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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 229, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2014/229\hfil Existence of solutions]
{Existence of solutions to nonlinear
parabolic unilateral problems with an obstacle depending on time}

\author[N. Bellal \hfil EJDE-2014/229\hfilneg]
{Nabila Bellal}  % in alphabetical order

\address{Nabila Bellal \newline
Universit\'e 20 ao\"ut 1955,  Skikda,  Alg\'erie}
\email{nabilabellal@yahoo.fr}

\thanks{Submitted March 23, 2014. Published October 27, 2014.}
\subjclass[2000]{35K86, 35R35, 49J40}
\keywords{Parabolic variational inequalities;
Leray-Lions operator; \hfill\break\indent penalization; existence theorem}

\begin{abstract}
 Using the penalty method, we prove the existence of solutions to
 nonlinear parabolic unilateral problems with an obstacle depending on time.
 To find a solution, the original inequality is transformed into an
 equality by adding a positive  function on the right-hand side and a
 complementary condition. This result can be seen as a generalization of
 the results by  Mokrane in \cite{M} where the obstacle is zero.
\end{abstract}

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

\section{Introduction}

The main purpose of this article is to prove the existence of a solution to a
nonlinear parabolic inequality of obstacle type.
Our problem is associated to a second-order nonlinear operator of Leray-Lions type.
We prove that actually the solution satisfies an equation with a modification
of the right-hand side by a positive function and a complementary condition.
 This result can be seen as a generalization of the result obtained
Mokrane \cite{M} when the obstacle is zero.

\subsection*{Statement of the problem}

 Let $\Omega$ be a bounded Lipschitz open set of $\mathbb{R}^N$ with boundary
$\partial\Omega$ and $T$ a positive real number. Set $ Q = \Omega \times (0,T)$ and
 $\Sigma = \Gamma \times (0,T)$.
 Given functions $ u_0 $  and $\psi$ we look for a solution $u$ to the problem
\begin{gather}\label{ineq1}
\frac{\partial u}{\partial t}+A( u) +g( u,Du) -f=\mu
\quad \text{in }Q=\Omega \times ] 0,T[, \\
\label{ineq2}
u\geq \psi,\quad \mu \geq 0, \quad \mu( u-\psi) =0\quad \text{in }Q,\\
\label{ineq3}
u( x,t) =0\quad \text{on }\Sigma,\\
\label{ineq4}
u( x,0) =u_0( x) \quad \text{in }\Omega .
\end{gather}

Here $A$ is a Leray-Lions operator from $L^{p}( 0,T;W_0^{1,p}(
\Omega ) ) $ into its dual, $f$ belongs to $L^{p' }(Q)$ and
$g(x,t,u,Du)$ is a nonlinear term, the prototype of which is $u|Du|^{q}$
with $q<p-1$, we suppose that $p>2$.

When $g$ is equal to zero, the corresponding result has
been proved e.g. in \cite{D}.
On the other hand, the equation associated with the unilateral problem
\eqref{ineq1}, \eqref{ineq3}, \eqref{ineq4} (i.e. the case  where $\mu =0$
in \eqref{ineq1}, the
conditions \eqref{ineq2} being omitted) has been solved in \cite{BM1}.
Here we extend Mokrane's result \cite{M}, by utilizing different
techniques.
For $\psi= 0$, \cite{M}  proved the existence of a solution.


Considered  just  as an equation (without obstacle) or as a variational
inequality this problem, or very similair ones with various types of hypotheses
on the operator $A$ (or the function $a(x,t,s,\xi)$ see below),
$ g$ and the data have  been addressed by several authors,
\cite{AAR,ABBM,KKS}.

For some of these results,  an extra condition  on the form $a( x,t,s,.)$
applied to the positive part on any test function is added.
 It seems for us that it is more interesting and realistic,
to avoid  this condition,
and replace it by an extra regularity condition on the obstacle.
Moreover these authors did not deal with  the existence of the
function $\mu$ and the complementary condition $\mu(u-\psi)=0$ in $Q$.

 In this article we use a regularization-penalization procedure and a
compactness result analogous to the ones introduced \cite{M}, and some
other different techniques.

This article is organised as follows. The  first part is devoted to the
hypotheses and the setting of the main result. In the second one we proceed by the
regularization-penalisation method.
We construct a one parameter family of solutions and prove some estimates on these
approximate solutions.
 In the third part we prove the convergence of an extracted subsequence
of this family, to a solution of our problem.


\section{Hypotheses and the main result}

Let $\Omega $ be a bounded subset of $ \mathbb{R}^{N}$,  with Lipschitz boundary
$ \partial \Omega ,Q$ be $\Omega \times ] 0,T[ $ for a given
$T$, $ 0< T< \infty $ and $\Sigma =\partial \Omega \times ] 0,T[$.
 Let $p$ and $p' $ be fixed with $\frac{1}{p}+\frac{1}{p' }=1$,
$2< p < \infty$,
$W^{1,p}_0(\Omega)$ is the usual  Sobolev space equipped with the $L^p$ norm
of the gradients.
 Let $A$ be a nonlinear operator from $L^{p}( 0,T;W_0^{1,p}( \Omega ) ) $
into its dual $L^{p' }( 0,T;W^{-1,p' }( \Omega )) $ of
Leray-Lions type defined by
\begin{equation*}
A(u) =-\operatorname{div}(a(x,t,u,Du)),
\end{equation*}
where $a( x,t,s,\xi ) $ is a Carath\'{e}odory function such that
\begin{equation}\label{eqcoer}
\begin{gathered}
 a(x,t,s,\xi)\leq \beta [|s|^{p-1}+|\xi|^{p-1}+k(x,t)], \quad
  k(x,t)\in L^{p'}(Q),\;  \beta >0 \\
{ [a(x,t,s,\xi)- a(x,t,s,\eta)][\xi-\eta]>0, \quad \forall \xi\neq \eta}\\
a(x,t,s,\xi)\xi \geq \alpha |\xi|^{p}, \quad \alpha >0.
\end{gathered}
\end{equation}

Let $g( x,t,u,Du) $ be a nonlinear lower order term having
growth of order $q$, $(q<p-1)$ with respect to
$|Du| $  and of order $m$ $(1< m <p-q)$  with respect to $ |u|$  and satisfying
a sign condition.  To be more precise we assume that $g$ is a
Carath\'{e}odory function such that
\begin{equation}\label{eqcriosg}
| g( x,t,s,\xi ) | \leq b( |s| ) ( h( x,t) +| \xi |^{q})
\end{equation}
where $1<q<p-1$, $h\in L^{\infty }( Q)$, and $b:\mathbb{R}
^{+}\to \mathbb{R}^{+}$ is
a  continuous,  nonnegative increasing function, having growth of order $m$,
$({1<m<p-q})$ with respect to $| u|$:
\begin{gather}\label{eq2.3}
b( | u| ) \leq \rho +| u|^{m}, \quad \rho >0, \;  1< m<p-q; \\
\label{eq2.4}
g( x,t,s,\xi ) s\geq 0\quad \forall (x,t,s,\xi)\in
\Omega\times \mathbb{R}^2\times \mathbb{R}^N.
\end{gather}
We have the following assumptions on $u_0$, $f$  and $\psi$:
\begin{gather}\label{eq2.5}
u_0\in L^{2}(\Omega), \\
\label{eq2.6}
f\in L^{p' }( Q), \\
\label{eq2.7}
\psi\in L^{p}( 0,T;W^{1,p}( \Omega ) ) \quad\text{with }\psi\leq 0\text{ on }\Sigma,\\
\label{eq2.7bis}
\psi(0)\le u_0\quad \text{a.e.  in } \Omega, \\
\label{eq2.8}
\psi^{+}\in L^{\infty }( Q), \\
\label{eq2.9}
\frac{\partial \psi}{\partial t} \in L^{p' }( Q)
\end{gather}
Also we assume a complementary condition on $a$ and $\psi$,
\begin{equation}\label{eq2.9bis}
\operatorname{div}(a(x,t,u, D\psi)\in L^{p'}(Q)\quad \text{for }
 u\in L^p(0,T, W^{1,p}_0(\Omega))
\end{equation}
and is bounded in $L^{p'}(Q)$ on bounded sets of $L^p(0,T, W^{1,p}_0(\Omega))$.


Our main result is the following.

\begin{theorem} \label{thm2.1}
Under  assumptions \eqref{eqcoer}--\eqref{eq2.9}  there exist at
least one  pair of functions $u$ and $\mu$  which are a solution of
\eqref{ineq1}--\eqref{ineq4} and satisfy
\begin{gather}\label{eq2.10}
u\in L^{\infty }( 0,T;L^{2}( \Omega ) ) \cap
L^{p}( 0,T;W_0^{1,p}( \Omega ) ), \\
\label{eq2.11}
\frac{\partial u}{\partial t}= \lambda_1+ \lambda _2\quad \text{with }
 \lambda _1\in L^{p' }( 0,T;W^{-1,p' }( \Omega ) ),\;
\lambda _2\in L^{1}( Q), \\
\label{eq2.12}
u\geq \psi\quad\text{in } Q, \\
\label{eq2.13}
\mu \in L^{p' }( Q), \\
\label{eq2.14}
\mu \geq 0, \\
\label{eq2.15}
g(x,t, u,Du) \in L^{1}( Q) \quad \text{and } ug(x,t,u,Du) \in L^{1}( Q),\\
\label{eq2.16}
\frac{\partial u}{\partial t}+A( u) +g(x,t, u,Du) -f=\mu
\quad \text{in } Q,\\
\label{eq2.17}
\mu( u-\psi) =0\quad \text{in } Q, \\
\label{eq2.18}
u\in C^{0}( 0,T;W^{-1,r}( \Omega ) ) \quad \text{for }
r<\inf ( p,\frac{p}{p-1},\frac{N}{N-1}), \\
\label{eq2.19}
u( x,0) =u_0( x) \quad \text{in } \Omega .
\end{gather}
\end{theorem}

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


\subsection{Approximate solutions}
For $\varepsilon >0$, we define
\begin{equation}\label{eq3.1}
g_{\varepsilon }( x,t,s,\xi ) =\frac{g( x,t,s,\xi ) }{
1+\varepsilon | g( x,t,s,\xi ) | }
\end{equation}
and we denote by $u_{\varepsilon }$ the solution of the approximate and
penalized problem
\begin{equation}\label{eq3.2}
\begin{gathered}
\begin{aligned}
&\frac{\partial u_{\varepsilon }}{\partial t}-\operatorname{div}(a(x,t,u_{
\varepsilon },Du_{\varepsilon }))+g_{\varepsilon }(x,t,u_{\varepsilon},
Du_{\varepsilon }) \\
&-\frac{1}{\varepsilon ^{p-1}} | ( u_{\varepsilon }-\psi)
^{-}| ^{p-2}( u_{\varepsilon }-\psi) ^{-}=f,\quad \text{in } Q,
\end{aligned} \\
 u_{\varepsilon }(x,0)=u_0(x), \quad x\in \Omega, \\
 u_{\varepsilon }=0\text{ on }\Sigma,  \\
 u_{\varepsilon }\in L^{p}( 0,T;W_0^{1,p}( \Omega ) )
\end{gathered}
\end{equation}
which has a weak solution by the classical result of   Lions  \cite{L},
 Donati \cite{D}, where $v^{-}$ denotes the negative part of $v$, i.e.
$v^{-}=\sup (0,-v) $, for any function $v$.

The function $u_\varepsilon$ is a solution of \eqref{eq3.2}
in the following sense:
\begin{equation}\label{eq3.2a}
\begin{gathered}
u_\varepsilon\in L^p(]0,T[, W^{1,p}_0(\Omega))\cap
\mathcal{C}([0,T], L^2(\Omega)),\\
{ \frac{\partial u_\varepsilon}{\partial t}\in L^{p'}(0,T; W^{-1,p'}(\Omega) ) },\quad
 u_\varepsilon(x,0)=u_0(x),\\
\begin{aligned}
&\int_0^T\langle\frac{\partial u_\varepsilon}{\partial t}, v\rangle dt
+\int_Qa(x,t,u_\varepsilon, Du_\varepsilon)Dv\,dx\,dt
+\int_Qg_\varepsilon(x,t,u_\varepsilon,Du_\varepsilon)v\,dx\,dt\\
&-\frac{1}{\varepsilon^{p-1}}\int_Q(( u_{\varepsilon }-\psi)
^{-}) ^{p-2}( u_{\varepsilon }-\psi) ^{-}v\,dx\,dt\\
&=\int_Q fv\,dx\,dt, \quad \forall v\in L^p(]0,T[, W^{1,p}_0(\Omega))
\end{aligned}
\end{gathered}
\end{equation}

\subsection{$L^p(0,T; W^{1,p}_0(\Omega))$ - estimate of $u_\varepsilon$}
Recall that since  $\psi\in L^p(]0,T[, W^{1,p}(\Omega))$, $p> 2$ and
${ \frac{\partial \psi}{\partial t}}\in L^{p' }( Q), $  we have
$\psi\in W^{1,p' }(Q)$. From this and by a slight modifaction of the
\cite[Lemma 1.1]{St}, we deduce that
${ \frac{\partial \psi^+}{\partial t}}\in L^{p' }( Q) $ and
$(u_\varepsilon-\psi^+)$ is a possible test function.
We use it in \eqref{eq3.2a}.

Multiplying \eqref{eq3.2}  by the test function
$( u_{\varepsilon }-\psi ^{+}) $ we get, denoting by
$\langle ,\rangle $ the duality
pairing between $W_0^{1,p}( \Omega ) $ and its dual
\begin{equation}\label{eq3.3}
\begin{aligned}
&\int_0^{t}\big\langle \frac{\partial ( u_{\varepsilon }-\psi
^{+}) }{\partial t},u_{\varepsilon }-\psi^{+}\big\rangle dt'
+\int_0^{t}\int_{\Omega }\ a( x,t' ,u_{\varepsilon
},Du_{\varepsilon })D( u_{\varepsilon }-\psi^{+})
\,dx\,dt'  \\
&+\int_0^{t}\int_{\Omega }
g_{\varepsilon }( x,t' ,u_{\varepsilon }, Du_{\varepsilon })( u_{\varepsilon }-\psi^{+})
\,dx\,dt'  \\
&-\frac{1}{\varepsilon ^{p-1}} \int_0^{t}\int_{\Omega } | (
u_{\varepsilon }-\psi) ^{-}| ^{p-2}( u_{\varepsilon
}-\psi) ^{-}( u_{\varepsilon }-\psi^{+}) \,dx\,dt'
 \\
&=\int_0^{t}\int_{\Omega } ( f-\frac{\partial \psi^+}{\partial t}
) ( u_{\varepsilon }-\psi^+)\,dx\,dt' .
\end{aligned}
\end{equation}
which implies
\begin{equation}\label{eq3.4}
\begin{aligned}
&\frac{1}{2} \| u_{\varepsilon }( t) -\psi^{+}(t) \| _{L^2
( \Omega ) }^{2}
 +\int_0^{t}\int_{\Omega }a( x,t' ,u_{\varepsilon
},Du_{\varepsilon })  Du_{\varepsilon }\,dx\,dt'
 \\
&+\int_0^{t}\int_{\Omega }u_{\varepsilon }g_{\varepsilon }(
x,t' ,u_{\varepsilon },Du_{\varepsilon })\,dx\,dt'
  \\
&{ +\frac{1}{\varepsilon^{p-1}}\int_0^{t}\| {( u_{\varepsilon }-\psi)
^{-}( t' ) }{}\| ^{p}_{L^p( \Omega ) }dt'+\frac{1
}{\varepsilon^{p-1}} \int_0^{t}\int_{\Omega }| (
u_{\varepsilon }-\psi) ^{-}| ^{p-1} \psi^{-}\,dx\,dt'}
 \\
& = \frac{1}{2} \| ( u_0-\psi
^{+}( 0) ) \| _{L^2
( \Omega ) }^{2}+\int_0^{t}\int_{\Omega }
 ( f-\frac{\partial \psi^{+}}{\partial t})  u_{\varepsilon }
\,dx\,dt'
-\int_0^{t}\int_{\Omega }( f-\frac{\partial \psi^{+}}{\partial t})
 \psi^{+}\,dx\,dt' \\
&\quad +\int_0^{t}\int_{\Omega }a( x,t' ,u_{\varepsilon
},Du_{\varepsilon }) D\psi^{+}\,dx\,dt'
+\int_0^{t}\int_{\Omega }\psi^{+}g_{\varepsilon }( x,t' ,u_{\varepsilon },
Du_{\varepsilon })\,dx\,dt'\,.
\end{aligned}
\end{equation}

Using the conditions \eqref{eqcoer}, \eqref{eqcriosg}, \eqref{eq2.3},
\eqref{eq2.4}, \eqref{eq2.8},  Poincar\'e and   H\"{o}lder inequalities we obtain
\begin{equation}\label{eq3.5}
\begin{aligned}
&\int_Q |a(x,t,u_\varepsilon, Du_\varepsilon)D\psi^+|\,dx\,dt\\
&\le  \beta\int_Q |u_\varepsilon|^{p-1} |D\psi^+|\,dx\,dt
+\beta\int_Q|Du_\varepsilon|^{p-1} |D\psi^+|\,dx\,dt
+\int_Q|k(x,t)|\,|D\psi^+|\,dx\,dt \\
&\leq \theta\int_Q|Du_\varepsilon|^{p}\,dx\,dt+M_1+M_2,
\end{aligned}
\end{equation}
and
$$
\big|\int_Q \psi^+g_\varepsilon(x,t,u_\varepsilon,Du_\varepsilon)\,dx\,dt\big|
\le 3\theta\int_0^t|Du_\varepsilon|^p_{L^p(\Omega)}\,dt' +M_3,
$$
where $\theta$ is any positive real number and  $M_1$, $M_2$  and $M_3$
depend on the data $\theta$ and $T$.

By \eqref{eqcoer},  we obtain
\begin{equation}\label{eq3.6}
 \int_0^{t}\int_{\Omega }a( x,t' ,u_{\varepsilon
},Du_{\varepsilon })Du_{\varepsilon }\,dx\,dt' \geq \alpha
\int_0^{t}\int_{\Omega }| Du_{\varepsilon }|
^{p}\,dx\,dt'
=\alpha \int_0^{t}\| Du_{\varepsilon }\| _{L^{p}(
\Omega ) }^{p}dt' .
\end{equation}
Moreover, since $f, \frac{\partial \psi^{+}}{\partial t}\in L^{p' }(
Q) $ and $u_0\in L^2( \Omega ) $ we deduce from \eqref{eq2.8}  and
 H\"{o}lder inequality that
\begin{equation}\label{eq3.8}
\begin{aligned}
&\int_0^{t}\int_{\Omega }( f-\frac{\partial \psi^{+}}{\partial t}
) u_{\varepsilon }\,dx\,dt' -\int_0^{t}\int_{\Omega }( f-
\frac{\partial \psi^{+}}{\partial t}) \psi^{+}\,dx\,dt' +\frac{1
}{2}\| ( u_0-\psi^{+}( 0) ) \| _{L^2( \Omega ) }^{2} \\
&\leq M_4+\theta \int_0^{t}\| Du_{\varepsilon}\| _{L^{p}( \Omega ) }^{p}dt' .
\end{aligned}
\end{equation}

Now we deduce from \eqref{eq3.4}  and  inequalities \eqref{eq3.5}, \eqref{eq3.6}
and \eqref{eq3.8} that
\begin{equation}\label{eq3.9}
\begin{aligned}
&\frac{1}{2}\| u_{\varepsilon }( t) -\psi^{+}(t) \| _{L^2 ( \Omega ) }^{2}
+( \alpha -5\theta ) \int_0^t\| u_{\varepsilon }\| _{W_0^{1,p}( \Omega )
}^{p}dt' \\
&+\int_0^{t}\int_{\Omega }u_{\varepsilon }g_{\varepsilon }(
x,t' ,u_{\varepsilon },Du_{\varepsilon })  \,dx\,dt'
+\frac{1}{\varepsilon ^{p-1}}\int_0^{t}\| {( u_{\varepsilon }-\psi)
^{-}( t' ) }\| _{L^{p}( \Omega) }^{p}dt' \\
&+\frac{1}{\varepsilon ^{p-1}} \int_0^{t}\int_{\Omega }| (
u_{\varepsilon }-\psi) ^{-}| ^{p-2}(u_{\varepsilon }-\psi)^{-}\psi^{-}\,dx\,dt'
\\
&\leq M_1+M_2+M_3+M_4.
\end{aligned}
\end{equation}

Choosing $\theta$ small enough (for example $\theta= \frac{\alpha}{10})$
it results that
\begin{gather}\label{eq3.10}
\| u_{\varepsilon }\| _{L^{p}( 0,T;W_0^{1,p}(\Omega ) ) }\leq C_1,\\
\label{eq3.11}
\| u_{\varepsilon }\| _{L^{\infty }( 0,T;L^2( \Omega ) ) }\leq C_2, \\
\label{eq3.12}
\int_{Q}u_{\varepsilon } g_{\varepsilon }( x,t,u_{\varepsilon
},Du_{\varepsilon }) \, dx\, dt\leq C_3.
\end{gather}
Note that $\theta, M_i$ and $C_i$ denote nonnegative constants
which do no depend on $\varepsilon $.
Then by extracting a subsequence also denoted by $u_{\varepsilon }$,
we see that there exists
\begin{equation}\label{eq3.13}
u_\varepsilon\in L^{p}( 0,T;W_0^{1,p}( \Omega) ) \cap L^{\infty
}( 0,T;L^{2}( \Omega ) )
\end{equation}
such that
\begin{gather}\label{eq3.14}
u_{\varepsilon }\rightharpoonup u  \quad\text{weakly in }L^{p}(0,T;W_0^{1,p}( \Omega ) ),\\
\label{eq3.15}
u_{\varepsilon }\rightharpoonup u  \quad \text{weakly star in }L^{\infty }(
0,T;L^{2}( \Omega ) )
\end{gather}
Then  \eqref{eq2.10}  is proved.

\subsection{$L^p(Q) $-estimate of $ \frac{(u_\varepsilon-\psi)^-}{\varepsilon}$}

The equation \eqref{eq3.2} can be written as
\begin{equation}\label{eq3.222}
\begin{aligned}
&\frac{\partial( u_\varepsilon-\psi) }{\partial t}
-\operatorname{div}[(a(x,t,u_\varepsilon ,Du_\varepsilon )
-a(x,t,u_\varepsilon,D\psi))]+g_\varepsilon(x,t,u_\varepsilon ,Du_\varepsilon) \\
&-\frac{1}{\varepsilon^{p-1}}| ( u_\varepsilon-\psi)
^{-}| ^{p-2}( u_\varepsilon-\psi) ^{-}\\
&=f-  \frac{\partial \psi }{\partial t}
+\operatorname{div}(a(x,t,u_\varepsilon,D\psi )),\quad \text{in }Q.
\end{aligned}
\end{equation}
Multiplying  \eqref{eq3.222}  by the test function  $-\frac{( u_{\varepsilon
}-\psi ) ^{-}}{\epsilon }$,  we obtain
\begin{equation}\label{eq3.333}
\begin{aligned}
&-\frac{1}{\varepsilon } \int_0^{T}\big\langle \frac{\partial (
u_{\varepsilon }-\psi ) }{\partial t},( u_{\varepsilon
}-\psi ) ^{-}\big\rangle dt\\
&-\frac{1}{\varepsilon }\int_{Q}[(a(x,t,u_\varepsilon ,Du_\varepsilon )
-a(x,t,u_\varepsilon,D\psi))]D( u_{\varepsilon }-\psi ) ^{-}\,dx\,dt \\
&-\frac{1}{\varepsilon }\int_{Q}( u_{\varepsilon }-\psi )
^{-}g_{\varepsilon }( x,t,u_{\varepsilon },Du_{\varepsilon })\,dx\,dt
 +\frac{1}{\varepsilon ^{p}}\int_{Q}|
( u_{\varepsilon }-\psi ) ^{-}| ^{p}\,dx\,dt\\
&=-\frac{1}{\varepsilon } \int_0^{T} \big\langle f-\frac{\partial \psi }{\partial t}
+ \operatorname{div}(a(x,t,u_\varepsilon, D\psi)),( u_{\varepsilon }-\psi ) ^{-}
\big\rangle dt.
\end{aligned}
\end{equation}
Using \eqref{eq2.6},  \eqref{eq2.9}, \eqref{eq2.9bis}, we have
$ f-\frac{\partial \psi }{\partial t}
+ \operatorname{div}(a(x,t,u_\varepsilon, D\psi))\in L^{p'}(0,T; L^{p'}(\Omega))$,
 then using  Young inequality the right hand side  of  \eqref{eq3.333} is absorbed
 by the fourth term of the left hand side. On the set where $u_\varepsilon\le \psi$,
thanks to the strict monotony,   the second term is non negative.

Concerning the  third term of \eqref{eq3.333}, we can rewrite it in the form
\begin{align*}
I&= -\frac{1}{\varepsilon }\int_{\{u_\varepsilon\le \psi,
 u_\varepsilon<0\}}( u_\varepsilon-\psi)
^-g_\varepsilon( x,t,u_\varepsilon, Du_\varepsilon )\,dx\,dt  \\
&\quad - \frac{1}{\varepsilon }\int_{\{0\le u_\varepsilon\le \psi \}}
( u_\varepsilon-\psi) ^-g_\varepsilon( x,t,u_\varepsilon, Du_\varepsilon )\,dx\,dt
=I_1+I_2,
\end{align*}
by the sign condition on $g$, $I_1$ is non negative.

For $I_2$ using the growth condition on $g, h, b$ and $\psi^+$, we can easily
obtain two positive constants $K_1$ and $K_2$ such that
$|g(x,t,u_\varepsilon, Du_\varepsilon)|\le K_1+K_2|Du_\varepsilon|^q$.
Then $I_2$ can be estimated as follows
\begin{align*}
|I_2|&\le K_1\int_{\{0\le u_\varepsilon\le \psi \}}
\frac{(u_\varepsilon-\psi)^-}{\varepsilon}\,dx\,dt
+ K_2\int_{\{0\le u_\varepsilon\le \psi \}}|Du_\varepsilon|^q
\frac{(u_\varepsilon-\psi)^-}{\varepsilon}\,dx\,dt\\
&=A_1+A_2.
\end{align*}
It is clear that $|A_1|\le C\|\frac{(u_\varepsilon-\psi)^-}{\varepsilon}\|_{L^p(Q)}$.
For $A_2$ we use \eqref{eq3.10} and H\"older inequality to obtain
\begin{align*}
A_2 &= K_2\int_{\{0\le u_\varepsilon\le \psi\}}|Du_\varepsilon|^q
\frac{(u_\varepsilon-\psi)^-}{\varepsilon}\,dx\,dt\\
&\le K_2\int_{\{0\le u_\varepsilon\le \psi \}}
\Bigr(|Du_\varepsilon|^{qr}\Bigl)^{\frac{1}{r}}
\Bigr((\frac{(u_\varepsilon-\psi)^-}{\varepsilon})^{r'}\Bigl)^{\frac{1}{r'}}\,dx\,dt
\end{align*}
with $\frac{1}{r}+\frac{1}{r'}=1$.
Choosing $r$ such that $qr=p$ and thus $r'=\frac{p}{p-q}$, one has
$A_2\le C\|\frac{(u_\varepsilon-\psi)^-}{\varepsilon}\|_{L^{r'}(Q)}$.
Since $q<p-1$ and thus  $r'<p$,  we get
$|A_2|\le C\|\frac{(u_\varepsilon-\psi)^-}{\varepsilon}\|_{L^p(Q)}$.
Therefore, we obtain
\begin{equation}\label{eq3.123}
\| \frac{( u_{\varepsilon }-\psi) ^{-}}{\varepsilon }
\| _{L^{p}( Q) }^{p} \leq C
\end{equation}
From \eqref{eq3.123} we infer that
\begin{equation}\label{eq3.17}
( u_{\varepsilon }-\psi) ^{-}\to 0\quad \text{strongly in } L^{p}( Q)
\end{equation}
and thus
\begin{equation}\label{eq3.18}
u\geq \psi \quad \text{a.e. on } Q
\end{equation}
which proves \eqref{eq2.12}.

\subsection{Equi-integrability of $g_{\varepsilon}( x,t,u_{\varepsilon },
Du_{\varepsilon }) $}
Now we adapt a method of \cite{W}  to prove the equi-integrability of
$g_{\varepsilon }( x,t,u_{\varepsilon },Du_{\varepsilon })$.
For $ \delta >0$,   define the sets
\begin{gather*}
F_{\delta }=\{ ( x,t) \in Q:| u| \leq \delta\},\\
G_{\delta }=\{ ( x,t) \in Q:| u| >\delta\}.
\end{gather*}
Using the estimates  \eqref{eq3.10}  on $u_{\varepsilon }$, the conditions
\eqref{eqcriosg}, \eqref{eq2.3}  and  \eqref{eq2.4}, for any measurable
 subset $E\subset Q$,   we have
\begin{equation}\label{eq3.19}
\begin{aligned}
&\int_{E} | g_{\varepsilon }( x,t,u_{\varepsilon},Du_{\varepsilon }) | \,dx\,dt\\
&=\int_{E \cap  F_{\delta}}| g_{\varepsilon }( x,t,u_{\varepsilon },
 Du_{\varepsilon}) | \,dx\,dt
 +\int_{E \cap  G_{\delta }}| g_{\varepsilon }(
x,t,u_{\varepsilon },Du_{\varepsilon }) | \,dx\,dt \\
&\leq \int_{E \cap  F_{\delta}}( \rho +| u_{\varepsilon }| ^{m})
( h( x,t) +| Du_{\varepsilon }|^{q})\,dx\,dt
 + \frac{1}{\delta }\int_{E \cap  G_{\delta }}u_{\varepsilon } g_{\varepsilon }(
x,t,u_{\varepsilon },Du_{\varepsilon }) \,dx\,dt\\
&\leq ( \rho +\delta ^{m}) \int_{E}( h( x,t) +| Du_{\varepsilon}| ^{q})\,dx\,dt
 +\frac{1}{\delta }\int_{E}u_{\varepsilon } g_{\varepsilon }(
x,t,u_{\varepsilon },Du_{\varepsilon }) \,dx\,dt\\
&\leq ( \rho +\delta ^{m}) ( \| h\| _{L^{\infty }( Q) }
| E|+C_1^{q/p}(| E|) ^{1-\frac{q}{p}}) +\ \frac{1}{\delta } C_3.
\end{aligned}
\end{equation}

From  \eqref{eq3.19},  by choosing first $\delta $ sufficiently
large and the measure of $E$ sufficiently small, we deduce that
\begin{equation}\label{eq3.20}
g_{\varepsilon }( x,t,u_{\varepsilon },Du_{\varepsilon })
\text{ is equi-integrable}.
\end{equation}
Note also that  \eqref{eq3.19}  with $E=Q$ implies
\begin{equation}\label{eq3.21}
g_{\varepsilon }( x,t,u_{\varepsilon },Du_{\varepsilon })
\text{ is bounded in }L^{1}( Q) .
\end{equation}

\subsection{Almost pointwise convergence of $u_{\varepsilon }$ and
$Du_{\varepsilon}$}
From \eqref{eq3.2} we can  write
$ \frac{\partial u_{\varepsilon }}{\partial t}=\lambda _1^{\varepsilon
}+\lambda _2^{\varepsilon } $, with
$\lambda_2^\varepsilon=g_\varepsilon(x,t, u_\varepsilon,Du_\varepsilon)$.
Since $u_{\varepsilon }$ is bounded in
$L^{p}( 0,T;W_0^{1,p}( \Omega )) $ (see \eqref{eq3.10} and
$\frac{( u_{\varepsilon }-\psi) ^{-}}{\varepsilon }$ is bounded in
$L^{p}( Q) $ (see  \eqref{eq3.123})  we deduce from \eqref{eq3.21} that
\begin{equation}\label{eq3.22}
 \frac{\partial u_{\varepsilon }}{\partial t}=\lambda _1^{\varepsilon
}+\lambda _2^{\varepsilon }
\end{equation}
with $\lambda _1^{\varepsilon}$ bounded in
$L^{p' }(0,T;W^{-1,p' }( \Omega ) )$ and
$\lambda _2^{\varepsilon }$ bounded in $L^{1}( Q)$.

Since $g_{\varepsilon}( x,t,u_{\varepsilon },Du_{\varepsilon
}) $ is equi-integrable in $L^{1}( Q) $ we can extract
subsequences (still denoted by $\lambda _1^{\varepsilon }$ and $\lambda
_2^{\varepsilon }$) such that
\begin{gather}\label{eq3.23}
\lambda _1^{\varepsilon }\rightharpoonup \lambda _1\quad \text{weakly in }
L^{p' }( 0,T;W^{-1,p' }( \Omega )),\\
\label{eq3.24}
\lambda _2^{\varepsilon }\rightharpoonup \lambda _2 \quad
\text{weakly in } L^{1}( Q)
\end{gather}
This implies 
\begin{equation}\label{eq3.25}
\frac{\partial u}{\partial t}=\lambda _1+\lambda _2\in L^{p'}( 0,T;W^{-1,p' }
( \Omega ) ) +L^{1}(Q)
\end{equation}
which proves \eqref{eq2.11}.

From \eqref{eq3.22}  and the estimate \eqref{eq3.10}
on $u_{\varepsilon }$ we have
\begin{equation}\label{eq3.26}
\parbox{95mm}{
$u_{\varepsilon }$ is bounded in $L^{p}( 0,T;W_0^{1,p}( \Omega))$
with $\frac{\partial u_{\varepsilon } }{\partial t}$ bounded in
$L^{p' }( 0,T;W^{-1,p' }( \Omega ))
+L^{1}( 0,T;L^{1}( \Omega ) ) \subset L^{1}(0,T;W^{-1,r}( \Omega ) )$
for all $r<\inf \{ \frac{N}{N-1},\frac{p}{p-1}\}$.
}\end{equation}
Since $W_0^{1,p}( \Omega ) \subset L^{p}( \Omega ) \subset W^{-1,r}( \Omega ) $
for $p>r$, the first injection being compact, a lemma of Aubin's type
(see eg. \cite[corollary 4]{S}) implies that
\begin{equation}\label{eq3.27}
u_{\varepsilon }\to u \quad \text{strongly in }L^{p}(0,T; L^{p}( \Omega ) )
\end{equation}
which also implies that at least for a subsequence; still denoted by
$u_{\varepsilon }$,
\begin{equation}\label{eq3.28}
u_{\varepsilon }\to u \quad \text{a.e in }Q.
\end{equation}

Then we apply a compactness result due to  Boccardo and  Murat
\cite{BM1, BM2}, and more precisely
\cite[Theorem 4.3 and Remark 4.1]{BM2}.
Since $u_{\varepsilon }$ is bounded in $L^{p}( 0,T;W_0^{1,p}(
\Omega )) $ and since
\begin{equation}\label{eq3.29}
\frac{\partial u_{\varepsilon }}{\partial t}-\operatorname{div}( a(
x,t,u_{\varepsilon },Du_{\varepsilon }) )
 =\lambda _1^{\varepsilon }+\lambda _2^{\varepsilon } \text{is bounded in }L^{p'
}( Q) +L^{1}( Q),
\end{equation}
in view of the approximation $g_{\varepsilon }( x,t,u_{\varepsilon
},Du_{\varepsilon }) $ which  is weakly compact in $L^{1}( Q) $
see  \eqref{eq3.20}, \eqref{eq3.21} and  \eqref{eq3.123},
we have (for a subsequence)
\begin{equation}\label{eq3.30}
Du_{\varepsilon }\to Du  \quad \text{strongly in }L^{q}( Q)  \forall q<p,
\end{equation}
which implies
\begin{equation}\label{eq3.31}
D u_{\varepsilon }\to Du \quad \text{a.e in }Q.
\end{equation}

\subsection{Passing to the limit in the equation}
Using  \eqref{eq3.1} and
\begin{equation}\label{eq3.32}
g_{\varepsilon }( x,t,u_{\varepsilon },Du_{\varepsilon })
\to g( x,t,u,Du)  \quad \text{a.e in }Q,
\end{equation}
which follows from  \eqref{eq3.28}, \eqref{eq3.31}  and
 \eqref{eq3.20}, we deduce, by Vitali's theorem, that
\begin{equation}\label{eq3.33}
g_{\varepsilon }(x,t, u_{\varepsilon },Du_{\varepsilon }) \to
g(x,t,  u,Du)  \quad \text{strongly in }L^{1}( Q).
\end{equation}
Moreover since $u_{\varepsilon }g_{\varepsilon }(x,t, u_{\varepsilon
},Du_{\varepsilon }) \geq 0$ a.e. in $Q $ and by \eqref{eq3.12},
Fatou's lemma implies
\begin{equation}\label{eq3.34}
ug(x,t, u;Du)  \text{ belongs to }L^{1}( Q) .
\end{equation}
which completes the proof of \eqref{eq2.15}.

Similarly since $u_{\varepsilon }$ is bounded in
 $L^{p}(0,T;W_0^{1,p}( \Omega )) $ (see  \eqref{eq3.10}) and since
 $ u_{\varepsilon }$ and $Du_{\varepsilon }$ tends to $u$ and $Du$ a.e in $Q$
we have
\begin{equation}\label{eq3.35}
a( x,t,u_{\varepsilon },Du_{\varepsilon }) \rightharpoonup
a( x,t,u,Du)  \quad \text{weakly in }L^{p' }(Q).
\end{equation}
Since $\frac{( u_{\varepsilon }-\psi) ^{-}}{\varepsilon }$ is bounded
in $L^{p}( Q) $ (see \eqref{eq3.123})
\begin{equation}\label{eq3.36}
\frac{1}{\varepsilon ^{p-1}}| ( u_{\varepsilon }-\psi)
^{-}| ^{p-2 }( u_{\varepsilon }-\psi)
^{-}\rightharpoonup \mu  \quad \text{weakly in }L^{p' }( Q)
\end{equation}
and we have $\mu \in L^{p' }( Q) $, $\mu \geq 0$ which
proves  \eqref{eq2.13},   \eqref{eq2.14}.
Therefore we can pass to the limit in each term of \eqref{eq3.2}
and thus prove that equation  \eqref{eq2.16}  holds.

Let us now prove \eqref{eq2.17}; i.e,
$$
\mu \cdot (u-\psi) = 0 \quad \text{a.e. in } Q\,.
$$
This follows from the equality
$$
\frac{1}{ \varepsilon^{p-1}} | (u_\varepsilon-\psi)^-|^{p-2}
(u_\varepsilon-\psi)^-  (u_\varepsilon-\psi)
= - \varepsilon |\frac{(u_\varepsilon-\psi)^-}{  \varepsilon }|^p
$$
since  $u_\varepsilon$ tends to $u$ strongly in $L^p(Q)$ (see \eqref{eq3.27})
while $\frac{1}{\varepsilon^{p-1}} |(u_\varepsilon-\psi)^-|^{p-2}
(u_\varepsilon-\psi)^-$ tends weakly to $\mu$ in $L^{p'}(Q)$ and
$\frac{(u_\varepsilon-\psi)^-}{\varepsilon}$ is bounded in $L^p(Q)$.


\subsection{Initial condition}
To complete the proof of the Theorem it remains to prove that
\eqref{eq2.18} and \eqref{eq2.19} hold.
We first  prove that for $r<\inf \{ \frac{N}{N-1},\frac{p}{p-1}\} $
\begin{equation}\label{eq3.38}
u_{\varepsilon }\to u\quad \text{strongly in }C^{0}(0,T;W^{-1,r}( \Omega ) ) .
\end{equation}
This allows us  to pass to the limit in
$u_{\varepsilon }( x,0) =u_0( x) $ and implies that $u $ satisfies the
initial condition.

Recalling that $g_{\varepsilon }(x,t, u_{\varepsilon },Du_{\varepsilon
}) $ converges in the strong topology of $L^{1}( Q) $, (see
\eqref{eq3.33}) we can improve \eqref{eq3.22}  to
\begin{equation}\label{eq3.39}
\frac{\partial u_{\varepsilon }}{\partial t}
=\lambda _1^{\varepsilon }+\lambda _2^{\varepsilon }
\end{equation}
with $\lambda _1^{\varepsilon }$ bounded in the space
 $L^{p' }( 0,T;W^{-1,p' }( \Omega ))$
and $\lambda _2^{\varepsilon }$ relatively compact in
$L^{1}( 0,T; L^{1}( \Omega ) )$.
Since
\begin{equation}\label{eq3.40}
W^{-1,p' }( \Omega ) +L^{1}( \Omega ) \subset
W^{-1,r}( \Omega ),
\end{equation}
for all $h>0$ we have
\begin{equation}\label{eq3.41}
\begin{aligned}
&\| u_{\varepsilon } ( t+h) -u_{\varepsilon }(t) \| _{W^{-1,r}( \Omega ) }\\
&=\| \int_{t}^{t+h}( \lambda _1^{\varepsilon }+\lambda
_2^{\varepsilon }) dt' \| _{W^{-1,r}( \Omega) }\\
&\leq C\int_{t}^{t+h}\| \lambda _1^{\varepsilon }\|
_{W^{-1,p' }( \Omega ) }dt' +C\int_{t}^{t+h}\|
\lambda _2^{\varepsilon }\| _{L^{1}( \Omega )}dt'\\
&\leq C h^{\frac{1}{p}}\| \lambda _1^{\varepsilon }\|
_{L^{p' }( 0,T;W^{-1,p' }( \Omega ) )
}+C\| \lambda _2^{\varepsilon }\| _{L^{1}(
t,t+h;L^{1}( \Omega ) ) },
\end{aligned}
\end{equation}
which  in view of \eqref{eq3.39} implies  that the function $u_{\varepsilon }$ 
is uniformly equicontinuous in $C^{0}( 0,T;W^{-1,r}( \Omega ) ) $.
Since $u_{\varepsilon }$ is bounded in $L^{\infty }( 0,T;L^{2}(
\Omega ) ) $, (see  \eqref{eq3.11}) we deduce from Ascoli's theorem
(see, eg \cite[Lemma 1]{S}) that  $u_{\varepsilon }$ is relatively compact in
$C^{0}( 0,T;W^{-1,r}( \Omega ) ) $ which proves \eqref{eq3.38}.


\subsection*{Remarks}
In this article, we assumed that $p>2$, and realized that does not seem to be easy
extending this method for the case $p<2$.

 It seems difficult to avoid a supplementary condition on $\psi$
like  \eqref{eq2.9bis}.  A similar condition is assumed for example  in
\cite[hypotheses (9), (10)]{D}.
The condition \eqref{eq2.9bis} can be seen as follows:
let us define for $u\in L^p(0,T, W^{1,p}_0(\Omega))$ the function
$G=f-\frac{\partial \psi}{\partial t}+\operatorname{div} a(x,t,u,D\psi)$.
The hypotheses on $a, \psi$ are set in order  to have $G\in L^{p'}(Q)$.
In the case where $a$ is independent of $u$, this is essentially a regularity
condition on the obstacle $\psi$. If $a$ depends on $u$, then with suitable
condition on the derivative of $a(x,t,s,\xi)$ with respect to $x, s, \xi$
one can see that\eqref{eq2.9bis} is satisfied by a function $a$ of the form
 $a(x,t,s,\xi)=b(x,t,s)|\xi|^{p-2}\xi$.

\subsection*{Acknowledgements}
The author is indebted to  the anonymous referees for their valuable comments
and suggestions that helped improving the original manuscript.

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\end{document}