.
\end{equation}
Define $A^2 : V \to V'$ by
$$
\langle A^2u,v\rangle =(Au,Av),   \quad  u,v\in V
$$
with $W=D(A^2)=\{ u\in V | A^2u \in H \}$.

\item[(H2)] $M(s)$ is a $C^1$ real function and there exist
 $\alpha, \beta >0$ such that
\begin{equation} \label{e2.2}%2.3
M(s)\geq \alpha +\beta s, \quad M'(s)\geq 0.
\end{equation}
Let $\bar{M}(t)$ be defined by
$$
\bar{M}(t)=\int^t_0M(s)ds.
$$ %2.4

\item[(H3)]
(1) $b\in L^{\infty}_{\rm loc}(\mathbb{R})$. \
(2) There exist $\mu_1, \mu_2 >0$ such that $b(s)s \geq \mu_1 |s|^2$
and $|b(s)|\leq \mu_2|s|$ for all $s\in \mathbb{R}$.

\end{itemize}
The multi-valued function $\varphi : \mathbb{R} \to 2^{\mathbb{R}}$ is
obtained by filling in jumps of a function $b :
\mathbb{R}\to \mathbb{R}$ by means of the functions
$\underline{b}_{\epsilon}$, $\overline{b}_{\epsilon}$, $\underline{b}$,
$\overline{b}$ from $\mathbb{R}$ to $\mathbb{R}$ as follows:
\begin{gather*}
\underline{b}_{\epsilon}(t)=\mathop{\rm ess\,inf}_{|s-t|\leq \epsilon}   b(s),
 \quad
\overline{b}_{\epsilon}(t)=\mathop{\rm ess\,sup}_{|s-t|\leq \epsilon}   b(s) ;
 \\
\underline{b}(t)=\lim_{\epsilon \to 0^+}\underline{b}_{\epsilon}(t), \quad
\overline{b}(t)=\lim_{\epsilon \to 0^+}\overline{b}_{\epsilon}(t);  \quad
\varphi(t)=[\underline{b}(t),\overline{b}(t)].
\end{gather*}
We shall need a regularization of $b$ defined by
\begin{equation*}
b^m(t)=m\int^{\infty}_{-\infty}b(t-\tau)\rho(m\tau)d\tau,
\end{equation*}
where $\rho \in C_0^{\infty}((-1,1)), \rho \geq 0$ and
$\int^1_{-1}\rho(\tau)d\tau=1$.
It is easy to show that $b^m$ is continuous for all $m\in \mathbb{N}$ and
$\underline{b}_{\epsilon},\overline{b}_{\epsilon},\underline{b},
\overline{b}, b^m$ satisfy the same condition (H3)(2) with a possibly different
constant if $b$ satisfies (H3)(2).

Now we are in a position to state our existence result.

\begin{theorem} \label{thm2.1}
Assume that the conditions (H1)-(H3) hold. Then given $u_0\in W$ and
$u_1\in V$, there exist $\Xi \in L^{\infty}(0,T;H)$ and a function
$u : [0,T]\times \Omega \to \mathbb{R}$ such that
$$
u\in L^{\infty}(0,T;V), u' \in L^{\infty}(0,T;H), u'' \in
L^{\infty}(0,T;V')
$$
and
\begin{gather}
u''+A^2u+M(\|A^{1/2}u\|^2)Au+\Xi = 0   \quad\text{in }   L^{\infty}(0,T;V'),
\label{e2.5}\\
\Xi(t,x) \in \varphi(u'(t,x))   \quad\text{a.e. }   (t,x)\in Q, \label{e2.6}\\
u(0)=u_0, \quad  u'(0)=u_1. \label{e2.7}
\end{gather}
\end{theorem}

Next, we establish the decay result. Let us define the energy of the
system \eqref{e1.1}-\eqref{e1.2} as
\begin{equation} \label{e}
E(t)=\frac{1}{2}\big \{ \|u'(t)\|^2+\|Au(t)\|^2
+\bar{M}(\|A^{1/2}u(t)\|^2) \big \}.
\end{equation}
Then we have

\begin{theorem} \label{thm2.2}
Assume that the conditions of Theorem \ref{thm2.1} hold. Then given $u_0 \in
W$ and $u_1\in V$, there exist $\Xi \in L^{\infty}(0,T;H)$  and a
function $u : [0, \infty)\times \Omega \to \mathbb{R}$ such
that
\begin{gather*}
u\in L^{\infty}(0,\infty;V), \quad u' \in L^{\infty}(0,T;H),
\quad u'' \in L^{\infty}(0,T;V') ,   \\
u''+A^2u+M(\|A^{1/2}u\|^2)Au+\Xi = 0  \quad \text{in }
 L^{\infty}(0,\infty;V'), \\
\Xi(t,x) \in \varphi(u'(t,x)) \quad  \text{a.e. }
(t,x)\in (0,\infty) \times \Omega, \\
u(0)=u_0, \quad  u'(0)=u_1
\end{gather*}
and $u$ satisfies the decay property
\begin{equation} \label{e2.9}
E(t)\leq Ce^{-\gamma t}, \quad  \forall t\geq 0
\end{equation}
for some positive constants $C$ and $\gamma$.
\end{theorem}

\section{Proof of Theorem \ref{thm2.1}}

The proof will be done by applying the Faedo-Galerkin method.

\subsection*{Step 1 : A priori estimate I}
Assume, for simplicity,
$V=D(A)$ is separable, then there is a sequence $(w_j)_{j\geq 1}$
consisting of eigenfunctions of the operator $A$ corresponding to
positive real eigenvalues $\lambda_j$ tending to $\infty$. Hence
$Aw_j=\lambda_j w_j, j\geq 1$. Let us define $W_m=Span\{
w_1,w_2,\dots ,w_m \}$. Note
that $(w_j)_{j\geq 1}$ is a basis of $H, V$ and $W$. \\
Consider a regularized Galerkin equation
\begin{equation} \label{e3.1}
(u_m''(t)+A^2u_m(t)+M(\|A^{1/2}u_m(t)\|^2)Au_m(t)+b^m(u_m'(t)),v) = 0,
 \quad \forall v\in W_m
\end{equation}
with the initial conditions
 \begin{gather}
u_m(0)=u_{0m}=\sum^m_{j=1}(u_0,w_j)w_j, \quad  u_{0m}\to u_0 \quad  \text{in }
   W,  \label{e3.2}\\
u'_m(0)=u_{1m}=\sum^m_{j=1}(u_1,w_j)w_j,  \quad u_{1m}\to u_1  \quad \text{in }
  V. \label{e3.3}
\end{gather}
Substituting  $u_m(t)=\sum^m_{j=1}g_{mj}(t)w_j$ in \eqref{e3.1} gives a
second-order ordinary
differential equations and its local solution $g_{mj}(t)$ exists on
$[0, t_m), 0<t_m< T$.
Replacing $v$ by $u'_m(t)$ in \eqref{e3.1}, we find
\begin{equation} \label{e3.4}
\frac{1}{2}\frac{d}{dt}\big \{ \|u'_m(t)\|^2+\|Au_m(t)\|^2
+\bar{M}(\|A^{1/2}u_m(t)\|^2) \big \}=-(b^m(u'_m(t)),u'_m(t)).
\end{equation}
Integrating in $(0,t),   t\leq t_m$,
\begin{equation} \label{e3.5} \begin{aligned}
&\frac{1}{2}\big \{ \|u'_m(t)\|^2+\|Au_m(t)\|^2
+\bar{M}(\|A^{1/2}u_m(t)\|^2) \big \}   \\
&=\frac{1}{2}\big \{ \|u_{1m}\|^2+\|Au_{0m}\|^2
+\bar{M}(\|A^{1/2}u_{0m}\|^2) \big \}-\int^t_0(b^m(u'_m(s)),u'_m(s))ds.
\end{aligned}
\end{equation}
By (H3)(2) and H\"{o}lder inequality,
\begin{equation} \label{e3.6}
\begin{aligned}
\int^t_0(b^m(u'_m(s)),u'_m(s))ds
&\leq \big( \int^t_0\|b^m(u_m'(s))\|^2ds \big)^{1/2}
\big(\int^t_0\|u_m'(s)\|^2ds \big)^{1/2}  \\
&\leq \mu_2 \int^t_0\|u_m'(s)\|^2ds.
\end{aligned}
\end{equation}
 From \eqref{e3.5} and \eqref{e3.6} we have
\begin{equation} \label{e3.7}
 \begin{aligned}
&\frac{1}{2}\big \{ \|u'_m(t)\|^2+\|Au_m(t)\|^2
+\bar{M}(\|A^{1/2}u_m(t)\|^2) \big \}   \\
&\leq \frac{1}{2}\big \{ \|u_{1m}\|^2+\|Au_{0m}\|^2
+\bar{M}(\|A^{1/2}u_{0m}\|^2) \big \} +\mu_2\int^t_0\|u'_m(s)\|^2ds.
\end{aligned}
\end{equation}
In what follows, we use $C$ to denote a generic positive constant
independent of $m$.
Using  \eqref{e3.2}, \eqref{e3.3}, \eqref{e3.7} and Gronwall's inequality
we obtain
\begin{equation} \label{e3.8}
\|u'_m(t)\|^2+\|Au_m(t)\|^2+\bar{M}(\|A^{1/2}u_m(t)\|^2)\leq C.
\end{equation}
Using \eqref{e2.2}, we deduce that
\begin{equation} \label{e3.9}
\|u'_m(t)\|^2+\|Au_m(t)\|^2+\alpha\|A^{1/2}u_m(t)\|^2
+\frac{\beta}{2}\|A^{1/2}u_m(t)\|^4\leq C.
\end{equation}
By (H3)(2) and \eqref{e3.8} we also obtain
\begin{equation} \label{e3.10}
 \begin{aligned}
\|b^m(u_m'(t))\|& =\big( \int_{\Omega}(b^m(u'_m(t,x)))^2dx\big)^{1/2}  \\
&\leq \mu_2 \big( \int_{\Omega}|u'_m(t,x)|^2dx\big)^{1/2}\\
&=\mu_2\|u'_m(t)\|\leq C.
\end{aligned}
\end{equation}
So we can extend the approximated solutions $u_m(t)$ to the whole
interval $[0,T]$ and we get
 \begin{gather}
(u'_m)   \text{ is bounded in } L^{\infty}(0,T;H), \label{e3.11}\\
(Au_m)   \text{ is bounded in } L^{\infty}(0,T;H), \label{e3.12}\\
(b^m(u'_m))   \text{ is bounded in } L^{\infty}(0,T;H). \label{e3.13}
\end{gather}
Moreover by  assumption \eqref{e2.1},
\begin{equation} \label{e3.14}
(u_m)   \text{ is bounded in } L^{\infty}(0,T;H).
\end{equation}

\subsection*{Step 2 : A priori estimate II}
It follows from \eqref{e3.1} that for all $v$ in $W_m$,
\begin{equation} \label{e3.15}
\begin{aligned}
&|\langle u_m''(t),v\rangle|\\
&\leq \|Au_m(t)\| \|Av\|+|M(\|A^{1/2}u_m(t)\|^2)| \|Au_m(t)\| \|v\|
 +\|b^m(u'_m(t))\| \|v\|  \\
&\leq (\|Au_m(t)\|+|M(\|A^{1/2}u_m(t)\|^2)| \|Au_m(t)\|
 +\|b^m(u'_m(t))\|)\|v\|_V \,.
\end{aligned}
\end{equation}
Since $M$ is a $C^1$ real function, we have from \eqref{e3.9} and
\eqref{e3.10}, by a density argument, that
\begin{equation} \label{e3.16}
(u_m'')   \text{ is bounded in } L^{\infty}(0,T;V').
\end{equation}

\subsection*{Step 3: Passage to the limit}
From the priori estimates \eqref{e3.11}-\eqref{e3.14} and \eqref{e3.16},
we have subsequences (in the sequence we denote subsequences by the same
symbols as original sequences) such that
 \begin{gather}
u_m\to u   \text{ weakly star in } L^{\infty}(0,T;V), \label{e3.17}\\
Au_m\to Au   \text{ weakly star in } L^{\infty}(0,T;H), \label{e3.18}\\
u'_m\to u'   \text{ weakly star in } L^{\infty}(0,T;H), \label{e3.19}\\
u''_m\to u''   \text{ weakly star in } L^{\infty}(0,T;V'), \label{e3.20}\\
b^m(u'_m)\to \Xi   \text{ weakly star in } L^{\infty}(0,T;H), \label{e3.21}\\
M(\|A^{1/2}u_m\|^2)Au_m \to  \Psi   \text{ weakly star in }
L^{\infty}(0,T;H). \label{e3.22}
\end{gather}
By the Aubin-Lions compactness lemma \cite{l1},
from \eqref{e3.17}, \eqref{e3.19} and \eqref{e3.20} that
 \begin{gather}
u_m\to u   \text{ strongly in } L^2(0,T;H), \label{e3.23}\\
u'_m\to u'   \text{ strongly in } L^2(0,T;V'). \label{e3.24}
\end{gather}
Now we shall prove that $\Psi = M(\|A^{1/2}u\|^2)Au$.
For $v\in L^2(0,T;H)$, we have
\begin{equation} \label{e3.25}
\begin{aligned}
&\int^T_0(\Psi(t)-M(\|A^{1/2}u(t)\|^2)Au(t),v)dt  \\
&=\int^T_0(\Psi(t)-M(\|A^{1/2}u_m(t)\|^2)Au_m(t),v)dt  \\
&\quad +\int^T_0(M(\|A^{1/2}u(t)\|^2)(Au_m(t)-Au(t)),v)dt  \\
&\quad+\int^T_0(M(\|A^{1/2}u_m(t)\|^2)-M(\|A^{1/2}u(t)\|^2))(Au_m(t),v)dt.
\end{aligned}
\end{equation}
On the other hand, the fact that $M$ is $C^1$ and \eqref{e3.9} give
\begin{equation} \label{e3.26}
\begin{aligned}
&\int^T_0(M(\|A^{1/2}u_m(t)\|^2)-M(\|A^{1/2}u(t)\|^2))(Au_m(t),v)dt \\
&\leq C\int^T_0 \big|  \|A^{1/2}u_m(t)\|^2-\|A^{1/2}u(t)\|^2 \big|
\|Au_m(t)\| \|v\|dt  \\
&\leq C\int^T_0 \big| (A(u_m(t)+u(t)),u_m(t)-u(t)) \big| dt  \\
&\leq C\big( \int^T_0 \|u_m(t)-u(t)\|^2dt \big)^{1/2}.
\end{aligned}
\end{equation}
Considering \eqref{e3.18}, \eqref{e3.22}, \eqref{e3.23} and \eqref{e3.26}, we deduce from \eqref{e3.25}
that
\begin{equation} \label{e3.27}
M(\|A^{1/2}u_m\|^2)Au_m \to  M(\|A^{1/2}u\|^2)Au \quad
  \text{weakly star in } L^{\infty}(0,T;H).
\end{equation}
Now we may take the limit $m\to \infty$ in the approximated
\eqref{e3.2}. Therefore, we obtain
\begin{equation} \label{e3.28}
\langle u''(t)+A^2u(t)+M(\|A^{1/2}u\|^2)Au+\Xi,v\rangle =0.   \forall v\in V.
\end{equation}

\subsection*{Step 4: $(u, \Xi)$ is a solution of \eqref{e2.5}-\eqref{e2.7}}
Let $\phi
\in C^1[0,T]$ with $\phi(T)=0$. By replacing $v$ by $\phi(t)w_j$ in
\eqref{e3.1} and integrating by parts the result over $(0,T)$, we have
\begin{equation} \label{e3.29}
 \begin{aligned}
&(u'_m(0),\phi(0)w_j)+\int^T_0(u'_m(s),\phi'(s)w_j)ds  \\
&=\int^T_0(Au_m(s),\phi(s)Aw_j)ds+\int^T_0(M(\|A^{1/2}u_m\|^2)Au_m(s)
+b^m(u'_m(s)),\phi(s)w_j)ds.
\end{aligned}
\end{equation}
Similarly, from \eqref{e3.28},
\begin{equation} \label{e3.30}
\begin{aligned}
&(u'(0),\phi(0)w_j)+\int^T_0(u'(s),\phi'(s)w_j)ds  \\
&=\int^T_0(Au(s),\phi(s)Aw_j)ds+\int^T_0 (M(\|A^{1/2}u\|^2)Au(s)
+\Xi(s),\phi(s)w_j)ds.
\end{aligned}
\end{equation}
Comparing  \eqref{e3.29} and \eqref{e3.30}, we infer that
$$
\lim_{m\to \infty}(u'_m(0)-u'(0),w_j)=0, \quad  \forall j\in \mathbb{N}.
$$
This implies that $u'_m(0)\to u'(0)$ weakly in $H$. By the
uniqueness of limit, $u'(0)=u_1$. Analogously, taking
$\phi \in C^2[0,T]$ with $\phi(T)=\phi'(T)=0$, we may obtain
that $u(0)=u_0$.

Next we will show that $\Xi(t,x)\in \varphi(u'(t,x))$   a.e.
  $(t,x)\in Q$. For this purpose, we show the conclusion
\begin{equation} \label{e3.31}
u'_m(t,x)\to u'(t,x) \quad\text{a.e. }  (t,x)\in Q.
\end{equation}
First we mention the following lemmas.

\begin{lemma}[{\cite[11, p. 221]{z2}}] \label{lem3.1}
 Assume that  $(g_m(t))$ is an absolutely
continuous sequence defined on $[a,b]$ and $|g'_m(t)|\leq F(t)$ a.e.
($m=1,2,3,\dots$), $F\in L(a,b)$. If
$\lim_{m\to \infty}g_m(t)=g(t)$ and
$\lim_{m\to \infty}g'_m(t)=f(t)$  a.e.
$t\in [a,b]$, then $g'(t)=f(t)$ a.e. $t\in [a,b]$.
\end{lemma}

\begin{lemma}[{\cite[p. 152]{z1}}] \label{lem3.2}
If $(u_m)\subset C[0,T]$,   $u\in C[0,T]$ and $u_m\to u$
weakly, then $\lim_{m\to \infty}u_m(t)=u(t)$, $t\in [0,T]$.
\end{lemma}

Set $u(t,x)=\sum^{\infty}_{j=1}g_j(t)w_j$ and
$u'(t,x)=\sum^{\infty}_{j=1}f_j(t)w_j$. Let $j\in \mathbb{N}$ be fixed.
Since $u_m(t)\to u(t)$ in $H$ for a.e. $t\in [0,T]$,
$$
\lim_{m\to \infty}(u_m(t,x)-u(t,x),w_j)=0, \quad
\lim_{m\to \infty}(g_{mj}(t)-g_j(t))=0
$$
a.e.   $t \in [0,T]$.
Since $|g'_{mj}(t)|$ is bounded a.e. $t\in [0,T]$ (see \eqref{e3.11}),
there exists a function
$\xi_j(t)$ defined on $[0,T]$ such that
$$
\lim_{m\to \infty}g'_{mj}(t)=\xi_j(t) \quad\text{a.e. }  t\in [0,T].
$$
By Lemma \ref{lem3.1}, $\xi_j(t)=g'_j(t)$   a.e.   $t\in [0,T]$.
Hence $\lim_{m\to \infty}g'_{mj}(t)=g'_j(t)$.
Let
\[
v(s,x)=\begin{cases}
w_j   &\text{if } s\in[0,t], \\
0   &\text{if }s\in [t,T].
\end{cases}
\]
Since $u'_m\to u'$ weakly star in $L^{\infty}(0,T;H)$,
$\int_Q u'_m(s,x)v\,dx\,ds \to \int_Q u'(s,x)v\,dx\,ds$
as $m\to \infty$, and hence
$$
g_j(t)=g_j(0)+\int^t_0f_j(s)ds.
$$
This implies that $g'_j(t)=f_j(t)$ a.e. $t\in [0,T]$. Since
$u'_m\to u'$ weakly in $H$ for a.e. $t\in [0,T]$ and
$u'\in C(0,T;V'), u'(t,x)=\sum^{\infty}_{j=1}g'_j(t)w_j \in C[0,T]$
and $u'_m(t,x)\to u'(t,x)$ weakly in $C[0,T]$ for a.e. $x\in \Omega$.
Thus by Lemma \ref{lem3.2}, we conclude that \eqref{e3.31}.
Thus, for given $\eta >0$, using the
theorems of Lusin and Egoroff, we can choose a subset $\omega
\subset Q$ such that $\mathop{\rm meas}(\omega)<\eta$,
$u'\in L^{\infty}(Q\setminus \omega)$ and $u'_m\to u'$ uniformly
on $Q\setminus \omega$. Thus,
for each $\epsilon>0$, there is an $N>\frac{2}{\epsilon}$ such that
\begin{equation*}
|u'_m(t,x)-u'(t,x)|<\frac{\epsilon}{2},  \quad
 \forall (t,x)\in Q\setminus \omega.
\end{equation*}
Then, if $|u'_m(t,x)-s|<1/m$, we have $|u'(t,x)-s|<\epsilon $ for all
$m>N$ and $(t,x)\in Q\setminus \omega$. Therefore we have
\[
\underline{b}_{\epsilon}(u'(t,x))\leq b^m(u'_m(t,x))
\leq \overline{b}_{\epsilon}(u'(t,x)), \quad
  \forall m>N, (t,x)\in Q\setminus \omega.
\]
Let $\phi \in L^{\infty}(Q)$, $\phi \geq 0$. Then
\begin{equation} \label{e3.32}
\begin{aligned}
\int_{Q\setminus \omega}\underline{b}_{\epsilon}(u'(t,x))\phi(t,x) \,dx\,dt
& \leq  \int_{Q\setminus \omega}b^m(u'_m(t,x))\phi(t,x) \,dx\,dt    \\
& \leq  \int_{Q\setminus
\omega}\overline{b}_{\epsilon}(u'(t,x))\phi(t,x) \,dx\,dt.
\end{aligned}
 \end{equation}
Letting $m\to \infty$ in \eqref{e3.32} and using \eqref{e3.21}, we obtain
\begin{equation} \label{e3.33}
\begin{aligned}
\int_{Q\setminus \omega}\underline{b}_{\epsilon}(u'(t,x))\phi(t,x) \,dx\,dt
& \leq  \int_{Q\setminus \omega}\Xi(t,x) \phi(t,x) \,dx\,dt   \\
& \leq  \int_{Q\setminus
\omega}\overline{b}_{\epsilon}(u'(t,x))\phi(t,x) \,dx\,dt.
\end{aligned}
 \end{equation}
Letting $\epsilon \to 0^+$ in \eqref{e3.33}, we infer that
$$
\Xi(t,x)\in \varphi(u'(t,x))\quad \text{a.e. in }   Q\setminus \omega,
$$
and letting $\eta \to 0^+$, we obtain
$$
\Xi(t,x)\in \varphi(u'(t,x))  \quad\text{a.e. in}   Q.
$$
Therefore, the proof of Theorem \ref{thm2.1} is complete.

\begin{remark} \label{rmk3.1} \rm
Even if we replace the condition (H3)(2) by the weaker linear
growth condition:
$$
|b(s)|\leq \mu_2(1+|s|), \quad   \forall s\in \mathbb{R},
$$
we obtain the same results as in Theorem \ref{thm2.1}.
\end{remark}

\begin{remark} \label{rmk3.2} \rm
If in Theorem \ref{thm2.1} we impose the  condition that $b$ is
nondecreasing, then we obtain the stronger results. In other words, the
solution $u$ of \eqref{e2.5}-\eqref{e2.7} satisfies
$$
u\in W^{1,\infty}(0,T;V)\cap W^{2,\infty}(0,T;H).
$$
Since the proof of this result is similar to that of \cite[Theorem 1.1]{p1},
we omit it here.
\end{remark}

\section{Energy decay of solutions}

In this section we shall prove Theorem \ref{thm2.2} by applying the following
lemma by Nakao \cite{n1}.

\begin{lemma} \label{lem4.1}
Let $\phi : \mathbb{R}^+\to \mathbb{R}$ be a bounded nonnegative function
for which there exist constant $\delta>0$ such that
$$
\sup_{t\leq s\leq t+1}\phi(s)\leq \delta(\phi(t)-\phi(t+1)), \quad
  \forall t\ge 0.
$$
Then there exist positive constants $C$ and $\gamma$ such that
$$
\phi(t)\leq Ce^{-\gamma t}, \quad \forall t\ge 0.
$$
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm2.2}]
 The existence part of solution of
Theorem \ref{thm2.2} is a consequence of the proof of Theorem \ref{thm2.1}. To prove
the decay property, we first obtain uniform estimates for the
approximated energy,
\begin{equation} \label{e4.1}
E_m(t)=\frac{1}{2}\big( \|u'_m(t)\|^2+\|Au_m(t)\|^2
 +\bar{M}(\|A^{1/2}u_m\|^2) \big)
\end{equation}
and then pass to the limit. Note that $E_m(t)$ is non-negative and
uniformly bounded. Let us fix an arbitrary $t>0$. From the
approximated problem \eqref{e3.1} with $v=u'_m(t)$,  by (H3)(2) we have
\begin{equation} \label{e4.2}
\frac{d}{dt}E_m(t)=-(b^m(u'_m(t)),u'_m(t))\leq -\mu_1\|u'_m(t)\|^2.
\end{equation}
This implies that $E_m(t)$ is a non-increasing function.
Setting $F^2_m(t)=E_m(t)-E_m(t+1)$ and integrating \eqref{e4.2} over
$(t,t+1)$ we have
\begin{equation} \label{e4.3}
F^2_m(t)\geq \mu_1 \int^{t+1}_t\|u'_m(s)\|^2ds.
\end{equation}
By applying the mean value theorem, there exist
$t_1\in [t,t+\frac{1}{4}]$ and $t_2\in [t+\frac{3}{4},t+1]$ such that
\begin{equation} \label{e4.4}
\|u'_m(t_i)\|\leq \frac{2}{\sqrt {\mu_1}}F_m(t), \quad  i=1,2.
\end{equation}
Now, replacing $v$ by $u_m(t)$ in the approximated problem we have
\begin{equation} \label{e4.5}
\begin{aligned}
&(A^2u_m(t),u_m(t))+(M(\|A^{1/2}u_m(t)\|^2)Au_m(t),u_m(t))  \\
&=-(u''_m(t),u_m(t))-(b^m(u'_m(t)),u_m(t)).
\end{aligned}
\end{equation}
Integrating \eqref{e4.5} over $(t_1,t_2)$ and using (H3)(2), we obtain
\begin{equation} \label{e4.6}
\begin{aligned}
&\int^{t_2}_{t_1}\|Au_m(s)\|^2+M(\|A^{1/2}u_m\|^2)\|A^{1/2}u_m(s)\|^2 ds  \\
&=-(u'_m(t_2),u_m(t_2))+(u'_m(t_1),u_m(t_1))   \\
&\quad +\int^{t_2}_{t_1}\|u'_m(s)\|^2ds-\int^{t_2}_{t_1}
\int_{\Omega}b^m(u'_m(s,x))u_m(s,x)\,dx\,ds  \\
&\leq \|u'_m(t_2)\| \|u_m(t_2)\|+\|u'_m(t_1)\| \|u_m(t_1)\|    \\
&\quad +\int^{t_2}_{t_1}\|u'_m(s)\|^2ds
 +\mu_2\int^{t_2}_{t_1}\|u'_m(s)\| \|u_m(s)\|ds.
\end{aligned}
\end{equation}
Using H\"{o}lder's inequality and  \eqref{e2.1},
from  \eqref{e4.3}, \eqref{e4.4} and
\eqref{e4.6}, we have
\begin{equation} \label{e4.7}
\begin{aligned}
& \int^{t_2}_{t_1}E_m(s)ds \\
& \leq \frac{3}{2\mu_1}F^2_m(t)+\frac{2}{\mu_0
\sqrt{\mu_1}}F_m(t)\|Au_m(t_2)\|   \\
& \quad +\frac{2}{\mu_0 \sqrt{\mu_1}}F_m(t)\|Au_m(t_1)\|
+\frac{\mu_2}{\mu_0} \big( \int^{t_2}_{t_1}\|u'_m(s)\|^2ds
\big)^{1/2} \sup_{t_1\leq s\leq t_2}\|Au_m(s)\|
\end{aligned}
 \end{equation}
and then we have
\begin{equation} \label{e4.8}
\int^{t_2}_{t_1}E_m(s)ds\leq C_1F^2_m(t)+C_2F_m(t)E_m(t)^{1/2},
\end{equation}
where $C_1,C_2$ are a generic positive constant independent of $m$.
Noting that
$E_m(t+1)\leq 2\int^{t_2}_{t_1}E_m(s)ds$ and $E_m(t+1)=E_m(t)-F^2_m(t)$,
from \eqref{e4.8} we have
\begin{equation} \label{e4.9}
\begin{aligned}
E_m(t) & \leq  2\int^{t_2}_{t_1}E_m(s)ds+F^2_m(t)  \\
& \leq  (2C_1+1)F_m^2(t)+2C_2F_m(t)E_m(t)^{1/2}.
\end{aligned}
\end{equation}
Young's inequality implies
\begin{equation} \label{e4.10}
E_m(t)\leq C_3F^2_m(t)
\end{equation}
for some positive constant $C_3$.
Since $E_m$ is non-increasing, from \eqref{e4.10}, we have
\[
\sup_{t\leq s\leq t+1}E_m(s)\leq C_3(E_m(t)-E_m(t+1)), \quad  \forall t\geq 0.
\]
Applying Lemma \ref{lem4.1}, there exist
positive constants $C$ and $\gamma$ such that
\[
E_m(t)\leq Ce^{-\gamma t},  \forall t\ge 0.
\]
Passing to the limit $m\to \infty$, we get \eqref{e2.9}.
This completes the proof of Theorem~\ref{thm2.2}.
\end{proof}


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\end{document}