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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 134, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/134\hfil
Asymptotic behaviour for a diffusion equation]
{Asymptotic behaviour for a diffusion equation governed
by nonlocal interactions}

\author[A. Andami Ovono \hfil EJDE-2010/134\hfilneg]
{Armel Andami Ovono}

\address{Armel Andami Ovono \newline
L.A.M.F.A CNRS UMR 6140 \\
33 rue Saint Leu \\
80039 Amiens cedex 1, France}
\email{andami@u-picardie.fr}

\thanks{Submitted March 9, 2010. Published September 20, 2010.}
\subjclass[2000]{35B35, 35B40, 35B51}
\keywords{Comparison principle; nonlocal diffusion;
 branch of solutions; \hfill\break\indent asymptotic behaviour}

\begin{abstract}
 In this article, we study the asymptotic behaviour of a nonlocal
 nonlinear parabolic equation governed by a parameter.
 After giving the  existence of unique branch of solutions composed
 by stable solutions in stationary case, we gives for the parabolic
 problem  $L^\infty$ estimates  of solution based on using the
 Moser iterations and existence of global attractor. We finish our
 study by the issue of asymptotic behaviour in some cases when
 $t\to \infty$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
%\allowdisplaybreaks

\section{Introduction}

The non-local problems are important in studying the behavior  of
certain physical phenomena and population dynamics. A major
difficulty in studying these problems often lie in the absence of
well-known properties as maximum principle, regularity and
properties of Lyapunov (see \cite{c1,c2}) and also the
difficulty to characterize and determine the stationary solutions
associated thus making study the asymptotic behavior of these
solutions very difficult.

In this article, we study the solution to the nonlocal equation
\begin{equation}\label{eq0}
\begin{gathered}
 u_t -\operatorname{div}(a(l_r( u(t)))\nabla u)=f \quad\text{in } \mathbb{R}^+\times\Omega\\
  u(x,t)=0 \quad\text{on }\mathbb{R}^+\times\partial\Omega\\
  u(.,0)=u_0 \quad\text{in }\Omega.
\end{gathered}
\end{equation}
In the above problem $u_0$ and $f$ are such that
\begin{equation} \label{e2}
u_0\in L^2(\Omega),\quad f\in  L^2(0,T, L^2(\Omega)),
\end{equation}
with $T$ an arbitrary positive number, $a$ is a continuous
function for which there exist $ m,M$ such that
\begin{equation} \label{p1}
0<m\leq a(\epsilon)\leq M\quad \forall\epsilon\in \mathbb{R}.
\end{equation}
The nonlocal functional $l_r(.)(x): L^2(\Omega)\to\mathbb{R}$,
is defined by
\begin{equation} \label{e4}
 u\to l_r(u(t))(x)=\int_{\Omega\cap B(x,r)} g(y)u(t,y) dy.
\end{equation}
Here  $B(x,r)$ is the closed ball of $\mathbb{R}^n$ with radius
$r$ and $g\in L^2(\Omega)$.
It is sometimes possible to consider $g$ more generally,
especially when one is interested in the study of stationary
solutions (see \cite{a3}).

 From the physical point of view problem \eqref{eq0} gives
many applications especially where $g=1$ in population dynamics.
Indeed, in this situation $u$ may represent a population density
and $l_r(u)$ the total mass of the subdomain $\Omega\cap B(x,r)$
of  $\Omega$. Hence \eqref{eq0} can describe the evolution of a
population whose diffusion velocity depends on the total mass of
a subdomain of $\Omega$. For more details of modelling we
refer the reader to \cite{c3}.
This type of equations can be applied more generally to other
models including the study of propagation of mutant gene
(see \cite{g1,g2,p1}). A very recent study of this
propagation was made by Bendahmane and Sep\'ulveda
\cite{b1} in which they analyze using a finite volume scheme
adapted, the transmission of this gene through 3 types of
people: susceptible, infected and recovered.

 From the mathematical point of view, when $r=d$ where $d$
is the diameter of $\Omega$, problem \eqref{eq0} has been
studied in various forms
\cite{c2,c4,c5,s1}.

However, when $0<r<d$, several questions from the theory of
bifurcations have arisen concerning the structure of stationary
solutions including the existence of a principle of comparison
of different solutions depending on the parameter $r$ and the
existence of branches (local and global) of solutions.
A large majority of these issues has been resolved in \cite{a3}.
It shows that when $a$ is decreasing the existence of a unique
global branch of solutions and existence of branch of solutions
that are purely local. Some questions may then arise:
\begin{itemize}
\item[(i)]  The unique branch described in \cite{a3} it is
 composed of stable solutions?
\item[(i)]  What about stability properties of the corresponding
 parabolic problem?
\end{itemize}
The plan for this work is the following. In section 2 we give
some existence and uniqueness results.  Section 3 is devoted
 to stationary problem corresponding to \eqref{eq0}.
In particular, we study in a radial case, a generalization of
Chipot-Lovat results about determination of the number of solutions.
 We also establish that the unique global branch of solutions
described in \cite{a3} is composed by stable solutions
(theorem \ref{bab2}). In section 4 firstly we address an
$L^{\infty}$ estimate taking to account $L^p$ estimate based
on Moser iterations. Secondly we prove existence of absorbing
set in $H^1_0$, which allows us to prove the existence of a
global attractor associated to \eqref{eq0} (see remark \ref{bab}).
Finally we obtain a result of stability properties of the
corresponding parabolic problem.

\section{Existence and uniqueness results}

In this section we show a result of existence.
We set $ V= H_{0}^{1}(\Omega)$ and  $V'$ its dual.
The norm in $V$ is
$$
\|u\|^2_V=\int_{\Omega}|\nabla u|^2dx,
$$
and the duality bracket of $V'$ and  $ V$ is
$\langle \cdot,\cdot\rangle$.

\begin{theorem} \label{terib1}
Let $T>0$,  $f\in  L^2(0,T,V')$ and $u_0\in L^2(\Omega)$, we assume
that $a$ is a continuous function and the assumption \eqref{p1}
checked then for every $r$ fixed,
$r\in[0,\operatorname{diam}(\Omega)]$,
there exists a function $u$ such that
\begin{equation}     \label{eq5}
\begin{gathered}
 u\in L^2(0,T,V), \quad u_t\in L^2(0,T,V')\\
 u(0,.)=u_0\quad\text{in }\Omega \\
 \frac{d}{dt}(u,\phi)+\int_{\Omega} a(l_{r}( u(t)))\nabla u\nabla
 \phi \,dx=\langle f,\phi\rangle\quad\text{in }
 D'(0,T)\; \forall\phi\in H_{0}^{1}(\Omega).
\end{gathered}
\end{equation}
Moreover, if  $a$ is locally Lipschitz; i.e,
 for each $c$ there exists $\gamma_c$ such that
 \begin{equation}  \label{eq6}
|a(\epsilon)-a(\epsilon')|\leq \gamma_c|\epsilon-\epsilon'|\quad
\forall\epsilon,\epsilon'\in [-c,c],
\end{equation}
then the solution of \eqref{eq5} is unique.
\end{theorem}


\begin{remark}\label{rmk1} \rm
Before to do the proof, we note that for
$r=0$, problem \eqref{eq5} is linear and the proof follows
a well-known result \cite{d1}. It is also valid when
$r=\operatorname{diam}(\Omega)$ (see \cite{c3}).
Therefore, we will focus in the case
$r\in]0,\operatorname{diam}(\Omega)[$.
\end{remark}


\begin{proof}
For the  proof of existence, we will use the Schauder fixed point
theorem. Let $w\in L^2(0,T,L^2(\Omega))$. Then the mapping
$t\mapsto l_r(w(t))$
is measurable.
As $a$ is continuous then
$t\mapsto a( l_r(w(t)))$ is also continuous.
The problem of finding  solution $u=u(x,t)$ of
\begin{equation}  \label{eq7}
 \begin{gathered}
 u\in L^2(0,T,V)\cap C([0,T],L^2(\Omega))\quad  u_t\in L^2(0,T,V')\\
         u(0,.)=u_0 \\
  \frac{d}{dt}(u,\phi)+\int_{\Omega} a(l_{r}( w(t)))\nabla u\nabla\phi
dx=\langle f,\phi\rangle \quad\text{in } D'(0,T)\;
 \forall\phi\in H_{0}^{1}(\Omega),
 \end{gathered}
 \end{equation}
is linear, and admits a unique solution $u=F_r(w)$
\cite{d1,c3}. Thus we show that the application
   \begin{equation}   \label{eq8}
   w\mapsto F_r(w)=u,
  \end{equation}
admits a fixed point. Taking $w=u$ in \eqref{eq7},
using  \eqref{p1} and using the Cauchy-Schwarz inequality, we have
\begin{equation}
     \label{eq9}
\frac{1}{2}\frac{d}{dt}|u|^2_2+m\| u\|_V^2\leq|f|_{\star}\|u\|_V,
    \end{equation}
where $ \|\cdot\|_V$ is the usual norm in $V$ and
$ |f|_{\star}$ is the dual norm of $f$. We take
$$
|u|_{L^2(0,T,V)}=\Big\{\int_{0}^T\|u\|^2_V dt\Big\}^\frac{1}{2}.
$$
Using Young's inequality to the right-hand side of \eqref{eq9}, 
it follows that
\begin{equation}
\label{eq10}
       \frac{1}{2}\frac{d}{dt}|u|^2_2+\frac{m}{2}\| u\|_V^2
\leq\frac{1}{2m}|f|_{\star}^2.
\end{equation}
By integrating  \eqref{eq10} on $(0,t)$ for $t\leq T$, we obtain
\begin{equation}
\label{eq11}
    \frac{1}{2}|u(t)|^2_2+\frac{m}{2}\int_{0}^t\| u\|_V^2dt
\leq\frac{1}{2}|u_0|^2_2+\frac{1}{2m}\int_{0}^t|f|_{\star}^2.
\end{equation}
We deduce that there exists a constant $ C=C(m,u_0,f)$ such that
\begin{equation} \label{eq12}
|u|_{L^2(0,T,V)}\leq C
\end{equation}Moreover
$$ < u_t,v\rangle + \langle  -\operatorname{div}(a(l_{r}( u(t)))
\nabla u),v\rangle =\langle f,v\rangle\quad \forall v\in V,
$$
This gives us
\begin{equation}
\label{eq13}
 |u_t|_{\star}\leq M\|u\|_V+|f|_{\star}.
\end{equation}
By squaring both sides and using the Young inequality, we have
\begin{equation}
\label{eq14}
  |u_t|_{\star}^2\leq 2M^2\|u\|_V^2+2|f|_{\star}^2.
\end{equation}
By integrating on $ (0,t)$ and assuming \eqref{eq12} we obtain
 \begin{equation} \label{eq15}
|u_t|_{L^2(0,T,V')}\leq C',
\end{equation}
with  $C'=C'(m,M,f,u_0)$, independent to $w$.
It follows from \eqref{eq12} and \eqref{eq15} that
 \begin{equation} \label{eq16}
|u_t|^2_{L^2(0,T,V')}+|u|^2_{L^2(0,T,V)}\leq R,
\end{equation}
with $R=C^2+C'^2$. From  \eqref{eq12} and the
Poincar\'e inequality it follows that
\begin{equation} \label{e1q}
|u|_{L^2(0,T,L^2(\Omega))}\leq R',
\end{equation}
By setting
\begin{equation}        \label{eq20}
R_1=\max(R',R),
\end{equation}
and associating  \eqref{e1q} and \eqref{eq20}, it follows that the
application  $F$ maps the ball     $B(0,R_1)$ of
$L^2(0,T,L^2(\Omega))$ into itself. Moreover the balls of
$ H^1(0,T,V,V')$  are relatively compact in $L^2(0,T,L^2(\Omega))$
(see \cite{d1} for more details). \eqref{eq16} clearly shows us
that  $F(B(0,R_1)$ is relatively compact in
 $B(0,R_1)$ with
$$
B(0,R_1)=\{u\in L^2(0,T,L^2(\Omega)): |u|_{L^2(0,T,L^2(\Omega))}
\leq R_1 \}.
$$
To apply the  Schauder fixed point theorem, as announced, we just
need to show that $F$ is continuous from $B(0,R_1)$ to itself.
This is actually the case and completes the proof of existence.

We will now discuss the uniqueness assuming of course that
assumption \eqref{eq6} be verified.  Consider  $u_1$ and $u_2$
two solutions \eqref{eq5}, by subtracting one obtains in
$\mathit{D}'(0,T)$
\begin{equation} \label{eq32}
\frac{d}{dt}(u_1-u_2,v)+ \int_{\Omega}( a(l_{r}( u_1(t))
\nabla u_1(t)-a(l_{r}( u_2(t)))\nabla u_2(t))\nabla\phi dx=0
\end{equation}
for all $\phi\in H_{0}^{1}(\Omega)$.
Since
\begin{align*}
&a(l_{r}( u_1(t)))\nabla u_1-a(l_{r}( u_2(t)))\nabla u_2(t) \\
&=( a(l_{r}( u_1(t)))-a(l_{r}( u_2(t)))\nabla u_1(t)
+ a(l_{r}( u_2(t)))\nabla(u_1(t)- u_2(t)),
\end{align*}
we obtain
\begin{equation} \label{p2}
\begin{aligned}
&\frac{d}{dt}(u_1-u_2,v)+ \int_{\Omega}a(l_{r}( u_2(t)))
 \nabla(u_1(t)- u_2(t))\nabla\phi dx\\
&=- \int_{\Omega}( a(l_{r}( u_1(t)))-a(l_{r}( u_2(t)))\nabla u_1
 \nabla\phi dx\quad\forall\phi\in H_{0}^{1}(\Omega).
\end{aligned}
\end{equation}
Moreover, $u_1,u_2\in C([0,T],L^2(\Omega))$ and there exist $z>0$
 such that $l_r(u_1(t))$ and $l_r(u_2(t))$ are in $[-z,z]$.
Taking $v=u_1-u_2$ in \eqref{p2},  by Cauchy-Schwarz
inequality and \eqref{eq6}, ti follows that
\begin{equation} \label{eq35}
\frac{1}{2}\frac{d}{dt}|u_1-u_2|^2_2+m\|u_1-u_2\|^2_V
\leq \gamma|l_{r}( u_1(t))-l_{r}( u_2(t))|\| u_1\|_V\|u_1-u_2\|_V .
\end{equation}
Also we have \cite{a3},
\begin{equation} \label{eq36}
|l_{r}( u(t))\leq C|B(x,r)\cap\Omega|^{1/(n\vee 3)}| g|_2| u(t)|_2
\leq |\Omega|^{1/(n\vee 3)}| g|_2| u(t)|_2,
\end{equation}
 where   $C$ a constant, $|\Omega|$  represents the measure of
$\Omega$ and  $n\vee 3$ the maximum between the dimension $n$
of $\Omega$ and 3. By using \eqref{eq36}, \eqref{eq35} and
the Young inequality
$$
ab\leq\frac{1}{2m}b^2+\frac{m}{2}a^2.
$$
We deduce
\begin{equation} \label{eq39}
\frac{d}{dt}|u_1-u_2|^2_2+m\|u_1-u_2\|^2_V\leq p(t)| u_1-u_2|_2^2,
\end{equation}
with
$$
p(t)=\frac{1}{m} (\gamma C |\Omega|^{1/(n\vee 3)}
| g|_2\,\| u_1\|_V\,)^2\in L^1(0,T),
$$
which leads to
\begin{equation} \label{eq40}
\frac{d}{dt}|u_1-u_2|^2_2\leq\ p(t)| u_1-u_2|_2^2.
\end{equation}
Multiplying \eqref{eq40} by  $e^{-\int^t_0 p(s)ds}$ it follows that
\begin{equation} \label{eq41}
e^{-\int^t_0 p(s)ds}\frac{d}{dt}|u_1-u_2|^2_2
- p(t)e^{-\int^t_0 p(s)ds}| u_1-u_2|_2^2\leq 0.
\end{equation}
Hence
\begin{equation}
\label{eq43}
\frac{d}{dt}\{e^{-\int^t_0 p(s)ds}|u_1-u_2|^2_2\}\leq 0.
\end{equation}
This shows that  $t\mapsto e^{-\int^t_0 p(s)ds}|u_1-u_2|^2_2$
is non-increasing. Since for $t=0$,
$$
u_1(0,.)=u_2(0,.)=u_0.
$$
This function vanishes at $0$  and nonnegative, we conclude
that it is identically zero. This concludes the proof.
\end{proof}


\section{Stationary solutions}

Consider the weak formulation to the stationary problem associated
with \eqref{eq0},
\begin{equation} \label{Pr}
\begin{gathered}
   -\operatorname{div}(a(l_{r}( u))\nabla u)=f \quad\text{in } \Omega\\
               u\in H^1_0(\Omega).
            \end{gathered}
\end{equation}
\subsection{The case $r=d$}
By taking $\phi$ the weak solution of the problem
\begin{gather*}
 -\Delta\phi=f \quad \text{in } \Omega\\
              \phi\in H^1_0(\Omega).
\end{gather*}
Due to a Chipot-Lovat \cite{c4}  results we obtain the following result.

\begin{theorem}\label{chi}
Let $a$ be a mapping from $\mathbb{R}$ into $(0, \infty)$.
The problem \eqref{Pr} with $r=d$ has as many solutions
as the problem,
in $\mathbb{R}$,
\begin{equation} \label{tot}
\mu a(\mu)=l_d(\phi),
\end{equation}
with $\mu=l_d(u_d)$.
\end{theorem}

\begin{remark} \label{rmk2} \rm
Theorem \ref{chi} allows us to see where $a$ is increasing that
the problem \eqref{Pr} with $r=d$ admits a unique solution and determine
for a given $a$  the exact number of solutions \eqref{Pr}.
However it is difficult or impossible to adapt the proof of
the theorem \ref{chi} in the case $0<r<d$.
\end{remark}

\subsection{The case $0<r<d$}
As announced in the introduction we focus our study to the
case of radial solutions of \eqref{Pr} with $r=d$. We will assume
$\Omega$ is the open ball of $\mathbb{R}^n$ with radius $d/2$
centered at zero. We set
\[
L^2_{\rm rad}(\Omega)=\{u\in L^2(\Omega):
\exists\tilde{u}\in L^2(]0,d/2[) \text{ such that }u(x)=\tilde{u}(\|x\|) \},
\]
and  assume that
\begin{equation} \label{cond}
\begin{gathered}
 f\in L^2_{\rm rad}(\Omega),\quad
 g\in L^2_{\rm rad}(\Omega),\\
 a\in W^{1,\infty}(\mathbb{R}),\quad \inf_{\mathbb{R}}a>0,\\
 f\geq 0\quad\text{a.e in}\quad\Omega,\quad
 g\geq 0\quad\text{a.e. in }\Omega.
\end{gathered}
\end{equation}
We start by giving in some sense in a linear case a result that
will be used later to explain the asymptotic behavior.

\begin{proposition} \label{marche}
Let $A,B\in C(\overline{\Omega})$ be positive radial functions
such that $A\leq B$ in $ \overline{\Omega}$ and also
$f,h\in  L^2(\Omega)$ two positive radial functions.
Let $u\in  H^1_0(\Omega)$ the radial solution to
\begin{gather} \label{comp1}
-\operatorname{div}(A(x)\nabla u)=f\quad\text{in }\Omega,\\
\label{comp2}
-\operatorname{div}(B(x)\nabla u)=h\quad\text{in }\Omega.
\end{gather}
 Then $f\leq h$ a.e. in $\Omega$.
\end{proposition}

\begin{proof}
We proved in \cite{a3} that if $u$ is a the radial solution
of \eqref{comp1} then for a.e. $t$ in $[0,d/2]$,
\begin{equation} \label{comp3}
\tilde{u}'(t)=-\frac{1}{\tilde{A}(t)}\int_0^t
\big(\frac{s}{t}\big)^{n-1}\tilde{f}(s)\,ds.
\end{equation}
From \eqref{comp1}, \eqref{comp2} and \eqref{comp3}, we obtain
\begin{equation*}
\frac{\tilde{B}(t)}{\tilde{A}(t)}\int_0^t \big(\frac{s}{t}\big)^{n-1}
\tilde{f}(s)\,ds=\int_0^t \big(\frac{s}{t}\big)^{n-1}\tilde{h}(s)\,ds.
\end{equation*}
Since  $A\leq B$ in $\overline{\Omega}$ and $f,h\geq 0$ with
$f\not\equiv 0,\,h\not\equiv 0$ hence $f\leq g$.
\end{proof}

In a nonlocal case, some results of existence of radial solutions
and comparison principle between  $u_r$, $u_d$ and $u_0$ have been
demonstrated in \cite{a3}. It is also proved when for
$r\in[0,d]$ we set
\begin{equation}
   I_r:=[{\inf_{\Omega} l_r(\phi)},{\sup_\Omega l_r(\phi)}].
\end{equation}
Here $\phi$ denotes the solution of
\begin{equation}
\begin{gathered}
 -\Delta\phi=f \quad\text{in } \Omega\\
              \phi\in H^1_0(\Omega).
\end{gathered}
\end{equation}
By the inclusion or not of $I_r $ at an interval of $\mathbb{R}$
we somehow generalize the theorem \ref{chi}.

\begin{lemma} \label{ler1}
Let  $r\in[0,d]$. Assume that \eqref{cond} holds  and there exist
$0\leq m_1\leq m_2$ such that
\begin{gather}
 a(m_1)={\max_{[m_1,m_2]} a}\quad
 a(m_2)={\min_{[m_1,m_2]} a}, \\
 I_r\subset [m_1a(m_1),m_2a(m_2)].
\end{gather}
Then \eqref{Pr} admits a radial solution $u$, and
\begin{equation}
m_1\leq l_r(u)\leq m_2\quad\text{a.e. in }\Omega.
\end{equation}
\end{lemma}

For the proof of the above lemma, we refer the reader to \cite{a3}.
Generalizing this construction type of the diffusion coefficient
$a$ we obtain

\begin{proposition} \label{babe1}
Let  $r\in[0,d]$. Assume that \eqref{cond} holds  and there exist
an odd integer  $n_1$ and $n_1+1$ positive real numbers
$\{m_i\}_{i=0\dots n_1}$,  with $m_0=0$ and for all
$i\in\{0,\dots,n_1-1\}$ we have $m_{i}<m_{i+1}$. Moreover
\begin{equation} \label{con1}
\begin{gathered}
a(m_i)={\max_{[m_i,m_{i+1}]} a},\quad
a(m_{i+1})={\min_{[m_i,m_{i+1}]} a}\quad\forall
i\in\{0,2,\dots,n_1-3,n_1-1\}\\
I_r\subset  \cap_{i=0,2,\dots,n_1-3,n_1-1}
[m_ia(m_{i}),m_{i+1}a(m_{i+1})]
\end{gathered}
\end{equation}
Then \eqref{Pr} admits at least $(n_1+1)/2$ radial solutions
$\{u_i\}_{i\in\{0,2,\dots,n_1-1\}}$ such that
\begin{equation*}
m_i\leq l_r(u_i)\leq m_{i+1}\quad \forall i\in\{0,2,\dots,n_1-3,n_1-1\}.
\end{equation*}
\end{proposition}

\begin{proof}
By induction, we set
\[
\mathcal{P}_{n_1}=\{\text{If condition \eqref{con1} is satisfied then
\eqref{Pr}  admits at least $\frac{n_1+1}{2}$ solutions.}\}
\]
By using lemma \ref{ler1} with $m_1=0$ and $m_2=m_1$, it is easy
to prove that for $n_1=1$, $\mathcal{P}_{n_1}$ is true.
For $n_1>1$, This procedure can be repeated to prove that
if  $\mathcal{P}_{n_1-2}$  holds true then $\mathcal{P}_{n_1}$
 holds too.
\end{proof}

\begin{example} \label{exa1} \rm
Let us see a function $a$ satisfying proposition \ref{babe1}.
For this, we consider the case $n_1=3$ and $r\in(0,d]$.
Considering \eqref{cond} and the strong maximum principle
we get  $\min I_r>0$. Taking
$$
m_1:= 2\,\frac{\max I_r}{a(0)},\quad
a(m_1):=\frac{a(0)}{2}
$$
 with $a(0)>0$ and also $a$ decreasing on $[0,m_1]$ then we
prove the conditions of lemma \ref{ler1}.

By repeating this process with $m_2>m_1$ and setting
$$
a(m_2):= \frac{\min I_r}{m_2},\quad m_3:=2\frac{\max I_r}{a(m_2)}
$$
with $a(m_3):=\frac{a(m_2)}{2}$ and also $a$ is decreasing on
$[m_2,m_3]$. This shows the existence of such $a$.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1} % art1.eps
\end{center}
\caption{The case $n_1=3$} \label{fig1}
\end{figure}

In the representation of $a$ in Figure \ref{fig1},
we have deliberately left, on solid line parts of the curve
satisfying the conditions of proposition  \ref{babe1}, and
dotted line one without constraints.
\end{example}

\begin{remark} \label{rmk3} \rm
As previously announced,  proposition \ref{babe1} generalizes
a  point of view Theorem \ref{chi}. However it does not accurately
determine the exact number of solutions of \eqref{Pr} and the
bifurcation points of branch of solutions. We have shown in
\cite{a2} a way to solve this problem using the linearized problem,
the principle of comparisons obtained in \cite{a3} and
the Krein-Rutman theorem.
\end{remark}

\subsection{Stable solutions of \eqref{Pr}}

\begin{definition} \label{def3.5} \rm
Given a domain $\Omega\subset\mathbb{R}^n$, a solution
$u_r\in H^1_0(\Omega)$ of \eqref{Pr} is stable if
for all $\phi\in H^1_0(\Omega)$,
\begin{equation} \label{sta}
 G_{u_r}(\phi):=\int_{\Omega}a(l_r(u_r))|\nabla\phi|^2
-\int_{\Omega}a'(l_r(u_r))l_r(\phi)\nabla u_r\nabla\phi\geq 0.
\end{equation}
\end{definition}

\begin{definition} \label{def3.6} \rm
Given $u:[0,d]\to H^1_0(\Omega)$, the graph of $u$ is called a
(global) branch of solutions if
\begin{itemize}
\item[(i)]
 $u\in C([0,d], H^1_0(\Omega))$,
\item[(ii)] $u(r)$ is solution to \eqref{Pr} for all $r$ in $[0,d]$.
\end{itemize}
The function $u$ is called a local branch if it is defined only
on a subinterval of $[0,d]$ with positive measure.
\end{definition}

Before concluding this section, we will focus on the case
$a$ non-increasing, to prove the stability of the global branch
of solutions.
Assume for all $r\in [0,d]$, $u_r$ is a solution to \eqref{Pr} and
\begin{equation} \label{oli1}
0\leq l_r(u_r)(x)\leq\mu_d\quad\text{for a.e. } x\in\Omega.
\end{equation}
Assume that there exists a solution $\mu_d$ to \eqref{tot} such that
\begin{equation} \label{tot1}
a(\mu_d)={\min_{[0,\mu_d]} a}\quad\text{and}\quad
a(0)={\max_{[0,\mu_d]} a}.
\end{equation}

\begin{theorem}[\cite{a3}] \label{tata1}
Assume \eqref{cond}, \eqref{oli1}, \eqref{tot1} and \eqref{tot} holds.
Assume in addition that $a\in W^{1,\infty}(\mathbb{R})$ and for some
positive constant $\epsilon$, it holds that
\begin{equation} \label{const}
C_1|g|_2|f|_2|a'|_{\infty,[-\epsilon,\mu_d+\epsilon]}
\frac{1}{a(\mu_d)^2}<1,
\end{equation}
where $C_1$ is a constant dependent to $\Omega$. Then
\begin{itemize}
\item[(i)] For all $r$ in $[0,d]$, \eqref{Pr} possesses a unique
 radial solution $u_r$ in $[u_0,u_d]$;
\item[(ii)] $\{(r,u_r):r\in[0,d]\}$ is a branch of solutions
 without bifurcation point;
\item[(iii)] it is only global branch of solutions;
\item[(iv)] if in addition, $a$ is non-increasing on
$[0,\mu_d]$ then $r\mapsto u_r$ is  nondecreasing.
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk4} \rm
It is very difficult to obtain property (iv) for any $a$.
However when $a$ is non-increasing provide us important
information for studying the stability of this branch of solutions.
\end{remark}

\begin{corollary} \label{bab2}
Let $u_d^1$ the smallest solution to \eqref{Pr}.
Assume \eqref{cond} and \eqref{tot} holds true and there exists
a solution $\mu_d$ to \eqref{tot} satisfied \eqref{tot1}.
Assume in addition that $a\in W^{1,\infty}(\mathbb{R})$,
$u_d^1$ satisfied \eqref{oli1} and for some positive constant
$\epsilon$, it holds that
\begin{equation} \label{oublie}
C_1|g|_2|f|_2|a'|_{\infty,[-\epsilon,\mu_d+\epsilon]}
\frac{1}{a(\mu_d)^2}<1,
\end{equation}
where $C_1$ is a constant dependent to $\Omega$.
Then $\{(r,u_r):r\in[0,d]\}$ is the only global branch of solutions
starting to $u_d^1$.
\end{corollary}

\begin{proof}
The fact that $\{(r,u_r):r\in[0,d]\}$ is the only global branch
of solutions results from theorem \ref{tata1}.
We will now show that this unique branch of solutions is stable
and start at $r=d$ by $u_d^1$. For this we consider without loss
of generality \eqref{Pr} admits two solutions $u_d^1$ and $u_d^2$
such that $u_d^1\leq u_d^2$.
We denote by $\mu_1$ and $\mu_2$ respectively solutions
of \eqref{tot} corresponding to $u_d^1$ and $u_d^2$
(see figure \ref{fig2}). It is easy to see that $\mu_1$ and
$\mu_2$ satisfied \eqref{tot1}.

Assume $\{(r,u_r):r\in[0,d]\}$ is the only global branch of
solutions starting to $u_d^2$. Then we get
$ C_1|g|_2|f|_2|a'|_{\infty,[-\epsilon,\mu_2+\epsilon]}
\frac{1}{a(\mu_2)^2}<1$. In this case, using theorem \ref{tata1}
 we get \eqref{Pr} possesses a unique radial solution
$u_r$ in $[u_0,u_d^2]$ and the mapping $r\mapsto u_r$ is
nondecreasing. By continuity of this mapping, we can find
a $r_0\in ]0,d[$ such that $u_{r_0}=u_d^1$ for a.e $x\in\Omega$.
This means that $u_d^1$ is a solution of ($P_{r_0}$).
This gives us a contradiction and concludes the proof.
\end{proof}

We are now able to prove the following result.

\begin{proposition} \label{prop3.9}
Under assumptions and notation of corollary \ref{bab2},
the global branch of solutions described in theorem \ref{tata1}
is composed by stable solutions.
\end{proposition}

\begin{proof}
For all $r\in[0,d]$, let $u_r$ be a solution belonging to
the global branch of solutions described in theorem \ref{tata1}.
By using the linearized problem of \eqref{Pr}, we get
for all $\phi\in H^1_0(\Omega)$,
\begin{equation} \label{oli2}
\begin{aligned}
&\int_{\Omega}a(l_r(u_r))|\nabla\phi|^2
-\int_{\Omega}a'(l_r(u_r))l_r(\phi)\nabla u_r\nabla\phi\\
&\geq {\inf_{\Omega}a(l_r(u_r))}|\nabla\phi|_2^2
-C|g|_2|a'|_{\infty,[-\epsilon,\mu_1+\epsilon]}
|\nabla u_r|_2|\nabla\phi|_2^2.
\end{aligned}
\end{equation}
Taking into account that
$|\nabla u_r|_2\leq C(\Omega)\frac{|f|_2}{{\inf_{\Omega}a(l_r(u_r))}}$
where  $C(\Omega)$ designed the Poincar\'e Sobolev constant.
We obtain
\begin{equation} \label{oli3}
\begin{aligned}
&\int_{\Omega}a(l_r(u_r))|\nabla\phi|^2
 -\int_{\Omega}a'(l_r(u_r))l_r(\phi)\nabla u_r\nabla\phi\\
&\geq |\nabla\phi|_2^2\Big({\inf_{\Omega}a(l_r(u_r))}
-C_1 |g|_2|a'|_{\infty,[-\epsilon,\mu_1+\epsilon]}
\frac{|f|_2}{{\inf_{\Omega}a(l_r(u_r))}}\Big).
\end{aligned}
\end{equation}
Moreover by assumptions \eqref{oli1} and \eqref{tot1} we get
$a(\mu_1)\leq{\inf_{\Omega}a(l_r(u_r))}$.
Thus \eqref{oublie} becomes
\begin{equation}
C_1|g|_2|f|_2|a'|_{\infty,[-\epsilon,\mu_d+\epsilon]}
\frac{1}{{\inf_{\Omega}a(l_r(u_r))}^2}<1.
\end{equation}
We obtain
\[
\int_{\Omega}a(l_r(u_r))|\nabla\phi|^2
-\int_{\Omega}a'(l_r(u_r))l_r(\phi)\nabla u_r\nabla\phi\geq 0.
\]
This concluded the proof.
\end{proof}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig2} % art2.eps
\end{center}
\caption{case of 2 solutions}
\label{fig2}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig3} % art3.eps
\end{center}
\caption{Branch of solutions}
\label{fig3}
\end{figure}


\section{Parabolic problem}

\subsection{$L^{\infty}$ estimate}

In what follows we obtain $L^{\infty}$ estimate of the solution
 \eqref{eq0} from $L^q$ estimate. The method we use is based
on iterations Moser type, for more details on the method
see \cite{q1}.

\begin{theorem} \label{terib3}
Let $n\geq 3$ and $u$ a classical solution of \eqref{eq0} defined
on $[0,T)$. Assume that $p>1$ and $q>1$ such that
$\frac{1}{p}+\frac{1}{q}=1$. Suppose further
that $U_q=\sup_{t<T}|u(t)|_q<\infty$,
$f\in L^\infty(0,\infty,L^q(\Omega))$.
If $p<\frac{n}{n-2}$ then $U_\infty<\infty$.
\end{theorem}

To prove this theorem we need the following result.

\begin{lemma} \label{moser1}
Consider $u$ a classical solution of \eqref{eq0} on $[0,T)$,
$r\geq 1$ and $p>1$  such that $\frac{1}{p}+\frac{1}{q}=1$
with $p<\frac{n}{n-2}$. We take
$\tilde U_r=\max\{1,|u_0|_\infty$, $U_r=sup_{t<T}|u(t)|_r\}$ and
let
$$
\sigma(r)=\frac{p(n+2)}{2[r(2p-pn+n)+np]}.
$$
Then there exists a constant $C_2=C_2(\Omega,m)$ such that
$$
\tilde U_{2r}\leq [C_2\,\|f\|_{L^\infty(0,\infty,L^q(\Omega))}
]^{\sigma(r)}r^{\sigma(r)}\tilde U_{r}.
$$
\end{lemma}

\begin{proof}
Multiplying  \eqref{eq0} by $u^{2r-1}$ and then using
the H\"{o}lder inequality  yields
\begin{equation} \label{it2}
\frac{1}{2r}\frac{d}{dt}\int_{\Omega}u^{2r}dx+m\frac{2r-1}{r^2}
\int_{\Omega}|\nabla (u^r)|^2dx\leq|f|_q|u^{2r-1}|_p.
\end{equation}
As
\begin{equation} \label{it3}
|u^{2r-1}|_p=|u^r|^{\frac{2r-1}{r}}_{p\frac{2r-1}{r}},
\end{equation}
by taking $w=u^r$ in \eqref{it2} and \eqref{it3}, we get easily
\begin{equation} \label{it4}
\frac{1}{2r}\frac{d}{dt}|w|^2_2+m\frac{2r-1}{r^2}|\nabla w|^2_2
\leq|f|_q|w|^{\alpha}_{\alpha p},
\end{equation}
with $\alpha=(2r-1)/r$. Let $\beta$ be such that
\begin{equation} \label{it5}
\frac{1}{\alpha p}=\beta+ \frac{1-\beta}{2^\star},
\end{equation}
with $2^\star=\frac{2n}{n-2}$.
We claim that $\beta\in(0,1)$.
In fact
$$
\beta=\frac{2nr-(n-2)(2r-1)p}{(n+2)(2r-1)p}.
$$
Since $p<\frac{2r}{2r-1}\frac{n}{n-2}$, it follows that $\beta>0$.
Also  $(n+2)(2r-1)p>2nr-(n-2)(2r-1)p$ implies
$\beta<1$. This prove that $\beta\in(0,1)$.
Using an interpolation inequality
in \eqref{it4} and \eqref{it5} (see \cite{q1}), we obtain
\begin{equation} \label{it6}
\frac{1}{2r}\frac{d}{dt}|w|^2_2+m\frac{2r-1}{r^2}|\nabla w|^2_2
\leq|f|_q\Big(|w|^{\beta}_1 |w|^{1-\beta}_{2^\star}\Big)^{\alpha }.
\end{equation}
Applying Sobolev injections in \eqref{it6}, we have
\begin{equation} \label{it7}
\begin{aligned}
&\frac{1}{2r}\frac{d}{dt}|w|^2_2+m\frac{2r-1}{r^2}|\nabla w|^2_2\\
&\leq\Big[|f|_q\big(\frac{2r}{m}\big)^{\frac{\alpha(1-\beta)}{2}}
|w|^{\beta\alpha}_1\,C^{(1-\beta)\alpha}\Big]
\Big[\big(\frac{m}{2r}\big)^{\frac{\alpha(1-\beta)}{2}}
|\nabla w|^{(1-\beta)\alpha}_2\Big],
\end{aligned}
\end{equation}
and
\begin{equation}\label{it8}
\begin{aligned}
&\frac{1}{2r}\frac{d}{dt}|w|^2_2+m\frac{2r-1}{r^2}|\nabla w|^2_2\\
&\leq\Big[|f|_q\big(\frac{2r}{m}\big)^{\frac{\alpha(1-\beta)}{2}}
|w|^{\beta\alpha}_1\,C^{(1-\beta)\alpha}\Big]
\Big[\big(\frac{m}{2r}\big)|\nabla w|^2_2\Big]
^{\frac{\alpha(1-\beta)}{2}}.
\end{aligned}
\end{equation}
Since $\beta\in(0,1)$ and $\frac{\alpha}{2}\in (0,1)$ it is clear
that $\frac{\alpha(1-\beta)}{2}\in (0,1)$. Applying Young's
inequality to \eqref{it8} with
$\frac{\alpha(1-\beta)}{2}+1-\frac{\alpha(1-\beta)}{2}=1$. We obtain
\begin{equation} \label{it9}
\begin{aligned}
&\frac{1}{2r}\frac{d}{dt}|w|^2_2+m\frac{2r-1}{r^2}|\nabla w|^2_2\\
&\leq\delta\Big[|f|_q^{\frac{1}{\delta}
}\big(\frac{2r}{m}\big)^{\frac{\alpha(1-\beta)}{2\delta}}
|w|^{\frac{\beta\alpha}{\delta}}_1\,C^{2/\delta}\Big]
+\frac{\alpha(1-\beta)}{2}\Big[\big(\frac{m}{2r}\big)
|\nabla w|^2_2\Big],
\end{aligned}
\end{equation}
with  $\delta=1-\frac{\alpha(1-\beta)}{2}$.
Joining the fact that $\frac{\alpha(1-\beta)}{2}\in (0,1)$ and
$\delta<1$ to  \eqref{it9}, we deduce
\begin{equation} \label{it10}
\frac{1}{2r}\frac{d}{dt}|w|^2_2+m\frac{3r-2}{2r^2}|\nabla w|^2_2
\leq|f|_q^{1/\delta}\big(\frac{2r}{m}\big)^{\frac{\alpha(1-\beta)}{2\delta}}
|w|^{\frac{\beta\alpha}{\delta}}_1\,C^{2/\delta}.
\end{equation}
We set
$$
2r\sigma(r)-1=\frac{\alpha(1-\beta)}{2\delta}\quad \text{and}\quad
2\rho(r)=\frac{\beta\alpha}{\delta}.
$$
Then \eqref{it10} becomes
\begin{equation} \label{it11}
\frac{1}{2r}\frac{d}{dt}|w|^2_2+m\frac{3r-2}{2r^2}|\nabla w|^2_2
\leq|f|_q^{1/\delta}\big(\frac{2r}{m}\big)^{2r\sigma(r)-1}
|w|^{2\rho(r)}_1\,C^{2/\delta}.
\end{equation}
Taking into account that $\frac{3r-2}{r}>1$, this gives us
\begin{equation} \label{it13}
\frac{d}{dt}|w|^2_2+m|\nabla w|^2_2\leq|f|_q^{1/\delta}
\big(\frac{2r}{m}\big)^{2r\sigma(r)}|w|^{2\rho(r)}_1
mC^{2/\delta}.
\end{equation}
By a calculation we can verify that
\[
\rho(r)=\frac{2nr-(n-2)(2r-1)p}{2r(p(n+2)+n)-2n(2r-1)p},
\]
and  that  $\rho(r)\in(0,1)$.

Using the Poincar\'e Sobolev  inequality and that
$\rho(r)<1$ in \eqref{it13} yields
\begin{equation} \label{it14}
\frac{d}{dt}|w|^2_2+\frac{m}{C_1(\Omega)}|w|^2_2
\leq|f|_q^{1/\delta}\big(\frac{2r}{m}\big)^{2r\sigma(r)}|w|^2_1
 mC^{2/\delta},
\end{equation}
where  $C_1(\Omega)$ designed the Poincar\'e Sobolev constant.
 Noticing that
\begin{equation} \label{it15}
\begin{aligned}
e^{-\frac{m}{C_1(\Omega)}t}\frac{d}{dt}
\Big(e^{\frac{m}{C_1(\Omega)}t}|w|^2_2\Big)
&=\frac{d}{dt}|w|^2_2+\frac{m}{C_1(\Omega)}|w|^2_2\\
&\leq|f|_q^{1/\delta}\big(\frac{2r}{m}\big)^{2r\sigma(r)}
|w|^2_1 mC^{2/\delta}.
\end{aligned}
\end{equation}
and integrating  \eqref{it15} on  $[0,t)$ we obtain
  \begin{equation} \label{it16}
|w(t)|^2_2\leq|w(0)|^2_2+\|f\|_{L^\infty(0,\infty,L^q(\Omega))}
^{1/\delta}\big(\frac{2r}{m}\big)^{2r\sigma(r)}m\,C^{2/\delta}|w|^2_1.
\end{equation}
Since
\begin{equation} \label{it17}
|w(0)|^2_2=\int_\Omega w(0)^2dx
=\int_\Omega u(0)^{2r}dx\leq|\Omega||u(0)|^{2r}_\infty
\leq|\Omega|\tilde{U}_r^{2r},
\end{equation}
\eqref{it16} and \eqref{it17} gives us
\begin{equation}
\label{it18}
\tilde{U}_{2r}^{2r}\leq|\Omega|\tilde{U}_r^{2r}
+\|f\|_{L^\infty(0,\infty,L^q(\Omega))}^{1/\delta}
\big(\frac{2r}{m}\big)^{2r\sigma(r)} mC^{2/\delta}\tilde{U}_r^{2r}.
\end{equation}
Whereas  $1/\delta >1$, $2r\sigma(r)>0$  and
$\sigma(r)=1/(2r\delta)$  it follows that
\begin{equation} \label{it20}
\tilde{U}_{2r}\leq C_2^{\sigma(r)}
\|f\|_{L^\infty(0,\infty,L^q(\Omega))}^{\sigma(r)}
r^{\sigma(r)}\tilde{U}_r,
\end{equation}
with $C_2=C_2(\Omega,m)$. This completes the proof of Lemma.
\end{proof}

\begin{lemma} \label{moser2}
Let $r>1$, $n\geq 3$, $p<\frac{n}{n-2}$ and
$\sigma(r)=\frac{p(n+2)}{2[r(2p-pn+n)+np]}$  then we have
$$
\sigma(2^kr)\leq\theta^k\sigma(r)\quad\forall k\in\mathbb{N},
$$
with  $\theta\in(0,1)$.
\end{lemma}

\begin{proof}
Setting $c_1=\frac{p(n+2)}{2}$, $c_2=(2p-pn+n)$ and
$c_3=np$ yields $\sigma(r)=\frac{c_1}{rc_2+c_3}$ with
$c_1,c_2,c_3\in \mathbb{R}^{\star}_{+}$.
By taking  $\theta=1-\frac{c_2}{2c_2+c_3}$ the proof of this
lemma is deduced by reasoning by induction.
\end{proof}

Returning now to the proof of the theorem.

\begin{proof}[Proof of Theorem \ref{terib3}]
Using lemma \ref{moser1} we have
$$
\tilde U_{2r}\leq [C_2\,\|f\|_{L^\infty(0,\infty,L^q(\Omega))}
]^{\sigma(r)}r^{\sigma(r)}\tilde U_{r}.
$$
By iterating this equation and taking $r=h,r=2h,r=2^2h,etc$, we obtain
$$
\tilde U_{2^{k+1}h}\leq [C_2\,\|f\|_{L^\infty
(0,\infty,L^q(\Omega))}]^{\lambda1}2^{\lambda2}
h^{\lambda1}\,\tilde U_{h},
$$
with
\begin{gather*}
\lambda_1:=\sigma(h)+\sigma(2h)+\sigma(2^2h)+\dots
+\sigma(2^{k-1}h)+\sigma(2^kr),\\
\lambda_2:=\sigma(2h)+2\sigma(2^2h)+3\sigma(2^3h)+\dots
+(k-1)\sigma(2^{k-1}h)+k\sigma(2^kr).
\end{gather*}
To complete the proof we just need to show that
$\lambda_1,\lambda_2<+\infty$. Indeed by lemma \ref{moser2}
$$
\lambda_1\leq\sum_{\mu=0}^k\alpha^{\mu}\sigma(h)
\leq\sum_{\mu=0}^\infty\alpha^{\mu}\sigma(h)
=\frac{\sigma(h)}{(1-\alpha)}<\infty.
$$
Noting also that
$$
\sigma(2^kh)\leq\theta^{k-1}\sigma(2h)\quad\forall k\in\mathbb{N}^\star,
$$
it follows that
$$
\lambda_2\leq\sum_{\mu=1}^k\mu\alpha^{\mu-1}\sigma(2h)
\leq\sum_{\mu=1}^\infty\mu\alpha^{\mu-1}\sigma(2h)
=\frac{\sigma(2h)}{(1-\alpha)^2}<\infty.
$$
This completes the proof of the theorem.
\end{proof}

\subsection{Uniform estimate in time}

We prove an estimate for $u$ in
$L^{\infty}(\mathbb{R}^+,H^1_0(\Omega))$.

\begin{theorem} \label{th1}
Assume that $f\in L^2(\Omega)$, $g\in H^1(\Omega)$,
$u_0\in H^1_0(\Omega)$  and $a\in W^{1,\infty}(\mathbb{R})$
with ${\inf_{\mathbb{R}} a> 0}$. Then a solution $u$ of \eqref{eq0}
is such that  $u\in L^{\infty}(\mathbb{R}^+,H^1_0(\Omega))$.
\end{theorem}

\begin{proof}
 Taking a spectral basis related to the Laplace operator in
the Galerkin approximation (see \cite{t1}) we find that
 $-\Delta u$ can be regarded  as test function in
$L^2(0,T,L^2(\Omega))$ for all $T>0$. By multiplying
\eqref{eq0} by  $-\Delta u(t)$ and integrating over $\Omega$,
\begin{equation} \label{22oct1}
( u_t,-\Delta u)+( -\operatorname{div}(a(l_r( u))\nabla u),
-\Delta u)=(f,-\Delta u),
\end{equation}
and
\begin{equation} \label{23oct1}
\frac{1}{2}\frac{d}{dt}\|u\|^2_V+(-a(l_r(u))\Delta u ,
-\Delta u)+(-a'(l_r(u))\nabla l_r(u).\nabla u ,
-\Delta u)=(f,-\Delta u).
\end{equation}
Here $(.,.)$ is the usual scalar product on $L^2(\Omega)$.
Taking into account
\begin{equation}
|\nabla l_r(u)|_2\leq K\,\|g\|_{H^1(\Omega)}|\nabla u|_2,
\end{equation}
where $K$ is a constant depending of $\Omega$.
the above equality yields
\begin{equation} \label{23oct3}
|(-a'(l_r(u))\nabla l_r(u).\nabla u,-\Delta u)|
\leq K \|g\|_{H^1(\Omega)}|a'|_{\infty}\| u\|^2_V|\Delta u|_2
\end{equation}
Now  from  \eqref{23oct3} and \eqref{23oct1}, we have
\begin{equation} \label{23oct6}
\frac{1}{2}\frac{d}{dt}\|u\|^2_V+ m|\Delta u|_2^2-K
\|g\|_{H^1(\Omega)}|a'|_{\infty}\| u\|^2_V|
\Delta u|_2\leq|f|_2|\Delta u|_2.
\end{equation}
Using Young's inequality $ab\leq \frac{1}{2m}a^2+\frac{m}{2}b^2$,
we obtain
\begin{equation} \label{23oct8}
\frac{d}{dt}\|u\|^2_V\leq\frac{1}{m}|f|_2^2
+\frac{1}{m} (K\,\|g\|_{H^1(\Omega)})^2\|a'\|_{\infty}^2\| u\|^4.
\end{equation}
To apply the uniform Gronwall lemma to \eqref{23oct8},
 we start with a small estimate. Recall that
\begin{equation} \label{29oct3}
\frac{d}{dt}|u|^2_2+\,m\|u\|^2_V\leq\frac{1}{\lambda\,m}|f|_{2}^2,
\end{equation}where $\lambda$ is the principal eigenvalue of the Laplacian operator with Dirichlet boundary conditions.

 By integrating on $[t,t_0)$ we have
\begin{equation}
|u(t+t_0)|^2_2+\,m\int_t^{t+t_0}\|u\|^2_V\,ds\leq\int_t^{t+t_0}
\frac{1}{\lambda\,m}|f|_{2}^2\,ds+|u(t)|^2_2,
\end{equation}
and
\begin{equation} \label{bazo}
\int_t^{t+t_0}\|u\|^2_V\,ds\leq\frac{t_0}{\lambda\,m^2}|f|_{2}^2\,ds
+\frac{1}{m}|u(t)|^2_2.
\end{equation}
Let $\rho_0>0$ such that $|u(t)|^2_2\leq\rho_0^2 $. By setting
\[
a_1=\frac{1}{m}c_1(\Omega)^2|a'|_{\infty}^2a_3,\quad
a_2=\frac{t_0}{m}|f|_2^2,\quad
a_3=\frac{t_0\lambda}{m^2}|f|_{2}^2+\frac{1}{m}\rho_0^2,
\]
and using uniform Gronwall lemma to \eqref{23oct8}, we
obtain
\begin{equation} \label{29oct1}
\|u(t+t_0)\|_V\leq (\frac{a_3}{t_0}+a_2)\exp(a_1)\quad
\forall t\geq 0,\quad t_0>0.
\end{equation}
Hence  $u\in L^{\infty}( t_0,+\infty,H^1_0(\Omega))$.
Using \eqref{23oct8} and the classical Gronwall lemma it is
easy to see that
$u\in L^{\infty}(0,t_0,H^1_0(\Omega))$.
 This completes the proof of the theorem.
\end{proof}

\begin{remark} \label{bab} \rm
This theorem shows us the existence of absorbing set in
$H^1_0(\Omega)$.  By considering $S(t)$ the semigroup associated
with the equation \eqref{eq0} defined by
\[
S(t):L^2(\Omega)\to L^2(\Omega),\quad   u_0 \mapsto u(t),
\]
with $u(t)$ a solution of \eqref{eq0}. As a result of
the theorem \ref{th1} and the compact embedding of
$H^1_0(\Omega)$ into $L^2(\Omega)$  we deduce that the
semigroup $S(t)$  possesses a global attractor.
Indeed it is easy to show the existence of absorbing set
in $L^2(\Omega)$, the main difficulty here is to show that
$S(t)$ is such that for all $B\subset L^2(\Omega)$
bounded, there exists $t_0=t_0(B)$ such that
\begin{equation}
\cap_{t\geq t_0}\cup S(t)B \quad\text{is relatively compact in }
 L^2(\Omega).
\end{equation}
This property known and that $S(t)$ is uniformly compact for $t$
large can be proved by using theorem \ref{th1} and the compact
embedding of $H^1_0(\Omega)$ into $L^2(\Omega)$.
\end{remark}


\subsection{Asymptotic behaviour}

In this part we are interested in the asymptotic behaviour of
a weak solutions of \eqref{eq0}. Our main interest here is
radial solutions. By radial solutions we
mean $\tilde{u}(|x|,t)=u(x,t)$. As in the stationary case
$\Omega$ is a open ball of $\mathbb{R}^n$. Remember that
\[
L^2_{\rm rad}(\Omega)=\{v\in L^2(\Omega):
\exists\tilde{v}\in L^2(]0,d/2[) \text{ such that }
 v(x)=\tilde{v}(\|x\|) \}.
\]
Not to confuse  $u_0$,  the solution
to \eqref{Pr} with $r=0$, and the initial value of \eqref{eq0},
we will take $u^0$ the initial value of \eqref{eq0}.

\begin{theorem} \label{thm4.5}
Assume that $f, g\in L^2_{\rm rad}(\Omega) $, $a$ is a continuous
function and the assumption \eqref{p1} checked then \eqref{eq0}
admits a radial solution.
\end{theorem}

\begin{proof}
Let $w\in L^2(0,t, L^2_{\rm rad}(\Omega))$ it is clear that
$l_r(w)$ is radial and also $a(l_r(w))$. Thus by \eqref{eq8} $F_r$
maps $L^2(0,t, L^2_{\rm rad}(\Omega))$  into itself.
The proof now follows by using arguments similar to those used
in theorem \ref{terib1}.
\end{proof}

Assume now
\begin{gather} \label{inf1}
f,g\geq 0\quad\text{in }\Omega, \\
\label{inf2}
u_0\leq u^0\leq u_d,
\end{gather}
with $u^0$ the initial value to \eqref{eq0} and $u_0$ and
$u_d$ respectively the solution of \eqref{Pr} with $r=0$ and of
 \eqref{Pr}.

We can now give a stability result assuming that \eqref{eq0}
admits a unique solution.

\begin{theorem} \label{thm4.6}
Assume \eqref{inf1} and $f, g\in L^2_{\rm rad}(\Omega)$.
Let $u$, $u_d$ and $u_0$ respectively the solution of \eqref{eq0},
$(P_d)$ and  $(P_0)$. If
$u_0\leq u^0\leq u_d$,
then
\[
u_0\leq u\leq u_d\quad \forall t.
\]
 \end{theorem}

\begin{proof}
Let
\begin{equation}
\mathcal{S}=\{t :l(u(s))\in [0,l_d(u_d)],\; \forall s\leq t\}.
\end{equation}
It is easy to prove that $\mathcal{S}$ contains 0
(see \ref{inf2}). By setting
\begin{equation}
t^{\star}=\sup\{t :t\in\mathcal{S}\}.
\end{equation}
By continuity of the mapping  $t\mapsto l_d(u(t))$, we have
\begin{equation}
 l_d(u(t^{\star}))\in [0,l_d(u_d)].
\end{equation}
By using \eqref{eq0} and \eqref{Pr} we get
in $\mathcal{D}(0,t^{\star})$
\begin{equation} \label{inf4}
 \frac{d}{dt}(u_d-u,\phi)+\int_{\Omega}a(l_d(u))\nabla(u_d-u)\nabla\phi
=-\int_{\Omega}(a(l_d(u_d))-a(l_d(u)))\nabla u_d\nabla\phi
\end{equation}
for all $\phi\in H^1_0(\Omega)$.
Choosing $\phi=(u_d-u)^-$, \eqref{inf4} becomes
\begin{equation} \label{inf5}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}|(u_d-u)^-|^2_2
 +\int_{\Omega}a(l_d(u_d))|\nabla(u_d-u)^-|^2\\
&=\int_{\Omega}(a(l_d(u))-a(l_d(u_d)))\nabla u_d\nabla(u_d-u)^-.
\end{aligned}
\end{equation}
Since $a$ is non-increasing ($a(l_d(u))-a(l_d(u_d))\geq 0$
for all $t\leq t^{\star}$) hence proposition \ref{marche} yields
\begin{equation}
\int_{\Omega}(a(l_d(u))-a(l_d(u_d)))\nabla u_d\nabla(u_d-u)^-\leq 0.
\end{equation}
Thus
\begin{equation} \label{inf6}
\frac{1}{2}\frac{d}{dt}|(u_d-u)^-|^2_2+a(l_d(u_d))|\nabla(u_d-u)^-|^2_2\leq 0
\end{equation}
Applying Poincarré Sobolev inequality, we have
\begin{equation}
\frac{1}{2}\frac{d}{dt}|(u_d-u)^-|^2_2+C_2|(u_d-u)^-|^2_2\leq 0,
\end{equation}
this proves
\[
\frac{d}{dt}\{e^{2t\,C_2}|(u_d-u)^-|^2_2\}\leq 0.
\]
Moreover, $(u_d-u)^-(0)=(u_d-u^0)^-=0$ it follows that
 $u_d\geq u\quad\forall t\in [0,t^{\star}]$.
In the same way we can also prove $u_0\leq u$
for all $t\in [0,t^{\star}]$. It follows that
\begin{equation} \label{inf9}
u_0\leq u\leq u_d\quad\forall t\in [0,t^{\star}]
\end{equation}
To finish we just need to prove that $t^{\star}$ is very large,
this is typically the case. Indeed if $t^{\star}<\infty$ then
\begin{equation}
l(u(t^{\star}))=0\quad\text{or}\quad l_d(u_d).
\end{equation}
 From \eqref{inf1} and \eqref{inf9} we deduce
\begin{equation} \label{inf10}
u(t^{\star})= u_0\quad\text{or}\quad u(t^{\star})=u_d.
\end{equation}
Due to the uniqueness of \eqref{eq0}, we deduce that $t=\infty$.
This shows that
\[
u_0\leq u\leq u_d\quad \forall t,
\]
 and completes the proof.
\end{proof}

\begin{remark} \label{rmk6} \rm
The fact that $|u(t)|^2_2$ is not a Lyapunov function that is
to say decreases in time, makes very complex the study of
certain asymptotic properties of our problem. Indeed under
our study it is tempting to show that for $r$ fixed $r\in]0,d[$
\[
u(t)\to u^1_r\quad\text{in}\quad L^2(\Omega),
\]
where $u$ is the solution of \eqref{eq0} and $u^1_r$
the solution belonging to the stable global branch described
previously. A numerical study would be a great contribution
to try to carry out some of our theoretical intuitions.
\end{remark}

\begin{thebibliography}{00}

\bibitem{a1}  Ambrosetti, A and Prodi, G.;
\emph{A primer of nonlinear analysis}, Cambridge University Press, 1995.

\bibitem{a2}  Andami Ovono, A;
\emph{About the counting stationary solutions of a nonlocal
problem by revisiting the Krein-Rutman theorem}, in preparation.

\bibitem{a3} Andami Ovono, A and Rougirel, A.;
\emph{Elliptic equations with diffusion parameterized by the range
of nonlocal interactions}, Ann. Mat. Pura Appl, 1 (2010), p. 163-183.

\bibitem{b1}  Bendahmane, M and Sep\'ulveda, M. A.;
\emph{Convergence of a finite volume scheme for nonlocal
reaction-diffusion systems modelling an epidemic disease},
Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), p. 823--853.

\bibitem{c1}  Caffarelli, L. and Silvestre, L.;
\emph{Regularity theory for fully nonlinear integro-differential
  equations}, Comm. Pure Appl. Math., 62 (2009), p. 597-638.

\bibitem{c2}  Chang, N.-H. and Chipot, M.;
\emph{On some mixed boundary value problems with nonlocal diffusion},
Adv. Math. Sci. Appl, 14 (2004), p. 1-24.

\bibitem{c3}  Chipot, M.;
\emph{Elements of nonlinear analysis}, Birkh\"auser Verlag, 2000.

\bibitem{c4}  Chipot, M. and Lovat, B.;
\emph{Existence and uniqueness results for a class of nonlocal
     elliptic and parabolic problems}, Dyn. Contin. Discrete Impuls.
 Syst. Ser. A Math. Anal. , 8 (2001), p. 35-51.

\bibitem{c5}  Chipot, M. and Rodrigues, J.-F.;
\emph{On a class of nonlocal nonlinear elliptic problems},
RAIRO Mod\'el. Math. Anal. Num\'er. , 26 (1992), p. 447--467. 

\bibitem{d1}  Dautray, R. and Lions, J.-L.;
\emph{Analyse math\'ematique et calcul num\'erique pour les sciences 
et les techniques. {V}ol. 8}, Masson, 1988.

\bibitem{g1}  Genieys, S. and Volpert, V. and Auger, P.;
\emph{Pattern and waves for a model in population dynamics with
nonlocal consumption of resources}, Math. Model. Nat. Phenom. , 1
(2006), p. 65-82

\bibitem{g2}  Gourley, S. A.;
\emph{Travelling front solutions of a nonlocal {F}isher equation},
J. Math. Biol. , 41 (2000), p. 272-284

\bibitem{p1} Perthame, B. and Genieys, S.;
\emph{Concentration in the nonlocal Fisher equation:
the Hamilton-Jacobi limit}, Math. Model. Nat. Phenom. , 2 (2007),
 p. 135-151.

\bibitem{q1} Quittner, P. and Souplet, Ph.;
\emph{Superlinear parabolic problems}, Birkh\"auser Verlag, 2007.

\bibitem{s1} Siegwart, M.;
\emph{Asymptotic behavior of some nonlocal parabolic problems},
Adv. Diff Equations, 2 (2006), p. 167-199.

\bibitem{t1}  Temam, R.;
\emph{Infinite-dimensional dynamical systems in mechanics and physics},
Springer-Verlag, 1997.

\end{thebibliography}

\end{document}