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\markboth{\hfil Existence of solutions of a variational unilateral system 
\hfil EJDE--2002/22}
{EJDE--2002/22\hfil M. R. Clark \&  Osmundo A. Lima \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 22, pp. 1--18. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Existence of solutions for a variational unilateral system
 %
\thanks{ {\em Mathematics Subject Classifications:} 35L85, 49A29.
\hfil\break\indent
{\em Key words:} weak solutions, variational unilateral nonlinear problem,
Galerkin method,  \hfil\break\indent penalization method.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted November 17, 2000. Published February 21, 2002.
\hfil\break\indent
M. R. Clark was a visiting professor of State University of Piau\'{\i}
} }
\date{}
%
\author{Marcondes R. Clark \& Osmundo A. Lima}
  

\maketitle

\begin{abstract}
 In this work the authors study the existence of weak solutions of the
 nonlinear unilateral mixed problem associated to the inequalities
 $$ \displaylines{
 u_{tt}-M(| \nabla u| ^{2})\Delta u+\theta \geq f, \cr
 \theta_t-\Delta \theta +u_t\geq g ,}
 $$
 where $f$, $g$, $M$ are given real-valued functions with $M$ positive.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
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\section{Introduction}

Let $\Omega $ be a bounded and  open set of $\mathbb{R}^n$, with smooth
boundary $\Gamma =\partial \Omega $, and let $T$ be a positive real number.
Let $\mathbb{Q}=\Omega \times ]0,T[$ be the cylinder with lateral boundary
$\Sigma =\Gamma \times ]0,T[$.

We study the variational nonlinear system
\begin{gather}
u_{tt}-M(| \nabla u| ^{2})\Delta u+\theta \geq f \quad\text{ in }\quad Q,  \\
\theta_t-\Delta \theta +u_t\geq g \quad \text{in } \quad Q, \\
u=\theta =0 \quad\text{in }\quad\Sigma  \\
u(0) =u_{0}, \quad u'(0) =u_1, \quad  \theta(0) =\theta_{0}.
\end{gather}
The  above system  with $M(s)=m_{0}+m_1s$
($m_{0}$ and $m_1$ positive constants) and  $\theta =0$ is a nonlinear
perturbation of the canonical Kirchhof model
\begin{equation}
u_{tt}-\big(m_{0}+ m_1\int_{\Omega }| \nabla u^{2}dx\big)\Delta u=f\,.  \tag{1.5}
\end{equation}
This model describes small vibrations of a stretched string when only
the transverse component of the  tension is considered, see for example,
 Arosio \& Spagnolo \cite{a1}, Pohozaev \cite{p1}.

Several authors have studied (1.5). For $\Omega $ bounded,
we can cite: D'ancona \& Spagnolo  \cite{d1}, Medeiros \&  Milla Miranda  \cite{m1},
Hosoya \& Yamada \cite{h1}, Lions \cite{l1}, Medeiros \cite{m2}, and
Matos \cite{m1}. For $\Omega $ unbounded, we can cite Bisiguin \cite{b1},
Clark \& Lima \cite{c2}, and Matos \cite{m1}.
The system (1.1)--(1.4) was studied also in the case
when (1.1) and (1.2) are equations, see  for example \cite{c1}.

In the present work we show the existence of a weak solution for the
variational nonlinear system (1.1)--(1.4), under appropriate
assumptions on $M$, $f$ and $g$. We employ Galerkin's approximation method
and the penalization method used by Frota \& Lar'kin \cite{f1}.

\section{Notation and main result}

We represent the Sobolev space of order
$m$ on $\Omega $ by
$$W^{m,p}(\Omega )=\{u\in L^{p}(\Omega);\,D^{\alpha }u\in L^{p}(\Omega ),
\forall \;| \alpha | \leq m\}$$
and its associated norm by
\begin{equation*}
\| u\|_{m,p}=\Big(\sum_{| \alpha | \leq m}|
D^{\alpha }u|_{L^{p}(\Omega ) }^{p}\Big)^{1/p},\quad u\in
W^{m,p}(\Omega ) , \quad 1\leq p<\infty\,.
\end{equation*}
When $p=2$, we have the usual Sobolev space $H^{m}(\Omega )$.
Let $D(\Omega )$ be the space of the test functions on $\Omega $,
and  let $W_{0}^{m,p}(\Omega )$ be the closure of $D(\Omega )$ in
$W^{m,p}(\Omega )$.
When $p=2$, we have $W_{0}^{2,p}(\Omega)=H_{0}^{m}(\Omega )$.
The dual space of $W^{m,p}_{0}(\Omega)$ is denoted by $W^{-m,p'}(\Omega)$,
with $p'$ such that $\frac{1}{p} + \frac{1}{p'} = 1$.
For the rest of this paper we use the symbol $(\cdot,\cdot)$ to indicate
the inner product in $L^2(\Omega)$, and $((\cdot,\cdot))$ to indicate the inner
product in $H_0^1(\Omega)$.

Let $\mathbb{K}=\{\psi \in W_{0}^{2,4}(\Omega );|\Delta \psi |\leq 1$ and
$\psi \geq 0$ a. e. in $\Omega $ \}.
Then we have the following proposition whose proof can be found in \cite{f1}

\begin{proposition} \label{prop2.1}
The set $\mathbb{K}$ is a closed and connected in $W_{0}^{2,4}(\Omega )$.
\end{proposition}

\paragraph{Definition} Let $V$ be a Banach space and $V'$ its dual. An  operator
$\beta$ from $V$ to $V'$ is called hemicontinous if the function
\begin{equation*}
\lambda \to \langle \beta (u+\lambda v),w\rangle
\end{equation*}
is continuous for all $\lambda \in \mathbb{R}$. The operator $\beta $ is called
monotone if
\begin{equation*}
\langle \beta (u)-\beta (v),u-v\rangle \geq 0,\quad \forall u,v\in V.
\end{equation*}

We consider the penalization operator
$\beta:W_{0}^{2,4}(\Omega )\to W^{-2,4/3}(\Omega )$ such that
$\beta (z)=\beta_{1}(z)+\beta_{2}(z)$, $z\in W_{0}^{2,4}(\Omega )$,
where $\beta_{1}(z)$ and $\beta_{2}(z)$ are defined by
\begin{gather*}
\langle \beta_{1}(z) ,v\rangle =-\int_{\Omega }z^{-}(x) v(x) dx, \\
\langle \beta_{2}(z) ,v\rangle =-\int_{\Omega }(1-| \Delta
z(x) | ^{2})^{-}\Delta z(x) \Delta v(x) dx
\end{gather*}
for all $v$ in $W_{0}^{2,4}(\Omega )$.

\begin{proposition} \label{prop2.2}
The operator $\beta$ defined above satisfies the following coditions:
\begin{itemize}
\item[i)]  $\beta$ is monotone and hemicontinous

\item[ii)]  $\beta$ is bounded; this is, $\beta(S)$ is bounded in
$W^{2,4/3}(\Omega )$ for all bounded set $S$ in $W^{2,4}_{0}(\Omega)$.

\item[iii)]  $\beta(u) = 0$ if only if $u$ belongs to $\mathbb{K}$.
\end{itemize}
\end{proposition}

The proof of this proposition can be found in \cite{f1}.

In this article, we assume the following hypotheses:
\begin{enumerate}
\item[A1)] $M\in C^{1}[0,\infty )$, $M(s) \geq 0$  for $s\geq 0$,
and $\int_{0}^{\infty }M(s) ds=\infty$

\item[A2)] $f,g$ belong to $H^{1}(0,T;L^{2}(\Omega )$.
\end{enumerate}
The main result of this paper is stated as follows.

\begin{theorem} \label{thm2.1}
 Assume A1) and A2). For $u_{0}\in H_{0}^{1}(\Omega )\cap
H^{2}(\Omega )$, $u_1,\theta_{0}$ in the interior of $\mathbb{K}$,
there exist functions
$u,\theta :\mathbb{Q}\to \mathbb{R}$ such that
\begin{gather}
u\in L^{\infty }(0,T;\;H_{0}^{1}(\Omega ) \cap H^{2}( \Omega ) )  \\
u'\in L^{1}(0,T;\;W_{0}^{2,4}(\Omega ) )\text{ and }
u'(t) \in \mathbb{K}\text{ a.e. in }[0,T] \\
u''\in L^{\infty }(0,T;\;L^{2}(\Omega ) ) \\
\theta \in L^{\infty }(0,T;\;H_{0}^{1}(\Omega ) )\;\text{ and }
\theta(t) \in \mathbb{K} \text{ a.e. in }[0,T]\,.
\end{gather}
Also
\begin{gather}
(u''(t) -M(\| u(t) \|^{2})\Delta u(t) +\theta (t) -f(t)
,v-u'(t) \geq 0, \;\forall v\in \mathbb{K\;}\text{ a.e. in }[0,T]  \\
(\theta '(t) -\Delta \theta (t)
+u'(t) -g(t) , v-\theta (t) )\geq 0\; \forall
v\in \mathbb{K\;}\text{ a.e. in }[0,T]  \\
u(0) =u_{0}, \; u'(0) =u_1, \;\theta(0) =\theta_{0}\,.
\end{gather}
\end{theorem}

To obtain the solution $\{u,\theta \}$ of problem (2.1)--(2.4) in
Theorem \ref{thm2.1}, we consider the following associated penalized
problem. For $0<\varepsilon <1$, consider
\begin{gather}
u_{\varepsilon }''(t) -M(\|u_{\varepsilon }(t) \| ^{2}) \Delta
u_{\varepsilon }(t) +\theta_{\varepsilon }(t) +
\frac{1}{\varepsilon }\beta (u_{\varepsilon }'(t)
) =f(t) \text{ \ in }Q  \tag{2.8} \\
\theta_{\varepsilon }'(t) -\Delta \theta
_{\varepsilon }(t) +u_{\varepsilon }'+\frac{1}{
\varepsilon }\beta (\theta_{\varepsilon }(t) )
=g(t) \text{ \ in }Q  \tag{2.9} \\
u_{\varepsilon }(0) =u_{0\varepsilon },
u_{\varepsilon}'(0) =u_{1\varepsilon },
\theta_{\varepsilon }(0) =\theta_{0\varepsilon }\text{ in }\Omega  \tag{2.10}
\end{gather}
Here $\beta $ is a penalization operator, $M$, $f$, and $g$ are as above.
The solution $\{u_{\varepsilon },\theta_{\varepsilon }\}$ of the
penalized problem (2.8)--(2.10) are guaranteed by the following theorem.

\begin{theorem} \label{thm2.2}
Suppose the hypotheses of the  Theorem \ref{thm2.1} hold, and for
\linebreak $0<\varepsilon <1$, then there exist functions
$\{u_{\varepsilon },\theta_{\varepsilon }\}$ such that
\begin{gather}
u_{\varepsilon },\theta_{\varepsilon }\in L^{\infty }(
0,T;H_{0}^{1}(\Omega ) \cap H^{2}(\Omega ) ) \tag{2.11} \\
u_{\varepsilon }'\in L^{4}(0,T;W_{0}^{2,4}(\Omega ))  \tag{2.12} \\
u_{\varepsilon }''\in L^{\infty }(0,T;L^{2}(\Omega ) )  \tag{2.13} \\
\theta_{\varepsilon }\in L^{4 }(0,T;W_{0}^{2,4}(\Omega) )  \tag{2.14} \\
(u_{\varepsilon }''(t) ,v) +M(
\| u_{\varepsilon }(t) \| ^{2}) ((
u_{\varepsilon}(t) ,v) ) +(\theta_{\varepsilon }(t) ,v)+
\frac{1}{\varepsilon }\langle\beta (u_{\varepsilon
}'(t) ) ,v\rangle \nonumber \\
=(f(t) ,v) \text{ a.e. in $[0,T]$ for all $v\in W^{2,4}_{0}(\Omega )$,}\tag{2.15} \\
%
(\theta_{\varepsilon }'(t) ,v) +(
(\theta_{\varepsilon}(t) ,v) ) +(u_{\varepsilon}'(t) ,v)
+\frac{1}{\varepsilon }\langle \beta (\theta_{\varepsilon }(t) ) ,v\rangle \nonumber\\
=(g(t) ,v) \text{a.e. in $[0,T]$ for all $v\in W^{2,4}_{0}(\Omega ) $,}
\tag{2.16}\\
%
u_{\varepsilon }(0) =u_{0\varepsilon}, u_{\varepsilon}'(0)
=u_{1\varepsilon }, \theta_{\varepsilon
}(0) =\theta_{0\varepsilon } . \tag{2.17}
\end{gather}
\end{theorem}

\paragraph{Proof}
We will use Galerkin's method and a compactness argument.

\noindent \textbf{First step} (Approximated system)
Let $w_1,\ldots ,w_{m},\ldots $
be an orthonormal base of  $W_{0}^{2,4}(\Omega ) $ consisting of eigenfunctions
of the Laplacian operator. Let \linebreak  $V_{m}=[w_1,\ldots ,w_{m}]$ the
subspace of $W_{0}^{2,4}(\Omega ) $, generated by the first $m$
vectors $w_{j}$.
We look for a pair of functions
$$u_{\varepsilon m}(t)
=\sum_{j=1}^{m}g_{jm}(t) w_{j}, \quad \theta_{\varepsilon
m}(t) =\sum_{j=1}^{m}h_{jm}(t) w_{j}\quad \text{ in}\quad 
V_{m}$$
with $g_{jm}\in C^{2}([0,T]) $ and $h_{jm}\in C^{1}([0,T])$, for all
$j=1,\ldots ,m$. Which are solutions of
the following system of ordinary differential equations
\begin{gather}
(u_{\varepsilon m}''(t) ,w_{j})
+M(\| u_{\varepsilon m}(t) \| ^{2})
((u_{\varepsilon m}(t) ,w_{j}) ) +(
\theta_{\varepsilon m}(t) ,w_{j}) +
\nonumber\\
\frac{1}{\varepsilon }\langle \beta (u_{\varepsilon m}'(t) ) ,w_{j}\rangle
=(f(t) ,w_{j}), \tag{2.18} \\
(\theta_{\varepsilon m}'(t) ,w_{j})
+((\theta_{\varepsilon m}(t) ,w_{j}) )
+(u'_{\varepsilon m}(t) ,w_{j}) +
\nonumber\\
\frac{1}{\varepsilon } \langle \beta (\theta_{\varepsilon m}(
t) ) ,w_{j}\rangle =(g(t) ,w_{j}), \tag{2.19}
\end{gather}
for $j=1,\ldots ,m$, with the initial conditions:
$u_{\varepsilon m}(0) =u_{0\varepsilon m}$,
$u_{\varepsilon m}'(0) =u_{1\varepsilon m}$,
$\theta_{\varepsilon m}(0) =\theta_{0\varepsilon m}$,
where
\begin{equation}
\begin{gathered}
u_{0\varepsilon m}=\sum_{j=1}^{m}(u_{0\varepsilon },w_{j} ) w_{j}\to u_{0}
\text{ strongly in }H_{0}^{1}(\Omega ) \cap H^{2}(\Omega ), \\
u_{1\varepsilon m}=\sum_{j=1}^{m}(u_{1\varepsilon},w_{j}) w_{j}\to u_1
\text{ strongly in } H_{0}^{1}(\Omega ), \\
\theta_{0\varepsilon m}=\sum_{j=1}^{m}(\theta_{0\varepsilon},w_{j}) w_{j}
\to \theta_{0}\text{ strongly in } W_{0}^{2,4}(\Omega).
\end{gathered}
\tag{2.20}
\end{equation}
The system (2.18)--(2.20) contains $2m$ unknowns functions
$g_{jm}(t), h_{jm}(t)$; \linebreak $j=1,2,\ldots ,m$. By Caratheodory's Theorem it
follows that (2.18)--(2.20) has a local solution $\{u_{\varepsilon m}(t)
, \theta_{\varepsilon m}(t) \}$ on $[0,t_{m}[$. In order to
extend these local solution to the interval $[0,T[$ and to take the
limit in $m$, we must obtain some a priori estimates.

\noindent{\bf Estimate (i)} Note that  finite linear combinations of the
$w_{j}$ are dense in $W_{0}^{2,4}(\Omega ) $, then we can take
$w\in W_{0}^{2,4}(\Omega) $ in (2.18) and (2.19) instead of $w_{j}$.
Taking $w=2u_{\varepsilon m}'(t) $ in (2.18) and $w=2\theta_{\varepsilon m}(t) $
 in (2.19) we obtain
\begin{gather}
\frac{d}{dt}| u_{\varepsilon m}'(t) | ^{2}
+\frac{d}{dt}\widehat{M}(\| u_{\varepsilon m}(t)\| ^{2})
+\frac{2}{\varepsilon }\langle \beta (u_{\varepsilon m}'(t) ) ,u_{\varepsilon m}'(t) \rangle \nonumber\\
=2(f(t) ,u_{\varepsilon m}'(t)) -2(\theta_{\varepsilon m}(t),
u_{\varepsilon m}'(t) ), \tag{2.21} \\
\frac{d}{dt}| \theta_{\varepsilon m}(t) |^{2}
+\| \theta_{\varepsilon m}(t) \| ^{2}
+\frac{2}{\varepsilon }\langle \beta (\theta_{\varepsilon m}(t) )
,\theta_{\varepsilon m}(t) \rangle \nonumber\\
=-2(u_{\varepsilon m}'(t) ,\theta_{\varepsilon m}(t) )
+2\langle g(t) ,\theta_{\varepsilon m}(t) \rangle\,, 
\tag{2.22}
\end{gather}
where $\widehat{M}(\lambda ) =\int_{0}^{\lambda }M(s) ds$.
Adding (2.21) and (2.22), and integrating from $0$ to $t\leq t_{m}$ we have
\begin{equation}
\begin{aligned}
| u_{\varepsilon m}'(t) |^2+| \theta_{\varepsilon m}(t) |^2+
\int_{0}^{\| u_{\varepsilon m}(t) \|^2}M(s)ds+
\int_{0}^{t}\|\theta_{\varepsilon m}(s) \|^2ds+
 \\
\frac{2}{\varepsilon }\int_{0}^{t}\langle \beta (u_{\varepsilon m}'(s) ) ,u_{\varepsilon m}'(s) \rangle ds+
\frac{2}{\varepsilon }\int_{0}^{t}\langle \beta (\theta_{\varepsilon m}(s) ) ,\theta_{\varepsilon m}(s) 
\rangle ds \leq 
\\ 
\int_{0}^{T}| f(t) |^{2}ds+3\int_{0}^{t}| u_{\varepsilon m}'(s)|^2ds+
3\int_{0}^{t}| \theta_{\varepsilon m}(s) |^2ds+
\\
\int_{0}^{T}| g(t) |^2dt+| \theta_{0\varepsilon m}|^2+| u_{1\varepsilon m}|^2.
\end{aligned} 
\tag{2.23}
\end{equation}
From (2.20) and hypothesis (A2) there exists a positive constant $C$,
independently of $\varepsilon >0$ and $m$ such that
\begin{equation}
\begin{aligned}
| u_{\varepsilon m}'(t) | ^{2}+| \theta_{\varepsilon m}(t) |
^{2}+\int_{0}^{\| u_{\varepsilon m}(t) \|^{2}}M(s) ds+
\int_{0}^{t}\| \theta_{\varepsilon m}(s) \| ^{2}ds+ \\
\frac{2}{\varepsilon }\Big[ \int_{0}^{t}\langle \beta (
u_{\varepsilon m}'(s) ) ,u_{\varepsilon m}'(s) \rangle ds
+\int_{0}^{t}\langle \beta (\theta _{\varepsilon m}(s) )
,\theta_{\varepsilon m}(s) \rangle ds\Big] \leq\\
 C+3\int_{0}^{t}| u_{\varepsilon m}'(s)
| ^{2}ds+3\int_{0}^{t}| \theta_{\varepsilon m}(s) | ^{2}ds.
\end{aligned} 
\tag{2.24}
\end{equation}

Next we analyze the sign of the term $\int_{0}^{t}\langle \beta (
u_{\varepsilon m}'(s) ) ,u_{\varepsilon m}'(s) \rangle ds$. Note that 
$-u_{\varepsilon m}'(t) \leq u_{\varepsilon m}'(t) ^{-}$. Then,
by the definition of $\beta $, we have
$$
\begin{aligned}
\langle \beta (u_{\varepsilon m}'(t) ),u_{\varepsilon m}'(t) \rangle
=& \langle \beta_{1}( u_{\varepsilon m}'(t) ) ,u_{\varepsilon m}'(t) \rangle+
\langle \beta_{2}(u_{\varepsilon m}'(t) ) ,u_{\varepsilon m}'(t) \rangle  \\
=& -\int_{\Omega }(u_{\varepsilon m}'(x,t)) ^{-}u_{\varepsilon m}'(x,t)dx+\\
& \int_{\Omega }(1-| \Delta u_{\varepsilon m}'(t) | ^{2})^{-}
 (\Delta u_{\varepsilon m}'(t) ) ^{2}dx\geq 0.
\end{aligned}
$$
Similarly, we have,
$$\langle \beta (\theta_{\varepsilon m}(t) ) ,\theta
_{\varepsilon m}(t) \rangle \geq 0\,.
$$
Because $M(s) \geq 0$ for all $s$, from (2.24) and
Gronwall's inequality it follows that
$$
| u_{\varepsilon m}'(t) | ^{2}+|\theta_{\varepsilon m}(t) | ^{2}\leq C_1,
\quad \forall \varepsilon ,m, \forall t\in [ 0,t_{m}[.
$$
Returning to (2.24), we obtain
\begin{equation}
\begin{aligned}
| u_{\varepsilon m}'(t) | ^{2}+| \theta _{\varepsilon m}(t) | ^{2}
+\int_{0}^{\|u_{\varepsilon m}(t) \| ^{2}}M(s) ds+\int_{0}^{t}
\| \theta_{\varepsilon m}(s) \| ^{2}ds +\\
\frac{2}{\varepsilon }[\int_{0}^{t}\langle \beta (u_{\varepsilon
m}'(s) ) ,u_{\varepsilon m}'(s) \rangle ds
+\int_{0}^{t}\langle \beta (\theta_{\varepsilon m}(s) ) ,
\theta_{\varepsilon m}(s) \rangle ds] \leq C+3C_1T\,.
\end{aligned}
\tag{2.25}
\end{equation}
Since $\int_{0}^{\infty }M(s) ds=\infty $, by
(2.25) we can find $C_1$ such that
\begin{equation*}
\| u_{\varepsilon m}(t)
\| ^{2}\leq C_1, \quad \forall \varepsilon ,m, \forall t\in [0,t_{m}[.
\end{equation*}
Thus there exists, other constant $C=C(T) $ independently of $
\varepsilon ,m$ and $t\in [ 0,t_{m}[$ such that
\begin{equation}
\begin{aligned}
| u_{\varepsilon m}'(t) | ^{2}+| \theta_{\varepsilon m}(t) | ^{2}
+\| u_{\varepsilon m}(t) \| ^{2}+\int_{0}^{t}\| \theta_{\varepsilon
m}(s) \| ^{2}ds+\\
\frac{2}{\varepsilon }\int_{0}^{t}\langle \beta (u_{\varepsilon
m}'(s) ) ,u_{\varepsilon m}'(s) \rangle ds
+\frac{2}{\varepsilon }\int_{0}^{t}\langle \beta (\theta
_{\varepsilon m}(s) ) ,\theta_{\varepsilon m}(s) \rangle ds
\leq C
\end{aligned}
\tag{2.26}
\end{equation}
{\bf Estimate (ii)} We will obtain a bound for
$|u_{\varepsilon m}''(0)| $. For this, we note that $u_1$ being in the
interior of $\mathbb{K}$ and $u_{1\varepsilon m}\to u_1$ imply that
 $u_{1\varepsilon m}$ is in the interior of $\mathbb{K}$, for $m$ large.
 Therefore, $|\Delta u_{1\varepsilon m}| \leq 1$ and
$u_{1\varepsilon m}\geq 0$ a. e. in $\Omega $. Also we have
$(u_{1\varepsilon m}) ^{-}=0$ and $(1-| \Delta u_{1\varepsilon m}| ^{2}) ^{-}=0$
a. e. in $\Omega $. Thus
\begin{equation}
\langle \beta (u_{1\varepsilon m}) ,u_{\varepsilon m}''(0) \rangle =0  \tag{2.27}
\end{equation}
Taking $t=0$ and $v=u_{\varepsilon m}''(0) $ in
(2.14), and observing (2.27), we obtain
\begin{equation*}
| u_{\varepsilon m}''(0) |^{2}+M(\| u_{0\varepsilon m}\| ^{2}) (
(u_{0\varepsilon m},u_{\varepsilon m}''(0) ) )
+(\theta_{\varepsilon m},u_{\varepsilon m}''(0) ) =(f(0),
u_{\varepsilon m}''(0) )
\end{equation*}
which implies
\begin{equation*}
| u_{\varepsilon m}''(0) | ^{2}\leq | f(0) | | u_{\varepsilon m}''(0) |
+M(\| u_{0\varepsilon m}\| ^{2}) | \Delta u_{0\varepsilon m}|
| u_{\varepsilon m}''(0) | +|\theta_{0\varepsilon m}|
| u_{\varepsilon m}''(0) | .
\end{equation*}
 From $u_{0\varepsilon m}\to u_{0}$ in
$H_{0}^{1}(\Omega ) \cap H^{2}(\Omega ) $,
 $\theta_{0\varepsilon m}\to \theta_{0}$ in $H_{0}^{1}(\Omega ) $,
$M\in C^{1}[0,\infty )$, and  $f\in H^{1}(0,T;L^{2}(\Omega ) $, we obtain
\begin{equation}
| u_{\varepsilon m}''(0) | \leq C,
\tag{2.28}
\end{equation}
with $C$ independent of $\varepsilon , m$,  and $t\in [0,T[$.

\noindent{\bf Estimate (iii)} We obtain estimates for
$| \Delta u_{\varepsilon m}'(t) |$, $| \Delta \theta_{\varepsilon
m}(t) | $, $\int_{0}^{t}| u_{\varepsilon m}'(s) | ^{3}ds$, and
$\int_{0}^{t}| \theta_{\varepsilon m}'(s) | ^{3}ds$.
For this, we need the following lemma whose proof can be found in
\cite{f1}.
\begin{lemma} \label{lm2.1}
Let $h:\Omega \to \mathbb{R}$ be an arbitrary function. Then
\begin{equation*}
h^{4}-1\leq 2(1-h^{2}) ^{-}h^{2}.
\end{equation*}
\end{lemma}

 By  this lemma, we have
\begin{equation*}
(\Delta u_{\varepsilon m}') ^{4}-1\leq 2[1-(
\Delta u_{\varepsilon m}') ^{2}]^{-}(\Delta
u_{\varepsilon m}') ^{2}\,.
\end{equation*}
Therefore,
$$\begin{aligned}
\| \Delta u_{\varepsilon m}'\|_{L^{4}(Q)}^{4}
= &\int_{0}^{T}\int_{\Omega}| \Delta u_{\varepsilon m}'(x,t) | ^{4}dx\,dt\\
\leq & 2\int_{0}^{T}\int_{\Omega}(1-\Delta |u_{\varepsilon m}'(x,t)| ^{2})
^{-}(\Delta u_{\varepsilon m}'(x,t) )
^{2}dx\,dt+ \mathop{\rm meas}(Q)\\
=& 2\int_{0}^{T}\langle \beta_{2}(\Delta u_{\varepsilon m}'(t) ) ,
u_{\varepsilon m}'(t) \rangle dx\,dt+\mathop{\rm meas}(Q)\\
 \leq & 2\int_{0}^{T}(\beta (u_{\varepsilon m}'(t)) ,u_{\varepsilon m}'(t) )dt
 +\mathop{\rm meas}(Q)\,.
\end{aligned}$$
Using (2.26), we obtain
\begin{equation}
\| \Delta u_{\varepsilon m}'\|_{L^{4}(Q)
}^{4}\leq C\varepsilon +\mathop{\rm meas}(Q) <C+\mathop{\rm meas}(Q)  \tag{2.29}
\end{equation}
with $C$ independent of $\varepsilon,m$ and $t\in [0,T[$.
Analogously, using the  Lemma \ref{lm2.1} with $h=\Delta \theta_{\varepsilon m}$
and (2.26), we obtain
\begin{equation}
\| \Delta \theta_{\varepsilon m}\|_{L^{4}(Q)
}^{4}\leq C+\mathop{\rm meas}(Q)  \tag{2.30}
\end{equation}
On the other hand, from (2.18) and (2.19), we obtain
\begin{align*}
\frac{1}{\varepsilon }\langle \beta (u_{\varepsilon m}'(t) ) ,v\rangle +
\frac{1}{\varepsilon }\langle \beta (\theta_{\varepsilon m}(t) ) ,v\rangle \leq C(| f(t) | ) + | g(t) | )\| v\|+
&\\ M(\| u_{\varepsilon m}(t) \| ^{2})\| u_{\varepsilon m}(t) \| .\| v\|+
C(| \theta_{\varepsilon m}(t) | ) +| u_{\varepsilon m}'(t) | )\leq
&\\
 |f(t) | | v| +| g(t)| | v|+ M(\| u_{\varepsilon m}(t) \| ^{2})
\| u_{\varepsilon m}(t) \| .\| v\|  |u_{\varepsilon m}''(t) | | v|
+ | \theta_{\varepsilon m}(t) | | v|+&\\
| \theta_{\varepsilon m}'(t) | |v|
+\| \theta_{\varepsilon m}(t) \| .\| v\|+| u_{\varepsilon m}'(t) | | v|\leq &\\
C\{| f(t) | +| g(t)| +|u_{\varepsilon m}''(t) |
 +| \theta_{\varepsilon m}(t) |
+ | \theta_{\varepsilon m}'(t) | +|u_{\varepsilon m}'(t) | \}\| v\|+&\\
(M(\| u_{\varepsilon m}(t) \|^{2}) \| u_{\varepsilon m}(t) \|
+\|\theta_{\varepsilon m}(t) \| ) \| v\|.\hspace{15pt}
\end{align*}
Since $f,g\in C^{0}([ 0,T] ;L^{2}(\Omega ))$, from
the inequality above we obtain
\begin{gather}
\frac{1}{\varepsilon }| \langle \beta (u_{\varepsilon m}'(
t) ) , v\rangle | \leq C_1\| v\| \quad \forall v\in
W_{0}^{2,4}(\Omega ) \,, \tag{2.31} \\
\frac{1}{\varepsilon }| \langle \beta (\theta_{\varepsilon m}(
t) ) , v\rangle | \leq C_1\| v\| \quad \forall v\in
W_{0}^{2,4}(\Omega ),  \tag{2.32}
\end{gather}
independent of $\varepsilon , m$ and $t\in [0,T]$; this is,
\begin{gather}
\| \beta (u'_{\varepsilon m}) \|_{L^{\infty }(
0,T;W^{2,4/3}(\Omega ) ) }\leq C_1\,, \tag{2.33} \\
\| \beta (\theta_{\varepsilon m}) \|_{L^{\infty
}(0,T;W^{2,4/3}(\Omega ) ) }\leq C_1\,.  \tag{2.34}
\end{gather}
To estimate $| \Delta u_{\varepsilon m}(t) |$, we note that
\begin{align*}
| \Delta u_{\varepsilon m}(t) | ^{2}
=&|\Delta u_{0\varepsilon m}| ^{2}+\int_{0}^{t}\frac{d
}{ds}| \Delta u_{\varepsilon m}(s) | ^{2}ds\\
=&|\Delta u_{0\varepsilon m}| ^{2}+
2C\int_{0}^{t}| \Delta u_{\varepsilon m}(s)| \| \Delta u_{\varepsilon m}'(s)
\| \\
\leq &| \Delta u_{0\varepsilon m}| ^{2}
+C \int_{0}^{t}(| \Delta u_{\varepsilon m}(s)
| ^{2}+\| \Delta u_{\varepsilon m}'(s)\| ^{2})ds\,,
\end{align*}
where $C$ is the  constant of the embedding from $H_{0}^{1}(\Omega ) $
into $L^{2}(\Omega )$.
From (2.20),\ (2.29) and Gronwall's inequality, we obtain
\begin{equation}
| \Delta u_{\varepsilon m}(t) | ^{2}<C\,,  \tag{2.35}
\end{equation}
where $C$ is a constant independent \ of$\;\varepsilon , m$ and $t\in
[ 0,T[$.

Next, we obtain an estimate for $\int_{0}^{t}\|
\Delta u_{\varepsilon m}'(s) \| ^{3}ds$.
Let $C$ represent various positives constants of the embedding in the
sequence
\begin{equation*}
W_{0}^{2,4}(\Omega ) \hookrightarrow H_{0}^{2}(\Omega
) \hookrightarrow H_{0}^{1}(\Omega )\,.
\end{equation*}
Observing that $W_{H^{2}(\Omega ) }\leq C| \Delta w| $ we obtain
\begin{equation}
\int_{0}^{t}\| u_{\varepsilon m}'(s)
\| ^{3}ds  \leq   C\int_{0}^{t}\| u_{\varepsilon
m}'(s) \|_{H^{2}(\Omega )}^{3}ds
 \leq   C\int_{0}^{t}| \Delta u_{\varepsilon m}'(s) | ^{3}ds,
\tag{2.36}
\end{equation}
independently of $\varepsilon $ and $m$.
It follows from H\"oder's inequality that
\begin{equation*}
\int_{0}^{t}| \Delta u_{\varepsilon m}'(
s) | ^{3}ds\leq (\int_{0}^{T}1^{1}ds)^{1/4}(\int_{0}^{t}\|
\Delta u_{\varepsilon m}'(s)\| ^{4}ds)^{3/4}
\end{equation*}
and substituting in (2.36) and observing (2.29), we obtain
\begin{equation}
\int_{0}^{t}\| u_{\varepsilon m}'(s)
\| ^{3}ds\leq C,  \tag{2.37}
\end{equation}
independent of $\varepsilon $, $m$ and $t\in [ 0,T[.$

\noindent{\bf Estimate (iv)} We will obtain the estimative for
$| u_{\varepsilon m}'' (t) | $.
Let us consider the functions
\begin{gather*}
\Psi_{h}(t) =\tfrac{1}{h}[u_{\varepsilon m}(t+h)-u_{\varepsilon m}(t) ]\,,\\
M_{h}(t) =\tfrac{1}{h}[M(\| u_{\varepsilon m}(t+h) \| ^{2})
- M(\| u_{\varepsilon m}(t) \| ^{2}) ]\,,\\
f_{h}(t) =\tfrac{1}{h}[f(t+h)-f(t)].
\end{gather*}
Setting $w=2\Psi_{h}'(t) $ in (1.14), we obtain
\begin{equation}
\begin{aligned}
2(u_{\varepsilon m}''(t) ,\Psi_{h}'(t) )+2M(\| u_{\varepsilon m}(t)
\| ^{2}) ((u_{\varepsilon m}(t) ,\Psi_{h}'(t) ) ) +\\
\frac{2}{\varepsilon }\langle \beta (u_{\varepsilon m}'(t)) ,\Psi_{h}'(t) )
\rangle =2(f(t) ,\Psi_{h}'(t) )
\end{aligned}. \tag{2.38}
\end{equation}
Substituting $t$ by $\ t+h$ $\in [ 0,T]$ in (2.18)  and
taking $w=2\Psi_{h}'(t) $, we set
\begin{equation}
\begin{aligned}
2(u_{\varepsilon m}''(t+h) ,\Psi_{h}'(t) ) +2M(\| u_{\varepsilon
m}(t+h) \| ^{2}) ((u_{\varepsilon m}(t+h) ,\Psi_{h}'(t) ) )+ \\
\frac{2}{\varepsilon }\langle \beta (u_{\varepsilon m}'(t+h))
,\Psi_{h}'(t) ) \rangle 
=2(f(t+h) ,\Psi_{h}'(t) ).
\end{aligned} \tag{2.39}
\end{equation}
Now, from (2.38) and (2.39) it follows, for $h\neq 0$, that
$$\begin{aligned}
2(\frac{u_{\varepsilon m}''(t+h)
-u_{\varepsilon m}''(t) }{h},\Psi_{h}'(t) )
+\frac{2}{h}M(\| u_{\varepsilon m}(t+h) \| ^{2}) ((u_{\varepsilon m}(t+h)
,\Psi_{h}'(t) ) ) -\\
\frac{2}{h}M(\| u_{\varepsilon m}(t) \|^{2})
((u_{\varepsilon m}(t) ,\Psi_{h}'(t) ) )
+\frac{2}{h\varepsilon }\langle \beta (u_{\varepsilon m}'(t+h) )
-\beta (u_{\varepsilon m}'(t)) ,\Psi_{h}'(t) \rangle =\\
2(\frac{f(t+h) -f(t) }{h},\Psi_{h}'(t) ),
\end{aligned}
$$
which implies
\begin{equation}
\begin{aligned}
\frac{d}{dt}| \Psi_{h}'(t) | ^{2}+\frac{2}{h}M(\| u_{\varepsilon m}(t+h)\| ^{2})
(u_{\varepsilon m}(t+h) ,\Psi_{h}'(t) )-  \\
\frac{2}{h}M(\| u_{\varepsilon m}(t) \|^{2}) ((u_{\varepsilon m}(t) ,\Psi_{h}'(t) ) ) 
+\\
\frac{2}{h\varepsilon }\langle \beta (u_{\varepsilon m}'(
t+h) ) -\beta (u_{\varepsilon m}'(t)) ,\Psi_{h}'(t) \rangle =
2(f_{h}(t) ,\Psi_{h}'(t) ).
\end{aligned} 
\tag{2.40}
\end{equation}
Nothing that
\begin{equation*}
\begin{aligned}
\frac{2}{h}M(\| u_{\varepsilon m}(t+h) \|^{2}) ((u_{\varepsilon m}(t+h)
,\Psi_{h}'(t) ) ) -\frac{2}{h}M(\| u_{\varepsilon m}(t) \|^{2})
((u_{\varepsilon m}(t) ,\Psi_{h}'(t) ) )=\\
2M(\| u_{\varepsilon m}(t+h) \| ^{2})((\Psi_{h}(t) ,\Psi_{h}'(t)) ) +
\frac{2M(\| u_{\varepsilon m}(t+h) \|
^{2}) }{h}((u_{\varepsilon m}(t) ,\Psi_{h}'(t) ) ) -\\
\frac{2M(\| u_{\varepsilon m}(t) \| ^{2})
}{h}((u_{\varepsilon m}(t) ,\Psi_{h}^{\prime}(t) ) ) 
=\\M(\| u_{\varepsilon m}(t+h) \| ^{2})
\frac{d}{dt}(\| \Psi_{h}(t) \| ^{2}) +
2M_{h}(t) ((u_{\varepsilon m}(t) ,\Psi_{h}'(t) ) ).
\end{aligned}
\end{equation*}
From (2.40) it follows that
\begin{equation*}
\begin{aligned}
\frac{d}{dt}| \Psi_{h}'(t) | ^{2}+M(\| u_{\varepsilon m}(t+h) \| ^{2})
\frac{d}{dt}(\| \Psi_{h}(t) \| ^{2}) +\\
\frac{2}{h^{2}\varepsilon }\langle \beta (u_{\varepsilon m}'(
t+h) ) -\beta (u_{\varepsilon m}'(t)) ,u_{\varepsilon m}'(t+h)
-u_{\varepsilon m}'(t) \rangle = \\
-2M_{h}(t) ((u_{\varepsilon m}(t) ,\Psi_{h}'(t) ) ) +2(f_{h}(t)
,\Psi_{h}'(t) ).
\end{aligned}
\end{equation*}
By the monotonicity of the operator $\beta $, we obtain
\begin{equation}
\begin{aligned}
\frac{d}{dt}| \Psi_{h}'(t) | ^{2}+M(
\| u_{\varepsilon m}(t+h) \| ^{2}) \frac{d}{dt}(\| \Psi_{h}(t) \| ^{2}) \\
\quad \leq 2| M_{h}(t) (\Delta u_{\varepsilon m}(
t) ,\Psi_{h}'(t) ) | +2| (f_{h}(t) ,\Psi_{h}'(t) ) |.
\end{aligned}
\tag{2.41}
\end{equation}
Integrating (2.41) in $t$ we have
$$\begin{aligned}
| \Psi_{h}'(t) |^{2}+\int_{0}^{t}M(\| u_{\varepsilon m}(s+h)
\| ^{2}) \frac{d}{ds}(\| \Psi_{h}(s)\| ^{2}) ds\leq \\
 | \Psi_{h}'(0) |^{2}+2\int_{0}^{t}| M_{h}(s) (\Delta
u_{\varepsilon m}(s) ,\Psi_{h}'(s) )| ds
+ 2\int_{0}^{t}| (f_{h}(s) ,\Psi_{h}^{\prime}(s) ) | ds\,.
\end{aligned}
$$
Taking the limit as $h\to 0$, it follows
\begin{equation}
\begin{aligned}
| u_{\varepsilon m}''(t) |^{2}+\int_{0}^{t}M(\| u_{\varepsilon m}(s)
\| ^{2}) \frac{d}{ds}\| u_{\varepsilon m}'(s) \| ^{2}ds\leq \\
 | u_{\varepsilon m}''(0) |^{2}+  2\int_{0}^{t}
[M'(\| u_{\varepsilon m}(s) \| ^{2}) \frac{d}{ds}\| u_{\varepsilon m}(
s) \| ^{2}]| \Delta u_{\varepsilon m}(s) ,u_{\varepsilon m}''(s) | ds+ \\
2\int_{0}^{t}| (f'(s) ,u_{\varepsilon m}''(s) ) | ds\,.
\end{aligned}\tag{2.42}
\end{equation}
Using Assumption (A2) and (2.28), we obtain, from (2.42),
\begin{equation}
\begin{aligned}
| u_{\varepsilon m}''(t) |^{2}+\int_{0}^{t}M(\| u_{\varepsilon m}(s)
\| ^{2}) \frac{d}{ds}\| u_{\varepsilon m}'(s) \| ^{2}ds\leq\\
 C+4\int_{0}^{t}| M'(\| u_{\varepsilon m}(
s) \| ^{2}) | \| u_{\varepsilon m}^{\prime}(s) \| \| u_{\varepsilon m}(s) \|
\| \Delta u_{\varepsilon m}(s) \| \|u_{\varepsilon m}''(s) \| ds+\\
\int_{0}^{t}| u_{\varepsilon m}''(s)|^{2} ds.
\end{aligned}
\tag{2.43}
\end{equation}
 From  (2.26), (2.35) and (2.37) it follows that there exists a positive
constant $C$ such that
\begin{equation}
\| u_{\varepsilon m}(t) \| ^{2}+| \Delta
u_{\varepsilon m}(t) | ^{2}+\int_{0}^{t}\|
u_{\varepsilon m}^{\prime  }(s) \|^{2} ds\leq
C, \quad \forall \varepsilon ,m,t.  \tag{2.44}
\end{equation}
Since $M\in C^{1}([0,\infty )) $, we also obtain from (2.44),
\begin{equation}
| M'(\| u_{\varepsilon m}(s) \|
^{2}) | \leq C, \quad \forall \varepsilon ,m,t .  \tag{2.45}
\end{equation}
On the other hand, using integration by parts, we get
$$\begin{aligned}
\int_{0}^{t}M(\| u_{\varepsilon m}(s) \|
^{2}) \frac{d}{ds}\| u_{\varepsilon m}'(s)\| ^{2}ds =\\
M(\| u_{\varepsilon m}(s) \|^{2}) \| u_{\varepsilon m}'(s) \| ^{2}
-M(\| u_{0\varepsilon m}(s) \|^{2}) \| u_{1\varepsilon m}(s) \|^{2}-\\
\int_{0}^{t}M'(\| u_{\varepsilon m}(s) \| ^{2})
\frac{d}{ds}\| u_{\varepsilon m}^{\prime}(s) \| ^{2}\| u_{\varepsilon m}'(
s) \| ^{2}ds.
\end{aligned}
$$
Estimates (2.37), (2.44), and (2.45) together imply
\begin{equation*}
-\int M'(\| u_{\varepsilon m}(s) \|^{2}) \frac{d}{ds}
\| u_{\varepsilon m}(s) \|^{2}\| u_{\varepsilon m}'(s) \| ^{2}ds\geq
-C\int_{0}^{t}\| u_{\varepsilon m}'(s)\| ^{3}\geq -C,
\end{equation*}
independently of $\varepsilon$, $m$, and $t$. Therefore,
\begin{equation}
\int_{0}^{t}M(\| u_{\varepsilon m}(s) \|
^{2}) \frac{d}{ds}\| u_{\varepsilon m}'(s)
\| ^{2}ds\geq M(\| u_{\varepsilon m}(t)
\| ^{2}) \| u_{\varepsilon m}'(t)
\| ^{2}-C,  \tag{2.46}
\end{equation}
independently of $\varepsilon$, $m$ and $t$.
Here, $C$ denote various positive constants.
Making use of inequalities (2.44)--(2.46) in (2.43) we obtain
\begin{equation}
| u_{\varepsilon m}''(t) |^{2}+M(\| u_{\varepsilon m}(t) \| ^{2})
\| u_{\varepsilon m}'(s) \| ^{2}\leq C+C\int_{0}^{t}| u_{\varepsilon m}''(
s) | ^{2}ds,  \tag{2.47}
\end{equation}
independently of  $\varepsilon$, $m$, and $t$.
From (2.47) and using Gronwall's inequality, we have
\begin{equation}
| u_{\varepsilon m}''(t) | ^{2}\leq C,
\tag{2.48}
\end{equation}
independently of $\varepsilon$, $m$ and $t$.

\paragraph{Passage to the limit}
By estimates (2.26) and (2.35) we obtain
\begin{equation*}
\begin{array}{c}
(u_{\varepsilon m})\quad \text{ is bounded in }\quad L^{\infty }(0,T;H_{0}^{1}(
\Omega ) \cap H^{2}(\Omega ) ), \\
(u_{\varepsilon m}')\quad \text{ is bounded in }\quad L^{\infty
}(0,T;L^{2}(\Omega ) ) ,\\
(\theta_{\varepsilon m})\quad \text{ is bounded in }\quad L^{\infty }(0,T;L^{2}(
\Omega ) ).
\end{array}
\end{equation*}
Therefore, we can get subsequences, if necessary,  denoted by
$(u_{\varepsilon m})$ and $(\theta_{\varepsilon m})$, such that
\begin{gather}
u_{\varepsilon m}\to u_{\varepsilon }\quad\text{ weak star in }\quad 
L^{\infty }(0,T;H_{0}^{1}(\Omega ) \cap H^{2}(\Omega) ),  \tag{2.49} \\
u_{\varepsilon m}'\to u_{\varepsilon }'\quad\text{ weak star in }\quad L^{\infty }(0,T;L^{2}(\Omega ) ),  \tag{2.50} \\
\theta_{\varepsilon m}\to \theta_{\varepsilon }\quad\text{ weak star in }\quad 
L^{\infty }(0,T;L^{2}(\Omega ) ).  \tag{2.51}
\end{gather}
Similarly by (2.48), we obtain
\begin{equation}
u_{\varepsilon m}''\to u_{\varepsilon }^{\prime
\prime }\quad\text{ weak star in }\quad L^{\infty }(0,T;L^{2}(\Omega ) ).
\tag{2.52}
\end{equation}
Also, by (2.33) and (2.34), there exist functions $\mathcal{X}_{\varepsilon
},\phi_{\varepsilon }\in L^{4/3}(0,T;W^{2,4/3}(
\Omega ) )$ such that
\begin{gather}
\beta (u_{\varepsilon m}') \to \mathcal{X}
_{\varepsilon }\quad\text{ in }\quad L^{4/3}(0,T;W^{2,4/3}(\Omega ) ) , \tag{2.53} \\
\beta (\theta_{\varepsilon m}) \to \phi
_{\varepsilon }\quad\text{ in }\quad L^{4/3}(0,T;W^{2,4/3}(\Omega ) ).  \tag{2.54}
\end{gather}
It follows from the  embeding  $W_{0}^{2,4}(\Omega ) $ into
$L^{4}(\Omega ) $ and of (2.29) that
\begin{equation*}
| u_{\varepsilon m}'|_{L^{4}(0,T;W_{0}^{2,4}(\Omega
) )}^{4}\leq C\| \Delta u_{\varepsilon m}'\|
_{L^{4}(\Omega ) }^{4}\leq K.
\end{equation*}
Therefore, there exists a subsequence of $\ (u_{\varepsilon m})$ such that
\begin{equation}
u_{\varepsilon m}'\to u_{\varepsilon }'\quad\text{
weak star in }\quad L^{4}(0,T;W_{0}^{2,4}(\Omega ) ).  \tag{2.55}
\end{equation}
Analogously, by (2.30) we obtain
\begin{equation}
\theta_{\varepsilon m}\to \theta_{\varepsilon }\quad\text{ weak star in}\quad L^{4}(0,T;W_{0}^{2,4}(\Omega ) ) . \tag{2.56}
\end{equation}
Being the embedding from $H_{0}^{1}(\Omega ) \cap H^{2}(
\Omega ) $ into $H_{0}^{1}(\Omega ) $ compact, we can
set a subsequence, again denoted by $(u_{\varepsilon m})$, such that:
\begin{equation}
u_{\varepsilon m}\to u_{\varepsilon }\quad \text{ strong in }\quad L^{2}(
0,T;H_{0}^{1}(\Omega ) ).   \tag{2.57}
\end{equation}
By assumption (A1) we obtain
\begin{equation}
M(\| u_{\varepsilon m}(t) \| ^{2})
\to M(\| u_{\varepsilon }(t) \|^{2}).   \tag{2.58}
\end{equation}
From the compactness of the embedding  $H_{0}^{1}(\Omega ) \hookrightarrow
L^{2}(\Omega ) $ we obtain
\begin{equation}
u_{\varepsilon m}'\to u_{\varepsilon }'\quad \text{
strong in }\quad L^{2}(0,T;L^{2}(\Omega ) ).   \tag{2.59}
\end{equation}
Then taking limit in the system (2.18)--(2.20), when $m\to
\infty $, with $w=v\varphi (t) $, $v\in W_{0}^{2,4}(\Omega
) $, $\varphi (t) \in \mathcal{D}(0,T) $ instead of $w_{j}$,
and using the fact that $\beta $ is monotone and
hemicontinous operator, we obtain that $\{u_{\varepsilon },\theta
_{\varepsilon }\}$ is a weak solution of the system (2.18)--(2.20).

The initial conditions (2.19) can be \ obtained by observing the
convergence above and the definition of weak solution; this is,
\begin{gather*}
u_{\varepsilon }'(0) =\lim_{m\to \infty }u
_{0\varepsilon m}=\lim_{m\to \infty
}\sum_{j=1}^{m}(u_{0\varepsilon},w_{j}) w_{j}=u_{0}\,,  \\
u_{\varepsilon }'(0) =\lim u_{1\varepsilon
m}=\lim_{m\to \infty }\sum_{j=1}^{m}(u
_{1\varepsilon},w_{j}) w_{j}=u_{1}\,,\\
\phi_{\varepsilon }(0) =\lim_{m\to \infty
}\theta_{0\varepsilon m}=\lim_{m\to \infty
}\sum_{j=1}^{m}(\theta_{0\varepsilon},w_{j}) w_{j}=\theta_{0}.
\end{gather*}
This concludes the proof of Theorem \ref{thm2.2}

\section{Main Result}

In this section, we will prove the Theorem \ref{thm2.1}.
By Theorem \ref{thm2.2},  there exists functions $u_{\varepsilon}, \theta_{\varepsilon }:\mathbb{Q}\to
\mathbb{R}$ such that
\begin{gather*}
u_{\varepsilon}\in L^{\infty }(0,T;H_{0}^{1}(\Omega ) \cap H^{2}(\Omega ) ), \\
u_{\varepsilon}',\theta_{\varepsilon }\in L^{4}(0,T;W_{0}^{2,4}(\Omega ) ) ,\\
u_{\varepsilon}''\in L^{\infty }(0,T;L^{2}(\Omega ) )  ,\\
\theta_{\varepsilon}\in L^{\infty }(0,T;L^{2}(\Omega) ),
\end{gather*}
satisfying the system
\begin{gather*}
(u_{\varepsilon}''(t) ,w)+M[\| u_{\varepsilon}(t) \| ^{2}]((
u_{\varepsilon}(t) ,w) ) + \frac{1}{\varepsilon}\langle \beta (u_{\varepsilon}'(
t) ,w) \rangle =(g(t) ,w)  ,\\
(\theta_{\varepsilon}(t) ,w) +((\theta_{\varepsilon }(t) ,w) ) +(
u_{\varepsilon}'(t) ,w) +\frac{1}{\varepsilon }\langle \beta
(\theta_{\varepsilon}(t),w) \rangle =(g(t) ,w),
\end{gather*}
a.e. in $[0,T]$, for all $w\in W_{0}^{2,4}(\Omega ) $.
$u_{\varepsilon }(0) =u_{0}$; $u_{\varepsilon}'(0) =u_{1}$, and
$\theta_{\varepsilon}(0) =\theta_{0}$.

Being the estimates (2.26), (2.29), (2.30), (2.33), (2.34), (2.32) and (2.44)
independently of $\varepsilon $, $m$ and $t$ we obtain by Uniform Boundedness
Theorem that there exists a positive constant $C$ such that
\begin{equation*}
\begin{aligned}
| u'_{\varepsilon}(t) | ^{2}+| \theta_{\varepsilon}(t) | ^{2}
+\| u_{\varepsilon}(t) \| ^{2}+\int_{0}^{T}\| \theta_{\varepsilon}(t)\| ^{2}ds+\\
\frac{2}{\varepsilon}\int_{0}^{T}\langle \beta (u_{\varepsilon}'(s) ,
u_{\varepsilon}'(s) ) \rangle ds+\frac{2}{\varepsilon }\int_{0}^{T}
\langle \beta(\theta_{\varepsilon }(s)) ,\theta_{\varepsilon
}(s) ) \rangle ds\leq \\
C \| \Delta u_{\varepsilon}'\|_{L^{4}(Q) }^{4} \leq &C\,,
\end{aligned}
\end{equation*}
and
\begin{gather*}
\| \Delta \theta_{\varepsilon }\|_{L^{4}(Q) }^{4}\leq C\,,\quad
\| \beta (u_{\varepsilon}') \|_{L^{\frac{4}{3}}(0,T;W^{2,4/3}(\Omega ) )}
\leq C\,, \\
\| \beta (\theta_{\varepsilon }) \|_{L^{4/3}(0,T;W^{2,4/3}(\Omega ) )}\leq C\,,
\quad
| \Delta u_{\varepsilon}(t) | ^{2}\leq C\,, \quad
| u_{\varepsilon }''(t) | ^{2}\leq C.
\end{gather*}
Consequently, we can find a subnet, which we still represent by
$(u_{\varepsilon})$,\ $(\theta_{\varepsilon}) $ such that
\begin{gather*}
u_{\varepsilon}\to u\quad \text{ weak star in}\quad L^{\infty }(
0,T;H_{0}^{1}(\Omega ) \cap H^{2}(\Omega ) ),\\
u_{\varepsilon}'\to u'\quad \text{ weak star in }\quad L^{\infty }(0,T;L^{2}(\Omega ) ),  \\
u_{\varepsilon}''\to u''\quad \text{ weak star in }\quad L^{\infty }(0,T;L^{2}(\Omega ) ),  \\
\beta (u_{\varepsilon }') \to \beta
(u') \quad \text{ weak in }\quad L^{4/3}(0,T;W^{-2,\frac{4}{3}}(\Omega ) ),  \\
\beta (\theta_{\varepsilon }) \to \beta (
\theta )\quad \text{ weak in }\quad L^{4/3}(0,T;W^{-2,\frac{4}{3}}(\Omega ) ),  \\
u_{\varepsilon}'\to u'\quad \text{ weak in }\quad L^{4}(0,T;W_{0}^{2,4}(\Omega ) ),  \\
\theta_{\varepsilon}\to \theta \quad \text{ weak in }\quad L^{4}(0,T;W_{0}^{2,4}(\Omega ) ).
\end{gather*}
By the compactness theorem of Aubin-Lions \cite{l1}, we obtain
\begin{gather}
u_{\varepsilon}\to u\quad \text{ strongly }\quad L^{2}(
0,T;H_{0}^{1}(\Omega ) ),   \tag{3.1} \\
u_{\varepsilon}'\to u'\quad \text{ strongly }\quad 
L^{2}(0,T;L^{2}(\Omega ) ) .  \tag{3.2}
\end{gather}
We observe that
\begin{gather*}
(u_{\varepsilon}''(t) ,v(t) ) +M(\| u_{\varepsilon}(t)
\| ^{2}) ((u_{\varepsilon }(t),v(t) ) ) +(\theta_{\varepsilon}(t) ,v(t) )
+\\ \frac{1}{\varepsilon}\langle \beta (u_{\varepsilon }'(t) ),v(t) \rangle 
=(f(t) ,v(t))\,,\\
(\theta_{\varepsilon}'(t) ,v(t)) +((\theta_{\varepsilon}(t) ,v(t) ) )
+(u_{\varepsilon}'(t) ,v(t) )
+ \frac{1}{\varepsilon}\langle \beta (\theta_{\varepsilon }(t)),v(t) \rangle
=(g(t) ,v(t) )\,.
\end{gather*}
is true for all $v\in L^{4}(0,T;W_{0}^{2,4}(\Omega )) $.

On the other hand, being $u_{\varepsilon}', \theta
_{\varepsilon }\in L^{4}(0,T;W_{0}^{2,4}(\Omega )
) $ implies
\begin{gather*}
(u_{\varepsilon}''(t) ,u_{\varepsilon}'(t) ) +M(\| u_{\varepsilon
}(t) \| ^{2}) ((u_{\varepsilon}(t) ,u_{\varepsilon }'(t) ))
+(\theta_{\varepsilon }(t) ,u_{\varepsilon}'(t) ) +\\ \frac{1}{\varepsilon }
\langle \beta(u_{\varepsilon }'(t) ),u_{\varepsilon}'(t) \rangle
=(f(t) ,u_{\varepsilon}'(t) )\,,  \\
(\theta_{\varepsilon }'(t) ,\theta_{\varepsilon }(t) ) +((\theta
_{\varepsilon }(t) ,\theta_{\varepsilon }(t) ) ) +(u_{\varepsilon }'(
t) ,\theta_{\varepsilon }(t) )
+\frac{1}{\varepsilon }\langle \beta (\theta_{\varepsilon }(t)
),\theta_{\varepsilon }(t) \rangle =(g(t),\theta_{\varepsilon }(t) ).
\end{gather*}
Subtracting the equations of the system above, we obtain
\begin{align*}
 \tag{3.3}
(u_{\varepsilon }''(t) ,v(t) -u_{\varepsilon }'(t) ) +M(\| u_{\varepsilon }(t)
\| ^{2}) ((u_{\varepsilon }(t) ,v(t) -u_{\varepsilon}(t) ) )  +&
\nonumber \\  
(\theta_{\varepsilon }(t) ,v-u_{\varepsilon}'(t) ) + \frac{1}{\varepsilon }
\langle \beta(u_{\varepsilon }'(t) ),v(t)-u_{\varepsilon }'(t) \rangle
=&(f(t),v(t) -u_{\varepsilon }'(t) )\,, 
\end{align*}
\begin{equation}
\begin{aligned}
(\theta_{\varepsilon }'(t) ,v(t)-\theta_{\varepsilon }(t) ) +
((\theta_{\varepsilon }(t) ,v(t) -\theta_{\varepsilon}(t) ) ) +& \\
 (u_{\varepsilon }^{\prime}(t) ,v(t) -\theta_{\varepsilon }(t) ) +
\frac{1}{\varepsilon }\langle \beta (\theta_{\varepsilon }(t)),v(t) 
-\theta_{\varepsilon }(t) \rangle
&=  (g(t) ,v(t) -\theta_{\varepsilon }(t) ), 
\end{aligned}
\tag{3.4}
\end{equation}
for all $v\in W_{0}^{2,4}(\Omega ) $.

Let us consider $v(t) \in K$ a. e. in $[0,T]$. Then we obtain
$\beta(v(t) )=0$ and being $\beta $ a monotone operator, we have
\begin{gather*}
\langle\beta (u_{\varepsilon }'(t) )-\beta (v(t) ) ,v(t) -u_{\varepsilon }'(
t) \rangle \leq 0\,,\\
\langle \beta (\theta_{\varepsilon }(t) )-\beta (v(t) ) ,v(t)
-\theta_{\varepsilon }(t) \rangle \leq 0\,.
\end{gather*}
Therefore,
\begin{gather}
\int_{0}^{T}(u_{\varepsilon }''(t) -M(\| u_{\varepsilon }(t) \|
^{2}) \Delta u_{\varepsilon }(t) +\theta_{\varepsilon }(t) -f(t) ,v(t)
-u_{\varepsilon }'(t) )dt \geq 0,  \tag{3.5} \\
\int_{0}^{T}(\theta_{\varepsilon }(t)-\Delta \theta_{\varepsilon }
+u_{\varepsilon }'-g(t) ,v(t) -\theta_{\varepsilon }(t))dt \geq 0,  \tag{3.6}
\end{gather}
for all $v\in L^{4}(0,T;W_{0}^{2,4}(\Omega ))$ with $v(t)\in K$ a.e. in $[0,T]$.
Now, taking the limit in (3.6) and (3.7), when $\varepsilon \to 0 $
and using (3.1)--(3.3) and observing that
$\Delta u_{\varepsilon }\to \Delta u$  weak in $L^{2}(0,T;L^{2}(\Omega ) )$
it follows that $u,\theta $ satisfy (1.5) and (1.6) in Theorem \ref{thm2.1}.

To conclude the proof of the existence of a solution, we show that \linebreak
$u'(t), \theta (t) \in \mathbb{K}$ a.e. in $[0,T]$. In fact, by (2.33)
and (2.34) we have
\begin{gather*}
\| \beta (u_{\varepsilon }') \|_{L^{\infty }(0,T;W^{2,\frac{4}{3}}(\Omega ) )}
\leq C\varepsilon ,\\
\| \beta (\theta_{\varepsilon }) \|_{L^{\infty
}(0,T;W^{2,\frac{4}{3}}(\Omega ) )}\leq C\varepsilon .
\end{gather*}
Therefore, as $\varepsilon \to 0$,
$\beta (u_{\varepsilon }') \to 0$ and $\beta (\theta_{\varepsilon }) \to 0$
strong $L^{\infty }(0,T;W^{2,\frac{4}{3}}(\Omega ) )$.

On the other hand we have
$\beta (u_{\varepsilon }') \to \beta (u')$  and
$\beta (\theta_{\varepsilon }) \to \beta (\theta )$
weak in $L^{4/3}(0,T;W^{2,4/3}(\Omega) ) $. Then,
$\beta (u'(t)) =\beta (\theta(t) ) =0$ in $L^{\infty}(0,T;W^{2,4/3}(\Omega
) ) $. Therefore, $u'(t) , \theta (t) \in \mathbb{K}$ a.e.
in $[0,T]$.

The initial conditions (1.7) can be verified easily. This concludes the
proof of Theorem \ref{thm2.1}.

\section{Uniqueness}

For proving  uniqueness of solutions in Theorem \ref{thm2.1}, we consider the restriction
$$u_{0}\in H_{0}^{1}(\Omega ) \cap H^{2}(\Omega ), \quad
u_{0}(x) \geq 0 \;\;\text{a. e. in $\Omega $}, \;\;\text{and}\;\; \| u_{0}\| >0.
$$
Consequently  $\| u(t) \| >0$, for all $t\in [ 0,T]$.
In fact, if there exists $t_{0}\in [ 0,T]$ such that $\|u_{0}\| =0$, then
\begin{equation*}
\int_{\Omega }| u(x,t_{0}) | ^{2}dx\leq C\| u(t_{0} ) \| ^{2}=0,
\end{equation*}
where $C$ is the constant of the embedding $H_{0}^{1}(
\Omega ) \hookrightarrow L^{2}(\Omega )$. Therefore, \linebreak $u(x,t_{0}) =0$, a.e.
in $\Omega $.

Since $u'(t) \in K$ a.e. in $[0,T]$, we have $u'(t) \geq 0$ a.e. in $\Omega $.
This implies that
\begin{equation}
u(x,t) \geq u(x,0) =u_{0}(x) \quad \text{in }
\Omega \text{ a.e. in }[0,T].  \tag{4.1}
\end{equation}
Being $\| u_{0}\| >0$, there exists $\Omega '\subset\Omega $ with
$\| \Omega '\| >0$ such that that $u_{0}(x) >0$.
By (3.1) it follows that $u(x,t_{0}) >0$ in $\Omega $. This
is a contradiction.

\begin{theorem} \label{thm3.1}
Under the hypotheses of Theorem \ref{thm2.1}, if
\begin{description}
\item[i)]  $M(\lambda ) >0$ for all $\lambda >0$, and $M(0) =0$.

\item[ii)]  $u_{0}\in H_{0}^{1}(\Omega ) \cap H^{2}(\Omega) $,
$u_{0}(x) \geq 0$ a.e. in $\Omega $, and $\|u_{0}\| >0$,
\end{description}
Then the solution $\{u,\theta \}$ of Theorem \ref{thm2.1} is unique.
\end{theorem}

\paragraph{Proof.}
From (i) and (ii) it follows that
$$m_{0}=\min \{M(\| u(t) \| ^{2}) ;\;t\in [ 0,T]\}>0.$$
Suppose we have two pairs of solutions $\{u,\theta \}$ and $\{w,\varphi \}$
satisfying the conditions of Theorem \ref{thm2.1}. Let $\Psi =u-w$ and
$\phi =\theta-\varphi $. Thus, $\Psi $ and $\phi $ satisfy
\begin{gather*}
(\Psi ''(t) -M(\| u(t) \| ^{2}) \Delta \Psi (t) +\{M(\| w(t) \| ^{2})
-M(\| u(t) \| ^{2}) \}\Delta w+\phi (t) ,\Psi '(t) )\leq 0,\\
(\phi '(t) -\Delta \phi (t) +\Psi '(t) ,\phi (t) )\leq 0,
\end{gather*}
which implies
\begin{align*}
\frac{1}{2}\frac{d}{dt}\{| \Psi '(t) |
^{2}+| \phi (t) | ^{2}+\| \phi (t) \| ^{2}\}+M(\| u(t) \|
^{2}) \frac{1}{2}\frac{d}{dt}\| \Psi (t) \|^{2}+2(\phi (t) ,\Psi '(t) )& \leq \\
 \{M(\| u(t) \| ^{2}) -M(\| w(t) \| ^{2}) \}(\Delta w(t) ,\Psi '(t) )\,.&
\end{align*}
Since
$$M(\| u(t) \| ^{2}) \frac{d}{dt}\|\Psi (t) \| ^{2}=\frac{d}{dt}\{M(\| u(
t) \| ^{2}) \| \Psi (t) \| ^{2}\}-\frac{d}{dt}[M(\| u(t) \| ^{2})
]\| \Psi (t) \| ^{2}$$  we obtain
\begin{align*}
\frac{1}{2}\frac{d}{dt}\{| \Psi '(t) |^{2}+| \phi (t) | ^{2}+\| \phi (
t) \| ^{2}\}+\frac{d}{dt}\{M(\| u(t)\| ^{2}) \| \Psi (t) \| ^{2}\}+
2(\phi (t) ,\Psi '(t) )\leq & \\
\{M(\| u(t) \| ^{2}) -M(\| w(t) \| ^{2}) \}(\Delta w(t) ,\Psi '(t) )+&\\
M'(\| u(t) \| ^{2}) ((u'(t) ,u(t) ) \| \Psi (t) \| ^{2}.&
\end{align*}
Now, integrating this inequality form $0$ to $t<T$, we obtain
\begin{equation}
\begin{aligned}
\frac{1}{2}\{| \Psi '(t) | ^{2}+|\phi (t) | ^{2}+\| \phi (t)
\| ^{2}\}+M(\| u(t) \| ^{2})\| \Psi (t) \| ^{2}+\\
2\int_{0}^{t}(\phi (t) ,\Psi '(t) ) ds\leq  \\
\int_{0}^{t}\{M(\| u(s) \| ^{2}) -M(\| w(s) \|
^{2}) \}(\Delta w(s) ,\Psi '(s) ) ds+\\
\int_{o}^{t}M'(\| u(s) \|^{2}) ((u'(s) ,u(s) )
\| \Psi (s) \| ^{2}ds.  \end{aligned}\tag{4.2}
\end{equation}
Note that $\| u(t) \| $ and $\| u^{\prime}(t) \| \in L^{\infty }(0,T) $.
Then there exists a positive constant $C_{0}$ such that
$$\| u(t) \| \leq C_{0}\quad \text{and}\quad  \| u^{\prime}(t) \| \leq C_{0}
\quad \text{a.e. in $[0,T]$.}
$$
Since $M\in C^{1}([0,\infty ))$, it follows $| M'(\xi ) | \leq C_1$, for all
$\xi \in [ 0,C_{0}]$.

Now, by the Mean Value Theorem, for each $s\in [ 0,T]$, there exists
$\xi_{s}$ between $\| u(s) \| ^{2}$ and $\|
w(s) \| ^{2}$ such that
\begin{equation}
\begin{aligned}
| M(\| u(s) \| ^{2}) -M(\| w(s) \| ^{2}) |
\leq C_1|\| u(s) \| ^{2}-\| w(s) \|^{2}|\leq \\
 C_{2}\| u(s) -w(s) \| =C_{2}\| \Psi(s) \|.
\end{aligned} \tag{4.3}
\end{equation}
Observing that $| \Delta w(s) | \leq C_{3}$,  from (4.2) and (4.3) we obtain
that
\begin{align*}
& | \Psi '(t) | ^{2}+\| \phi (t) \| ^{2} +M(\| u(t)
\| ^{2}\|) \Psi (t) \| ^{2}\leq\\
& C_{4}\int_{o}^{t}\left\{ | \Psi '(s)
| ^{2}+\| \Psi (s) \| ^{2}+\| \phi (s) \| ^{2}\right\} ds,
\end{align*}
which implies
$$
| \Psi '(t) | ^{2}+\| \phi (t) \| ^{2}+\| \Psi (t) \| ^{2}\leq
C_{5}\int_{o}^{t}\left\{ | \Psi '(t)| ^{2}+\| \Psi (t) \| ^{2}
+\| \phi(s) \| ^{2}\right\} ds.
$$
where $C_{5}=C_{4}/\min \{1,m_{0}\}$. From the above inequality and
Gronwall inequality  if follows that
$\| \phi (t) \| =\| \Psi (t) \| =0$, i.e., $\phi $ and $\Psi $ are zero
almost everywhere. This completes the proof of uniqueness.

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\end{thebibliography}
\newpage
\noindent\textsc{Marcondes R. Clark }\\
Federal University of Piau\'{\i} - CCN - DM
\\
Av. Ininga  S/N - 64.049-550 - Teresina - PI - Brazil\\
e-mail: mclark@ufpi.br\\

\noindent\textsc{Osmundo A. Lima }\\
State University of Para\'{\i}ba-DM \\
CEP 58.109-095 - Campina Grande - PB- Brazil\\
e-mail: osmundo@openline.com.br

\end{document}