\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 $$\label{inte1} \begin{gathered} u_t -\Delta u = -u^n |\nabla v |^p=:-f(u,v), \\ v_t -d \Delta v = u^n |\nabla v|^p, \end{gathered}$$ posed on $\mathbb{R}^+ \times \Omega$ with initial data $$\label{condini} u( 0;x)=u_0(x), \quad v(0;x)=v_0(x) \quad \text{in }\Omega$$ and boundary conditions (in the case $\Omega$ is a bounded domain in $\mathbb{R}^{n}$) $$\label{bound} \frac{\partial u}{\partial \eta}=\frac{\partial v}{\partial \eta}=0, \quad \text{on } \mathbb{R}^+ \times \partial \Omega.$$ 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 % $$\label{inte10} \begin{gathered} L_{\lambda}\omega := \omega_{t}-\lambda\Delta\omega= b\nabla\omega, \quad t>0,\; x\in \mathbb{R}^N \\ \omega(0,x) = \omega_{0}(x)\in L^{\infty}, \end{gathered}$$ 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, $$\label{hyp1b} L_{\lambda}\omega\leq 0,\quad (\geq)\quad (0,\infty)\times \mathbb{R}^N$$ 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 $x\geq (\frac{c}{\lambda d})^{1/(2-p)}$, $$\label{inte14} \phi(x)\leq 0$$ and hence $\omega \leq M$ where $$\label{inte15} M=C \Big(\frac{pC}{2d\lambda}\Big)^{p/2-p}(\frac{2-p}{2}).$$ 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 $$\label{inte16} L_{d}v+ kv=kv+u^{n}|\nabla v|^{p}$$ 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 $$\label{inte17} v=e^{-kt}v^{0}+\int_{Q_t}e^{-k(t-\tau)}G_{d}(t-\tau;x-\xi)(kv+u^{n}|\nabla v|^{p})d\xi d\tau,$$ 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 $$\label{inte18} \nabla v=e^{-kt}\nabla v^{0}+\int_{Q_t}e^{-k(t-\tau)}\nabla _{x}G_{d}(t-\tau;x-\xi)(kv+u^{n}|\nabla v|^{p})d\xi d\tau.$$ 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 $$\label{inte18b} \nu_{1}=\nu_{1}^{0}+(kC_{2}+C_{1}^{n}\nu_{1}^{p}) \frac{\chi}{\sqrt{d}}\int_{0}^{t}e^{-k(t-\tau)}\frac{d\tau}{\sqrt{t-\tau}}.$$ 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 $$\label{inte19} \nu_{1}\leq\nu_{1}^{0}+\Big(sC_{2}+\frac{C_{1}^{n}}{s}\nu_{1}^{p}\Big) \chi\sqrt{\frac{\pi}{d}}.$$ Now we minimize the right hand side of (\ref{inte19}) with respect to $s$ to obtain $$\label{inte20} \nu_{1}\leq\nu_{1}^{0}+\frac{2\chi\sqrt{\pi}}{d} \Big(C_{2}C_{1}^{n}\nu_{1}^{p}\Big)^{1/p}.$$ Note that $\nu_{1}^{0}=C_{2}$. We have two cases: Case (i) $1\leq p<2$. In this case \eqref{inte20} implies $$\label{inte21} |\nabla v|\leq\nu_{1}\leq\overline{\nu}(p)=D,\quad \text{in } Q_{t},$$ where $D$ is a positive constant. Case (ii) $p=2$. In this case \eqref{inte20} holds under the additional condition $$\label{inte22} C_{2}C_{1}^{n}\leq\frac{d}{4\pi\chi}.$$ Similarly we obtain from \eqref{inte1}$_1$, $$\label{inte23} U_{1}:=\sup_{Q_{T}}|\nabla u|\leq C_{1}+C_{1}\frac{2\sqrt{\pi}\chi}{\sqrt{d}}\nu_{1}^{p/2}\leq Constant.$$ 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 $$\label{inte24} \begin{gathered} u_t- \Delta u = -u^n |\nabla v |^2 \\ v_t-d \Delta v = u^n |\nabla v|^2 \\ \end{gathered}$$ where $\Omega$ be a bounded domain in $\mathbb{R}^{N}$, with the homogeneous Neumann boundary condition $$\label{inte25} \frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial\nu}=0, \quad \text{on } \mathbb{R}^+ \times \partial \Omega$$ subject to the initial conditions $$\label{inte26} u( 0;x)=u_0(x); \quad v(0;x)=v_0(x)\quad \text{in }\Omega.$$ 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 $0 \max_{\Omega}u^{n}$, 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 $$\label{inte27} \partial_{t}\omega_{\sigma}-d\Delta\omega_{\sigma}=u^{n}(t,x)\frac{\epsilon+|\nabla v_{\sigma}|^{2}}{1+\epsilon|\nabla v\sigma|^{2}}-2\sigma u^{n}\frac{(\epsilon^{2}-1)\nabla v_{\sigma}.\nabla \omega_{\sigma}}{(1+\epsilon|\nabla v_{\sigma}|^{2})^{2}}.$$ 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 $p\geq2$. 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