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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 130, pp. 1--18.\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{7mm}}

\begin{document}
\title[\hfilneg EJDE-2006/130\hfil Weak solutions]
{Weak solutions for a strongly-coupled nonlinear system}

\author[O. A. Lima, A. T. Lour\^edo, A. O. Marinho\hfil EJDE-2006/130\hfilneg]
{Osmundo A. Lima, Aldo T. Lour\^edo, Alexandro O. Marinho}  % in alphabetical order

\address{Osmundo A. Lima\newline
Universidade Estadual da Para\'iba, DME, Campina Grande - PB,
Brazil} \email{osmundo@hs24.com.br}

\address{Aldo T. Lour\^edo\newline
Universidade Estadual da Para\'iba, DME,
Campina Grande - PB, Brazil}
\email{aldotl@bol.com.br}

\address{Alexandro O. Marinho\newline
Universidade Federal da Para\'iba, DM, Jo\~ao Pessoa - PB, Brazil}
\email{nagasak@ig.com.br}

\date{}
\thanks{Submitted March 3, 2006. Published October 16, 2006.}
\thanks{O. A. Marinho is partially supported by CNPq-Brazil}
\subjclass[2000]{35L85, 35L05, 35L20, 35L70, 49A29}
\keywords{Weak solutions; coupled system; monotonic operator}

\begin{abstract}
 In this paper the authors study the existence of local weak
 solutions of the strongly nonlinear system
 \begin{gather*}
 u''+\mathcal{A}u +f(u,v)u = h_1 \\
 v''+\mathcal{A}v +g(u,v)v = h_2
 \end{gather*}
 where $ \mathcal{A}$ is the pseudo-Laplacian operator and $f$,
 $g$, $h_1 $ and $h_2 $ are given functions.
\end{abstract}

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

\section{Introduction}

Let $\Omega $ be an open and bounded subset in ${\mathbb{R}^n} $
with smooth boundary $\Gamma $ and let $T $ be a positive real
number. In the cylinder $Q=\Omega \times ]0, T[$, with lateral
boundary $\sum = \Gamma \times ]0, T[$, we consider the nonlinear
system
\begin{equation} \label{e*}
 \begin{gathered}
 u''+ \mathcal{A}u + f(u,v)u=h_1 \\
 v''+ \mathcal{A}v + g(u,v)v=h_2 \\
 u(0)=u_0,\quad v(0)=v_0, \quad u'(0)=u_1, \quad v'(0)=v_1 \\
 u=v=0 \quad \mbox{on } \Sigma = \Gamma \times ]0, T[
 \end{gathered}
\end{equation}
where
$$
\mathcal{A}u=- \sum_{i=1}^n\frac{\partial}{\partial x_i}
\Big(\big |\frac{\partial u}{\partial x_i}|^{p-2}\frac{\partial
u}{\partial x_i} \Big), \quad p>2,
$$
is the pseudo-Laplacian operator, $f$ is a continuous function in
the first variable and Lipschitz in the second variable and $g $ is
a Lipschitz's function in the first variable and continuous in the
second variable, with $f(0,0)=g(0,0)=0 $ and $u_0$, $v_0$, $u_1$,
$v_1$, $h_1 $ and $h_2 $ are given functions.

When $p\geq 2$, many authors studied the system \eqref{e*}. For
instance, we can mention: Segal \cite{segal}, where the physical
meaning of
 \eqref{e*} is presented, Medeiros and Menzala \cite{luis1}, Medeiros
and M. Miranda \cite{luis2}, Castro
\cite{castro}, Biazutti \cite{biazutti} and more recently, Clark
and Lima \cite{clark1} showed the existence, a local solution and
its uniqueness for the system
\begin{gather*}
u'' -\Delta u + f(u,v)u= h_1 \quad \mbox{in } Q=\Omega\times (0,T)\\
v'' -\Delta u + g(u,v)v= h_2 \quad \mbox{in } Q\\
u(0)=u_0,\quad u'(0)=u_1 \quad \mbox{in } \Omega\\
v(0)=v_0,\quad v'(0)=v_1 \quad \mbox{in } \Omega \\
u=0,\quad v=0 \quad \mbox{on }\Sigma=\Gamma\times (0,T) ,
\end{gather*}
where the functions $f$ and $g$ satisfying the same
conditions of the problem \eqref{e*}. Castro \cite{castro} showed
the existence of solution for the system
\begin{gather*}
u'' + \mathcal{A}u -\Delta u' + |v|^{\rho + 2}|u|^{\rho}u = f_1 \quad
 \mbox{in }Q\\
v'' + \mathcal{A}v -\Delta v' + |u|^{\rho + 2}|v|^{\rho}v = f_2 \quad
 \mbox{in }Q\\
u(0)=u_0,\quad u'(0)=u_1 \quad  \mbox{in } \Omega \\
v(0)=v_0, \quad v'(0)=v_1 \quad \mbox{in } \Omega \\
u=0,\quad v=0 \quad \mbox{on }\Sigma,
\end{gather*}
 where $\mathcal{A}$ is the pseudo-Laplacian operator.
We can show that the functions $f(u,v)=|u|^{\rho +2}|v|^{\rho}$ and
$g(u,v)=|v|^{\rho + 2}|u|^{\rho},\rho \geq -1$, satisfy the
conditions of the system \eqref{e*}. Consequently the above system,
without the dissipations $\Delta u'$ and $\Delta v'$, is a
particular case of $(\ast)$. Thus, we see that \eqref{e*}
generalizes the above mentioned problems.

To show the existence of a {\it local} solution for
\eqref{e*}, we encounter following technical difficulties:
\begin{itemize}
\item[(i)] The choices of the functional spaces;
\item[(ii)] In the a priori estimate for $u''_m$, we had that to use
the projection operator, since, to derive the approximated equation
we will have much technical difficulties because of the
pseudo-Laplacian operator in the equation;
\item[(iii)] In the passage to the limit, we use strongly the fact
that $\mathcal{A}$ is a monotonic and hemicontinuous operator.
\end{itemize}
We remark that these difficulties do not appear in \cite{clark1}.

\subsection*{Notation}
We represent the Sobolev space of order $m $ in $\Omega $ by
$$
W^{m,p}(\Omega)=\{u \in L^p(\Omega): D^{\alpha}u \in L^p(\Omega) \forall
| \alpha | \leq m \},
$$
with the norm
$$
\| u \|_{m,p}=\Big(\sum_{|\alpha | \leq m} |D^{\alpha}u | ^p_{L^p(\Omega)}
 \Big) ^ {1/p}, u \in W^{m,p}(\Omega), 1\leq p <\infty.
$$
Let $\mathcal{D}(\Omega) $ be the space of test functions in
$\Omega$ and by $W_0^{m,p}(\Omega) $ we represent the closure of
$\mathcal{D}(\Omega) $ in $ W^{m,p}(\Omega)$. The dual space of
$W_0^{m,p}(\Omega) $ is denoted by $W^{-m,p '} (\Omega) $ with $p '
$ is such that $  \frac{1}{p} + \frac{1}{p '} =1$. We
use the symbols $(\cdot ,\cdot ) $ and $|\cdot|$, to indicate the inner
product and the norm in $L^2(\Omega)$. We use
$\langle \cdot,\cdot \rangle _{W^{-1,p}(\Omega),W_0^{1,p}(\Omega)} $ to
indicate the duality between $W^{-1,p '}(\Omega) $ and
$W_0^{1,p}(\Omega) $ and $ \|\cdot\|_0 $ to indicate the norm
$W_0^{1,p}(\Omega). $ The pseudo-Laplacian operator $\mathcal{A} $
is such that
$$
\begin{array}{cccc}
 \mathcal{A}: & W_0^{1,p}(\Omega) & \to & W^{-1,p '}(\Omega) \\
 & u & \mapsto & {\mathcal{A}u}
\end{array}
$$
and it satisfies the following properties:
\begin{itemize}
\item $\mathcal{A} $ is monotonic, that is, $\langle \mathcal{A}u - \mathcal{A}v, u-v
\rangle \geq 0, \forall u, v \in W_0^{1,p}(\Omega)$;

\item $\mathcal{A} $ is hemicontinuous, that is, for each
$u, v, w \in W_0^{1,p}(\Omega) $ the function
$\lambda \mapsto \langle \mathcal{A}(u+ \lambda
v),w \rangle $ is continuous in $\mathbb{R}$;
\item $\langle \mathcal{A}u(t),u(t)\rangle_{W^{-1,p '}(\Omega)
\times W_0^{1,p}(\Omega)} = \|u\|_0^p $;
\item $\langle \mathcal{A}u(t),u'(t)\rangle _{W^{-1,p '}(\Omega)
\times W_0^{1,p}(\Omega)} = \frac{1}{p}\frac{d}{dt}\|u\|_0^p $,
 $\frac{d}{dt} = ' $;
\item $\| \mathcal{A}u(t)\|_{W^{-1,p '}(\Omega)} \leq C \|u\|_0^{p-1}$, where
$C $ is a constant;
\end{itemize}
We will use the same notation for the operator $P $ and its
restrictions, as well as for the operator $P^{\ast}$.

 The next lemma plays a central role in the proof of the Existence Theorem.
Its proof can be found  in \cite{clark1}.

\begin{lemma} \label{mainlemma}
Let $\phi $ be a positive real function, $\alpha, \beta $ and $\gamma $,
positive real constants, with $\gamma>1 $, such that
$$
\phi(t)\leq \alpha + \beta\int_0^t\big\{\phi(s) + \phi^{\gamma}(s)\big \}ds.
$$
Then, there exists $T_0\in\mathbb{R}$, with $0 <T_0 <T $, such
that $\phi $ is bounded in $[0,T_0[$.
\end{lemma}


 \noindent \textbf{Definition.} A local weak solution
of the problem \eqref{e*} is a pair of functions
$u=u(x,t)$, $v=v(x,t)$ defined for all
$(x,t)\in Q_{T_0}=\Omega\times (0,T_0)$,  and $T_0>0 $ fixed, satisfying
\begin{gather*}
u,v \in L^{\infty}(0,T_0;W_0^{1,p}(\Omega)); \\
 u',v' \in L^{\infty}(0,T_0;L^2(\Omega)); \\
 \frac{d}{dt}(u',w) + \langle \mathcal{A}u,w\rangle
 + \langle f(u,v)u,w \rangle =\big( h_1,w\big), \forall
w \in W_0^{1,p}(\Omega) \mbox{in } D'(0,T_0); \\
 \frac{d}{dt}(v',w) + \langle \mathcal{A}v,w\rangle
 + \langle g(u,v)v,w \rangle =\big(
h_2,w\big), \quad \forall w \in W_0^{1,p}(\Omega) \mbox{ in }
D'(0,T_0);\\
 u(0)=u_0, \quad u'(0)=u_1,\quad  v(0)=v_0, \quad
 v'(0)=v_1.
\end{gather*}

\section{Existence Results}

\begin{theorem} \label{thm31}
Let $f $ and $g $ be functions of two variables
such that $f $ is continuous in the first variable and Lipschitz in
the second variable and $g $ is Lipschitz in the first and
continuous in the second variable, with $f(0,0)=g(0,0)=0 $.
\begin{gather} \label{e1}
h_1,h_2 \in L^{2}(0,T;L^2(\Omega)); \\
 \label{e2} u_0,v_0 \in W_0^{1,p}(\Omega); \\
 \label{e3} u_1,v_1 \in L^2(\Omega).
\end{gather}
Then it exists $T_0>0$, $T_0 \in \mathbb{R}$ and functions
$u:Q_{T_0}\to \mathbb{R}$ and $v:Q_{T_0}\to \mathbb{R}$ satisfying
\begin{gather} \label{e4}
u,v \in L^{\infty}(0,T_0;W_0^{1,p}(\Omega));\\
\label{e5} u',v' \in L^{\infty}(0,T_0;L^2(\Omega)); \\
\label{e6}  \frac{d}{dt}(u',w) + \langle \mathcal{A}u,w\rangle +
\langle f(u,v)u,w\rangle =\big(h_1,w\big), \quad
 \forall   w  \in W_0^{1,p}(\Omega) , \mbox{ in } D'(0,T_0); \\
\label{e7}  \frac{d}{dt}(v',w) + \langle \mathcal{A}v,w\rangle
 + \rangle g(u,v)v,w \rangle = \big(h_2,w\big), \quad
 \forall   w  \in W_0^{1,p}(\Omega), \mbox{ in }   D'(0,T_0);\\
 \label{e8} u(0)=u_0,  \quad  v(0)=v_0; \\
 \label{e9} u'(0)=u_1, \quad  v'(0)=v_1.
\end{gather}
\end{theorem}

The main tools in the proof of this theorem are the Faedo-Galerkin method and compactness arguments.
Let $H_0^s(\Omega)$, with $s>m=n\big(\frac{1}{2}-\frac{1}{p}\big) + 1 $
a separable Hilbert space such that
$H_0^s(\Omega)\hookrightarrow W_0^{1,p}(\Omega)$,
is a continuous and dense immersion. In $H_0^s(\Omega) $, there
exists a complete orthonormal hilbertian base $ \{w_j \}_{j\in N} $
in $L^2(\Omega)$. We consider $V_m=[w_1,\dots ,w_m] $ the
subspace of $H_0^s(\Omega) $ generated by the $m $ first vectors of
the base $ \{w_j \}_{j\in \mathbb{N}}$. Also, we have the following
chain of continuous and dense immersions.
\begin{equation} \label{e10}
H_0^s(\Omega)\hookrightarrow W_0^{1,p}(\Omega)\hookrightarrow
L^2(\Omega)\hookrightarrow W^{-1,p'}(\Omega)\hookrightarrow
H^{-s}(\Omega).
\end{equation}
We will divide the proof in three steps: $(i)$ Approximated Problem,
$(ii)$ A Priori Estimates $I$ and $(iii)$ A Priori Estimates $II$.

\subsection*{Approximated Problem}
We want to find $u_m(t),v_m(t) $ in $V_m $ satisfying the
approximated problem.
\begin{gather}
(u_{m}''(t),w) + \langle \mathcal{A}u_{m}(t),w \rangle +
\langle f(u_m(t),v_m(t))u_m(t),w \rangle
= (h_{1}(t),w), \label{e11}\\
(v_{m}''(t),w) + \langle \mathcal{A}v_{m}(t),w\rangle +
 \langle g(u_m(t),v_m(t))v_{m}(t),w\rangle
= (h_{2}(t),w), \label{e12}
\end{gather}
for all $w\in V_{m}$;  and
\begin{equation} \label{e13}
\begin{gathered}
u_{m}(0) = u_{0m} , \quad u_{m}'(0) = u_{1m}, \\
v_{m}(0) = v_{0m}, \quad  v_{m}'(0) = v_{1m};
\end{gathered}
\end{equation}
So that
\begin{gather*}
u_{0m}\to u_{0},\quad v_{0m}\to v_{0},
\quad \mbox{in } W_{0}^{1,p}(\Omega); \label{e14} \\
u_{1m}\to u_{1},\quad v_{1m}\to v_{1},\quad \mbox{in } L^{2}(\Omega).\label{e15}
\end{gather*}
It can be shown that the above system satisfies the Caracthéodory's
conditions; therefore there exists solutions $u_m(t),v_m(t) $ in
$[0,t_m)$,  $t_m <T $ satisfying \eqref{e11}--$(2.13)$.

\subsection*{A priori estimates I}
Let us consider $w=2u'_m(t) $ in \eqref{e11}. It follows that
\begin{align*}
&2(u''_m(t),u'_m(t)) + 2\langle \mathcal{A}u_m(t),u'_m(t)\rangle
+ 2\langle f(u_m(t),v_m(t))u_m(t),u'_m(t)\rangle \\
&= (h_1(t),u'_m(t)).
\end{align*}
Thus
\begin{equation*}%\label{16}
\frac{d}{dt}|u'_m(t)|^2 +\frac{2}{p}\frac{d}{dt}\|u_m(t)\|_0^p
=2(h_1(t),u'_m(t)) - 2\langle f(u_m(t),v_m(t))u_m(t),u'_m(t)\rangle .
\end{equation*}
Similarly, setting $w=2v'_m(t) $ in \eqref{e12} it follows that
\begin{equation*}%\label{17}
\frac{d}{dt}|v'_m(t)|^2 +\frac{2}{p}\frac{d}{dt}\|v_m(t)\|_0^p
=2(h_2(t),v'_m(t)) - 2\langle g(u_m(t),v_m(t))u_m(t),v'_m(t)\rangle .
\end{equation*}
Summing the two equalities above, then integrating from $0$ to $t$,  $t<t_m$,
and using the Cauchy-Schwarz's inequality
 and $ab\leq\frac{a^2 + b^2}{2}$, we obtain
\begin{align*} %\label{19}
&|u'_m(t)|^2 +|v'_m(t)|^2 +\frac{2}{p}\|u_m(t)\|_0^p
+\frac{2}{p}\|v_m(t)\|_0^p \\
&\leq |u'_m(0)|^2 + |v'_m(0)|^2 +
\frac{2}{p}\|u_m(0)\|_0^p +\frac{2}{p}\|v_m(0)\|_0^p\\
&\quad + 2\int_{0}^t \int_{\Omega}|f(u_m(s),v_m(s))| |u_m(s)| |u'_m(s)|ds \\
&\quad + 2\int_{0}^t\int_{\Omega}|g(u_m(s),v_m(s))| |v_m(s)| |v'_m(s)|ds  \\
&\quad + \int_{0}^t\big(|u'_m(s)|^2 +
|v'_m(s)|^2\big)ds +  \int_{0}^T\big(|h_1(t)|^2 +
|h_2(t)|^2\big)dt.
\end{align*}
 From \eqref{e1}, \eqref{e14}, and \eqref{e15}, it follows that
\begin{equation}  \label{e20}
\begin{aligned}
&|u'_m(t)|^2 +|v'_m(t)|^2 +\frac{2}{p}\|u_m(t)\|_0^p
+\frac{2}{p}\|v_m(t)\|_0^p \\
&\leq   C+ \int_{0}^t\big(|u'_m(s)|^2 + |v'_m(s)|^2\big)ds \\
&\quad  + 2\int_{0}^t |f(u_m(s),v_m(s))| |u_m(s)| |u'_m(s)|ds  \\
&\quad + 2\int_{0}^t|g(u_m(s),v_m(s))| |v_m(s)| |v'_m(s)|ds.
\end{aligned}
\end{equation}
 From the Sobolev immersions it is well known that
\begin{equation*} \label{e21}
W_0^{1,p}(\Omega)\hookrightarrow L^q(\Omega), \quad \forall  1\leq
q\leq\frac{np}{n-p}.
\end{equation*}
Let $\alpha, \beta> 0 $, such that
$  \frac{1}{\alpha}+ \frac{1}{\beta} + \frac{1}{2}=1 $, with
$1\leq \alpha, \beta\leq  \frac{np}{n-p}$.

Now, using Holder and Young  inequalities,
the inequality $ab\leq\frac{a^2+b^2}{2}$ and the hypothesis over $f$, we have
\begin{align*} %\label{22}
&2\int_{0}^t\int_{\Omega} |f(u_m(s),v_m(s))| |u_m(s)| |u'_m(s)|ds\\
& \leq  C\int_{0}^t \int_{\Omega}|v_m(s)| |u_m(s)| |u'_m(s)|ds\\
& \leq C\int_{0}^t \Big(\int_{\Omega}|v_m(s)|^{\alpha}\Big)^{\frac{1}{\alpha}}
\Big(\int_{\Omega}|u_m(s)|^{\beta}\Big)^{\frac{1}{\beta}}
\Big(\int_{\Omega}|u'_m(s)|^2\Big)^{2}\\
&= C\int_{0}^t|v_m(s)|_{L^{\alpha}(\Omega)}
|u_m(s)|_{L^{\beta}(\Omega)}|u'_m(s)|_{L^{2}(\Omega)}ds\\
&\leq C\int_{0}^t\Big\{\frac{1}{p}|v_m(s)|_{L^{\alpha}(\Omega)}^p+
\frac{p-1}{p}|u_m(s)|_{L^{\beta}(\Omega)}^{\frac{p}{p-1}}\Big\}
|u'_m(s)|_{L^{2}(\Omega)}ds \\
&\leq C\int_{0}^t\Big\{\frac{1}{p}|v_m(s)|_{L^{\alpha}(\Omega)}^p+
\frac{1}{p}|u_m(s)|_{L^{\beta}(\Omega)}^{\frac{p}{p-1}(p-1)}
+ \frac{p-2}{p-1}\Big\} |u'_m(s)|_{L^{2}(\Omega)}ds\\
&= C\int_{0}^t\Big\{\frac{1}{p}|v_m(s)|_{L^{\alpha}(\Omega)}^p+
\frac{1}{p}|u_m(s)|_{L^{\beta}(\Omega)}^{p} + \frac{p-2}{p-1}\Big\}
|u'_m(s)|_{L^{2}(\Omega)}ds\\
&\leq C\int_{0}^t\Big\{\frac{1}{p}|v_m(s)|_{L^{\alpha}(\Omega)}^p+
\frac{1}{p}|u_m(s)|_{L^{\beta}(\Omega)}^{p} +
\frac{p-2}{p-1}\Big\}^2 + |u'_m(s)|_{L^{2}(\Omega)}^2ds\\
&\leq C\int_{0}^t\Big\{\frac{1}{p^2}|v_m(s)|_{L^{\alpha}(\Omega)}^{2p}+
\frac{1}{p^2}|u_m(s)|_{L^{\beta}(\Omega)}^{2p} +
\big(\frac{p-2}{p-1}\big)^2+
|u'_m(s)|_{L^{2}(\Omega)}^2\Big\}ds \\
&\leq  C\int_{0}^t\Big\{\frac{1}{p^2}|v_m(s)|_{L^{\alpha}(\Omega)}^{2p}+
\frac{1}{p^2}|u_m(s)|_{L^{\beta}(\Omega)}^{2p} + 1 +
|u'_m(s)|_{L^{2}(\Omega)}^2\Big\}ds.
\end{align*}
Since $W_0^{1,p}(\Omega)\hookrightarrow L^{\alpha}(\Omega)$ and
$W_0^{1,p}(\Omega)\hookrightarrow L^{\beta}(\Omega)$, it follows that
\begin{equation} \label{e23}
\begin{aligned}
&2\int_{0}^t\int_{\Omega} |f(u_m(s),v_m(s))| |u_m(s)| |u'_m(s)|ds \\
&\leq C\int_{0}^t\Big\{\frac{1}{p^2}\|v_m(s)\|_0^{2p}+
\frac{1}{p^2}\|u_m(s)\|_0^{2p} + 1 +
|u'_m(s)|_{L^{2}(\Omega)}^2\Big\}ds.
\end{aligned}
\end{equation}
Similarly, we have
\begin{equation} \label{e24}
\begin{aligned}
&2\int_{0}^t\int_{\Omega} |g(u_m(s),v_m(s))| |v_m(s)| |v'_m(s)|ds \\
&\leq C\int_{0}^t\Big\{\frac{1}{p^2}\|u_m(s)\|_0^{2p}+
\frac{1}{p^2}\|v_m(s)\|_0^{2p} + 1 +
|v'_m(s)|_{L^{2}(\Omega)}^2\Big\}ds.
\end{aligned}
\end{equation}
Substituting, \eqref{e23} and \eqref{e24} in \eqref{e20},
\begin{equation} \label{e25}
\begin{aligned}
&|u'_m(t)|^2 +|v'_m(t)|^2 +\frac{2}{p}\|u_m(t)\|_0^p
+\frac{2}{p}\|v_m(t)\|_0^p \\
&\leq C + C\int_{0}^t\big(|u'_m(s)|^2 + |v'_m(s)|^2\big)ds  +
  C\int_{0}^t\big\{\|u_m(s)\|_0^{2p} + \|v_m(s)\|_0^{2p}\big\} \\
&\quad + C\int_{0}^t2\,ds \\
&\leq  C  + C\int_{0}^t\big(|u'_m(s)|^2 + |v'_m(s)|^2\big)ds  +
  C\int_{0}^t\big\{\|u_m(s)\|_0^{2p} + \|v_m(s)\|_0^{2p} \big\} \\
&\quad + C\int_{0}^T2\,ds \\
&\leq  C  + C\int_{0}^t\big(|u'_m(s)|^2 + |v'_m(s)|^2\big)ds
+ C\int_{0}^t\big\{\|u_m(s)\|_0^{2p} + \|v_m(s)\|_0^{2p} \big\}.
\end{aligned}
\end{equation}
Note that
\begin{align*}
&\frac{2}{p}|u'_m(t)|^2 +\frac{2}{p}|v'_m(t)|^2
+\frac{2}{p}\|u_m(t)\|_0^p +\frac{2}{p}\|v_m(t)\|_0^p\\
&\leq |u'_m(t)|^2 +|v'_m(t)|^2 +\frac{2}{p}\|u_m(t)\|_0^p
+\frac{2}{p}\|v_m(t)\|_0^p,
\end{align*}
 with $p>2$, It follows that
 %\label{e26}
\begin{align*}
&|u'_m(t)|^2 +|v'_m(t)|^2 +\|u_m(t)\|_0^p +\|v_m(t)\|_0^p\\
&\leq C  + C\int_{0}^t\big(|u'_m(s)|^2 + |v'_m(s)|^2\big)ds
+ C\int_{0}^t\big\{\|u_m(s)\|_0^{2p} + \|v_m(s)\|_0^{2p} \big\}\\
&\leq  C + C\int_{0}^t\Big\{\big(|u'_m(s)|^2 + |v'_m(s)|^2\big)^2 +
\big(\|u_m(s)\|_0^p + \|v_m(s)\|_0^p\big)^2 \\
&\quad + 2\big(|u'_m(s)|^2 + |v'_m(s)|^2\big)\big(\|u_m(s)\|_0^p
+ \|v_m(s)\|_0^p\big)  \Big\}ds  \\
&\quad + C\int_{0}^t\big\{|u'_m(s)|^2 +
|v'_m(s)|^2 + \|u_m(s)\|_0^{p} + \|v_m(s)\|_0^{p} \big\}ds\\
&= C +  C\int_{0}^t\big\{|u'_m(s)|^2 + |v'_m(s)|^2 +
\|u_m(s)\|_0^{p} + \|v_m(s)\|_0^{p}\big\}^2ds \\
&\quad + C\int_{0}^t\big\{|u'_m(s)|^2 +
|v'_m(s)|^2 + \|u_m(s)\|_0^{p} + \|v_m(s)\|_0^{p} \big\}ds.
\end{align*}
By setting
$$
\phi(t)=|u'_m(t)|^2 +|v'_m(t)|^2 +\|u_m(t)\|_0^p +\|v_m(t)\|_0^p,
$$
the above inequality can be rewritten as
\begin{equation} \label{e27}
\phi(t)\leq C + C\int_0^{t}\big\{\phi(s) + \phi^2(s)\big\}ds.
\end{equation}
Then, by Lemma \ref{mainlemma}, there exists $T_0 \in \mathbb{R}$, with
$0<T_0<T$, such that $\phi$ is bounded in $[0,T_0)$.
From this, we have
\begin{equation} \label{e28}
|u'_m(t)|^2 +|v'_m(t)|^2 +\|u_m(t)\|_0^p +\|v_m(t)\|_0^p\leq C  \quad
\forall  t \in [0,T_0), \quad \forall  m  \in \mathbb{N}.
\end{equation}
Therefore, by prolongation results, we can extend the
solutions
$u_m(t), v_m(t) $, to the interval $[0,T_0]$.

We will estimate, now, the second derivatives $u ''_m(t)$, $v ''_m(t)$.
Since the procedure, to estimates $u ''_m(t) $ and $v ''_m(t) $ are
similar, we will fix our attention only on bounding $u''_m(t)$.


\subsection{A priori Estimates II}
Let $P_m:L^2(\Omega)\to V_m \subset L^2(\Omega) $ be
$$
P_m(h)= \sum_{j=1}^m(h,w_j)w_j ,
$$
the projection operator on $L^2(\Omega)$. Observe that $P_m=P_m ^ * $ and
$P_m \in \mathcal{L}(H_0^s(\Omega)) $.
Now, by the approximate equation \eqref{e12},
\begin{equation} \label{e29}
(u''_m(t),w) + \langle \mathcal{A}u_m(t),w\rangle +
 \langle f(u_m(t),v_m(t))u_m(t),w\rangle
 = (h_1(t),w)
\end{equation}
for all  $w  \in V_m $. By the chain of immersions \eqref{e10} we
have
$$
\langle u''_m(t)+ \mathcal{A}u_m(t)+
 f(u_m(t),v_m(t) )u_m(t)-h_1(t),w\rangle _{H^{-s}(\Omega),H_0^s(\Omega)} =0,
$$
 for all $w \in V_m$.
 From this equality and the fact that $P_mw=w,  \forall w \in V_m$, we have
 $$
P_m^*(u ''_m(t)+ \mathcal{A}u_m(t) +
 f(u_m(t),v_m(t) )u_m(t)-h_1(t)) =0
$$
in $V_m$. From this, by the linearity of $P_m ^ *$, the fact that
$u''_m \in V_m$, and by the continuous and dense immersions, we have
$$
 u ''_m(t)=-P_m^*(\mathcal{A}u_m(t)) -
 P_m^*(f(u_m(t),v_m(t) )u_m(t) )+P_m^*(h_1(t))
$$
in $H^{-s}(\Omega)$. Thus
\begin{align*}  %\label{e30}
\|u''_m(t)\|_{H^{-s}(\Omega)}
&\leq  \|P_m^*(f(u_m(t),v_m(t))u_m(t))\|_{H^{-s}(\Omega)} \\
&\quad +  \|P_m^*(\mathcal{A}u_m(t))\|_{H^{-s}(\Omega)} +
 \|P_m^*(h_1(t))\|_{H^{-s}(\Omega)}
\end{align*}
With $P_m \in \mathcal{L}(H_0^s(\Omega))$ which implies
$P_m^* \in \mathcal{L}(H^{-s}(\Omega))$. Since
$W^{-1,p'}(\Omega)\hookrightarrow H^{-s}(\Omega)$,  it
follows that $P_m^* \in \mathcal{L}(W^{-1,p'}(\Omega),H^{-s}(\Omega))$,
 Then
\begin{equation} \label{e31}
 \|P_m^*(Au_m(t))\|_{H^{-s}(\Omega)}\leq C
 \|(\mathcal{A}u_m(t))\|_{W^{-1,p'}(\Omega)}\leq C \|u_m(t)\|_0^{p-1}.
\end{equation}
Since, $L^2(\Omega)\hookrightarrow H^{-s}(\Omega)$, we have
$P_m^* \in \mathcal{L}(L^2(\Omega),H^{-s}(\Omega)$.
Furthermore,
\begin{equation} \label{e32}
 \|P_m^*(h_1(t))\|_{H^{-s}(\Omega)}\leq C  |h_1(t)|_{L^2(\Omega)}.
\end{equation}
Now, to bound the term $\|P_m^*(f(u_m(t),v_m(t) )u_m(t))
\|_{H^{-s}(\Omega)} $, it is necessary to place $f(u_m(t),v_m(t)
)u_m(t) $ in some space contained in
$H^{-s}(\Omega)$.
Let $\gamma, \theta \in  [1, \frac{np}{n-p}] $, such
that $  \frac{1}{\gamma} + \frac{1}{\theta}=1 $.
Since $W_0^{1,p}(\Omega)\hookrightarrow L^q(\Omega)$  for
$1\leq q\leq\frac{np}{n-p}$,
we have, in particular
$W_0^{1,p}(\Omega)\hookrightarrow L^{\gamma}(\Omega)$.
Therefore,
$$
\big(L^{\gamma}(\Omega)\big) ' \hookrightarrow W^{-1,p '} (\Omega).
$$
 From the chain of immersions \eqref{e10}, we have
$W^{-1,p '}(\Omega)\hookrightarrow H^{-s}(\Omega)$, from where
\begin{equation} \label{e33}
 L^{\theta}(\Omega)=\big(L^{\gamma}(\Omega)\big)'\hookrightarrow
H^{-s}(\Omega)
\end{equation}
Now, it is sufficient to show that
$f(u_m(t),v_m(t))u_m(t) \in L^{\theta}(\Omega)$.
From the H\"older inequality and the hypothesis on $f$ we
have
\begin{equation} \label{e34}
\begin{aligned}
\int_{\Omega} |f(u_m(s),v_m(s))u_m(s)|^{\theta}dx
&=\int_{\Omega} |f(u_m(s),v_m(s))|^{\theta}|u_m(s)|^{\theta}dx\\
&\leq C_f^{\theta}\int_{\Omega} |v_m(s))|^{\theta}|u_m(s)|^{\theta}dx \\
&\leq C_f^{\theta}\Big(\int_{\Omega}|v_m(s))|^{\alpha'\theta}
\Big)^{1/\alpha'}
\Big(\int_{\Omega}|u_m(s))|^{\beta'\theta}\Big)^{\frac{1}{\beta'}},
\end{aligned}
\end{equation}
where $C_f $ is the Lipschitz constant, associated $f $ and
$  \frac{1}{\alpha'} + \frac{1}{\beta'}=1$.

If $  \theta\alpha'\leq \frac{np}{n-p}$ and
$  \theta\beta'\leq \frac{np}{n-p}$, then
$$
\theta\leq \frac{1}{\alpha'}\frac{np}{(n-p)}, \quad  \mbox{and} \quad
\theta\leq \frac{1}{\beta'}\frac{np}{(n-p)},
$$
from which,
$$
2\theta\leq \big(\frac{1}{\alpha'}+\frac{1}{\beta'}\big)\frac{np}{n-p}.
$$
Then, we have
$$
1\leq\theta\leq\frac{np}{2(n-p)}<\frac{np}{n-p}.
$$
Noticing that $W_0^{1,p}(\Omega)\hookrightarrow L^{\theta\alpha'}(\Omega)$
and $W_0^{1,p}(\Omega)\hookrightarrow L^{\theta\beta'}(\Omega)$, we have
\begin{equation*} %\label{35}
\int_{\Omega} |f(u_m(s),v_m(s))u_m(s)|^{\theta}dx
\leq C_f^{\theta}|v_m(t)|_{L^{\alpha'\theta}}^{\theta}
  |u_m(t)|_{L^{\beta'\theta}}^{\theta}
\leq C\|u_m(t)\|_0^{\theta}\|v_m(t)\|_0^{\theta}.
\end{equation*}
 From this estimate and \eqref{e28}, it follows
\begin{equation} \label{e36}
\int_{\Omega} |f(u_m(s),v_m(s))u_m(s)|^{\theta}dx< \infty;
\end{equation}
that is,
\begin{equation} \label{e37}
f(u_m(t),v_m(t))u_m(t)  \in
 L^{\theta}(\Omega)=\big(L^{\gamma}(\Omega)\big)',  \quad \mbox{for }
1\leq \theta \leq \frac{np}{2(n-p)},
\end{equation}
and
\begin{equation} \label{e38}
\|f(u_m(t),v_m(t))u_m(t)\|_{L^{\theta}(\Omega)}\leq
C,  \quad \forall   m, \;   t \in [0,T_0]
\end{equation}
Similarly, we have
\begin{equation} \label{e39}
\|g(u_m(t),v_m(t))v_m(t)\|_{L^{\theta}(\Omega)}\leq
C,  \quad \forall   m,   \;   t \in [0,T_0]
\end{equation}

We will also need that $f(u_m(t),v_m(t) )u^2_m(t)   \in
 L^{\theta}(\Omega)$.
In fact, by H\"older inequality,
\begin{align*} %\label{e40}
&\int_{\Omega} |f(u_m(s),v_m(s))u^2_m(s)|^{\theta}dx\\
&=\int_{\Omega} |f(u_m(s),v_m(s))|^{\theta}|u^2_m(s)|^{\theta}dx\\
&\leq C_f^{\theta}\int_{\Omega}
|v_m(s))|^{\theta}|u_m(s)|^{\theta}|u_m(s)|^{\theta}dx \\
&\leq C_f^{\theta}\Big(\int_{\Omega}|v_m(s))|^{\xi\theta}
 \Big)^{\frac{1}{\xi}}
\Big(\int_{\Omega}|u_m(s))|^{\delta\theta}\Big)^{1/\delta}
\Big(\int_{\Omega}|u_m(s))|^{\omega\theta}\Big)^{1/\omega},
\end{align*}
where $C_f$ is the Lipschitz constant, associated to $f$ and
$\frac{1}{\delta} + \frac{1}{\omega} + \frac{1}{\xi}=1$.
If $\theta\xi\leq \frac{np}{n-p}$,
$\theta\delta\leq \frac{np}{n-p}$ and
$\theta\omega\leq \frac{np}{n-p}$ then
$$
\theta\leq \frac{1}{\xi}\frac{np}{n-p},\quad
\theta\leq \frac{1}{\delta}\frac{np}{n-p}, \quad
\theta\leq \frac{1}{\omega}\frac{np}{n-p}
$$
which implies
$$
3\theta\leq \big(\frac{1}{\xi}+\frac{1}{\delta} +
\frac{1}{\omega}\big)\frac{np}{n-p}.
$$
Then
$$
1\leq\theta\leq\frac{np}{3(n-p)}<\frac{np}{n-p}.
$$
Observing that
$W_0^{1,p}(\Omega)\hookrightarrow L^{\theta\xi}(\Omega)$ ,
$W_0^{1,p}(\Omega)\hookrightarrow L^{\theta\delta}(\Omega)$ and
$W_0^{1,p}(\Omega)\hookrightarrow L^{\theta\omega}(\Omega)$,
it follows that
\begin{equation} \label{e41}
\begin{aligned}
\int_{\Omega} |f(u_m(s),v_m(s))u^2_m(s)|^{\theta}dx
&\leq C_f^{\theta}|v_m(t)|_{L^{\xi\theta}}^{\theta}|u_m(t)
|_{L^{\omega\theta}}^{\theta} |u_m(t)|_{L^{\delta\theta}}^{\theta} \\
&\leq C\|u_m(t)\|_0^{2\theta}\|v_m(t)\|_0^{\theta}.
\end{aligned}
\end{equation}
This estimate and \eqref{e28} lead us to
\begin{equation*} %\label{42}
\int_{\Omega} |f(u_m(s),v_m(s))u^2_m(s)|^{\theta}dx< \infty;
\end{equation*}
that is,
\begin{gather} \label{e43}
f(u_m(t),v_m(t))u^2_m(t)  \in
 L^{\theta}(\Omega)=\big(L^{\gamma}(\Omega)\big)',  \quad \mbox{for }
1\leq \theta \leq \frac{np}{3(n-p)},
\\
\label{e44}
\|f(u_m(t),v_m(t))u^2_m(t)\|_{L^{\theta}(\Omega)}\leq
C,  \quad  \forall   m,   \;  t \in [0,T_0]
\end{gather}
Similarly, we have
\begin{equation} \label{e45}
\|g(u_m(t),v_m(t))v^2_m(t)\|_{L^{\theta}(\Omega)}\leq
C,  \quad  \forall   m,   \;   t \in [0,T_0]
\end{equation}
Note that if $  \theta \leq \frac{np}{3(n-p)} $, we
still have \eqref{e37}  and \eqref{e43}, because
$ \frac{np}{3(n-p)}<\frac{np}{2(n-p)}$.
Thus, as $L^{\theta}(\Omega)\hookrightarrow H^{-s}(\Omega) $, we
have that $P_m^* \in \mathcal{L}(L^{\theta}(\Omega),H^{-s}(\Omega))
$. Therefore
\begin{equation} \label{e46}
 \|P_m ^ {*} (f(u_m(t),v_m(t) )u_m(t)) \|_{H^{-s}(\Omega)} \leq C
 \|f(u_m(t),v_m(t) )u_m(t)\|_{L^{\theta}(\Omega)}.
\end{equation}
Hence, from the estimates \eqref{e31}, \eqref{e32} and \eqref{e46}.
 we have
\begin{equation*} %47
 \|u''_m(t)\|_{H^{-s}(\Omega)}\leq C\big\{ \|u_m(t)\|_0^{p-1} +
 \|f(u_m(t),v_m(t))u_m(t)\|_{L^{\theta}(\Omega)} + |h_1(t)|\big\}.
\end{equation*}
 From this inequality, it results
\begin{align*} %48
 \int_0^{T_0}\|u''_m(t)\|^2_{H^{-s}(\Omega)}dt
&\leq  C\big\{\int_0^{T_0}\|u_m(t)\|_0^{2(p-1)}dt + \int_0^{T_0}|h_1(t)|^2dt\\
&\quad + \int_0^{T_0}\|f(u_m(t),v_m(t))u_m(t)\|^2_{L^{\theta}(\Omega)}dt
\big\}.
\end{align*}
Therefore, from (\ref{e25}), \eqref{e36} and \eqref{e1}, we conclude that
\begin{equation} \label{e49}
 \|u''_m(t)\|_{L^2(0,T_0;H^{-s}(\Omega)}\leq C,  \quad \forall
 m \in \mathbb{N}.
\end{equation}
Arguing in a similar way, one can deduce that
\begin{equation} \label{e50}
 \|v''_m(t)\|_{L^2(0,T_0;H^{-s}(\Omega)}\leq C, \forall  m \in \mathbb{N} .
\end{equation}
 From \eqref{e28}, we have
\begin{gather*} %51, 52
\|u_m(t)\|_0\leq C  \quad\mbox{and}\quad
   \|v_m(t)\|_0\leq C, \quad \forall m , \; t \in [0,T_0]. \\
|u'_m(t)|\leq C  \quad \mbox{and} \quad   |v'_m(t)|\leq C,\quad \forall
m ,\; t \in [0,T_0].
\end{gather*}
 From where, it follows that
$\mathop{\rm ess\,sup}_{t\in [0,T_0]} \|u_m(t)\|_0\leq C$;
that is
\begin{equation} \label{e53}
\|u_m\|_{L^{\infty}(0,T_0;W_0^{1,p}(\Omega))}\leq C,  \quad \forall
m \in \mathbb{N}.
\end{equation}
Similarly, we have
\begin{gather}
\|v_m\|_{L^{\infty}(0,T_0;W_0^{1,p}(\Omega))}\leq C, \quad
\forall  m \in \mathbb{N}; \label{e54}\\
\|u'_m\|_{L^{\infty}(0,T_0;L^2(\Omega))}\leq C, \quad \forall  m \in
\mathbb{N}; \label{e55}\\
\|v'_m\|_{L^{\infty}(0,T_0;L^2(\Omega))}\leq C, \quad \forall  m \in
\mathbb{N}; \label{e56}
\end{gather}
 Therefore, from \eqref{e38}, \eqref{e39}, \eqref{e44}, \eqref{e45},
\eqref{e49}, \eqref{e50}, \eqref{e53}, \eqref{e54}, \eqref{e55}, \eqref{e56},
we have
\begin{gather}
(u_m)_m,(v_m)_m  \quad \mbox{are bounded in }
 L^{\infty}(0,T_0;W_0^{1,p}(\Omega)); \label{e57} \\
(u'_m)_m,(v'_m)_m \quad \mbox{are bounded in }
L^{\infty}(0,T_0;L^2(\Omega)); \label{e58}\\
(u''_m)_m,(v''_m)_m \quad \mbox{are bounded in }
L^{2}(0,T_0;H^{-s}(\Omega)); \label{e59}\\
(f(u_m,v_m)u_m)_m,(g(u_m,v_m)v_m)_m  \quad \mbox{are bounded in }
L^{\infty}(0,T_0;L^{\theta}(\Omega)); \label{e60}\\
(f(u_m,v_m)u^2_m)_m,(g(u_m,v_m )v^2_m )_m  \quad
\mbox{are bounded in }   L^{\infty}(0,T_0;L^{\theta}(\Omega)); \label{e61}
\end{gather}
 Furthermore, since $\mathcal{A}$ is bounded, we have
\begin{equation*} %62
(\mathcal{A}u_m)_m,(\mathcal{A}v_m)_m  \quad \mbox{are bounded in }
L^{\infty}(0,T_0;W^{-1,p'}(\Omega)).
\end{equation*}

\subsection*{Taking Limits}
 From the estimates and Banach-Alaoglu-Boubarki theorem guarantee the
 existence of subsequences $(u_{\nu} )_{\nu},(v_{\nu} )_{\nu} $ of
$(u_m)_m,(v_m)_m $, respectively, such that
\begin{gather}
u_{\nu}\stackrel{*} {\rightharpoonup}u, \quad
v_{\nu}\stackrel{*} {\rightharpoonup}v \quad \mbox{in }
  L^{\infty}(0,T_0;W_0^{1,p}(\Omega)). \label{e63}\\
u'_{\nu}\stackrel{*} {\rightharpoonup}u', \quad
v'_{\nu}\stackrel{*} {\rightharpoonup}v' \quad \mbox{in }
 L^{\infty}(0,T_0;L^2(\Omega)). \label{e64}\\
u''_{\nu}\stackrel{*} {\rightharpoonup}u'', \quad
v''_{\nu}\stackrel{*} {\rightharpoonup}v'' \quad \mbox{in }
 L^{2}(0,T_0;H^{-s}(\Omega)). \label{e65}\\
Au_{\nu}\stackrel{*} {\rightharpoonup}\chi, \quad
\mathcal{A}v_{\nu}\stackrel{*} {\rightharpoonup}\eta \quad \mbox{in }
L^{\infty}(0,T_0;W^{-1,p'}(\Omega)). \label{e66}
\end{gather}
As $L^{2}(0,T_0;H^{-s}(\Omega)) $ is reflexive, the convergence
\eqref{e65} becomes
\begin{equation} \label{e67}
u''_{\nu}\rightharpoonup u'', v''_{\nu} \rightharpoonup v'' \quad
\mbox{in }  L^{2}(0,T_0;H^{-s}(\Omega)).
\end{equation}
Let us consider the approximate equation \eqref{e11} in the form
\begin{equation*} %\label{e68}
(u''_{\nu}(t),w) + \langle \mathcal{A}u_{\nu}(t),w\rangle
 + \langle f(u_{\nu,}(t),v_{\nu,}(t))u_{\nu}(t),w\rangle
= (h_1(t),w) \quad \forall  w  \in V_m, \; \nu\geq  m
\end{equation*}
Now, multiplying the above equality  by $\varphi \in D(0,T_0) $ and integrating
from $0 $ for $T_0 $ we obtain
\begin{align*} % 69
&\int_0^{T_{0}}(u''_{\nu}(t),w)\varphi  dt  +
\int_0^{T_{0}}\langle \mathcal{A}u_{\nu}(t),w\rangle \varphi  dt
+ \int_0^{T_{0}}\langle f(u_{\nu,}(t),v_{\nu,}(t))u_{\nu}(t),w\rangle
\varphi dt \\
&= \int_0^{T_{0}}(h_1(t),w)\varphi dt \quad  \forall  w \in V_m, \;
\nu \geq  m.
\end{align*}
Integrating by parts, we obtain
\begin{equation} \label{e70}
\begin{aligned}
& -\int_0^{T_{0}}(u'_{\nu}(t),w)\varphi'  dt  +
\int_0^{T_{0}}\langle \mathcal{A}u_{\nu}(t),w \rangle \varphi  dt
 + \int_0^{T_{0}}\langle f(u_{\nu,}(t),v_{\nu,}(t))u_{\nu}(t),w\rangle
 \varphi  dt \\
&= \int_0^{T_{0}}(h_1(t),w)\varphi dt  \quad \forall
 w \in V_m,\; \nu\geq  m.
\end{aligned}
\end{equation}
With $u'_{\nu}\stackrel{*} {\rightharpoonup}u'$ in
$L^{\infty}(0,T_{0};L^2(\Omega))=\big(L^1(0,T_{0};L^2(\Omega))\big)'$
then
\begin{equation} \label{e71}
\langle u'_{\nu},\phi\rangle \to
\langle u',\phi\rangle , \quad \forall \phi \in L^1(0,T_0;L^2(\Omega)).
\end{equation}
Convergence \eqref{e71} with
$\big<u'_{\nu},\phi\big>=
\int_0^{T_0}(u'_{\nu}(t),\phi(t))dt$, and assuming
$\phi(x,t)=w(x)\psi(t)$ imply hat
$$
\int_0^{T_0}(u'_{\nu}(t),\phi(t))dt=\int_0^{T_0}(u'_{\nu}(t),w(x))\psi(t)dt, \forall
 w \in L^2(\Omega), \quad \forall \psi \in L^1(0,T_0).
$$
Consequently, for all $w \in L^2(\Omega)$ and all $\psi \in L^1(0,T_0)$,
\begin{equation*} %72
\int_0^{T_0}(u'_{\nu}(t),w(x))\psi(t)dt\to
\int_0^{T_0}(u'(t),w(x))\psi(t)dt\,.
\end{equation*}
 In fact,
\begin{equation*} %73
\int_0^{T_0}(u'_{\nu}(t),w(x))\varphi'(t)dt\to
\int_0^{T_0}(u'(t),w(x))\varphi'(t)dt,
\end{equation*}
for all  $w \in V_m\subset W_0^{1,p}(\Omega)\subset
L^2(\Omega)$ and all $\psi=\varphi'$,
$\varphi \in D(0,T_0)\subset L^1(0,T_0)$.
In a similar way,
\begin{equation*} %74
\int_0^{T_0}<\mathcal{A}u_{\nu}(t),w(x)>\psi(t)dt\to
\int_0^{T_0}<\chi(t),w(x)>\psi(t)dt,
\end{equation*}
for all $w \in W_0^{1,p}(\Omega)$ and all $\psi \in L^1(0,T_0)$.
In fact,
\begin{equation*} %75
\int_0^{T_0}(\mathcal{A}u_{\nu}(t),w(x))\varphi(t)dt\to
\int_0^{T_0}(\chi(t),w(x))\varphi(t)dt,
\end{equation*}
for all  $w \in V_m\subset W_0^{1,p}(\Omega)$ and all
$\varphi \in D(0,T_0)\subset L^1(0,T_0)$.

 From \eqref{e34}, we have the existence of a subsequence
$(f(u_{\nu,},v_{\nu,})u_{\nu})_{\nu}$ such that
\begin{equation} \label{e76}
f(u_{\nu,},v_{\nu,})u_{\nu}\stackrel{*} {\rightharpoonup}
\lambda, \quad \mbox{in }  L^{\infty}(0,T_0;L^{\theta}(\Omega)).
\end{equation}
 Since $L^{\infty}(0,T_0;L^{\theta}(\Omega))\hookrightarrow
L^{\theta}(0,T_0;L^{\theta}(\Omega))$, we have from
\eqref{e41} that
\begin{equation*} %77
(f(u_m(t),v_m(t))u_m(t))_m,(g(u_m(t),v_m(t))v_m(t))_m
\end{equation*}
are bounded in $L^{\theta}(0,T_0;L^{\theta}(\Omega))$;
Thus we guarantee the existence of a subsequence, denoted as above,
such that
\begin{equation} \label{e78}
f(u_{\nu,},v_{\nu,})u_{\nu}\rightharpoonup \lambda, \quad
\mbox{in }  L^{\theta}(0,T_0;L^{\theta}(\Omega)).
\end{equation}
Since
\begin{gather*}
(u'_m)_m, \quad  \mbox{is bounded in }  L^{\infty}(0,T_0;L^2(\Omega)),\\
(u_m)_m,  \quad \mbox{is bounded in }  L^{\infty}(0,T_0;W_0^{1,p}(\Omega))
W_0^{1,p}(\Omega)\stackrel{c}{\hookrightarrow} L^2(\Omega),
\end{gather*}
we have by Aubin-Lions theorem, the existence of a subsequence
$(u_{\nu} )_{\nu} $ such that
\begin{gather} % \label{P} %79-80
u_{\nu}\to u, \quad \mbox{in} L^2(0,T_0;L^2(\Omega))\equiv L^2(Q_{T_0})
\label{e79}\\
u_{\nu}\to u, \quad \mbox{a.e. in } Q_{T_0} \label{e80}
\end{gather}
Since, the sequences $(v_m)_m,(v'_m)_m $ satisfy the same conditions,
it follows that, there exists a subsequence $(v_{\nu} )_{\nu} $ such
that
\begin{gather} %81-82
v_{\nu}\to v, \quad \mbox{in} L^2(0,T_0;L^2(\Omega))\equiv L^2(Q_{T_0})
\label{e81} \\
v_{\nu}\to v, \quad \mbox{a.e, in} Q_{T_0} \label{e82}
\end{gather}
 From \eqref{e80}, \eqref{e82}, and of the hypothesis on $f,g $,
we have
\begin{gather} \label{e83}
f(u_{\nu,},v_{\nu,})u_{\nu}\to f(u,v)u, \quad \mbox{a.e. in }  Q_{T_0}. \\
g(u_{\nu,},v_{\nu,})v_{\nu}\to g(u,v)v, \quad \mbox{a.e. in }  Q_{T_0}.
\label{e84}
\end{gather}
 From \eqref{e38}, we have
$$
\|f(u_m,v_m)u_m\|_{L^{\theta}(Q_{T_0})}\leq
C,  \quad \forall  m,
$$
where $L^{\theta}(Q_{T_0})\equiv L^{\theta}(0,T_0;L^{\theta}(\Omega))$.
From this and \eqref{e83}, by means of Lion's Lemma, it follows that
\begin{equation*} %85
f(u_{\nu,},v_{\nu,})u_{\nu}\rightharpoonup f(u,v)u, \mbox{in
}  L^{\theta}(Q_{T_0}),
\end{equation*}
for $  1\leq \theta\leq \frac{np}{3(n-p)}$. Therefore,
from \eqref{e78}, we have
$\lambda=f(u,v)u$ and from \eqref{e76}.
 This implies
\begin{equation} \label{e86}
f(u_{\nu,},v_{\nu,})u_{\nu}\stackrel{*} {\rightharpoonup}
f(u,v)u, \quad \mbox{in }  L^{\infty}(0,T_0;L^{\theta}(\Omega)).
\end{equation}
Similarly,
\begin{equation*} %87
g(u_{\nu,},v_{\nu,})v_{\nu}\stackrel{*} {\rightharpoonup}
g(u,v)v, \quad \mbox{in }  L^{\infty}(0,T_0;L^{\theta}(\Omega)).
\end{equation*}
The convergence in \eqref{e86}  implies
 \begin{equation*} %88
\int_0^{T_0}\big<f(u_{\nu}(t),v_{\nu}(t))u_{\nu}(t),w(x)\big>\psi(t)dt\to
\int_0^{T_0}\big<f(u(t),v(t))u(t),w(x)\big>\psi(t)dt,
\end{equation*}
 for all  $w \in W_0^{1,p}(\Omega)\subset L^{\gamma}(\Omega)$,
for all $\psi \in L^1(0,T_0)$.
In fact,
\begin{equation*}  %89
\int_0^{T_0}\big<f(u_{\nu}(t),v_{\nu}(t))u_{\nu}(t),w(x)\big>\varphi(t)dt\to
\int_0^{T_0}\big<f(u(t),v(t))u(t),w(x)\big>\varphi(t)dt,
\end{equation*}
for all  $w \in V_m\subset W_0^{1,p}(\Omega)\subset
L^{\gamma}(\Omega)$, for all $\varphi \in D(0,T_0)\subset L^1(0,T_0)$.
Taking the limit, as $\nu\to\infty $, in \eqref{e70} and using the
convergences obtained above, we have
\begin{equation} \label{e90}
\begin{aligned}
&-\int_0^{T_0}(u'(t),w)\varphi' dt +
\int_0^{T_{0}}\langle \chi(t),w\rangle \varphi  dt
+ \int_0^{T_0}\langle f(u(t),v(t))u(t),w \rangle\varphi  dt \\
&=\int_0^{T_{0}}(h_1(t),w) \varphi  dt  ,\quad  \forall
 w \in V_m, \; \varphi  \in D(0,T_0).
\end{aligned}
\end{equation}
Note that, with a similar reasoning for the approximate equation
\eqref{e12} we obtain
\begin{equation} \label{e91}
\begin{aligned}
&-\int_0^{T_0}(v'(t),w)\varphi' dt + \int_0^{T_0}\langle \eta(t),w\rangle \varphi
dt
+  \int_0^{T_0}\langle g(u(t),v(t))v(t),w\rangle \varphi  dt\\
& = \int_0^{T_{0}}(h_2(t),w)\varphi  dt , \quad \forall
 w \in V_m, \; \varphi  \in D(0,T_{0}).
\end{aligned}
\end{equation}
Now, using the basis definition and the fact that $V_m $ is dense in
$W_0^{1,p}(\Omega) $,  expressions \eqref{e90} and \eqref{e91} take the
form
\begin{equation} \label{e92}
\begin{aligned}
&-\int_0^{T_0}(u'(t),w)\varphi' dt    +
\int_0^{T_{0}}<\chi(t),w>\varphi  dt
+  \int_0^{T_0}\langle f(u(t),v(t))u(t),w\rangle \varphi   dt \\
&= \int_0^{T_{0}}(h_1(t),w) \varphi  dt, \quad \forall
 w \in W_0^{1,p}(\Omega),\; \varphi \in D(0,T_0),
\end{aligned}
\end{equation}
and
\begin{equation} \label{e93}
\begin{aligned}
&-\int_0^{T_0}(v'(t),w)\varphi' dt + \int_0^{T_0}\langle \eta(t),w\rangle
\varphi dt
 +  \int_0^{T_0}\langle g(u(t),v(t))v(t),w\langle \varphi dt\\
&=\int_0^{T_0}(h_2(t),w)\varphi  dt , \quad \forall
 w \in W_0^{1,p}(\Omega),\; \varphi  \in D(0,T_0).
\end{aligned}
\end{equation}
Note that, the mappings $t\mapsto (u'(t),w), t\mapsto (v'(t),w)
$ being functions in  $L^{\infty}(0,T_0) $, they define
distributions on $(0,T_0) $. Therefore, the first integrals of
\eqref{e92}, \eqref{e93}  are the derivative of these distributions.
Thus, from \eqref{e92} we have
\begin{equation*} %94
\int_0^{T_0}\big\{\frac{d}{dt}(u'(t),w) + \langle \chi(t),w \rangle
+ \langle f(u(t),v(t))u(t),w\rangle -(h_1(t),w)\big\}\varphi dt=0
\end{equation*}
for all $w \in W_0^{1,p}(\Omega)$ and all $\varphi \in D(0,T_0)$.
Thus,
\begin{equation*} %95
\frac{d}{dt}(u'(t),w) + \langle \chi(t),w\rangle
 +\langle f(u(t),v(t))u(t),w\rangle
=(h_1(t),w),
\end{equation*}
 for all $w \in W_0^{1,p}(\Omega)$, in $D'(0,T_0)$.
Similarly,
\begin{equation*} %96
\frac{d}{dt}(v'(t),w) + \langle \eta(t),w\rangle
+ \langle g(u(t),v(t))v(t),w \rangle =(h_2(t),w),
\end{equation*}
 for all $w \in W_0^{1,p}(\Omega)$, in $D'(0,T_0)$.

If one shows that $\mathcal{A}u(t)=\chi(t) $ and
$\mathcal{A}v(t)=\eta(t) $, the proof of the theorem will be complete;
since the verification of the initial conditions can be done in a
standard way.

The monotonocity of $\mathcal{A} $ implies that
$$
\int_0^{T_0}\langle \mathcal{A}u_{\nu}(t)-\mathcal{A}w, u_{\nu}
-w \rangle dt \geq 0,\quad  \forall w \in W_0^{1,p}(\Omega);
$$
that is,
$$
0 \leq \int_0^{T_0}\langle \mathcal{A}u_{\nu}(t), u_{\nu} \rangle dt -
\int_0^{T_0}\langle \mathcal{A}u_{\nu}(t), w \rangle dt -
\int_0^{T_0}\langle \mathcal{A}w, u_{\nu}(t)-w \rangle dt,
$$
for all $w \in W_0^{1,p}(\Omega)$.
\begin{equation*} %97  \label{eq86}
0 \leq \lim sup \int_0^{T_0}\langle \mathcal{A}u_{\nu}(t), u_{\nu}
\rangle dt - \int_0^{T_0}\langle \chi(t), w \rangle dt -
\int_0^{T_0}\langle \mathcal{A}w, u(t)-w \rangle dt,
\end{equation*}
for all $w \in W_0^{1,p}(\Omega)$.
Considering the approximate equation \eqref{e11} with $m=\nu $ and
$w=u_{\nu}(t) $ we have
$$
(u_{\nu}''(t), u_{\nu}(t)) + \langle \mathcal{A}u_{\nu}(t), u_{\nu}(t) \rangle
 + \langle f(u_{\nu},v_{\nu})u_{\nu}, u_{\nu} \rangle
 = ( h_1(t), u_{\nu}(t)).
$$
Therefore,
$$
\frac{d}{dt}(u_{\nu}'(t),u_{\nu}(t)) - | u_{\nu}'(t)|^2 +
 \langle \mathcal{A}u_{\nu}(t),u_{\nu}(t) \rangle +
\langle f(u_{\nu},v_{\nu})u_{\nu},u_{\nu} \rangle =(h_1(t), u_{\nu})
$$
Integrating  from $0$ the $ T_0 $ we have
\begin{equation} \label{e98}
\begin{aligned}
\int_0^{T_0} \langle \mathcal{A}u_{\nu}(t),u_{\nu}(t) \rangle dt
&=(u_{\nu}'(0),u_{\nu}(0)) - (u_{\nu}'(T_0),u_{\nu}(T_0))
  + \int_0^{T_0}| u_{\nu}'(t)|^2 dt \\
&\quad - \int_0^{T_0} \langle f(u_{\nu},v_{\nu})u_{\nu},u_{\nu} \rangle dt
+ \int_0^{T_0}(h_1(t), u_{\nu})dt
\end{aligned}
\end{equation}

Recall that  $W_0^{1,p}(\Omega)\hookrightarrow L^2(\Omega)$.
Since $u_{\nu}(0) \rightharpoonup u(0) $
in $W_0^{1,p}(\Omega) $ it implies
$u_{\nu}(0) \to u(0) in L^2(\Omega)$.
Since $u_{\nu} ' (0) \rightharpoonup u'(0)$ in $L^2(\Omega) $,
it implies
\begin{equation} \label{e99}
( u_{\nu}'(0),u_{\nu}(0)) \to ( u'(0),u(0)) \quad\text{in }
{\mathbb{R}}
\end{equation}

Recall that  $(u_m(T_0) )_m $ is bounded in
$W_0^{1,p}(\Omega) $ and $(u_m'(T_0) )_m $ is bounded in
$L^2(\Omega)$. Thus, there exists subsequences
$(u_{\nu}(T_0))_{\nu}$ and $(u_{\nu} ' (T_0) )_{\nu} $ such that
$$
u_{\nu}(T_0) \rightharpoonup u(T_0) \quad \text{in }
W_0^{1,p}(\Omega) \stackrel{c}\hookrightarrow L^2(\Omega),
$$
which implies
\begin{gather*}
u_{\nu}(T_0) \to u(T_0) , in L^2(\Omega), \\
u_{\nu}'(T_0) \rightharpoonup u'(T_0) in L^2(\Omega)
\end{gather*}
Consequently,
\begin{equation}  \label{e100}
( u_{\nu}'(0),u_{\nu}(T_0)) \to ( u'(T_0),u(T_0)) \quad\text{in }
\mathbb{R}.
\end{equation}
 We have that $(u_m')$ bounded in $L^{\infty}(0,T_0; L^{2}(\Omega))$.
Since
$$
L^{\infty}(0,T_0; L^{2}(\Omega)) \hookrightarrow
L^{2}(0,T_0; L^{2}(\Omega)),
$$
it follows that
$(u_m')$ is bounded in $L^{2}(0,T_0; L^{2}(\Omega))$.
We also have that
$(u_m'')$ is bounded in $L^{2}(0,T_0; H^{-s}(\Omega))$.
Therefore, by the Aubin-Lions Theorem, there exists a subsequence
$(u_{\nu} ' ) $ such that
$$
u_{\nu}' \to u' \quad\text{in } L^2(0,T_0;
L^2(\Omega))\equiv L^2(Q_{T_0}).
$$
 Hence
\begin{equation}  \label{e101}
\int_0^{T_0}|u_{\nu}'(t)|^2dt \to \int_0^{T_0}|u'(t)|^2dt
\end{equation}
Note that
$$
\langle f(u_m(t),v_m(t))u_m(t), u_m(t) \rangle _{L^{\theta},
L^{\gamma}}= \langle f(u_m(t),v_m(t))u_m^2(t), 1 \rangle
_{L^{\theta}, L^{\gamma}}.
$$
 From \eqref{e101} we have
$u_{\nu}^2 \to u^2$ a.e.  in $Q_{T_0}$.
Similarly
$$
\int_0^{T_0}|v_{\nu}'(t)|^2dt \to \int_0^{T_0}|v'(t)|^2dt
$$
hence, we have $ v_{\nu}^2 \to v^2$ a.e.  in $Q_{T_0}$,
From \eqref{e44}, we have
\begin{equation} \label{e102}
\|f(u_{\nu},v_{\nu})u_{\nu}^2\|_{L^{\theta}(0,T_0;L^{\theta}(\Omega))\equiv
L^{\theta}(Q_{T_0})}\leq C,  \quad  \forall   m.
\end{equation}
 From this inequality and \eqref{e61}, we guarantee the existence of a
subsequence such that
\begin{gather} \label{e103}
f(u_{\nu},v_{\nu})u_{\nu}^2 \stackrel{*}{\rightharpoonup} \sigma
\quad\text{in } L^{\infty}(0,T_0; L^{\theta}(\Omega)) \\
f(u_{\nu},v_{\nu})u_{\nu}^2 {\rightharpoonup} \sigma \quad\text{in }
 L^{\theta}(0,T_0; L^{\theta}(\Omega))  \label{e104}
\end{gather}
Thus, from \eqref{e80}, \eqref{e82} and the hypotheses on $f,g $,
 we have that
\begin{gather}
f(u_{\nu},v_{\nu})u_{\nu}^2 \to f(u,v)u^2 \quad\text{a.e. in } Q_{T_0},
\label{e105}\\
g(u_{\nu},v_{\nu})u_{\nu}^2 \to g(u,v)u^2 \quad\text{a.e in } Q_{T_0}
\label{e106}
\end{gather}
 From \eqref{e102}, \eqref{e105} and the Lions' Lemma it follows that
\begin{equation*} % 107
f(u_{\nu},v_{\nu})u_{\nu}^2 \rightharpoonup f(u,v)u^2 in
 L^{\theta}(Q_{T_0})\equiv L^{\theta}(0,T_0; L^{\theta}(\Omega)),
 \quad\text{for }1 \leq \theta \leq \frac{np}{3(n-p)}
\end{equation*}
 From this convergence and \eqref{e104}, we have
$ {\sigma} = f(u,v)u^2 $
and from \eqref{e103},
\begin{equation} \label{e108}
f(u_{\nu},v_{\nu})u_{\nu}^2 \stackrel{*}{\rightharpoonup} f(u,v)u^2
\quad\text{in } L^{\infty}(0,T_0; L^{\theta}(\Omega)).
\end{equation}
Similarly,
\begin{equation*} %109
g(u_{\nu},v_{\nu})v_{\nu}^2 \stackrel{*}{\rightharpoonup} g(u,v)u^2
 in L^{\infty}(0,T_0; L^{\theta}(\Omega)).
\end{equation*}
The convergence \eqref{e108}  implies
$$
\langle f(u_{\nu},v_{\nu})u_{\nu}^2, \psi \rangle \to
\langle f(u,v)u^2, \psi \rangle , \quad \forall \psi \in L^1(0,T_0;
L^{\gamma}(\Omega))
$$
 or better
$$
\int_0^{T_0}\langle f(u_{\nu},v_{\nu})u_{\nu}^2, w(x) \rangle
\varphi (t) dt \to \int_0^{T_0}\langle f(u,v)u^2,
w(x)\rangle \varphi (t)dt ,
$$
for all $w \in L^{\gamma}(\Omega)$ and all $\varphi \in L^1(0,T_0)$.
When fixing $w\equiv 1 $ and $ \varphi \equiv 1$, we have
$$
\int_0^{T_0}\langle f(u_{\nu}(t),v_{\nu}(t))u_{\nu}(t), u_{\nu}(t) \rangle
dt = \int_0^{T_0}\langle f(u_{\nu}(t),v_{\nu}(t))u_{\nu}^2(t), 1
\rangle dt
$$
which approaches
$$
\int_0^{T_0}\langle f(u(t),v(t))u^2(t), 1\rangle dt
= \int_0^{T_0}\langle f(u(t),v(t))u(t), u(t) \rangle dt.
$$
hence
\begin{equation} \label{e110}
 \int_0^{T_0}\langle f(u_{\nu}(t),v_{\nu}(t))u_{\nu}(t),
u_{\nu}(t) \rangle dt \to \int_0^{T_0}\langle
f(u(t),v(t))u(t), u(t) \rangle dt,
\end{equation}
as $ {\nu} \to {\infty}$.
Therefore, taking the limit in \eqref{e98}, using the convergence
\eqref{e99}, \eqref{e100}, \eqref{e101} and
\eqref{e110}, as ${\nu} \to + {\infty}$, we have
\begin{align*} %111
\limsup \int_0^{T_0} \langle Au_{\nu}(t),u_{\nu}(t) \rangle dt
&= (u'(0), u(0)) - (u'(T_0),u(T_0)) + \int_0^{T_0}|u'(t)|^2dt\\
&\quad -  \int_0^{T_0} \!\langle f(u(t),v(t))u(t),u(t)\rangle
dt + \int_0^{T_0}\!(h_1(t), u(t))dt
\end{align*}
 From this equality and \eqref{e110}, we have
\begin{equation} \label{e112}
\begin{aligned}
0&\leq (u'(0),u(0)) - (u'(T_0)-u(T_0)) + \int_0^{T_0} |u'(t)^2|dt -
\int_0^{T_0} \langle f(u,v)u,u \rangle dt \\
&\quad - \int_0^{T_0} \langle \chi(t), w \rangle dt
 - \int_0^{T_0} \langle Aw, u(t)-w \rangle dt
 + \int_0^{T_0}(h_1(t),u(t))dt,
\end{aligned}
\end{equation}
for all $w \in W_0^{1,p}(\Omega)$.
 From the approximate equation \eqref{e11}, we have
$$
(u''_{\nu}(t),w) + \langle Au_{\nu}(t), w \rangle +
\langle f(u_{\nu}(t),v_{\nu}(t))u_{\nu}(t), w \rangle = (h_1(t),w), \quad
\forall w \in V_m, {\nu} \geq m.
$$
Now, let $ \varphi \in C^1([0,T_0])$. Then
\begin{align*}
 &\int_0^{T_0}(u''_{\nu}(t),w)\varphi + \int_0^{T_0}\langle Au_{\nu}(t),
w \rangle \varphi + \int_0^{T_0}\langle f(u_{\nu}(t),v_{\nu}(t))
u_{\nu}(t), w \rangle \varphi \\
&= \int_0^{T_0}(h_1(t), w),
\end{align*}
for all $w \in V_m$ and all ${\nu} \geq m$.
 Setting
\begin{align*}
& (u_{\nu}'(t),w)\varphi(T_0) - (u_{\nu}'(0),w)\varphi(0) -
\int_0^{T_0}(u_{\nu}'(t),w){\varphi}' dt \\
&+ \int_0^{T_0}\langle Au_{\nu}(t), w \rangle \varphi    dt
 +  \int_0^{T_0}\langle f(u_{\nu}(t),v_{\nu}(t))u_{\nu}(t),
  w \rangle \varphi (t) dt\\
&= \int_0^{T_0}(h_1(t), w)\varphi (t)dt,\quad \forall w \in V_m,
 \; \varphi \in C^1([0,T_0]),\; {\nu} \geq m.
\end{align*}
Taking into account the previous convergence statements, it
follows that
\begin{align*}
&(u'(T_0),w)\varphi (T_0) - (u'(0),w)\varphi (0) -
\int_0^{T_0}(u'(t),w){\varphi}'dt \\
&+ \int_0^{T_0}\langle \chi(t), w \rangle \varphi dt
+  \int_0^{T_0}\langle f(u(t),v(t))u(t), w
\rangle \varphi (t)dt\\
&= \int_0^{T_0}(h_1(t), w)\varphi (t)dt, \quad
\forall w \in V_m ,\; \varphi \in C^1([0,T_0])
\end{align*}
Using a basis argument and the fact that $V_m $ is dense in
$W_0^{1,p}(\Omega)$, it follows that
\begin{equation} \label{e113}
\begin{aligned}
&(u'(T_0),w)\varphi (T_0) - (u'(0),w)\varphi (0) -
\int_0^{T_0}(u'(t),w){\varphi}'dt \\
& + \int_0^{T_0}\langle \chi(t), w \rangle \varphi dt
  + \int_0^{T_0}\langle f(u(t),v(t))u(t), w \rangle \varphi (t)dt\\
&= \int_0^{T_0}(h_1(t), w)\varphi (t)dt,
\quad \forall w \in W_0^{1,p}(\Omega), \; \varphi \in
C^1([0,T_0]).
\end{aligned}
\end{equation}
Observing that the set of the linear combinations of the type
$w \varphi $, with $w \in W_0^{1,p}(\Omega) $ and
$\varphi \in C^1([0,T_0])$, is dense in the space
$$
V = \{v \in L^2(0,T_0; W_0^{1,p}(\Omega)), v' \in L^2(0,T_0; L^2(\Omega))
\}.
$$
It follows that \eqref{e113} is true in the space
$V$.

Using the fact that,
\begin{gather*}
 u \in L^{\infty}(0,T_0;
W_0^{1,p}(\Omega))\hookrightarrow L^{2}(0,T_0; W_0^{1,p}(\Omega)), \\
u' \in L^{\infty}(0,T_0; L^2(\Omega))\hookrightarrow
L^{2}(0,T_0; L^2(\Omega)),
\end{gather*}
we obtain that $ u \in V$. So \eqref{e113}
takes the form
\begin{align*} %114
&(u'(T_0),w)\varphi (T_0) - (u'(0),w)\varphi (0) \\
&-\int_0^{T_0}(u'(t),u'(t))dt
+ \int_0^{T_0}\langle \chi(t),u(t) \rangle dt + \int_0^{T_0}\langle
f(u,v)u, u \rangle dt\\
&= \int_0^{T_0}(h_1(t),u(t)dt
\end{align*}
Substituting this expression in \eqref{e112}, it follows that
$$
0 \leq \int_0^{T_0}\langle \chi (t), u(t)-w \rangle dt
-\int_0^{T_0}\langle \mathcal{A}w, u(t)-w \rangle dt, \quad
\forall w \in W_0^{1,p}(\Omega).
$$
Let us take $w=u(t)+\lambda v(t), \lambda >0$. Thus
$$
0 \leq - \int_0^{T_0}\langle \chi (t), \lambda v(t) \rangle dt +
\int_0^{T_0}\langle \mathcal{A}u(t)+\lambda v(t), \lambda v(t)
\rangle dt, \forall w \in W_0^{1,p}(\Omega)
$$
which implies
$$
0 \leq -\int_0^{T_0}\langle \chi (t), \lambda v(t) \rangle dt +
\int_0^{T_0}\langle \mathcal{A}(u(t)+\lambda v(t)), \lambda v(t)
\rangle dt.
$$
Dividing the previous inequality by $\lambda$ and letting
 $\lambda \to 0^{+}$,  by the hemicontinuity of
$\mathcal{A}$,  we have
$$
 0 \leq -\int_0^{T_0}\langle \chi (t), v(t) \rangle dt +
\int_0^{T_0}\langle \mathcal{A}(u(t)), v(t) \rangle dt, \quad
\forall v \in W_0^{1,p}(\Omega).
$$
Hence
$$
0 \leq \int_0^{T_0}\langle \mathcal{A}u(t)- \chi (t), v(t) \rangle dt,
\quad \forall v \in W_0^{1,p}(\Omega).
$$
Now, for $\lambda<0$ it follows that
$$
\int_0^{T_0}\langle \mathcal{A}u(t)- \chi (t), v(t) \rangle dt \leq 0,
\quad  \forall v \in W_0^{1,p}(\Omega).
$$
Therefore,
$$
 0 \leq \int_0^{T_0}\langle \mathcal{A}u(t)- \chi (t), v(t) \rangle dt \leq 0,
\quad \forall v \in W_0^{1,p}(\Omega).
$$
Thus $ \mathcal{A}u(t)=\chi (t)$.
Similarly, $\mathcal{A}v(t)=\eta (t)$. This completes the proof of
the theorem.


\begin{thebibliography}{99}

\bibitem{biazutti} Biazutti. A;
\emph{Sobre uma Equa\c c\~ao n\~ao Linear de Vibra\c c\~oes:
Exist\^encia de Solu\c c\~oes Fracas e Comportamento Assint\'otico}.
Phd Thesis, IM-UFRJ, Rio-Brazil (1988).

\bibitem{brezis} Brezis, H; \emph{Analyse Fonctionelle-theorie et
Applications}. Masson, Paris (1983).

\bibitem{castro} Castro, N. N; \emph{Existence and asymptotic behavior
of solutions for a nonlinear evolution problem}. Appl. of
Mathematics, 42(1997), No. 6, 411-420.

\bibitem{clark} Clark, M.R., Lima, O.A; \emph{On a class of nonlinear
Klein-Gordon equations}.  Proceedings of 52 Semin\'ario
Brasileiro de An\'alise (2000), 445-451.

\bibitem{maciel} Clark, M.R., Maciel, A.B; \emph{On a mixed problem for
a nonlinear $K\times K$ system}. IJAM, 9, No. 2(2000), 207-219.

\bibitem{clark1} Clark, M.R., Clark, H.R., Lima, O.A;
\emph{On a Nonlinear Coupled System. International Journal of Pure
and Applied Mathematics}. Vol. 20 No. 1 (2005), 81-95.

\bibitem{lions} Lions, J.L; \emph{Quelque Methodes des Resolution des
Probl\'ems aux Limites non Lineaires}. Dunod, Paris (1969).

\bibitem{luis} Medeiros, L.A., Menzala, G.P; \emph{On a mixed problem
for a class of nonlinear Klein-Gordon equation}. Acta Math Hung.,
$52, No.  1-2(1988), 61-69$.

\bibitem{luis1} Medeiros, L.A., Menzala, G.P; \emph{On the existence of
global solutions for a coupled nonlinear Klein-Gordon equations}.
Funkcialaj Ekvacioj, 30 (1987), 147-161.

\bibitem{luis2} Medeiros, L.A., Miranda, M.M; \emph{Weak Solutions for
the system of nonlinear Klein-Gordon equation}. Annali di Matematica
Pura ed Applicata, IV, CXLVI (1987), 173-183.

\bibitem{segal} Segal, I; \emph{Nonlinear partial differential equations
in quantum field theory}. Proc. Symb. Appl. Math. A.M.S.,
(1965), 210-226.

\end{thebibliography}


\end{document}