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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 197, pp. 1--12.\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{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/197\hfil Image restoration]
{Image restoration using a reaction-diffusion process}

\author[N. Alaa, M. Aitoussous, W. Bouarifi, D. Bensikaddour 
\hfil EJDE-2014/197\hfilneg]
{Noureddine Alaa, Mohammed Aitoussous, \\ Walid Bouarifi, Djemaia Bensikaddour}  % in alphabetical order

\address{Noureddine Alaa \newline
Laboratory LAMAI,
Faculty of Science and Technology of Marrakech,
University Cadi Ayyad \\
B.P. 549, Street Abdelkarim Elkhattabi, Marrakech - 40000,
Morocco}
 \email{n.alaa@uca.ma}

\address{Mohammed Aitoussous \newline
Laboratory LAMAI,
Faculty of Science and Technology of Marrakech,
University Cadi Ayyad \\
B.P. 549, Street Abdelkarim Elkhattabi, Marrakech - 40000, Morocco}
\email{aitoussous.med@gmail.com}

\address{Walid Bouarifi \newline
Department of Computer Engineering \\
National School of Applied Sciences of Safi\\
Cadi Ayyad University\\
Sidi Bouzid, BP 63, Safi - 46000, Morocco}
\email{w.bouarifi@uca.ma}

\address{Djemaia Bensikaddour \newline
Department of Mathematics\\
University of Mostaganem,  Algeria}
\email{bensikaddour@yahoo.fr}

\thanks{Submitted June 8, 2014. Published September 22, 2014.}
\subjclass[2000]{35J55, 35J60, 35J70}
\keywords{Degenerate elliptic systems; quasilinear; chemotaxis;
\hfill\break\indent angiogenesis;  weak solutions}

\begin{abstract}
 This study shows how partial differential equations can be employed
 to restore a digital image. It is in fact a generalization of the work
 presented by Catt\'e \cite{12}, which modify the Perona-Malik Model
 by nonlinear diffusion. We give a demonstration of the consistency
 of the reaction-diffusion model proposed in our work.
\end{abstract}

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

\section{Introduction}

Image processing is always a challenging problem; this topic has become ``hot''
in recent years and a very active field of computer applications and research
\cite{14}. Various techniques have been developed in Image Processing during the
last four to five decades, the use of these techniques has exploded and they are
now used for all kinds of tasks in all kinds of areas: artistic effects,
medical visualization, industrial inspection, human computer interfaces, etc.
One of the most active topics in this field has been restoration of images,
as can be ascertained from recent survey papers \cite{4,5}.
A number of different techniques have been proposed for digital image restoration,
utilizing a number of different models and assumptions.
The restoration of degraded images is an important problem because it allow to
recovery lost information from the observed degraded image data.
Two kinds of degradations are usually encountered: spatial degradations
(e.g., the loss of resolution) caused by blurring and point degradation
(e.g., additive random noise), which affect only the gray levels of the
individual picture points. Image is restored to its original quality by
inverting the physical degradation phenomenon such as defocus, linear motion,
atmospheric degradation and additive noise.
Partial differential equation (PDE) methods in image processing have proven
to be fundamental tools for image diffusion and restoration \cite{4,5,6,9,23,34}.
The Perona-Malik equation \cite{24}, proposed in 1987, is one of the first attempts
to derive a model that incorporates local information from an image within a
 PDE framework. It has stimulated a great deal of interest in image processing
community \cite{5, 33}. A nonlinear diffusion model (which they called `anisotropic')
was conducted by Perona and Malik in order to avoid the blurring of edges and
other localization problems presented by linear diffusion models, they apply
a diffusion process whose diffusivity is steered by derivatives of the evolving
image. The model proposes a nonlinear diffusion method for avoiding the blurring
and localization problems of linear diffusion filtering \cite{23, 24} by applying
a process that reduces the diffusivity in places having higher likelihood of
being edges. This likelihood is measured by a function of the local gradient.
Unfortunately, it was shown by Kichenassamy \cite{19} that the basic Perona-Malik
PDE model is ill-posed in the sense of Hadamard. It was shown by Kawohl and Kutev
that the equation may have no global weak solutions in $C^1$ \cite{17}.
Zhang \cite{35} established that the one-dimensional Perona-Malik equation
admits infinitely many weak solutions. H\"ollig \cite{17} constructed a
forward-backward diffusion process which can have infinitely many solutions,
his study has become a pessimistic results about the well-posedness of
the Perona-Malik equation. In 1992, Lions and Alvarez \cite{4, 5} offered an
interesting nonlinear form of restoration equation with solving the
 Perona-Malik equation with a finite difference method.
Although the basic model is ill-posed, its discretizations are found to be stable,
this fact is sometimes referred to as the Perona-Malik paradox \cite{19}.
The explanation for these observations was given by Weickert and Benhamouda \cite{33},
 who investigated the regularizing effect of a standard finite difference
discretization. This observation motivated much research towards the introduction
of the regularization directly into the PDE to avoid the dependence on the
numerical schemes \cite{12,21}. A variety of spatial, spatio-temporal, and
temporal regularization procedures have been proposed over the years
\cite{10,12, 20, 27,31, 32}. The one that has attracted much attention is
the mathematically sound formulation in $1992$ by Catt\'e, Lions, Morel and
Coll \cite{12}. They suggested introducing the regularization in space and
time directly into the continuous equation in order to obtain a related
well-posed model which becomes more independent of the numerical implementation
which causes critically dependence between the dynamics of the solution and
the sort of regularization procedure. They proposed to replace the diffusivity
 $g(|\nabla u|^2)$by a slight variation
$g(| \nabla u_{\sigma }| ^2)$ in the Perona-Malik equation, with
$u_{\sigma }=G_{\sigma }\ast u$, where $G_{\sigma}$ is a smooth kernel
(Gaussian of variance $\sigma ^2$). Since differentiation is highly susceptible
to noise. They prove existence, uniqueness and regularity for the related model
and demonstrate experimentally that the related model gives similar results
to the Perona-Malik equation \cite{24}. In 2006, the study of Morfu \cite{36}
was focused on the contrast enhancement and noise filtering, he considers the
Fisher equation which generally allows simulating the transport mechanisms
in living cells, but also enhances the contrast and segmenting images.
The model proposed by Morfu is:
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}-\operatorname{div}(g(| \nabla u| )\nabla u)=f(u)
\quad\text{in } Q_T, \\
u( 0,x) =u_0(x)\quad\text{in } \Omega,  \\
\frac{\partial u}{\partial \upsilon }=0\quad\text{on } \Sigma _T,
\end{gathered}   \label{eq1}
\end{equation}
where $\Omega$ is the domain of the image, $T>0$, $u_0$ is the original image
to be processed and $f(s) = s(s-a)(1-s)$ with $0<a<1$. The Major defects of this
 model are: (1) Sensitivity to noise; If we increase slightly the noise,
the method gives unsatisfactory results because the image noise causes severe
oscillations of the gradient and the model keeps the noise that considers edges.
(2) No results of existence and consistency. To overcome this problem, we propose
an improved algorithm which will be able to resist to noise and which can improve
the contrast and noisy images.
The aim of our work is to modify the model of Morfu \cite{36} by applying a Gaussian
filter on the gradient of the noisy image during the calculation
of the coefficient of anisotropic diffusion. The proposed model is as follows:
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}-\operatorname{div}(g(| \nabla u_{\sigma
}| )\nabla u)=f( t,x,u) \quad\text{in }Q_T, \\
u( 0,x) =u_0\quad\text{in }\Omega,  \\
\frac{\partial u}{\partial \upsilon }=0\quad \text{on }\Sigma _T\,.
\end{gathered}  \label{eq1b}
\end{equation}
Here $\Omega =]0,1[\times ]0,1[$ denotes picture domain with boundary
$\partial \Omega $, with Neumann boundary conditions. Where $u(t,x)$
is the solution of this PDE (restored image) we are searching for,
this solution is depended on two parameters; the scale parameter denoted
by $t$ and the spatial coordinate $x$. $\upsilon$ is an outward Normal
to domain $\Omega $ and $u_0$ is the original image to be processed.
$Q_T=] 0,T[ \times \Omega$ and $\Sigma _T=] 0,T[\times \partial \Omega $ where
$T$ is a fixed reel number ($T>0$). Let $\sigma>0$, $G_{\sigma}$ is the Gaussian
filter where:
\begin{equation}
G_{\sigma }( x) =\frac{1}{\sqrt{2\pi \sigma }}e^{-( \frac{| x| ^2}{4\sigma }) },
\quad x\in \mathbb{R}^2.
\end{equation}
We consider the gradient norm of $w$ as:
$$
| \nabla w| =\Big( \sum_{i=1}^{i=2}(\frac{\partial w}{\partial x_{i}})^2\Big)^{1/2},
$$
$\nabla w_{\sigma }$ is the smoothed version of gradient norm where $w$:
$\nabla w_{\sigma}:=\nabla (w\ast G_{\sigma })=w\ast \nabla G_{\sigma }$.
The Diffusivity $g$ is a smooth decreasing function defined by
$g:[0,+\infty [ \to [ 0,+\infty [ $ where $g(0)=1$, and
$\lim_{s\to \infty }g(s)=0$, one of the diffusivities Perona and Malik proposed is
\begin{equation}
g(s)=\frac{d}{\sqrt{1+\eta (\frac{s}{\lambda })^2}}\,,
\end{equation}
where $\eta \geq 0$, $d>0$ and $\lambda $ is a threshold (contrast) parameter
that separates forward and backward diffusion \cite{32}. The nonlinearity $f$ has
no limitation of increasing. We assume that the initial data satisfy $0\leq u_0(x)$,
and for $f$ we introduce the following assumptions:
\begin{equation}
\text{$f:Q_T\to \mathbb{R}$ is measurable and  $f(t,x,.):\mathbb{R}\to \mathbb{R}$
 is continuous}. \label{geq00}
\end{equation}
In addition, we give here the following main properties of $f$:
\begin{itemize}
  \item the positivity of the solution $u$ of \eqref{eq1} is preserved over time,
which is ensured by:
  \begin{equation}
\text{for almost $(t,x)\in Q_T$, $f(t,x,0)\geq 0$}; \label{geq0}
\end{equation}

  \item the total mass is controlled in function of time:
  \begin{equation}
\text{for all $u\in \mathbb{R}$ and for almost $(t,x)\in Q_T$,
$uf(t,x,u)\leq 0$}. \label{geq01}
\end{equation}
\end{itemize}
The special case $f=0$ was treated by Catt\'e \cite{12}, where they considered
the problem
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}-\operatorname{div}(g(| \nabla u_{\sigma
}| )\nabla u)=0\quad\text{in }Q_T, \\
u( 0,x) =u_0\quad\text{in } \Omega,  \\
\frac{\partial u}{\partial \upsilon }=0\quad\text{on }\Sigma _T\,.
\end{gathered}
\end{equation}
They established the existence, uniqueness and regularity of a solution for
$\sigma > 0$ and $u_0\in L^2(\Omega)$. This study is devoted to a generalization
of their work in the case where $f$ is nonzero.
Note that if the diffusion coefficient is constant $g(s)=d$ (which corresponds
to the situation where  $\eta=0$), the existence of positive global solutions
have been obtained by several authors \cite{13,16,28}. When $u_0\in L^1(\Omega)$,
only Pierre \cite{25} proves the existence of global weak positive solutions.
In all these works, the hypothesis \eqref{geq01} plays an important role in study
of this diffusion-reaction equation. Indeed, if \eqref{geq0} is not satisfied,
\cite{22} proved the explosion in finite time of the solutions.

This work began with an introduction where we describe briefly the nonlinear
diffusion model proposed by Catt\'e \cite{12} applied in image processing for
restoration and which serves as background for our proposed model generalization.
This is followed by a concept definition of solution used here and we present
the main results of this work. The next section describes the global existence
of our reaction diffusion equation; this is done in three steps: the first
step is to truncate the equation and shows that the problem obtained has a solution.
In the second step we establish appropriate estimates on the approximate solutions.
In the last step, we show the convergence of the approximate system.
We use a new technique recently introduced by Pierre \cite{26} for study of
semi-linear isotropic systems. Our results are a generalization of these results
in the case of anisotopique reaction diffusion equation firstly introduced
by \cite{12} in the case of the equations without reaction term.

Now we will recall some functional spaces that will be used throughout this paper.
For all $k\in \mathbb{N}$, $H^{k}( \Omega ) $ is the set of functions $u$
defined in $\Omega $ such as $u$ and its order $D^{s}u$ derivatives where
$| s| =\sum_{j=1}^{n}s_{j}\leq k$ are in $L^2( \Omega )$. $H^{k}( \Omega ) $
is a Hilbert space for the norm
\begin{equation}
\| u\| _{H^{k}( \Omega ) }=\Big(\sum_{| s| \leq k}\int_{\Omega
}| D^{s}u| ^2dx\Big) ^{1/2}.
\end{equation}
We denote by $( H^1( \Omega ) ) '$ the dual of $H^1( \Omega )$.

$L^{p}( 0,T,H^{k}( \Omega ) ) $ is the set of functions $u$ such that,
for all every $t\in ( 0,T)$, $u(t)$ belongs to $H^{k}( \Omega ) $ with the norm
\begin{equation}
\| u\| _{L^{p}( 0,T;H^{k}( \Omega ) )
}=\Big( \int_0^{T}\| u(t)\| _{H^{k}( \Omega
) }^{p}dt\Big) ^{1/p},\quad 1<p<\infty ,\; k\in \mathbb{N}\,.
\end{equation}
$L^{\infty }( 0,T;L^2( \Omega ) )$ is the set of functions $u$ such that,
for all every $t\in ( 0,T))$, $u(t)$ belongs to $L^2( \Omega ) $ with the norm
\begin{equation}
\| u\| _{L^{\infty }( 0,T;L^2( \Omega )
) }=\Big( \sup_{0<t<T}\| u(t)\| _{L^2( \Omega) }^2\Big) ^{1/2}\,.
\end{equation}
$L^{\infty }( 0,T;\mathcal{C}^{\infty }( \Omega ) ) $ is the set of functions
$u$ such that, for all every $t\in (
0,T)$, $u(t)$ belongs to $\mathcal{C}^{\infty }( \Omega) $ with the norm
\begin{equation}
\| u\| _{L^{\infty }( 0,T;\mathcal{C}^{\infty }(
\Omega ) ) }=\inf \big\{ c,~\| u( t)
\| _{\mathcal{C}^{\infty }( \Omega ) }\leq c\text{ sur }
( 0,T) \big\} .
\end{equation}

\section{Consistency of the model: Existence and uniqueness results}

\subsection{Assumptions}
Firstly, it must be specified the direction in which we want to solve
the problem \eqref{eq1}.

\begin{definition} \label{def2.1} \rm
A function $u$ is a weak solution of \eqref{eq1} if
\begin{equation}
\begin{gathered}
u\in L^{\infty }( 0,T;L^2( \Omega ) ) \cap
L^2( 0,T;H^1( \Omega ) ) ,\quad f( t,x,u) \in L^1( Q_T),  \\
\text{for all $\varphi \in C^1(Q_T)$ such that
$\varphi (T,.) =0$}, \\
\int_{Q_T}-u\frac{\partial \varphi }{\partial t}+g(| \nabla
u_{\sigma }| )\nabla u\nabla \varphi
=\int_{Q_T}f( t,x,u( t) ) \varphi +\int_{\Omega }u_0\varphi (0,x)
\end{gathered}
\end{equation}\label{eq2}
If moreover $u\in C^1(Q_T)$  then we say that $u$ is a classical solution
of \eqref{eq1}.
\end{definition}

\subsection{Main result}

Our main result in this paper is the following existence theorem.

\begin{theorem}\label{thm2.1}
Assume that \eqref{geq00}--\eqref{geq01} and that
for all $R\geq 0$,
\begin{equation}
\sup_{| u| \leq R}(|f( t,x,u) | )\in L^1(Q_T)\,.\label{eq2.2}
\end{equation}
Then for all fixed  $T>0$ and $\sigma >0$ and for any
 $0\leq u_0$ $\in $ $L^2(\Omega ) $ such as $u_0\geq 0$,
 problem \eqref{eq2} admits a weak positive solution.

If moreover for all $r\geq 1$ $f(t,x,r)\leq 0$ and $u_0(x)\leq 1$,
 we have $0\leq u(t,x)\leq 1$ in $Q_T$.
\end{theorem}

\begin{remark} \label{rmk2.3} \rm
A typical example when the result of this paper can be applied is the
Ficher equation outcome the population dynamics
\begin{equation}
f(t,x,u)=-\beta u(u-a)^{2\alpha }(1-u)
\end{equation}
where $\alpha ,\beta >0$ and $0<a<1$.
\end{remark}

The proof of Theorem \eqref{thm2.1} is done in four steps:
\smallskip

\noindent\textbf{Step 1: Positivity of the solutions:}
Consider the function 
\begin{equation}
\operatorname{sign}^-( r) =\begin{cases}
-1 & \text{if } r<0 \\
0  & \text{if }r\geq 0
\end{cases}
\end{equation}
We  build a sequence of convex functions
$j_{\varepsilon }(r) $ such as $j'_{\varepsilon }( r) $ is bounded and
for all $r\in \mathbb{R}$, $j'_{\varepsilon }(r) \to \operatorname{sign}^-( r) $
when $\varepsilon \to 0$.

Let $u$ be a solution of \eqref{eq2}, we multiply both sides of the first
equation by $j'_{\varepsilon }( u) $
and by integrating  on $Q_{t}=]0,t[\times \Omega $ \ for $t$ $\in [0,T[ $, we obtain
\begin{equation}
\int_{Q_{t}}j'_{\varepsilon }( u) \frac{
\partial u}{\partial t}\,dx\,dt+\int_{Q_{t}}A\nabla u.\nabla
j'_{\varepsilon }( u) \,dx\,dt
=\int_{Q_{t}}f( s,x,u) j'_{\varepsilon }( u) \,dx\,ds
\end{equation}
where $A(t,x)=g(| \nabla u_{\sigma }| )\in L^{\infty
}( 0,T;\mathcal{C}^{\infty }( \Omega ) ) $ because
$u\in L^{\infty }( 0,T;L^2( \Omega ) ) $ and $g,G_{\sigma }$
are $C^{\infty }$ and we can show the existence of a  $C_0$ depends only on
$G_{\sigma },\| u_0\| _{\substack{ L^2( \Omega) }}$ such as:
\begin{equation}
\| \nabla u_{\sigma }\| _{L^{\infty }( Q_T)}\leq C_0\,.
\end{equation}
Moreover as $g$ is decreasing, then there $a=g(C_0)>0$ which depends only on
$\sigma $ and on $\| u_0\|_{\substack{ L^2( \Omega ) }}$ such as:
\begin{equation}
A(t,x)\geq a\quad  \forall (t,x)\in Q_T \,.\label{eq2.7}
\end{equation}
Consequently,
\begin{equation}
\int_{\Omega }[ j_{\varepsilon }( u) ( t)
-j_{\varepsilon }( u) ( 0) ] dx
+a\int _{Q_{t}}| \nabla u| ^2j''_{\varepsilon } ( u) \,ds\,dx
\leq \int_{Q_{t}}f(s,x,u)j_{\varepsilon}'( u) \,dx\,ds\,. \label{eq2.8}
\end{equation}
Since $\int_{\Omega }j_{\varepsilon }( u) (0) dx=0$ and
 $\int_{Q_{t}}| \nabla u| ^2j''_{\varepsilon }( u) \,dx\,ds\geq 0$ then we have
\begin{align*}
\int_{\Omega }j_{\varepsilon }( u) ( t) dx
&\leq \int_{Q_T}f(s,x,u)j'_{\varepsilon }( u)\,dx\,ds \\
&\leq \int_{[ u<0] }f(s,x,u)j'_{\varepsilon }( u) \,dx\,ds
+\int_{[ u\geq 0] }f( s,x,u) j'_{\varepsilon }( u) \,dx\,ds
\end{align*}
On the set where $u\geq 0$ we have $j'_{\varepsilon }( u) =0$ and
 $\int_{[ u\geq 0] }f(s,x,u) j'_{\varepsilon }( u) \,dx\,ds=0$; therefore
\begin{equation}
\int_{\Omega }j_{\varepsilon }( u) ( t) dx\leq
\int_{[ u<0] }f(s,x,u) j'_{\varepsilon }( u) \,dx\,ds\,.
\end{equation}
When $\varepsilon \to 0$, we obtain 
\begin{equation}
\int_{\Omega }( u) ^{-}( t) dx\leq
-\int_{[ u\leq 0] }f( s,x,u) \,dx\,ds\,.
\end{equation}
Using \eqref{geq01} and the fact that
$( u) ^{-}( t) \geq 0 $, we obtain $( u) ^{-}( t) =0$ on $\Omega $;
therefore $u\geq 0$ in $Q_T$.
\smallskip

\noindent\textbf{Step 2: An existence result when $f$ is bounded:}

\begin{theorem} \label{thm2.4}
Assume \eqref{geq0}--\eqref{geq00}, and that
there exists $M\geq 0$ such as for almost $(t,x)\in Q_T$ and all
$ r\in \mathbb{R}$, 
\begin{equation}
| f(t,x,r)| \leq M\,. \label{eq2.12}
\end{equation}
Then for all $u_0$ $\in $ $L^2(\Omega ) $,  problem \eqref{eq2} admits
a weak solution.
Moreover, there exists $C=C(M,a,T,\| u_0\| _{L^2( \Omega )})$ such that
\begin{equation}
\sup_{0<t<T}\| u(t)\| _{L^2( \Omega )
}+\| u\| _{L^2( 0,T;H^1( \Omega )) }\leq C\,.  \label{eq2.13}
\end{equation}
\end{theorem}

\begin{proof}
We will show the existence of a weak solution by the classical 
 Schauder fixed point theorem. Firstly we introduce the  space
\begin{equation}
\mathcal{W}( 0,T) =\big\{ v\in L^2( 0,T;H^1(
\Omega ) ):\frac{dv}{dt}\in L^2( 0,T;( H^1(\Omega ) ) ') \big\}
\end{equation}
which is a Hilbert space for the graph norm. Let
$v\in \mathcal{W}( 0,T) \cap L^{\infty }( 0,T;L^2( \Omega) ) $
and we consider $u(v)$ the solution of the linear problem
\begin{equation} \label{Lv}
\begin{gathered}
u(v)\in C( [ 0,T] ;L^2( \Omega ) ) \cap L^2( 0,T;H^1( \Omega ) ), \\
\text{for all $\varphi \in C^1(Q_T)$ such that $\varphi (T,.) =0$}, \\
\int_{Q_T}-u(v)\frac{\partial \varphi }{\partial t}+g(| \nabla
v_{\sigma }| )\nabla u(v)\nabla \varphi =\int_{Q_T}f(t,x,v( t) ) \varphi
+\int_{\Omega }u_0\varphi (0,x)
\end{gathered}
\end{equation}
According to the classical theory \cite{7,11}, equation
\eqref{Lv} admits a unique solution $u(v)\in \mathcal{W}( 0,T) $
moreover by applying a classic bootstrap argument, we have
$u(v)(t)\in H^1( \Omega ) $ for all $t>0$; since
$f(t,x,v( t) ) \in L^{\infty }(Q_T)$, then
$u(v)(t)\in H^1( \Omega ) $ for all $t>0$.
Therefore by iteration and by application the general classical theory
another time \cite{35}, we deduce that $u(v)$ is a classical solution and
$u(v)\in C^{\infty }( ]0,T[\times \Omega )$. We take
$\varphi =u(v)$ in \eqref{Lv}, and deduce that for all $0<t<T$,
\begin{equation}
\frac{1}{2}\int_{\Omega }u(v)^2(t)
+\int_{Q_{t}}g(| \nabla v_{\sigma }| )| \nabla u(v)|
^2=\int_{Q_{t}}f( t,x,v( t) ) u(v)
+\frac{1}{2} \int_{\Omega }u_0^2dx
\end{equation}
Using  \eqref{eq2.7} and the assumption \eqref{eq2.12} on $f$, we obtain
\begin{equation}
\frac{1}{2}\int_{\Omega }u(v)^2(t)+a\int_{Q_{t}}| \nabla
u(v)| ^2\leq M(1+\int_{Q_{t}}u(v)^2)+\frac{1}{2}
\int_{\Omega }u_0^2dx\,.
\end{equation}
Now by Gronwall's lemma, we obtain the estimation \eqref{eq2.13}.
These estimates lead us to introduce the space
\begin{align*}
\mathcal{W}_0( 0,T)
=\Big\{& v\in \mathcal{W}( 0,T) \cap L^{\infty }( 0,T;L^2(\Omega ) ):
v(0)=u_0 \text{and}\\
&\sup_{0<t<T}\| u(t)\| _{L^2( \Omega) }+\| u\| _{L^2( 0,T;H^1( \Omega
) ) }\leq C\,,
\end{align*}
where $C=C(M,a,T,\| u_0\| _{ L^2(\Omega ) })$ is  the constant obtained
in \eqref{eq2.13}.
\end{proof}

We can easily verify that $\mathcal{W}_0( 0,T) $
is a nonempty closed convex in $\mathcal{W}( 0,T) $, moreover it injects with
a compact way in $L^2(0,T;L^2( \Omega ) ) $. Then we define the application:
\begin{equation}
\begin{aligned}
F &: \mathcal{W}_0( 0,T)  \to  \mathcal{W}_0( 0,T) \\
 v &\mapsto F(v)=u(v),  \text{ where $u$ is a solution of \eqref{Lv}}\,.
\end{aligned}
\end{equation}
Estimate \eqref{eq2.12} shows that $F$ is well defined. To apply the Schauder
fixed point theorem, we show that $F$ is weakly continuous from
$\mathcal{W}_0( 0,T)$ in $\mathcal{W}_0( 0,T)$.

Then consider a sequence $(v_n)$ in $\mathcal{W}_0( 0,T) $, such as
$v_n\rightharpoonup v$ in $\mathcal{W}_0( 0,T) $, and let
 $u_n=F(v_n)$. According to the classical results of compactness,
 we can extract from the sequence $(u_n)$
a subsequence yet denoted $(u_n)$  such that
\begin{itemize}

\item $u_n\rightharpoonup u$  weakly in  $L^2( 0,T;H^1(\Omega ) )$

\item $u_n\rightharpoonup u$  strongly in $L^2( 0,T;L^2(\Omega ) )$
 and almost everywhere in $Q_T$

\item $\nabla u_n\rightharpoonup \nabla u$  weakly in $L^2(0,T;L^2( \Omega ) )$

\item $v_n\rightharpoonup v$ strongly in $L^2( 0,T;L^2(\Omega ) )$
 and almost everywhere in $Q_T$

\item $\nabla G_{\sigma }\ast v_n\rightharpoonup \nabla G_{\sigma }\ast v$
 strongly in $L^2( 0,T;L^2( \Omega ) )$ and almost everywhere in $Q_T$

\item $g(| \nabla G_{\sigma }\ast v_n| )\rightharpoonup
g(| \nabla G_{\sigma }\ast v| )$ strongly in $L^2( 0,T;L^2( \Omega ) )$

\item $f(t,x,v_n)\to f(t,x,v)$ strongly in $L^1(Q_T)$

\end{itemize}
The latter is obtained by applying the dominated convergence theorem.
We can then pass to the limit in \eqref{Lv}, with $v_n$ instead of $v$,
and obtain that $u=F(v)$.
By uniqueness of the solution of \eqref{Lv}, then the sequence
 $u_n=F(v_n)$ converges weakly to $u=F(v)$ in $\mathcal{W}_0( 0,T) $.
We deduce the existence of $u\in \mathcal{W}_0( 0,T) $ such as $u=F(u)$,
and thus the existence of $u\in \mathcal{W}( 0,T) $ such us $u=U$.
\smallskip

\textbf{Step 3: Approximate problem and a priori estimates}
Consider the  truncation function
$\Psi _n\in \mathcal{C}_0^{\infty }(\mathbb{R}) $ defined by
\begin{equation}
\Psi _n( r) =\begin{cases}
1 & \text{if }  | r| \leq n, \\
0 & \text{if }  | r| \geq n+1\,.
\end{cases}
\end{equation}
We truncate the nonlinearity $f$ by $\Psi _n$,
\begin{equation}
f_n( t,x,u) =\Psi _n( | u| )f( t,x,u).
\end{equation}
Thus, we can easily check that $f_n$ satisfies \eqref{geq0}, \eqref{geq00},
\eqref{geq01} with $M=M(n)$ and for almost $(t,x)\in Q_T$,
for all $r\in \mathbb{R}$ $f_n( t,x,u) \to f(t,x,r)$.

Since $u_0\in L^2( \Omega ) $ and
$| f_n(t,x,r)| \leq M_n$,  theorem \eqref{eq2.12} is applied,
then we can deduce the existence of a weak solution of the problem
\begin{equation} \label{eq2.22}
\begin{gathered}
\frac{\partial u_n}{\partial t}-\operatorname{div}( g(| \nabla
( u_n) _{\sigma }| )\nabla u_n) =f_n(
t,x,u_n)  \quad\text{in } Q_T, \\
u_n( 0,x) =u_0 \quad \text{on }\Omega, \\
\frac{\partial u_n}{\partial \upsilon }=0\quad\text{on }\Sigma _T\,.
\end{gathered}
\end{equation}

\begin{remark} \label{rmk2.5} \rm
Since $u_0\geq 0$ on $\Omega $, the (i) assures that $u_n\geq 0$ is in $Q_T$.
 Moreover, under the assumption \eqref{geq01} we have also $f_n( t,x,u_n) \leq 0$
in $Q_T$.
\end{remark}

Now we will show that a subsequence $u_n$ converges to the weak solution $u$
of problem \eqref{eq1}. For this we need to prove the following result:

\begin{lemma}
Let $(u_n)$ the sequence of weak solutions defined by \eqref{eq2.13}, then we have:
\begin{itemize}
\item[(i)] $\int_{Q_T}| f_n(t,x,u_n) | \leq \int_{\Omega }|u_0| dx$,

\item[(ii)] $(u_n) $ is bounded in $L^2(0,T;H^1( \Omega ) )$ and
 $$
\int_{Q_T}| u_nf_n( t,x,u_n) | \,dx\,dt\leq \frac{1}{2}\int_{\Omega }u_0^2dx,
$$

\item[(iii)] $( u_n) $ is relatively compact in $L^2( Q_T) $.
\end{itemize}
\end{lemma}

\begin{proof}
(i) By Remark \ref{rmk2.5}, $| f_n( t,x,u_n) | =-f_n(t,x,u_n) $.
 Thus by integrating the equation satisfied by
$u_n$ in $Q_T$ we obtain
\begin{equation}
\int_{\Omega }u_n( T)dx -\int_{Q_T}f_n( t,x,u_n) \,dx\,dt
=\int_{\Omega }u_0\,dx \,;
\end{equation}
therefore
\begin{equation}
\int_{Q_T}| f_n( t,x,u_n) |
\,dx\,dt\leq \int_{\Omega }| u_0| dx\,.
\end{equation}

(ii) Firstly we show that $u_n$ is bounded in
$ L^2( Q_T) $, for this we consider $\varphi =u_n$ as a function test
in \eqref{eq2.22}, we then deduce that
\begin{equation}
\frac{1}{2}\int_{\Omega }u_n^2(t)+\int_{Q_{t}}g(|
\nabla ( u_n) _{\sigma }| )| \nabla
u_n| ^2=\int_{Q_{t}}f( t,x,u_n) u_n+\frac{1}{2}
\int_{\Omega }u_0^2dx\,.
\end{equation}
Then we use \eqref{eq2.7} and the hypothesis \eqref{eq2.8} on $f$ to obtain
\begin{equation}
\frac{1}{2}\int_{\Omega }u_n^2(t)+a\int_{Q_{t}}| \nabla
u_n| ^2\leq \frac{1}{2}\int_{\Omega }u_0^2dx\,.
\end{equation}
We have also
\begin{equation}
\int_{Q_T}u_n| f_n( t,x,u_n) |\,dx\,dt
\leq \frac{1}{2}\int_{\Omega }u_0^2\,dx\,,\label{eq2.27}
\end{equation}
where we have
\begin{gather*}
\sup_{0<t<T}\| u_n(t)\| _{L^2( \Omega ) } \leq \| u_0\| _{L^2( \Omega ) }, \\
\| u_n\| _{L^2( 0,T;H^1( \Omega )) } \leq (1+\frac{1}{2a})\| u_0\| _{L^2(
\Omega ) }
\end{gather*}

(iii) Since $\frac{\partial u_n}{\partial t}=\operatorname{div}(A_n\nabla u_n)
+f_n( t,x,u_n) $ is bounded in
$L^1(0,T;( H^1( \Omega ) ) ')+L^1(\Omega )$. Since $u_n$ is also bounded in
$L^2( 0,T;H^1( \Omega ) ) $ and that the injection of $H^1(\Omega ) $ in
$L^2( \Omega ) $ is compact, it follows that $( u_n) $ is relatively compact in
$L^2(Q_T) $ \cite{30}.
\end{proof}


\noindent\textbf{Step 4: Convergence}
According to (iii), the sequence $(u_n) $ is relatively compact in $L^2( Q_T) $,
so we can extract a subsequence still denoted $( u_n) $ such that
\begin{itemize}
\item $u_n \rightharpoonup u$ strongly in $L^2( Q_T)$  and almost everywhere in
$Q_T$,

\item $\nabla G_{\sigma }\ast u_n \rightharpoonup \nabla G_{\sigma }\ast u$
 strongly in $L^2( Q_T)$ and almost everywhere
in $Q_T$.

\item $g(| \nabla G_{\sigma }\ast u_n| ) \rightharpoonup
g(| \nabla G_{\sigma }\ast u| )$ strongly in $L^2( Q_T)$

\item $f_n(t,x,u_n) \to \ f(t,x,u)$  for almost everywhere in
$Q_T$
\end{itemize}
To prove that $u$ is a weak solution of \eqref{eq1}, it suffices to prove that
$f_n(t,x,u_n)\to f(t,x,u)$ in $L^1(Q_T) $. Since $f_n(t,x,u_n)\to f(t,x,u)$
almost everywhere in $Q_T$, we will demonstrate that $(f_n(t,x,u_n))$ is uniformly
integrable in $L^1( Q_T) $. For this we show that:
for each $\varepsilon >0$, there exists $\delta >0$ such that for
 all $E\subset Q_T$ measurable with $| E| <\delta$,  we have 
\begin{equation}
\int_{E} | f_n(t,x,u_n)| dx\leq \varepsilon\,.
\label{eq2.29}
\end{equation}
Then for all $k\geq 0$,
\begin{equation}
\int_{E}| f_n(t,x,u_n)| dx\leq
\int_{E\cap [ u_n\leq k]}|
f_n(t,x,u_n)| dx+\int_{E\cap [
u_n>k]}| f_n(t,x,u_n)| dx\,. \label{eq2.30}
\end{equation}
For the first term on the right-hand side, we have
\begin{equation}
\int_{E\cap [ u_n\leq k]}| f_n(t,x,u_n)| dx
\leq \int_{E}\sup_{|r| \leq k}(| f(t,x,r)| dx \,.\label{eq2.31}
\end{equation}
According to  \eqref{eq2.2}, we have $\sup_{|
u| \leq k}(| f(t,x,u)| \in L^1(Q_T)$ is
uniformly integrable in $L^1(Q_T)$, therefore
for each $\varepsilon >0$ there exist $\delta >0$  such that if
$|E| <\delta$ then 
\begin{equation}
\int_{E}\sup_{|u| \leq k}(| f(t,x,u)| dx\leq \frac{
\varepsilon }{2}\,. \label{eq2.32}
\end{equation}
For the second term we have
\begin{equation}
\int_{E\cap [ u_n>k]}| f_n(t,x,u_n)|
dx\leq \frac{1}{k}\int_{Q_T}u_n|
f_n(t,x,u_n)| dx \,.\label{eq2.33}
\end{equation}
Then, using \eqref{eq2.27} we obtain
\begin{equation*}
\int_{E\cap [ u_n>k]}|
u_nf_n(t,x,u_n)| dx\leq \frac{1}{2k}\|
u_0\| _{L^2( \Omega ) }^2\,.
\end{equation*}
Now if we choose $k\geq \| u_0\|_{L^2( \Omega ) }^2/\varepsilon $, then we have
\begin{equation}
\int_{E\cap [ u_n>k]}| f_n(t,x,u_n)|
dx\leq \frac{\varepsilon }{2}; \label{eq2.34}
\end{equation}
consequently, \eqref{eq2.29} follows from \eqref{eq2.32} and \eqref{eq2.34}.

Using the following lemma, we complete the proof of Theorem \ref{thm2.1}.

\begin{lemma} \label{lem2.7}
Let $u$ be a weak solution of \eqref{eq2}, and assume that 
$0\leq u_0\leq 1$ in $\Omega $.
Then $0\leq u\leq 1$ in $Q_T$.
\end{lemma}

\begin{proof}
We have already obtained the positivity of weak solutions if the 
initial data is positive. So, we assume that $u_0\leq 1$ and proof that
$u\leq 1$.
For this,  we take $\bar{u}=1-u$, then we have $\nabla \bar{u}=\nabla u$,
we can verify that $\bar{u}$ satisfies
\begin{equation}
\begin{gathered}
\bar{u}\in L^{\infty }( 0,T;L^2( \Omega ) ) \cap
L^2( 0,T;H^1( \Omega ) ) ,\quad f( t,x,1-\bar{u}) \in L^1( Q_T), \\
\text{for all $\varphi \in C^1(Q_T)$ such that $\varphi (T,.) =0$}, \\
\int_{Q_T}-\bar{u}\frac{\partial \varphi }{\partial t}+g(|
\nabla \bar{u}_{\sigma }| )\nabla \bar{u}\nabla \varphi
=\int_{Q_T}f( t,x,1-\bar{u}( t) ) \varphi
-\int_{\Omega }u_0\varphi (0,x)\,.
\end{gathered} \label{eq2.35}
\end{equation}
Then we consider the sequence of convex functions 
$j_{\varepsilon }(r) $ such as $j'_{\varepsilon }( r) $ is bounded and
for all $r\in \mathbb{R}$, 
$j'_{\varepsilon }( r) \to \operatorname{sign}^-( r) $ when $\varepsilon \to 0$.
We take $\varphi =j'_{\varepsilon }( \bar{u}) $ as a test function in
\eqref{eq2.35} and integrating with respect to $t\in ]0,T[$,
we obtain
\begin{equation}
-\int_{\Omega }j_{\varepsilon }( \bar{u}) (
t,x) dx\leq \int_0^{t}\int_{\Omega }f( t,x,1-
\bar{u}) j'_{\varepsilon }( \bar{u}) \,dx\,dt\,.
\end{equation}
Passing to the limit as $\varepsilon \to 0$, we obtain
\begin{equation}
-\int_{\Omega }( \bar{u}) ^{-}( t,x) dx\leq
\int_0^{t}\int_{[ u\geq 1] }f(t,x,u) \,dx\,dt\,.
\end{equation}
Using that for all $r\geq 1$, $f(t,x,r)\leq 0$,
we deduce
\begin{equation}
\int_{\Omega }( \bar{u}) ^{-}( t,x) dx\geq 0;
\end{equation}
Therefore $\bar{u}(t)\geq 0$ which implies $u=1-\bar{u}\leq 1$.
\end{proof}

\subsection*{Acknowledgments}
We are grateful to the anonymous referee for the
corrections and useful suggestions that  improved this
article.


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\end{document}