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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 72, pp. 1--14.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/72\hfil Existence of solutions]
{Existence of solutions for a degenerate seawater intrusion problem}
\author[M. El Alaoui T. \& M. H. Tber\hfil EJDE-2005/72\hfilneg]
{Mohamed El Alaoui Talibi, Moulay Hicham Tber}   % in alphabetical order

\address{Mohamed El Alaoui Talibi \hfill\break
Facult\'{e} des Sciences Semlalia\\
 Marrakech, Morocco}
\email{elalaoui@ucam.ac.ma}

\address{Moulay Hicham Tber \hfill\break
Facult\'{e} des Sciences Semlalia\\
 Marrakech, Morocco}
\email{hicham.tber@ucam.ac.ma}

\date{}
\thanks{Submitted November  1, 2004. Published June 30, 2005.}
\subjclass[2000]{35K60, 35K65, 76S05, 76T05}
\keywords{Seawater intrusion; elliptic-parabolic system;
degenerate equations; \hfill\break\indent existence result}

\begin{abstract}
 We study a seawater intrusion problem in a confined aquifer.
 This process can be formulated as a coupled system of partial
 differential equations which includes an elliptic and a degenerate
 parabolic equation.
 Existence results of weak solutions, under realistic assumptions,
 are established through time discretization combined with parabolic
 regularization.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
 \numberwithin{figure}{section}
 \newtheorem{thm}{Theorem}[section]
\newtheorem{defn}[thm]{Definition}
 \newtheorem{lem}[thm]{Lemma}
 \newtheorem{prop}[thm]{Proposition}

\section{Introduction}

The motivation for the following mathematical problem arises from
the area of modelling groundwater in coastal aquifers. Groundwater
is a major source of water supply in many parts of the world. It
supports domestic consumption, irrigation, and industrial
processing. The use of groundwater has been rising steadily in the
last several decades. It has been exploited to sustain a growing
population and economy. The loss of surface water to pollution has
further increased the stress on groundwater extraction. By now, as
much as on third of the world's drinking water is derived from
groundwater. Although better protected than surface water,
groundwater can also be contaminated. Once contaminated--because
of its subsurface, hidden, and inaccessible nature--detection and
remediation are more difficult.

Despite its abundance, unregulated extraction of groundwater can
easily cause localized problems. In coastal zones, the intensive
extraction of groundwater has upset the long established balance
between freshwater and seawater potentials, causing encroachment
of seawater into freshwater aquifers. As a large proportion of the
world's population (about 70\%) dwells in coastal zones, the
optimal exploitation of fresh groundwater and the control of
seawater intrusion are the challenges for the present-day and
future water supply engineers and managers. The modelling of
groundwater in coastal aquifers is an important and difficult
issue in water resources. The primary difficulty resides in
efficient and accurate simulation of the movement of the saltwater
front. Freshwater and saltwater are miscible fluids and therefore,
the zone separating them takes the form of a transition zone
caused by hydrodynamic dispersion. For certain problems, the
simulation can be simplified by assuming that each liquid is
confined to a well defined portion of the flow domain with an
abrupt interface separating the two domains (cf. \cite{Bear},
\cite{BearVerrujit} and \cite{Essaid}). This modelling approach,
called sharp interface, does not give information concerning the
nature of the transition zone but does reproduce the regional flow
dynamics of the system and the response of the interface to
applied stresses, for more details about seawater intrusion
problem with sharp interface approach we refer to
\cite[Section 13.2]{BearVerrujit}.

In the present paper, we address the seawater intrusion problem
with  sharp interface model in a confined aquifer. The model to be
presented herein is formulated in terms of a  two-dimensional
coupled system consisting of an elliptic and a degenerate
parabolic  equations. The main difficulties related to the
analysis of this system are the coupling between equations and the
degeneracy due to the possibility to have no saltwater in some
zones of the aquifer. This type of system occurs in a variety of
physical situations such as in petroleum engineering and has been
studied by many authors (see, e.g.,
\cite{Alt,Amziane,Antonsev,Arbogast,Hidani,Ewing,Gagneux}). Let us
also mention that the steady state seawater problem has been
treated in \cite{ElalaouiTber}, however the inflow of the
saltwater is mostly a transient process, then the time-dependent
problem is of greater practical interest. In the present paper, we
use the technique developed by Alt and Luckauss \cite{Alt} to
derive an existence result for the transient system modeling
seawater interface problem with sharp interface under realistic
assumptions.

The outline of the paper is as follows. In section~2 all necessary
mathematical notations are defined, the equations of the problem
are formulated and the general assumptions are stated. The third
section is devoted to the presentation and analysis of a
regularized problem. We prove the existence of at least one weak
solution for the problem in the non degenerate case. The result is
obtained by time discretization and the technique developed in
\cite{Alt}. In the last section, we get the existence of weak
solutions for the degenerate case.

\section{Problem setting and assumptions}


\subsection*{The differential system}

We consider the flow of fresh and salt groundwater, separated by a sharp
interface, in a confined aquifer. The aquifer is bounded by two approximately
horizontal and impermeable layers (Figure \ref{Intrusion}). The lower
and upper surfaces of the aquifer are described by $z=-H_{2}$ and
$z=-H_{1}$, respectively.

\begin{figure}
\begin{center}
\includegraphics[width=10cm, height=5cm]{fig1}
\end{center}
\caption{Saltwater intrusion phenomena} \label{Intrusion}
\end{figure}

The substitution of Darcy's law into continuity equations of the two
fluids (fresh and salt), the continuity of flux and the pressure through
the interfacial boundary and the integral of aquations over the vertical
lead to the following system of coupled partial differential
 equations \cite{BearVerrujit}:
\begin{equation} \label{e2.1}
\begin{gathered}
S(x)\partial_{t}h-\mathop{\rm div}(\alpha k(x)T_{s}(h)\nabla h)
+\mathop{\rm div}(k(x)T_{s}(h)\nabla\varphi)  =  -I_{s}\\
-\mathop{\rm div}(k(x)T_{a}\nabla\varphi)+\mathop{\rm div}(\alpha
k(x)T_{s}(h)\nabla h)  =  I_{f}+I_{s}
\end{gathered}
\end{equation}
for $(x,t)\in\Omega_{T}:=\Omega\times J$ with $J=]0,T[$ and
$\Omega$ is an open bounded domain of $\mathbb{R}^{2}$, describing
the projection of the porous medium on the horizontal plane $z=0$,
with a smooth boundary $\Gamma=\Gamma_{D}\cup\Gamma_{N}$. Here
$T_{a}=H_{2}-H_{1}$ is the thickness of the aquifer,
$T_{s}=H_{2}-h$ is the thickness of saltwater zone, $k$ is the
hydraulic conductivity, $S$ is the storativity of the aquifer,
$\alpha$ is a positive constant representing the relative density
difference, $\varphi$ is the freshwater hydraulic head, $h$ is the
depth of the interface and where $I_{f}$ and $I_{s}$ are supply
functions, representing distributed surface supply of fresh and
saline water into the aquifer.

Introducing the new variables $f=\dfrac{\varphi}{\alpha}$ and $K=\alpha k$ leads to the following system:
\begin{equation}
\begin{gathered}
S(x)\partial_{t}h-\mathop{\rm div}(K(x)T_{s}(h)\nabla h)
+\mathop{\rm div}(K(x)T_{s}(h)\nabla f)  =  -I_{s}\\
-\mathop{\rm div}(K(x)T_{a}\nabla f)+\mathop{\rm
div}(K(x)T_{s}(h)\nabla h)  =  I_{f}+I_{s}
\end{gathered}\label{System}
\end{equation}
The boundary conditions are
\begin{equation}
\begin{gathered}
h=h_{D},\, f=f_{D} \quad \text{on }  \Gamma_{D}\\
(K(x)T_{s}(h)\nabla h-K(x)T_{s}(h)\nabla f)\cdot\overrightarrow{n}=0
\quad \text{on }  \Gamma_{N}\\
(K(x)T_{a}\nabla f-K(x)T_{s}(h)\nabla h)\cdot\overrightarrow{n}=0
\quad \text{on } \Gamma_{N}
\end{gathered}\label{SystemBC}
\end{equation}
 where $f_{D}$ and $h_{D}$ are given functions, and $\overrightarrow{n}$
is the outward unit normal to $\Gamma$. The initial condition is
\begin{equation}
h(x,0)=h_{0}(x),\quad x\in\Omega.\label{SystemIC}
\end{equation}

\subsection*{Notation and assumptions}

We introduce the Hilbert space
 \[
V=\{ \varphi\in H^{1}(\Omega):\varphi=0\text{ on }\Gamma_{D}\},
\]
under assumption (A1) below, the norm and semi-norm defined
on $H^{1}(\Omega)$ are equivalent in $V$. We denote by $V'$
the dual space of $V$ and by $\langle .,.\rangle $ the
duality pairing between $V$ and $V'$. $(.,.)_{Q}$ is the
$L^{2}(Q)$ inner product ($Q$ is omitted if $Q=\Omega$). $\delta$
is a small positive real number and we denote by
$T_{f}(h)=h-H_{1}=T_{a}-T_{s}(h)$
the thickness of freshwater zone. We make now the following assumptions:
\begin{itemize}
\item[(A1)] $\Omega\subset\mathbb{R}^{2}$ is an open bounded domain
with Lipschitz boundary $\Gamma$,
 \[
\Gamma=\Gamma_{D}\cup\Gamma_{N},\quad
\Gamma_{D}\cap\Gamma_{N}=\emptyset,\quad \text{and}\quad
\mathop{\rm meas}(\Gamma_{D})\neq0.
\]

\item[(A2)] $S=S(x)\in L^{\infty}(\Omega)$, $S(x)\geq S_{*}>0$,
and $K(x)$ is bounded, symmetric,  and uniformly positive definite matrix,
i.e.,
 \[
0<K_{*}\leq|\xi|^{-2}\sum_{i,j=1}^{2}K_{i,j}(x)\xi_{i}\xi_{j}\leq K^{*}
<\infty\quad x\in\Omega,\,\xi\neq0\in\mathbb{R}^{2}.
\]

\item[(A3)]  $H_{1}$, $H_{2}$ are positive constants such
that $H_{2}>H_{1}+\delta$.

\item[(A4)] $I_{s}\geq0$, $(T_{a}-\delta)I_{f}-\delta I_{s}\geq0$,
$I_{s}\in L^{\infty}(J;\, L^{2}(\Omega))$, and
$I_{f}\in L^{\infty}(J;\, V'(\Omega))$.

\item[(A5)] The boundary data satisfy
$f_{D},\, h_{D}\in L^{2}(J;\, H^{1}(\Omega))$,
 \[
\partial_{t}h_{D}\in L^{1}(\Omega_{T}),\quad
 S(x)\partial_{t}h_{D}\in L^{2}(J,V'),\quad
 H_{1}+\delta\leq h_{D}\leq H_{2}
\]
a.e. on $\Omega_{T}$.

\item[(A6)] $h_{0}$ satisfies $H_{1}+\delta\leq h_{0}\leq H_{2}$,
 a.e. on $\Omega$.
\end{itemize}
To study the system (\ref{System})-(\ref{SystemIC}), we
use the regularization technique described above.


\section{The regularized problem}

To solve problem (\ref{System})-(\ref{SystemIC}), we first consider
its parabolic regularization:
\begin{equation}
\begin{gathered}
S(x)\partial_{t}h_{\varepsilon}-\mathop{\rm div}(K(x)T_{s}
(h_{\varepsilon})\nabla h_{\varepsilon})
-\varepsilon\Delta h_{\varepsilon}+\mathop{\rm div}(K(x)T_{s}(h_{\varepsilon})
\nabla f_{\varepsilon})  =  -I_{s}\\
-\mathop{\rm div}(K(x)T_{a}\nabla f_{\varepsilon})+\mathop{\rm
div}(K(x)T_{s}(h_{\varepsilon})\nabla h_{\varepsilon})
 = I_{f}+I_{s},
 \end{gathered}\label{SysReg}
\end{equation}
\begin{equation}
\begin{gathered}
h_{\varepsilon}=h_{D},\quad f_{\varepsilon}=f_{D} \quad\text{on } \Gamma_{D}\\
(K(x)T_{s}(h_{\varepsilon})\nabla h_{\varepsilon}
+\varepsilon\nabla h_{\varepsilon}-K(x)T_{s}(h_{\varepsilon})
\nabla f_{\varepsilon})\cdot\overrightarrow{n}=0 \quad \text{on }
 \Gamma_{N}\\
(K(x)T_{a}\nabla f_{\varepsilon}-K(x)T_{s}(h_{\varepsilon})
\nabla h_{\varepsilon})\cdot\overrightarrow{n}=0 \quad \text{on }
 \Gamma_{N}
\end{gathered} \label{SysRegBC}
\end{equation}
and
\begin{equation}
h_{\varepsilon}(x,0)=h_{0}(x),\quad x\in\Omega.\label{SysRegIC}
\end{equation}
where $\varepsilon$ is a small positive parameter.

\begin{defn}\label{Def1} \rm
A pair of functions $(h_{\varepsilon},f_{\varepsilon})$,
is called a weak solution to the regularized problem $(\ref{SysReg})-(\ref{SysRegIC})$
if it satisfies the system:
\begin{gather}
H_{1}+\delta\leq h_{\varepsilon}\leq H_{2},\quad\text{ a.e. on }\Omega_{T};
\label{Def1Eq1}
\\
h_{\varepsilon}\in L^{2}(J,V)+h_{D},\quad
S(x)\partial_{t}h_{\varepsilon}\in L^{2}(J,V'),\quad
f_{\varepsilon}\in L^{2}(J,V)+f_{D},\label{Def1Eq2}
\\
-\int_{J}\langle S\partial_{t}h_{\varepsilon},\varphi\rangle dt
=\int_{J}(Sh_{\varepsilon},
\frac{\partial\varphi}{\partial t})+(S(x)h_{0}(x),\varphi(x,0))
\quad\forall\varphi\in\mathcal{D}(\Omega\times[0,T[),\label{Def1Eq3}
\\
\begin{aligned}
\int_{J}\langle S\partial_{t}h_{\varepsilon},v\rangle dt
+\int_{J}(K(x)T_{s}(h_{\varepsilon}) \nabla h_{\varepsilon},\nabla v)dt&\\
+\varepsilon\int_{J}(\nabla h_{\varepsilon}, \nabla v)
-\int_{J}(K(x)T_{s}(h_{\varepsilon})\nabla f_{\varepsilon},\nabla v)dt&
=-\int_{J}(I_{s},v)dt,\quad\forall v\in L^{2}(0,T,V),
\end{aligned}\label{Def1Eq4}
\\
\begin{aligned}
\int_{J}(K(x)T_{a}\nabla f_{\varepsilon},\nabla w)dt
&-\int_{J}(K(x)T_{s}(h_{\varepsilon})\nabla h_{\varepsilon},\nabla w)dt\\
&=\int_{J}(I_{s}+I_{f},w)dt, \quad \forall w\in L^{2}(0,T,V)\,.
\end{aligned} \label{Def1Eq5}
\end{gather}
\end{defn}

We now state the main result of this section.

\begin{thm} \label{Theo}
Under assumptions (A1)--(A6), the system \eqref{Def1Eq1}-\eqref{Def1Eq5}
has a weak solution in the sense of definition \ref{Def1}.
\end{thm}

To show this proposition we make use of a backward time difference
scheme:

For each positive integer $M$, divide $J$ into $m=2^{M}$ subintervals
of equal length $\Delta t=T/m=2^{-M}T$. Set $t_{i}=i\Delta t$ and
$J_{i}=(t_{i-1},t_{i}]$ for an integer $i$, $1\leq i\leq m$. Denote
the time difference operator by
\[
\partial^{\eta}v(t)=\frac{v(t+\eta)-v(t)}{\eta},
\]
 for any function $v(t)$ and constant $\eta\in\mathbb{R}$. Also
we define
\[
l_{\Delta t}(V)=\{ v\in L^{\infty}(J;V): v\text{ is constant in time on
 each subinterval }J_{i}\subset J\}.
\]
 For $v_{\Delta t}\in l_{\Delta t}(V)$, set $v^{i}=v_{\Delta t}|_{J_{i}}$
for notational convenience. Finally, let
\[
h_{D,\Delta t}=\frac{1}{\Delta t}\int_{J_{i}}h_{D}(x,\tau)d\tau,\quad
f_{D,\Delta t}=\frac{1}{\Delta t}\int_{J_{i}}f_{D}(x,\tau)d\tau,\quad
t\in J_{i}.
\]
 Now the discrete time solution is a pair of functions
 $h_{\Delta t}\in l_{\Delta t}(V)+h_{D,\Delta t}$,
$f_{\Delta t}\in l_{\Delta t}(V)+f_{D,\Delta t}$ satisfying
\begin{gather}
H_{1}+\delta\leq h_{\Delta t}\leq H_{2},\quad \text{a.e. on }\Omega_{T}.
\label{Def3Eq1}
\\
\begin{aligned}
&\int_{J}(S\partial^{-\Delta t}h_{\Delta t},v)dt
+\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla h_{\Delta t},\nabla v)dt\\
&+\varepsilon\int_{J}(\nabla h_{\Delta t},\nabla v)dt
-\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla f_{\Delta t},\nabla v)dt\\
&=-\int_{J}(I_{s},v)dt\quad\forall v\in l_{\Delta t}(V),
\end{aligned} \label{Def3Eq2}
\\
\begin{aligned}
&\int_{J}(K(x)T_{a}\nabla f_{\Delta t},\nabla w)dt
-\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla h_{\Delta t},\nabla w)dt \\
&=\int_{J}(I_{s}+I_{f},w)dt,\quad\forall w\in l_{\Delta t}(V).
\end{aligned}\label{Def3Eq3}
\end{gather}
This approximation scheme is extended such that $h_{\Delta t}=h_{0}$
for $t\leq0$.

In the following, $C$ indicates a generic constant independent of
$\Delta t$ which will probably take different values in different
occurrences.

\begin{lem} \label{Lemma1}
 The discrete scheme has at least one solution $(h_{\Delta t},f_{\Delta t})$.
\end{lem}

The proof of this lemma will be given in the end of this section.

\begin{lem} \label{Lemma2}
The solution of the discrete schemes also satisfies
\begin{equation}
\varepsilon\int_{J}\Vert h_{\Delta t}\Vert_{H^{1}(\Omega)}^{2}
+\int_{J}\Vert f_{\Delta t}\Vert_{H^{1}(\Omega)}^{2}\leq C
\label{EstLemma2}\end{equation}
 with a constant $C$ independent of $\Delta t$.
\end{lem}

\begin{proof}
Taking $v=h_{\Delta t}-h_{D,\Delta t}\in l_{\Delta t}(V)$ in (\ref{Def3Eq2})
and $w=f_{\Delta t}-f_{D,\Delta t}\in l_{\Delta t}(V)$ in (\ref{Def3Eq3}),
we have
\begin{equation}
\begin{aligned}
&\int_{J}(S\partial^{-\Delta t}h_{\Delta t},h_{\Delta t}-h_{D,\Delta t})
+\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla h_{\Delta t},\nabla h_{\Delta t}
-\nabla h_{D,\Delta t})\\
&+\varepsilon\int_{J}(\nabla h_{\Delta t},\nabla h_{\Delta t}
-\nabla h_{D,\Delta t})-\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla f_{\Delta t},
\nabla h_{\Delta t}-\nabla h_{D,\Delta t})\\
&=-\int_{J}(I_{s},h_{\Delta t}-h_{D,\Delta t}),
\end{aligned}\label{Proof1Eq1}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\int_{J}(K(x)T_{a}\nabla f_{\Delta t},\nabla f_{\Delta t}
-\nabla f_{D,\Delta t})dt-\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla h_{\Delta t},
\nabla f_{\Delta t}-\nabla f_{D,\Delta t})dt\\
&=\int_{J}(I_{s}+I_{f},f_{\Delta t}-f_{D,\Delta t})dt,
\end{aligned}\label{Proof1Eq2}
\end{equation}
 Summing these two equalities and noting
  that $T_{a}=T_{s}(h_{\Delta t})+T_{f}(h_{\Delta t}))$, we have
\begin{align*}
&\int_{J}(S\partial^{-\Delta t}h_{\Delta t},h_{\Delta t})dt
+\varepsilon\int_{J}(\nabla h_{\Delta t},\nabla h_{\Delta t})\\
&+\int_{J}(K(x)T_{f}(h_{\Delta t})\nabla f_{\Delta t},\nabla f_{\Delta t})dt
+\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla h_{\Delta t},\nabla h_{\Delta t})dt\\
&-2\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla f_{\Delta t},\nabla h_{\Delta t})dt
+\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla f_{\Delta t},\nabla f_{\Delta t})dt\ \\
&=\int_{J}(S\partial^{-\Delta t}h_{\Delta t},h_{D,\Delta t})dt
+\varepsilon\int_{J}(\nabla h_{\Delta t},\nabla h_{D,\Delta t})\\
&\quad +\int_{J}(K(x)T_{f}(h_{\Delta t})\nabla f_{\Delta t},\nabla f_{D,\Delta t})dt\\
&\quad +\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla h_{\Delta t},\nabla h_{D,
\Delta t})dt+\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla f_{\Delta t},
\nabla f_{D,\Delta t})dt\\
&\quad -\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla f_{\Delta t},\nabla h_{D,
\Delta t})dt-\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla h_{\Delta t},
\nabla f_{D,\Delta t})dt\\
&\quad -\int_{J}(I_{s},h_{\Delta t}-h_{D,\Delta t})
+\int_{J}(I_{s}+I_{f},f_{\Delta t}-f_{D,\Delta t})dt
\end{align*}
 Hence
\begin{align*}
&\int_{J}(S\partial^{-\Delta t}h_{\Delta t},h_{\Delta t})dt
+\varepsilon\int_{J}(\nabla h_{\Delta t},\nabla h_{\Delta t})
+\int_{J}(K(x)T_{f}(h_{\Delta t})\nabla f_{\Delta t},\nabla f_{\Delta t})dt\\
&+\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla(h_{\Delta t}
-f_{\Delta t}),\nabla(h_{\Delta t}-f_{\Delta t}))dt\\
&=\int_{J}(S\partial^{-\Delta t}h_{\Delta t},h_{D,\Delta t})dt
+\varepsilon\int_{J}(\nabla h_{\Delta t},\nabla h_{D,\Delta t})\\
&\quad +\int_{J}(K(x)T_{f}(h_{\Delta t})\nabla f_{\Delta t},\nabla f_{D,\Delta t})dt\\
&+\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla(h_{\Delta t}-f_{\Delta t}),
\nabla(h_{D,\Delta t}-f_{D,\Delta t}))dt\\
&-\int_{J}(I_{s},h_{\Delta t}-h_{D,\Delta t})dt+\int_{J}(I_{s}
+I_{f},f_{\Delta t}-f_{D,\Delta t})dt
\end{align*}
 Next it is easy to see that
 \begin{equation}
\int_{J}(S\partial^{-\Delta t}h_{\Delta t},h_{\Delta t})dt
=\sum_{i=1}^{i=m}(S(h^{i}-h^{i-1}),h^{i})
\geq\frac{1}{2}\{(Sh^{m},h^{m})-(Sh^{0},h^{0})\}.\label{Eq1}
\end{equation}
 Also, since $S=S(x)\in L^{\infty}(\Omega)$, $H_{1}+\delta\leq h_{\Delta t}\leq H_{2}$
and $H_{1}+\delta\leq h_{D}\leq H_{2}$, we have
\begin{equation}
\begin{aligned}
\int_{J}(S\partial^{-\Delta t}h_{\Delta t},h_{D,\Delta t})dt
&=(Sh^{m},h_{D}^{m})-(Sh^{0},h_{D}^{0})-\int_{0}^{T-\Delta t}(Sh_{\Delta t},
\partial^{\Delta t}h_{D,\Delta t})dt,\\
&\leq C+C\int_{0}^{T-\Delta t}\Vert\partial^{\Delta t}h_{D,\Delta t}
\Vert_{L^{1}(\Omega)},
\end{aligned}\label{Eq2}
\end{equation}
 and from
 \[
\int_{0}^{T-\Delta t}\Vert\partial^{\Delta t}h_{D,\Delta t}\Vert_{L^{1}
(\Omega)}=\sum_{i=1}^{m-1}\Vert h_{D}^{i}-h_{D}^{i-1}\Vert_{L^{1}(\Omega)},
\]
 it follows that
\[
\int_{0}^{T-\Delta t}\Vert\partial^{\Delta t}h_{D,\Delta t}\Vert_{L^{1}(\Omega)}
=\sum_{i=1}^{m-1}\frac{1}{\Delta t}\Vert\int_{t_{i-1}}^{t_{i}}
\int_{t-\Delta t}^{t}\partial_{t}h_{D}(.,\tau)d\tau dt\Vert_{L^{1}(\Omega)},
\]
 so \begin{equation}
\int_{0}^{T-\Delta t}\Vert\partial^{\Delta t}h_{D,\Delta t}\Vert_{L^{1}
(\Omega)}\leq\int_{J}\Vert\partial_{t}h_{D}\Vert_{L^{1}(\Omega)}dt.\label{Eq3}
\end{equation}
 Now, we use Young inequality and combine (A2), (A5), (\ref{Eq1})-(\ref{Eq3})
to taht for every $\mu>0$,
\begin{align*}
\varepsilon\int_{J}(\nabla h_{\Delta t},\nabla h_{\Delta t})
+\int_{J}((K(x)T_{f}(h_{\Delta t})-2\mu C)\nabla f_{\Delta t},
\nabla f_{\Delta t})dt&\\
+\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla(h_{\Delta t}-f_{\Delta t}),
\nabla(h_{\Delta t}-f_{\Delta t}))dt
& \leq  C
\end{align*}
 Then for $\mu$ small enough and by the fact that
$T_{f}(h_{\Delta t})\geq\delta>0$
(for all $h_{\Delta t}$ satisfying $(\ref{Def3Eq1})$) and $K$ satisfying
(A1) we obtain the desired result.
\end{proof}

\begin{lem} \label{lem3.5}
There is a subsequence such that, $h_{\Delta t}$ strongly converges
in $L^{2}(\Omega_{T})$.
\end{lem}

\begin{proof}
According to \cite[Lemma 2.6]{Ewing}, it suffices to show that there
exists a constant $C$ such
that, for any $\xi>0$,
\[
\frac{1}{\xi}\int_{\xi}^{T}\Vert S^{\frac{1}{2}}(h_{\Delta t}(.,t)
-h_{\Delta t}(.,t-\xi))\Vert_{L^{2}(\Omega)}^{2}dt\leq C.
\]
 Let $k$ be fixed $(1\leq k\leq m);$ for $\tau\in J_{i}$, we define
the interval $Q=Q(\tau)=((i-k)\Delta t,i\Delta t]$, and the characteristic
function $\chi_{Q}$.

Taking $v(x,t)=\chi_{Q}(t)\partial^{-k\Delta t}(h_{\Delta t}(x,\tau)
-h_{D,\Delta t}(x,\tau))\in l_{\Delta t}(V)$
in (\ref{Def3Eq2}), we obtain
 \begin{align*}
&\int_{J}(S\partial^{-\Delta t}h_{\Delta t},\chi_{Q}(t)
\partial^{-k\Delta t}h_{\Delta t})dt\\
&=\int_{J}(S\partial^{-\Delta t}h_{\Delta t},\chi_{Q}(t)
\partial^{-k\Delta t}h_{D,\Delta t})dt\\
&\quad -\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla h_{\Delta t},
\nabla\chi_{Q}(t)\partial^{-k\Delta t}(h_{\Delta t}-h_{D,\Delta t}))dt\\
&\quad +\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla f_{\Delta t},
\nabla\chi_{Q}(t)\partial^{-k\Delta t}(h_{\Delta t}-h_{D,\Delta t}))dt\\
&\quad -\varepsilon\int_{J}(\nabla h_{\Delta t},\nabla\chi_{Q}(t)
\partial^{-k\Delta t}(h_{\Delta t}-h_{D,\Delta t}))
-\int_{J}(I_{s},\chi_{Q}(t)\partial^{-k\Delta t}(h_{\Delta t}
-h_{D,\Delta t}))dt
\end{align*}
 applying the relation
  \[
\int_{J}\partial^{-\Delta t}h_{\Delta t}\chi_{Q}dt
=k\Delta t\partial^{-k\Delta t}h_{\Delta t}(.,\tau),
\]
 and integrating again from $k\Delta t$ to $T$, we claim that we obtain
\begin{align*}
&k\Delta t\int_{k\Delta t}^{T}\Vert S^{\frac{1}{2}}
\partial^{-k\Delta t}h_{\Delta t}(.,\tau)\Vert_{L^{2}(\Omega)}^{2}d\tau \\
&\leq C+k\Delta t\int_{k\Delta t}^{T}(S\partial^{-k\Delta t}h_{\Delta t}
(.,\tau),\partial^{-k\Delta t}h_{D,\Delta t}(.,\tau))d\tau.
\end{align*}
 In fact, if we take for example the term
  \[
\int_{k\Delta t}^{T}\int_{J}(K(x)T_{s}(h_{\Delta t})\nabla h_{\Delta t}(x,t),
\nabla\chi_{Q}(t)\partial^{-k\Delta t}h_{\Delta t}(x,\tau))dtd\tau
=I_{1}+I_{2},
\]
 where
\[
I_{1}=\frac{1}{k\Delta t}\int_{k\Delta t}^{T}\int_{J}(K(x)T_{s}(h_{\Delta t})
\chi_{Q}(t)\nabla h_{\Delta t}(x,t),\nabla h_{\Delta t}(x,\tau))dtd\tau
\]
 and
\[
I_{2}=\frac{1}{k\Delta t}\int_{k\Delta t}^{T}\int_{J}(K(x)T_{s}(h_{\Delta t})
\nabla h_{\Delta t}(x,t),\nabla\chi_{Q}(t)h_{\Delta t}(x,\tau-k\Delta t))dtd\tau.
\]
For $I_{1}$, we have $I_{1}\leq I_{1,1}+I_{1,2}$,
with
\begin{gather*}
I_{1,1}=\frac{1}{k\Delta t}\int_{k\Delta t}^{T}\int_{J}K(x)T_{s}
(h_{\Delta t})\chi_{Q}(t)|\nabla h_{\Delta t}(x,t)|^{2}dtd\tau,\\
I_{1,2}=\frac{1}{k\Delta t}\int_{k\Delta t}^{T}\int_{J}K(x)T_{s}
(h_{\Delta t})\chi_{Q}(t)|\nabla h_{\Delta t}(x,\tau)|^{2}dtd\tau.
\end{gather*}
 Hence
\begin{gather*}
I_{1,1}\leq C\frac{(\Delta t)^{2}}{k\Delta t}
\sum_{i=k}^{i=m}\sum_{j=i-k+1}^{j=i}\Vert\nabla h^{j}(x)
\Vert_{L^{2}(\Omega)}^{2}=C\frac{(\Delta t)^{2}}{k\Delta t}
\sum_{i=k}^{i=m}\sum_{j=1}^{j=k}\Vert\nabla h^{j+i-k+1}(x)
\Vert_{L^{2}(\Omega)}^{2}
\\
I_{1,1}\leq C\frac{\Delta t}{k\Delta t}\sum_{j=1}^{j=k}\sum_{i=k}^{i=m}
\Delta t\Vert\nabla h^{j+i-k}(x)\Vert_{L^{2}(\Omega)}^{2}
\leq C\Vert\nabla h_{\Delta t}\Vert_{L^{2}(\Omega_{T})}^{2}
\end{gather*}
 then by (\ref{EstLemma2}),
$I_{1,1}\leq C$.  For $I_{1,2}$, we have
\begin{gather*}
I_{1,2}=\frac{1}{k\Delta t}\int_{k\Delta t}^{T}\int_{J}K(x)T_{s}(h_{\Delta t})
\chi_{Q}(t)|\nabla h_{\Delta t}(x,\tau)|^{2}dt\,d\tau
 \\
I_{1,2}=\frac{1}{k\Delta t}\int_{k\Delta t}^{T}\int_{(i-k)\Delta t}^{i\Delta t}
K(x)T_{s}(h_{\Delta t})|\nabla h_{\Delta t}(x,\tau)|^{2}dt\,d\tau
 \\
I_{1,2}\leq C\frac{k\Delta t}{k\Delta t}\int_{k\Delta t}^{T}|
\nabla h_{\Delta t}(x,\tau)|^{2}dt\,d\tau\leq C.
\end{gather*}
Therefore,  $I_{1}\leq C$.
Similarly  we prove that $I_{2}$ and all the other diffusive terms are bounded.
 Moreover, as for (\ref{Eq3}), we obtain
 \begin{align*}
k\Delta t\int_{k\Delta t}^{T}(S\partial^{-k\Delta t}h_{\Delta t}(.,\tau),
\partial^{-k\Delta t}h_{D,\Delta t}(.,\tau))d\tau
&\leq C\int_{k\Delta t}^{T}\Vert\partial^{-k\Delta t}h_{D,\Delta t}
\Vert_{L^{1}(\Omega)}d\tau
 \\
&\leq C\Vert\partial_{t}h_{D}\Vert_{L^{1}(\Omega_{T})}.
\end{align*}
 Consequently, the estimation is valid and the strong convergence can be deduced.
\end{proof}

We are now ready to prove the main theorem.

\begin{proof}[Proof of Theorem \ref{Theo}]
By the lemmas above, there exists a subsequence, also denoted by
$(h_{\Delta t},f_{\Delta t})$,
and $(h,f)\in L^{2}(J,H^{1}(\Omega))^{2}$ such that
\begin{gather}
h_{\Delta t}-h_{D,\Delta t}\to h-h_{D},\quad\text{weakly in }L^{2}(J,V)
\label{Eq1PrPro} \\
f_{\Delta t}-f_{D,\Delta t}\to f-h_{D},\quad\text{weakly in }L^{2}(J,V)
\label{Eq2PrPop}\\
h_{\Delta t}\to h\quad\text{strongly in }L^{2}(\Omega_{T})\label{Eq3PrPro}\\
T_{s}(h_{\Delta t})\to T_{s}(h)\quad\text{strongly in }L^{2}(\Omega_{T})
\label{Eq4PrPro}\\
h_{\Delta t}\to h\quad\text{a. e. in }\Omega_{T}.\label{Eq4-5PrPro}
\end{gather}
 Next, for any $v\in L^{2}(J;V)$, $v_{\Delta t}\in l_{\Delta t}(V)$
for $\Delta t$ sufficiently small, where
$v_{\Delta t}(x,t)=\Delta t^{-1}\int_{J_{i}}v(x,\tau)d\tau$.
Observe that
\[
\int_{J}(S\partial^{-\Delta t}h_{\Delta t},v)dt
=\int_{J}(S\partial^{-\Delta t}h_{\Delta t},v_{\Delta t})dt\]
 and
\[
\Vert\nabla v_{\Delta t}\Vert_{L^{2}(\Omega_{t})}
\leq\Vert\nabla v\Vert_{L^{2}(\Omega_{t})}.
\]
 By taking $v_{\Delta t}$ as test function in $(\ref{Def3Eq2})$
and using $(\ref{EstLemma2})$ we get
\[
\int_{J}(S\partial^{-\Delta t}h_{\Delta t},v)dt
=\int_{J}(S\partial^{-\Delta t}h_{\Delta t},v_{\Delta t})dt
\leq C\Vert\nabla v_{\Delta t}\Vert_{L^{2}(\Omega_{t})}
\leq C\Vert\nabla v\Vert_{L^{2}(\Omega_{t})}.
\]
 Consequently, for a subsequence $S\partial^{-\Delta t}h_{\Delta t}$
converges weakly in $L^{2}(J,V')$, if $v\in\mathcal{D}(\Omega_{T})$,
we have
\begin{align*}
\langle S\partial^{-\Delta t}h_{\Delta t},v\rangle _{\mathcal{D}'(\Omega_{T}),
\mathcal{D}(\Omega_{T})}
&=\int_{J}(S\partial^{-\Delta t}h_{\Delta t},v)dt\\
&=-\int_{0}^{T-\Delta t}(Sh_{\Delta t},\partial^{\Delta t}v)dt\\
&\to -\int_{J}(Sh,\partial_{t}v)dt
=\langle S\partial_{t}h,v\rangle _{\mathcal{D}'(\Omega_{T}),
\mathcal{D}(\Omega_{T})},
\end{align*}
 Therefore,
 \begin{equation}
S\partial^{-\Delta t}h_{\Delta t}\rightharpoonup S\partial_{t}h
\quad\text{ weakly in }L^{2}(J,V').\label{Eq5PrPro}
\end{equation}
 Combining (\ref{Eq1PrPro})-(\ref{Eq5PrPro}), and since
 $\cup_{M=1}^{\infty}l_{\Delta t}(V)$
is dense in $L^{2}(J,V)$, we obtain (\ref{Def1Eq4}) and (\ref{Def1Eq5}).

 On other hand, if $v\in\mathcal{C}^{\infty}(\Omega_{T})$ with $v(x,T)=0$,
we find that
\[
\int_{J}(S\partial^{-\Delta t}h_{\Delta t},v)dt
+\int_{0}^{T-\Delta t}(S[h_{\Delta t}-h_{0}],\partial^{\Delta t}v)dt
=\frac{1}{\Delta t}\int_{T-\Delta t}^{T}(S[h_{\Delta t}-h_{0}],v)dt,
\]
 which yields (\ref{Def1Eq3}).
Finally by (\ref{Def3Eq1}) and (\ref{Eq4-5PrPro}) we find (\ref{Def1Eq1})
and thus the proof of the theorem is complete.
\end{proof}

\subsection*{Proof of Lemma \ref{Lemma1}}
In this subsection, we allow $h$ being outside $[H_{1}+\delta,H_{2}]$.
Here $T_{s}(h)$ is extended continuously and constantly outside
$[H_{1}+\delta,H_{2}]$.
Lemma \ref{Lemma1} is purely an elliptic result, and will follow
from the next proposition. For notational convenience the subscript
$\Delta t$ is omitted below.

\begin{prop} \label{prop3.6}
In addition to assumptions (A1)-(A6), suppose that
$0<\eta_{*}\leq\eta_{1}(x)\in L^{\infty}(\Omega)$
and $\eta_{2}(x)\in L^{\infty}(\Omega)$ such that
$\eta_{1}(x)(H_{1}+\delta)\leq\eta_{2}(x)\leq\eta_{1}(x)H_{2}$.
Then, the following problem has a weak solution
$(h,f)\in(V+h_{D})\times(V+f_{D})$:
\begin{gather}
H_{1}+\delta\leq h(x,t)\leq H_{2},\quad a.e.\,\text{on }\Omega_{T}.
\label{Def4Eq1}\\
\begin{aligned}
(\eta_{1}h,v)+(K(x)T_{s}(h)\nabla h,\nabla v)+\varepsilon(\nabla h,\nabla v)&&\\
-(K(x)T_{s}(h)\nabla f,\nabla v)&=-(I_{s},v)+(\eta_{2},v),\quad\forall
v\in V
\end{aligned}\label{Def4Eq2}
\\
(K(x)T_{a}\nabla f,\nabla w)-(K(x)T_{s}(h)\nabla h,\nabla w)
=(I_{s}+I_{f},w)\,\,\forall w\in V.\label{Def4Eq3}
\end{gather}
\end{prop}

\begin{proof}
Let $\{ v_{i}\}_{i=1}^{\infty}$ be a base for $V$, we set
$V_{m}=\mathop{\rm span}\{ v_{1},\dots ,v_{m}\}$.
With $V_{m}$ replacing $V$ in (\ref{Def4Eq2}) and (\ref{Def4Eq3}),
we obtain a Galerkin method.

For $v^{j}=\sum_{i=1}^{m}\beta_{i}^{j}v_{i}$, $j=1,2$, we introduce
the mapping $\Phi_{m}:\mathbb{R}^{2m}\rightarrow\mathbb{R}^{2m}$
by \[
\Phi_{m} \begin{pmatrix} \beta^{1}\\ \beta^{2}\end{pmatrix}
=\begin{pmatrix} \hat{\beta}^{1}\\ \hat{\beta}^{2}\end{pmatrix}\,,
\]
where \begin{align*}
\hat{\beta}_{i}^{1}
&=(\eta_{1}(v^{1}+h_{D}),v_{i})+(K(x)T_{s}(v^{1}+h_{D})\nabla(v^{1}+h_{D}),
\nabla v_{i})+\varepsilon(\nabla(v^{1}+h_{D}),\nabla v_{i})\\
&\quad -(K(x)T_{s}((v^{1}+h_{D}))\nabla(v^{2}+f_{D}),\nabla v_{i})-(I_{s},
v_{i})-(\eta_{2},v_{i}),
\\
\hat{\beta}_{i}^{2}
&=(K(x)T_{a}\nabla(v^{2}+f_{D}),\nabla v_{i})-(K(x)T_{s}(v^{1}
+h_{D})\nabla(v^{1}+h_{D}),\nabla v_{i})\\
&-(I_{s}+I_{f},v_{i})\,.
\end{align*}
 By assumptions (A1)-(A6), $\Phi_{m}$ is continuous. Also,
it can be shown  that, for any $\mu>0$,
\begin{align*}
\Phi_m\begin{pmatrix} \beta^{1}\\ \beta^{2}\end{pmatrix}
\cdot\begin{pmatrix} \beta^{1}\\ \beta^{2}\end{pmatrix}
&\geq\frac{1}{2}(\eta_{*}-\mu)\Vert v^{1}\Vert_{L^{2}(\Omega)}^{2}
+\frac{1}{2}(\varepsilon-\mu C')\Vert v^{1}\Vert_{V}^{2}\\
&\quad +[\frac{K_{*}\delta}{2}-C'\frac{\mu}{2}]\Vert v^{2}\Vert_{V}^{2}
-C(\mu,h_{D},f_{D},I_{f},I_{s})\\
&\quad +\frac{1}{2}(K(x)T_{s}(v^{1}+h_{D})\nabla(v^{1}-v^{2}),
\nabla(v^{1}-v^{2}))\,.
\end{align*}
 Therefore, for fixed $\mu$ small enough,
 $\Phi_m\begin{pmatrix} \beta^{1}\\ \beta^{2}\end{pmatrix} \cdot
 \begin{pmatrix} \beta^{1}\\ \beta^{2}\end{pmatrix}$ is strictly positive
for $|\beta^{1}|+|\beta^{2}|$ sufficiently large.  As result, $\Phi_{m}$
has a zero; i.e., there is a solution to the Galerkin approximation.

As in proof of Lemma $\ref{Lemma2}$ it can be seen that this Galerkin
solutions $h^{m}$ and $f^{m}$ are uniformly bounded in $H^{1}(\Omega)$
(independently of $m)$, so there exists a subsequence $h^{m}\rightharpoonup h$
and $f^{m}\rightharpoonup f$ weakly in $H^{1}(\Omega)$ with $h\in V+h_{D}$
and $f\in V+f_{D}$. Moreover, $h_{m}\rightarrow h$ strongly in $L^{2}(\Omega)$
and a.e. on $\Omega$. Therefore $(h,f)$ satisfy (\ref{Def4Eq2})
and (\ref{Def4Eq3}).\\
 Finally, a standard maximum principle argument on (\ref{Def4Eq2})
(with $v=(h-H_{2})^{+})$ can be applied to show that $h\leq H_2$.
To show that $h\geq H_{1}+\delta$ it suffices to set $v=(h-H_{1}-\delta)^{-}$
in (\ref{Def4Eq2}) and $w=\dfrac{(T_{a}-\delta)}{T_{a}}v$ in (\ref{Def4Eq3}).
Summing the two equations, using $(A4)$ and the fact that the extention
of $T_{s}$ is equal to $(T_{a}-\delta)$ on 
$\{ x\in\mathbb{R}: x\leq H_{1}+\delta\}$
we obtain the desired result.
\end{proof}

\section{Study of the degenerate problem}

In this section, we obtain the convergence of the solutions
of the regularized problem to a weak solution of the degenerate problem,
obtaining hence the main result of this paper.
We first define the weak formulation of the degenerate problem.

\begin{defn} \label{Def2} \rm
A pair of functions $(h,f)$, is called a weak solution
to the degenerate problem \eqref{System}-\eqref{SystemIC} if the
following proprieties are fulfilled
\begin{gather}
H_{1}+\delta\leq h\leq H_{2},\quad\text{a.e. on }\Omega_{T}.\label{Def2Eq1}
\\
h\in L^{2}(J,V)+h_{D},\quad S\frac{\partial h}{\partial t}\in L^{2}(J,V'),
\quad f\in L^{2}(J,V)+f_{D},\label{Def2Eq2}
\\
-\int_{J}\langle S\partial_{t}h,\varphi\rangle dt
=\int_{J}(Sh,\frac{\partial\varphi}{\partial t})+(S(x)h_{0}(x),\varphi(x,0))
\quad\forall\varphi\in\mathcal{D}(\Omega\times[0,T[),\label{Def2Eq3}
\\
\begin{aligned}
&\int_{J}\langle S\partial_{t}h,v\rangle dt
+\int_{J}(K(x)\sqrt{T_{s}(h)}\nabla\phi(h),\nabla v)dt-\int_{J}(K(x)T_{s}(h)
\nabla f,\nabla v)dt\\
&=-\int_{J}(I_{s},v)dt\quad\forall v\in L^{2}(0,T,V),
\end{aligned}\label{Def2Eq4}\\
\begin{aligned}
&\int_{J}(K(x)T_{a}\nabla f,\nabla w)dt
-\int_{J}(K(x)\sqrt{T_{s}(h)}\nabla\phi(h),\nabla w)dt\\
&=\int_{J}(I_{s}+I_{f},w)dt, \quad \forall w\in L^{2}(0,T,V),
\end{aligned}\label{Def2Eq5}
\end{gather}
 where $\phi(s)=\int_{H_{1}}^{s}\sqrt{T_{s}(\xi)}d\xi$ is introduced
to absorb the degeneracy of the equations.
\end{defn}

\begin{thm}\label{Theo02}
Under the assumptions (A1)-(A6), the system \eqref{System}-\eqref{SystemIC}
has a weak solution in the sense of definition \ref{Def2}.
\end{thm}

The proof of this theorem is based on the following lemmas

\begin{lem} \label{Lem-fin-1}
Let $(h_{\varepsilon},f_{\varepsilon})$ a solution
sequence to regularized problem. Then we have the following estimates:
\begin{gather*}
\Vert\nabla\phi(h_{\varepsilon})\Vert_{L^{2}(\Omega_{T})}^{2}\leq C\,,\quad
\varepsilon\Vert\nabla h_{\varepsilon}\Vert_{L^{2}(\Omega_{T})}^{2}\leq C\,,\\
\Vert\nabla f_{\varepsilon}\Vert_{L^{2}(\Omega_{T})}^{2}\leq C\,,\quad
\Vert S\partial_{t}h_{\varepsilon}\Vert_{L^{2}(J,V')}^{2}\leq C\,.
\end{gather*}
\end{lem}

\begin{proof} In this section $C$ is a generic constant independent of $\varepsilon$.
 Take $v=h_{\varepsilon}-h_{D}\in V$ in (\ref{Def1Eq4}) and
 $w=f_{\varepsilon}-f_{D}\in V$
in (\ref{Def1Eq5}) and summing the two equalities to have, for every $\mu>0$,
\begin{align*}
&\frac{1}{2}\int_{\Omega_{T}}\varepsilon|\nabla h_{\varepsilon}|^{2}
+\frac{1}{2}\int_{\Omega_{T}}(K(x)T_{f}(h_{\varepsilon})-\mu C)
|\nabla f_{\varepsilon}|^{2} \\
&+\frac{1}{2}\int_{\Omega_{T}}K(x)T_{s}(h_{\varepsilon})|
\nabla(h_{\varepsilon}-f_{\varepsilon})|^{2}\\
&\leq-\int_{J}\langle S\partial_{t}h_{\varepsilon},h_{\varepsilon}-h_{D}\rangle
+\frac{1}{2}\int_{\Omega_{T}}\varepsilon|\nabla h_{D}|^{2}+\frac{1}{2}
\int_{\Omega_{T}}K(x)T_{f}(h_{\varepsilon})|\nabla f_{D}|^{2}\\
&\quad +\frac{1}{2}\int_{\Omega_{T}}K(x)T_{s}(h_{\varepsilon})|\nabla(h_{D}
-f_{D})|^{2}+\int_{\Omega_{T}}|I_{s}(h_{\varepsilon}-h_{D})|\\
&\quad +\frac{1}{2\mu}\int_{\Omega_{T}}|I_{s}+I_{f}|^{2}+\int_{\Omega_{T}}|I_{s}
+I_{f}||f_{D}|
\end{align*}
Since
$H_{1}+\delta\leq h_{\varepsilon}\leq H_{2}$ and
$H_{1}+\delta\leq h_{D}\leq H_{2}$,
 we have
\begin{gather*}
T_{f}(h_{\varepsilon})=h_{\varepsilon}-H_{1}\geq\delta,\\
\int_{\Omega_{T}}|I_{s}(h_{\varepsilon}-h_{D})|
\leq C\Vert I_{s}\Vert_{L^{1}(\Omega_{T})}\,.
\end{gather*}
 Moreover, the following identity can be deduced as in \cite[Lemma 1.5]{Alt},
 \begin{align*}
&\int_{J}\langle S\partial_{t}h_{\varepsilon},h_{\varepsilon}-h_{D}\rangle\\
&=\int_{J}(S\partial_{t}h_{D},h_{\varepsilon}-h_{D})+\frac{1}{2}\Vert
S(h_{\varepsilon}-h_{D})(T)\Vert_{L^{2}(\Omega)}^{2}-\frac{1}{2}
\Vert S(h_{\varepsilon}-h_{D})(0)\Vert_{L^{2}(\Omega)}^{2}
\end{align*}
 also we have
\[
\int_{J}(S\partial_{t}h_{D},h_{\varepsilon}-h_{D})
\leq C\Vert\partial_{t}h_{D}\Vert_{L^{1}(\Omega_{T})}\,.
\]
 Therefore, for $\mu$ small enough,
\begin{gather*}
\varepsilon\Vert\nabla h_{\varepsilon}\Vert_{L^{2}(\Omega_{T})}^{2}\leq C\,,\\
\Vert\nabla f_{\varepsilon}\Vert_{L^{2}(\Omega_{T})}^{2}\leq C\,,\\
\int_{\Omega_{T}}K(x)T_{s}(h_{\varepsilon})|\nabla(h_{\varepsilon}
-f_{\varepsilon})|^{2}\leq C\,.
\end{gather*}
Since
\[
\Vert\nabla\phi(h_{\varepsilon})\Vert_{L^{2}(\Omega_{T})}^{2}
\leq C\int_{\Omega_{T}}K(x)T_{s}(h_{\varepsilon})|
\nabla h_{\varepsilon}|^{2}\,,
\]
 we have
\begin{align*}
&\Vert\nabla\phi(h_{\varepsilon})\Vert_{L^{2}(\Omega_{T})}^{2}\\
&\leq2C\int_{\Omega_{T}}K(x)T_{s}(h_{\varepsilon})|\nabla(h_{\varepsilon}
-f_{\varepsilon})|^{2}+2C\int_{\Omega_{T}}K(x)T_{s}(h_{\varepsilon})
|\nabla f_{\varepsilon}|^{2}\leq C.
\end{align*}
 To show the last estimate, let $v\in L^{2}(J,V)$, we have
 \[
\int_{J}\langle S\partial_{t}h_{\varepsilon},v\rangle dt
\leq\int_{J}(K(x)T_{s}(h_{\varepsilon})\nabla(f_{\varepsilon}
-h_{\varepsilon}),\nabla v)dt
+\varepsilon\int_{J}(\nabla h_{\varepsilon},\nabla v)dt
+\int_{J}(I_{s},v)dt,
\]
 Then
\[
\int_{J}\langle S\partial_{t}h_{\varepsilon},v\rangle
dt\leq C\Vert v\Vert_{L^{2}(J,V)}
\]
 which completes the proof.
 \end{proof}

\begin{lem} \label{lem-fin}
The sequence $(h_{\varepsilon},f_{\varepsilon})$,
also satisfies the following inequality
\begin{equation}
\int_{\xi}^{T}(S(h_{\varepsilon}(.,t)-h_{\varepsilon}(.,t-\xi)),
\phi(h_{\varepsilon}(.,t))-\phi(h_{\varepsilon}(.,t-\xi)))dt
\leq C\xi\quad\forall\xi\in[0,T],\label{Est}
\end{equation}
 and we can extract a subsequence, also denoted
$(h_{\varepsilon},f_{\varepsilon})$,
such that, $(h_{\varepsilon},\phi(h_{\varepsilon}))$
 converges strongly to $(h,\phi(h))$ in $L^{2}(\Omega_{T})$.
\end{lem}

\begin{proof}
Let $\xi\in[0,T]$, and $v\in L^{2}(J,V)$. We have
\[
\int_{\xi}^{T}(S(h_{\varepsilon}(.,t)-h_{\varepsilon}(.,t-\xi)),v)dt
\leq\int_{\xi}^{T}\Vert S(h_{\varepsilon}(.,t)-h_{\varepsilon}(.,t-\xi))
\Vert_{V'}\Vert v\Vert_{V}dt,
\]
 moreover we know that (see \cite[p. 155]{Brezis}), for all $\xi\in[0,T]$,
\[
\frac{1}{\xi^{2}}\int_{\xi}^{T}\Vert S(h_{\varepsilon}(.,t)-h_{\varepsilon}
(.,t-\xi))\Vert_{V'}^{2}dt\leq\int_{0}^{T}\Vert\partial_{t}Sh_{\varepsilon}
\Vert_{V'}^{2}dt\,.
\]
 Then
\[
\int_{\xi}^{T}(S(h_{\varepsilon}(.,t)-h_{\varepsilon}(.,t-\xi)),v)dt
\leq\xi\Vert v\Vert_{L^{2}(J,V)}\Vert S\partial_{t}h_{\varepsilon}
\Vert_{L^{2}(J,V')}\,;
\]
therefore, taking $v=\phi(h_{\varepsilon}(.,t))-\phi(h_{\varepsilon}(.,t-\xi)$
and by the previous lemma we have the estimate (\ref{Est}).

On the other hand we have $T_{s}(\xi)=H_{2}-\xi$. Then $\phi$ is
continuous strictly decreasing function on $[H_{1},H_{2}]$, and $\mathcal{C}^{1}$
on $]H_{1},H_{2}[$. Consequently $\phi^{-1}$ is lipschitz function
on $[H_{1},H_{2}]$.
 Hence,  since $H_{1}+\delta\leq h_{\varepsilon}\leq H_{2}$, we
obtain
\[
\int_{\xi}^{T}S(\phi^{-1}\circ\phi(h_{\varepsilon}(.,t))
-\phi^{-1}\circ\phi(h_{\varepsilon}(.,t-\xi)),\phi(h_{\varepsilon}(.,t))
-\phi(h_{\varepsilon}(.,t-\xi)))dt\leq C\xi
\]
 and
\[
\int_{\xi}^{T}(S(x)(\phi(h_{\varepsilon}(.,t))-\phi(h_{\varepsilon}(.,t-\xi)),
\phi(h_{\varepsilon}(.,t))-\phi(h_{\varepsilon}(.,t-\xi)))dt\leq C\xi
\]
 Moreover,
 \[
\Vert\nabla\phi(h_{\varepsilon})\Vert_{L^{2}(\Omega_{T})}^{2}\leq C\,.
\]
 Therefore, as in \cite[ Lemma 2.6]{Ewing}, we deduce that
$\phi(h_{\varepsilon})$ converges strongly in $L^{2}(\Omega_{T})$ to
$\phi(h)$.
\end{proof}


\begin{proof}[Proof of Theorem \ref{Theo02}]
By Lebesque's theorem, lemma \ref{Lem-fin-1} and lemma \ref{lem-fin}
we deduce that there exist $(h,f)\in(L^{2}(J,V)+h_{D})\times(L^{2}(J,V)+f_{D})$
and a subsequence $(h_{\varepsilon},f_{\varepsilon})$ such that
\begin{gather*}
h_{\varepsilon}\to h\quad\text{strongly in }L^{p}(\Omega_{T})\quad
\forall p\in[1,\infty[.
 \\
\phi(h_{\varepsilon})\to\phi(h)\quad\text{strongly in }
L^{p}(\Omega_{T})\,\,\forall p\in[1,\infty[\,,
\\
S\partial_{t}h_{\varepsilon}\to S\partial_{t}h\quad\text{weakly in }
L^{2}(J,V')\,,
\\
f_{\varepsilon}-f_{D}\to f-f_{D}\quad\text{weakly in }L^{2}(J,V),
 \\
\phi(h_{\varepsilon})-\phi(h_{D})\to\phi(h)-\phi(h_{D})\quad\text{weakly in }
L^{2}(J,V),
 \\
\sqrt{\varepsilon}(h_{\varepsilon}-h_{D})\to 0\quad\text{weakly in }L^{2}(J,V),
\\
h_{\varepsilon}\to h\quad\text{a. e. in }\Omega_{T}.
\end{gather*}
Since $\phi(h_{D})\in L^{2}(J,H^{1}(\Omega))$, as $\varepsilon$ approaches
$0$ in (\ref{Def1Eq4}) and in (\ref{Def1Eq5}), we obtain (\ref{Def2Eq4})
and (\ref{Def2Eq5}).
 To show (\ref{Def2Eq1}) and (\ref{Def2Eq3}) it suffices to proceed
as in theorem $\ref{Theo}$.
\end{proof}

\subsection*{Acknowledgments}
Our great thanks to Professor B. Amaziane for correcting the English of this paper. We would also like to thank the anonymous referee of this journal and Professor Julio G. Dix for  pointing out some corrections and their help in improving the presentation of an early version of this article.\\
This work was supported by the Action Integr\'{e}e CMIFM AI MA/04/94.


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