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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 158, pp. 1--7.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/158\hfil Nonlinear gradient dependent systems]
{Solution to nonlinear gradient dependent systems with a balance law}

\author[Z. Dahmani, S.  Kerbal\hfil EJDE-2007/158\hfilneg]
{Zoubir Dahmani, Sebti  Kerbal}

\address{Zoubir Dahmani \newline
 Department of Mathematics, Faculty of Sciences, 
 University of Mostaganem, Mostaganem, Algeria}
\email{zzdahmani@yahoo.fr}

\address{Sebti  Kerbal \newline
Department of Mathematics and Statistics,
 Sultan Qaboos Uiverstiy, Alkhod,
Muscat, Sultanate of Oman }
 \email{skerbal@squ.edu.om}

\thanks{Submitted April 15, 2007. Published November 21, 2007.}
\subjclass[2000]{35B40, 35B50, 35K57}
\keywords{Reaction-diffusion systems; global existence; asymptotic behavior;
\hfill\break\indent maximum principle}

\begin{abstract}
 In this paper, we are concerned with the initial boundary value
 problem (IBVP) and with the Cauchy problem to the reaction-diffusion
 system
 \begin{gather*}
 u_t-\Delta u    = -u^n |\nabla v |^p,\\
 v_t-d \Delta v  =  u^n |\nabla v|^p,
 \end{gather*}
 where $1\leq p\leq2$, $d$ and $n$ are positive real numbers.
 Results on the existence and large-time behavior
 of the solutions are presented.
\end{abstract}

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

 \section{Introduction}

In the first part of this article, we are interested in the
existence of global classical nonnegative solutions to the
 reaction-diffusion equations
\begin{equation} \label{inte1}
\begin{gathered}
 u_t -\Delta u = -u^n |\nabla v |^p=:-f(u,v), \\
 v_t -d \Delta v  = u^n |\nabla v|^p,
\end{gathered}
\end{equation}
posed on $\mathbb{R}^+ \times \Omega$ with initial data
\begin{equation}\label{condini}
 u( 0;x)=u_0(x), \quad  v(0;x)=v_0(x)
\quad \text{in }\Omega
\end{equation}
 and boundary conditions (in the case $\Omega$ is a bounded domain
in $ \mathbb{R}^{n}$)
\begin{equation}\label{bound}
 \frac{\partial u}{\partial \eta}=\frac{\partial
v}{\partial \eta}=0, \quad  \text{on } \mathbb{R}^+ \times
\partial \Omega.
\end{equation}
Here $\Delta$ is the Laplacian operator, $u_0$ and $v_0$ are given
 bounded nonnegative functions, $\Omega\subset
\mathbb{R}^n$ is a regular domain, $\eta$ is the outward normal to
$\partial\Omega$. The diffusive coefficient $d$ is a positive real.
One of the basic questions for (\ref{inte1})-(\ref{condini}) or
(\ref{inte1})-(\ref{bound}) is the existence of global solutions.
Motivated by extending known results on reaction-diffusion systems
with conservation of the total mass but with non linearities
depending only for the unknowns, Boudiba, Mouley and Pierre
succeeded in obtaining $L^{1}$ solutions only for the case
$u^{n}|\nabla v|^{p}$ with $p<2$.
In this article, we are interested essentially in classical
solutions in the case where $p=2$ ($\Omega$ bounded or
$\Omega=\mathbb{R}^n$ ; in the latter case, there are no boundary
conditions).

\section{Results}

The existence of a unique  classical solution over the whole time
interval $[0,T_{\rm max}[$ can be obtained by a known procedure: a local
solution is continued globally by using a priori estimates on
$\|u\|_{\infty}, \|v\|_{\infty}, \||\nabla u|\|_{\infty}$, and
$\||\nabla v|\|_{\infty}$.
\subsection{The Cauchy problem}
\subsubsection*{Uniform bounds for $u$ and $v$}
First, we consider the auxiliary problem
%
\begin{equation} \label{inte10}
\begin{gathered}
 L_{\lambda}\omega := \omega_{t}-\lambda\Delta\omega=
b\nabla\omega, \quad  t>0,\; x\in \mathbb{R}^N \\
 \omega(0,x) = \omega_{0}(x)\in L^{\infty},
\end{gathered}
\end{equation}
where $b=(b_{1}(t,x),\dots ,b_{N}(t,x)), b_{i}(t,x)$ are continuous
on $[0,\infty)\times\mathbb{R}^{N}$, $\omega$ is a classical
solution of (\ref{inte10}).

\begin{lemma}\label{lm2.1}
Assume that $ \omega_{t}, \nabla\omega, \omega_{x_{i}x_{i}}$, $i=1,\dots ,N$
are   continuous,
 \begin{equation}\label{hyp1b}
 L_{\lambda}\omega\leq 0,\quad (\geq)\quad (0,\infty)\times \mathbb{R}^N
  \end{equation}
and $\omega(t,x)$ satisfies \eqref{inte10}$_2$. Then
\begin{gather*}
\omega(t,x)\leq C:=\sup_{x\in \mathbb{R}^N}\omega_{0}(x),\quad
(0,\infty)\times   \mathbb{R}^N.\\
\omega(t,x)\geq C:=\inf_{x\in \mathbb{R}^N}\omega_{0}(x),\quad
(0,\infty)\times   \mathbb{R}^N.
\end{gather*}
\end{lemma}

The proof of the above lemma is elementary and hence is omitted.
Now, we consider the problem (\ref{inte1})-(\ref{condini}). It follows
by the maximun principle that
$$
u,v\geq 0,\quad \text{in } \mathbb{R}^{+}\times \mathbb{R}^{N}.
$$

\paragraph{Uniform bounds of $u$}
We have
$$
u\leq C_{1}:=\sup_{\mathbb{R}^N}u_{0}(x),
$$
thanks to the maximum principle.

\paragraph{Uniform bounds of $v$.}
Next, we derive an upper estimate for $v$.
Assume that $1\leq p < 2$. We transform \eqref{inte1}$_2$ by the
substitution $\omega=e^{\lambda v}-1$ into
$$
\omega_{t}-\lambda\Delta\omega
=\lambda e^{\lambda v}(v_{t}-d \Delta v-d\lambda\ |\nabla v |^{2} )
= \lambda e^{\lambda v}( u^{n}|\nabla v |^{p}
-d\lambda\ |\nabla v |^{2} ).
$$
Let
$$
\phi(x)\equiv Cx^{p}- d\lambda x^{2} ;\quad C >0 ,\;x\geq 0.
$$
By elementary computations,
$$
\phi(x)\geq 0  \quad \text{when } x\leq \Big(\frac{C}{\lambda d}
\Big)^{1/(2-p)} .
$$
But in this case
$$
|\nabla v| \leq \Big (\frac{c}{\lambda d}\Big)^{1/(2-p)}.
$$
In the case $x\geq (\frac{c}{\lambda d})^{1/(2-p)}$,
\begin{equation}\label{inte14}
 \phi(x)\leq 0
\end{equation}
and hence $ \omega \leq M$ where
\begin{equation}\label{inte15}
 M=C \Big(\frac{pC}{2d\lambda}\Big)^{p/2-p}(\frac{2-p}{2}).
\end{equation}
Then we have $v \leq C_2$.

\subsubsection{Uniform bounds for $|\nabla u|$ and $|\nabla v|$.}
 At first, we present the  uniform bounds for $|\nabla v|$.
We write \eqref{inte1}$_2$ in the form
\begin{equation}\label{inte16}
 L_{d}v+ kv=kv+u^{n}|\nabla v|^{p}
\end{equation}
and transform it by the substitutions $\omega=e^{kt}v$ to obtain
\begin{gather*}
L_{d}\omega=e^{kt}(L_{d}v+kv)=e^{kt}(kv+u^{n}|\nabla
v|^{p}),\quad t>0,\; x\in\mathbb{R}^{N}\\
\omega(0,x)=v_{0}(x).
\end{gather*}
Now let
$$
G_{\lambda}=G_{\lambda}(t-\tau;x-\xi)
=\frac{1}{[4\pi\lambda(t-\tau)]^{\frac{N}{2}}}
\exp\Big(\frac{|x-\xi|^{2}}{4\lambda(t-\tau)}\Big)
$$
be the fundamental solution related to the operator $L_{\lambda}$.
Then, with $Q_t=(0,t)\times \mathbb{R}^{N}$, we have
$$
\omega=e^{kt}v=v^{0}(t,x)+\int_{Q_t}G_{d}(t-\tau;x
-\xi)e^{k\tau}(kv+u^{n}|\nabla
v|^{p})d\xi d\tau
$$
or
\begin{equation}\label{inte17}
v=e^{-kt}v^{0}+\int_{Q_t}e^{-k(t-\tau)}G_{d}(t-\tau;x-\xi)(kv+u^{n}|\nabla
v|^{p})d\xi d\tau,
\end{equation}
where $v^{0}(t,x)$ is the solution of the homogeneous problem
$$
L_{d}v^{0}=0,\;\;\;\;v^{0}(0,x)=v_{0}(x).
$$
 From (\ref{inte17}) we have
\begin{equation}\label{inte18}
 \nabla v=e^{-kt}\nabla
v^{0}+\int_{Q_t}e^{-k(t-\tau)}\nabla
_{x}G_{d}(t-\tau;x-\xi)(kv+u^{n}|\nabla v|^{p})d\xi d\tau.
\end{equation}
Now we set $\nu_{1}=\sup|\nabla v|$ and
$\nu_{1}^{0}=\sup|\nabla v^{0}|$, in $Q_{t}$.
 From (\ref{inte17}), and using $v\leq C_{2}$, we have
$$
\nu_{1}=\nu_{1}^{0}+(kC_{2}+C_{1}^{n}\nu_{1}^{p})
\int_{0}^{t}e^{-k(t-\tau)}\Big(\int_{\mathbb{R}^{N}}
|\nabla_{x}G_{d}(t-\tau;x-\xi)|d\xi \Big)d\tau .
$$
We also have
$$
\int_{\mathbb{R}^{N}}|\nabla_{x}G_{d}(t-\tau;x-\xi)|d\xi=
\int_{\mathbb{R}^{N}}\frac{|x-\xi|}{2d(t-\tau)}|G_{d}(t-\tau,;x-\xi)|d\xi
$$
which is transformed by the substitution $\rho=2\sqrt{d(t-\tau)\nu}$
into
$$
\int_{\mathbb{R}^{N}}|\nabla_{x}G_{d}|d\rho=\frac{w_{N}}{\pi^{N/2}}
\int_{0}^{\infty}e^{-\nu^{2}}d\nu=\frac{\chi}{\sqrt{d(t-\tau)}}
$$
where
$\chi=\frac{w_{N}}{2\pi^{N/2}}\Gamma(\frac{N+1}{2})
=\frac{\Gamma(\frac{N+1}{2})}{\Gamma(\frac{N}{2})}$.
It follows  that
\begin{equation}\label{inte18b}
\nu_{1}=\nu_{1}^{0}+(kC_{2}+C_{1}^{n}\nu_{1}^{p})
\frac{\chi}{\sqrt{d}}\int_{0}^{t}e^{-k(t-\tau)}\frac{d\tau}{\sqrt{t-\tau}}.
\end{equation}
Recall that
$$
\int_{0}^{t}e^{-k(t-\tau)}\frac{d\tau}{\sqrt{t-\tau}}
=\frac{2}{\sqrt{k}}\int_{0}^{t}e^{-z^{2}}dz<\sqrt{\frac{\pi}{k}}.
$$
If we set $s=\sqrt{k}$  in (\ref{inte18b}) then we have
\begin{equation}\label{inte19}
\nu_{1}\leq\nu_{1}^{0}+\Big(sC_{2}+\frac{C_{1}^{n}}{s}\nu_{1}^{p}\Big)
\chi\sqrt{\frac{\pi}{d}}.
\end{equation}
Now we minimize the right hand side of (\ref{inte19}) with respect to
$s$ to obtain
\begin{equation}\label{inte20}
\nu_{1}\leq\nu_{1}^{0}+\frac{2\chi\sqrt{\pi}}{d}
\Big(C_{2}C_{1}^{n}\nu_{1}^{p}\Big)^{1/p}.
\end{equation}
Note that $\nu_{1}^{0}=C_{2}$.

We have two cases:
Case (i) $1\leq p<2$. In this case \eqref{inte20} implies
\begin{equation}\label{inte21}
|\nabla v|\leq\nu_{1}\leq\overline{\nu}(p)=D,\quad \text{in } Q_{t},
\end{equation}
where $D$ is a positive constant.

Case (ii) $p=2$. In this case \eqref{inte20} holds under the
additional condition
\begin{equation}\label{inte22}
 C_{2}C_{1}^{n}\leq\frac{d}{4\pi\chi}.
\end{equation}
Similarly we obtain from \eqref{inte1}$_1$,
\begin{equation}\label{inte23}
 U_{1}:=\sup_{Q_{T}}|\nabla u|\leq
C_{1}+C_{1}\frac{2\sqrt{\pi}\chi}{\sqrt{d}}\nu_{1}^{p/2}\leq Constant.
\end{equation}
The estimates \eqref{inte20} and (\ref{inte23}) are independent of
$t$, hence $T_{\rm max}=+\infty$.

Finally, we have the main result.

\begin{theorem}\label{thm2.2}
Let $  p=2$ and $(u_{0},v_{0})$ be bounded such that
\eqref{inte22} holds, then system \eqref{inte1}-\eqref{condini} admits
a global solution.
\end{theorem}

\subsection{The Neumann Problem}
In this section, we are concerned with the Neumann problem
\begin{equation} \label{inte24}
\begin{gathered}
 u_t- \Delta u = -u^n |\nabla v |^2 \\
 v_t-d \Delta v  = u^n |\nabla v|^2 \\
\end{gathered}
\end{equation}
where $\Omega$ be a bounded domain in $\mathbb{R}^{N}$, with the
homogeneous Neumann boundary condition
\begin{equation}\label{inte25}
 \frac{\partial u}{\partial\nu}=\frac{\partial
v}{\partial\nu}=0, \quad  \text{on }  \mathbb{R}^+ \times
\partial \Omega
\end{equation}
subject to the initial conditions
\begin{equation}\label{inte26}
 u( 0;x)=u_0(x);  \quad  v(0;x)=v_0(x)\quad \text{in }\Omega.
\end{equation}
The initial nonnegative functions $u_{0},\;v_{0}$ are assumed to
belong to the Holder space $C^{2,\alpha}(\Omega)$.


\subsubsection*{Uniform bounds for $u$ and $v$}
In this section a priori estimates on
$\|u\|_{\infty}$ and $\|v\|_{\infty}$ are presented.

\begin{lemma}\label{lem2.3}
  For each $ 0<t<T_{\rm max}$ we have
\begin{equation}\label{hyp1}
 0\leq u(t,x)\leq M,\quad 0\leq v(t,x)\leq M,
\end{equation}
for any $x\in\Omega$.
\end{lemma}


\begin{proof}
 Since $u_{0}(x)\geq 0$ and $f(0,v)=0$,  we first obtain $u\geq 0$
 and then $v\geq 0$ as  $v_{0}(x)\geq 0$.
Using the maximum principle,  we conclude that
$$
0\leq u(t,x)\leq M, \quad \text{on }  Q_{T}
$$
where
$$
M\geq M_{1} : = \max_{x\in\Omega}u_{0}(x) .
$$
Using  $\omega=e^{\lambda v}-1$, with  $ d\lambda \geq M_{1}^{n}$,
 from (\ref{inte24}), we obtain
\begin{gather*}
\omega_{t}-d\Delta\omega=\lambda|\nabla v|^{2}(u^{n}-d\lambda)
e^{\lambda v}, \quad \text{on }  Q_{T}\\
\frac{\partial u}{\partial v}=0 \quad \text{on }   \partial S_{T}.
\end{gather*}
Consequently as $d\lambda > \max_{\Omega}u^{n}$, we deduce from the
maximum principle that
$$
0\leq\omega(t,x)\leq \exp(\lambda|v_{0}|_{\infty})-1.
$$
Hence
$$
v(x,t)\leq\frac{1}{\lambda}\ln(|\omega|_{\infty}+1)\leq Constant<\infty.
$$
\end{proof}

\subsubsection*{Uniform bounds for $|\nabla v|$ and $|\nabla u| $}
To obtain uniform a priori estimates for $|\nabla v|$, we make use
of some techniques already used by Tomi \cite{Sou} and von Wahl
\cite{Sui}


\begin{lemma}\label{lem2.4}
  Let $ (u,v)$ be a solution to \eqref{inte20} -\eqref{inte22} in its
maximal interval of existence $[0,T_{\rm max}[$.
Then there exist a constant $C$ such that
$$
\|u\|_{L^{\infty}([0,T[,W^{2,q}(\Omega))}\leq C \quad \text{and} \quad
 \|v\|_{L^{\infty}([0,T[,W^{2,q}(\Omega))}\leq C.
$$
\end{lemma}

\begin{proof}
 Let us introduce the function
$$
f_{\sigma,\epsilon}(t,x,u,\nabla v)=\sigma u^{n}(t,x)
\frac{\epsilon+|\nabla v|^{2}}{1+\epsilon|\nabla v|^{2}}.
$$
It is clear that $|f_{\sigma,\epsilon}(t,x,u,\nabla v)|\leq
C(1+|\nabla v|^{2})$ and a global solution $v_{\sigma, \epsilon}$
differentiable in $\sigma$ for the equation
$$
v_{t}-d\Delta v=f_{\sigma,\epsilon}(t,x,u,\nabla v)
$$
exists.
Moreover,
$v_{\sigma, \epsilon}\to v$ as $\sigma\to 1$
and $\epsilon\to 0$,
uniformly on every compact of $[0,T_{\rm max}[$.

The function $\omega_{\sigma} := \frac{\partial
v_{\sigma,\epsilon}}{\partial \sigma}$ satisfies
\begin{equation}\label{inte27}
\partial_{t}\omega_{\sigma}-d\Delta\omega_{\sigma}=u^{n}(t,x)\frac{\epsilon+|\nabla v_{\sigma}|^{2}}{1+\epsilon|\nabla
v\sigma|^{2}}-2\sigma u^{n}\frac{(\epsilon^{2}-1)\nabla
v_{\sigma}.\nabla \omega_{\sigma}}{(1+\epsilon|\nabla
v_{\sigma}|^{2})^{2}}.
\end{equation}
Hereafter, we derive uniform estimates in $\sigma$ and $\epsilon$.
Using Solonnikov's estimates for parabolic equation \cite{Fig} we
have
$$
\|\omega_{\sigma}\|_{L^{\infty}([0,T(u_{0},v_{0})[,W^{2,p}(\Omega))}
\leq C[\|\nabla v_{\sigma}\|^{2}_{L^{p}(\Omega)}
+\|\nabla v_{\sigma}.\nabla \omega_{\sigma}\|^{2}_{L^{p}(\Omega)}].
$$
The Gagliardo-Nirenberg inequality \cite{Fig} in the in the form
$$
\|u\|_{W^{1,2p}(\Omega)}\leq C\|u\|^{1/2}_{L^{\infty}(\Omega)}
C\|u\|^{1/2}_{W^{2,p}(\Omega)}
$$
and the $\delta$-Young inequality (where $\delta >0$)
$$
\alpha\beta\leq \frac{1}{2}(\delta\alpha^{2}+\frac{\beta^{2}}{\delta}),
$$
 allows one to obtain the estimate
$$
\|\omega_{\sigma}\|_{L^{\infty}([0,T(u_{0},v_{0})[,W^{2,p}
(\Omega))}\leq C(1+\|\omega_{\sigma}\|_{W^{2,p}(\Omega)}).
$$
But $\omega_{\sigma}=\frac{\partial v_{\sigma}}{\partial \sigma}$,
hence by Gronwall's inequality we have
$$
\|v_{\sigma}\|_{L^{\infty}([0,T[,W^{2,p}(\Omega))}\leq C e^{C\sigma}.
$$
Letting $\sigma\to 1$ and $\epsilon\to 0$, we obtain
$$
\|v\|_{L^{\infty}([0,T[,W^{2,p}(\Omega))}\leq C.
$$
On the other hand, the Sobolev injection theorem allows to assert
that $u \in C^{1,\alpha}(\Omega)$. Hence in particular
$|\nabla u|\in C^{0,\alpha}(\Omega)$. Since $|\nabla v|$ is
uniformly bounded, it is easy then to bound $|\nabla u|$ in
$L^{\infty}(\Omega)$.
As a consequence, one can affirm that the solution $(u,v)$ to
problem (\ref{inte24}) -(\ref{inte26}) is global; that is
$T_{\rm max}=\infty$.
\end{proof}

\subsection{Large-time behavior}

In this section, the large time behavior of the global solutions
to (\ref{inte24})-(\ref{inte26}) is briefly presented.

\begin{theorem}\label{thm2.5}
Let $ (u_{0},v_{0})\in C^{2,\epsilon}(\Omega)\times C^{2,\epsilon}(\Omega)$
for some $0<\epsilon<1$. The system (\ref{inte24})-(\ref{inte26})
has a global classical solution. Moreover, as $t\to \infty$,
$u\to k_{1}$ and $v\to k_{2}$ uniformly in $x$,
and
$$
k_{1}+k_{2}=\frac{1}{|\Omega|}\int_{\Omega}[u_{0}(x)+v_{0}(x)]dx.
$$
\end{theorem}

\begin{proof}
 The proof of the first part of the Theorem is
presented above. Concerning the large time behavior, observe first
that for any $t\geq0$,
$$
\int_{\Omega}[u(t,x)+v(t,x)]dx=\int_{\Omega}[u_{0}(x)+v_{0}(x)]dx.
$$
Then, the function $t\to \int_{\Omega}u(x)dx$ is
bounded; as it is decreasing, we have
$$
\int_{\Omega}u(x)dx\to  k_1 \quad \text{as } t\to \infty;
$$
the function $t\to \int_{\Omega}v(x)dx$ is
increasing and bounded, hence admits a finite limit $k_{2}$ as
$t\to \infty$.
As $\bigcup_{t\geq0}\{(u(t),v(t))\}$ is relatively compact in
$C(\overline{\Omega})\times C(\overline{\Omega})$,
$$
u(\tau_{n})\to \widetilde{u} ,\quad v(\tau_{n})\to \widetilde{v}
\quad \text{in } C(\overline{\Omega}),
$$
through a sequence $\tau_{n}\to\infty$.
It is not difficult to show that in fact
$(\widetilde{u},\widetilde{v})$ is the stationary solution
to (\ref{inte24})-(\ref{inte26}) (see \cite{CM}).

As the stationary solution $(u_{s},v_{s})$ to
(\ref{inte24})-(\ref{inte26}) satisfies
\begin{gather*}
- \Delta u_{s}=-u_{s}^{n}|\nabla v_{s}|^{2}, \quad \text{in } \Omega,\\
-d \Delta v_{s}=u_{s}^{n}|\nabla v_{s}|^{2}, \quad \text{in } \Omega,
\frac{\partial u_{s}}{\partial\nu}=\frac{\partial v_{s}}{\partial\nu}=0,
\quad \text{on } \partial\Omega ,
\end{gather*}
we have
$$
- \int_{\Omega}\Delta u_{s}.u_{s}dx
=-\int_{\Omega}u^{n+1}_{s}|\nabla v_{s}|^{2}dx
$$
which in the light of the Green formula can be written
$$
\int_{\Omega}|\nabla u_{s}|^{2}dx
=-\int_{\Omega}u^{n+1}_{s}|\nabla v_{s}|^{2}dx
$$
hence
$|\nabla u_{s}|=|\nabla v_{s}|=0$ implies $u_{s}=k_{1}$ and $v_{s}=k_{2}$.
\end{proof}

\subsection*{Remarks}
(1) It is very interesting to address the question
of existence global solutions of the system
(\ref{inte24})-(\ref{inte26}) with a genuine nonlinearity of the
form $u^{n}|\nabla v|^{p}$ with $p\geq2$.

(2) It is possible to extend the results presented  here
for systems with nonlinear boundary conditions satisfying reasonable
growth restrictions.

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\end{document}