\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 141, pp. 1--9.\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 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/141\hfil Existence of global solutions] {Existence of global solutions for a system of reaction-diffusion equations having a triangular matrix} \author[E. H. Daddiouaissa\hfil EJDE-2008/141\hfilneg] {El Hachemi Daddiouaissa} \address{El Hachemi Daddiouaissa \newline Department of Mathematics University Kasdi Merbah, UKM Ouargla 30000, Algeria} \email{dmhbsdj@gmail.com} \thanks{Submitted August 19, 2007. Published October 16, 2008.} \subjclass[2000]{35K57, 35K45} \keywords{Reaction-diffusion systems; Lyapunov functional; global solution} \begin{abstract} We consider the system of reaction-diffusion equations \begin{gather*} u_{t}-a\Delta u=\beta-f(u,v)-\alpha u,\\ v_{t}-c\Delta u-d\Delta v=g(u,v)-\sigma v. \end{gather*} Our aim is to establish the existence of global classical solutions using the method used by Melkemi, Mokrane, and Youkana \cite{melk}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} In this manuscript, we consider a reaction-diffusion system that arises in the study of physical, chemistry, and various biological processes including population dynamics \cite{cast,cuss,fitz,hama,kira,webb,zeid}. The system of equations is \begin{gather} \label{e1.1} \frac{\partial u}{\partial t}-a\Delta u=\beta-f(u,v)-\alpha u \quad (x,t)\in\Omega \times R_{+}\\ \frac{\partial v}{\partial t}-c\Delta u-d\Delta v=g(u,v)-\sigma v\quad (x,t)\in\Omega \times R_{+}, \label{e1.2} \end{gather} with the boundary conditions $$\label{e1.3} \frac{\partial u}{\partial \eta}=\frac{\partial v}{\partial \eta}=0\quad\text{on } \partial \Omega \times R_{+}.$$ and the initial data $$\label{e1.4} 0\leq u(0,x)=u_{0}(x) ;\quad 0\leq v(0,x)=v_{0}(x)\quad\text{in } \Omega.$$ where $\Omega$ is a smooth open bounded domain in $R^{n}$, with boundary $\partial \Omega$ of class $C^{1}$ and $\eta$ is the outer normal to $\partial \Omega$. The constants of diffusion $a,c,d$ are such that $a>0$, $d>0$, $c>0$ and $a>d$, $c^{2}<4ad$ which is the parabolic condition, and $\alpha,\sigma$ are positive constants, $\beta\geq 0$, and $f,g$ are nonnegative functions of class $C(R_{+}\times R_{+})$, such that \begin{itemize} \item[(H1)] For all $\tau\geq 0$, $f(0,\tau)=0$; \item[(H2)] For all $\xi\geq 0$ and all $\tau\geq 0$, $0\leq f(\xi,\tau)\leq \varphi(\xi)(\mu+\tau)^{r}$; \item[(H3)] For all $\xi\geq 0$ and all $\tau \geq 0$, $g(\xi,\tau)\leq \psi(\tau)f(\xi,\tau)+\phi(\tau)$, \end{itemize} where $r,\mu$ are positive constants such that $r\geq 1$, $\mu\geq 1$, $\varphi,\psi$ and $\phi$ are nonnegative functions of class $C(R^{+})$, such that \begin{gather} \label{e1.5} \lim_{\tau\to +\infty}\frac{\psi(\tau)}{\tau} =\lim_{\tau\to +\infty}\frac{\phi(\tau)}{\tau}=0.\\ \phi(0)> \beta \frac{c}{a-d}. \label{e1.6} \end{gather} In addition we suppose that $$\label{e1.7} g(\xi,\frac{c}{a-d}\xi)+\frac{c}{a-d}f(\xi,\frac{c}{a-d}\xi) \geq\frac{c}{a-d}[(\sigma-\alpha)\xi+\beta],\quad \forall \xi\geq0.$$ Melkemi et al \cite{melk} established the existence of global solutions, (eventually uniformly bounded in time) using a novel approach that involved the use of a Lyapunov function for system \eqref{e1.1}--\eqref{e1.4} when $c=0$. Here, we apply the same method to the study of system \eqref{e1.1}--\eqref{e1.4} when $c>0$; that is, for a model that involves a triangular matrix. \section{Existence of local and positive solutions} First we convert system \eqref{e1.1}--\eqref{e1.4} into an abstract first order system in $X=C(\overline{\Omega})\times C(\overline{\Omega})$ of the form \begin{gather} \label{e2.1} U'(t)=AU(t)+F(U(t)), \quad t>0,\\ U(0)=U_{0}\in X, \label{e2.2} \end{gather} where \begin{gather*} AU(t)=( a\Delta u, c\Delta u+d\Delta v), \\ F(U(t))=(\beta-f(u,v)-\alpha u,g(u,v)-\sigma v ). \end{gather*} Since $F$ is locally Lipschitz in $U$ and $X$, for every initial data $U_{0}\in X$, system \eqref{e2.1}--\eqref{e2.2} admits a unique strong local solution on $]0,T^{*}[$, where $T^{*}$ is the eventual blowing-up time, (see Kirane \cite{kira}, Friedman \cite{frie}, Henry \cite{henr}, Pazy \cite{pazy}). Multiplying \eqref{e1.1} by $\frac{c}{a-d}$, and subtracting the resulting equation from \eqref{e1.2} leads to the system \begin{gather} \label{e2.3} \frac{\partial u}{\partial t}-a\Delta u=\Lambda(u,z), \quad (x,t)\in\Omega \times R_{+}\\ \frac{\partial z}{\partial t}-d\Delta z=\Upsilon(u,z),\quad (x,t)\in\Omega \times R_{+}. \label{e2.4} \end{gather} where \begin{gather*} \Lambda(u,z)=\beta-f(u,v)-\alpha u\\ \Upsilon(u,z)=g(u,v)+\frac{c}{a-d}f(u,v) +\frac{c}{a-d}(\alpha u-\beta)-\sigma v, \\ z=v-\frac{c}{a-d}u, \end{gather*} with the boundary conditions $$\label{e2.5} \frac{\partial u}{\partial \eta}=\frac{\partial z}{\partial \eta}=0,\quad \text{on } \partial \Omega \times R_{+}$$ and initial data \begin{gather} \label{e2.6} u(0,x)=u_{0}(x)\quad \text{in } \Omega,\\ z(0,x)=z_{0}(x)=v_{0}(x)-\frac{c}{a-d} u_{0}(x)\quad \text{in }\Omega. \label{e2.7} \end{gather} If we assume \eqref{e1.7} and (H1) then a simple application of a comparison theorem to system \eqref{e2.3}-\eqref{e2.4} implies (see \cite{kira}) that for positive initial data $u_{0}\geq0$ and $z_{0}\geq o$ we have that $u(t,x)\geq0, \quad v(t,x)\geq\frac{c}{a-d}u(t,x)\quad \forall (x,t)\in \Omega \times ]0,T^{*}[.$ \section{Existence of global solutions} Before we establish the existence of a global solution, we introduce some notation. Here, we let \begin{equation*} \| u\|_{p}^{p}=\frac{1}{|\Omega|}\int_{\Omega} |u(x)|^{p} dx \text{ and } \| u\|_{\infty}= \max_{x\in\Omega}|u(x)|. \end{equation*} denote the usual norms in spaces $L^{p}(\Omega)$, $L^{\infty}(\Omega)$ and $C(\overline{\Omega})$. Applying the comparison principle we get that $$\label{e3.1} u(t,x)\leq \max(\| u_{0}\|_{\infty},\frac{\beta}{\alpha})=K.$$ To establish the uniform boundedness of $v$, it is sufficient to show the uniform boundedness of $z$. This task is carried out using a result found in Henry \cite[pp. 35-62]{henr}, from which it sufficient to derive a uniform estimate for $\| \Upsilon(u,z) \|_{p}$; that is, finding a constant $C$ such that $$\label{e3.2} \| \Upsilon(u,z) \|_{p}\leq C.$$ Here C is a nonnegative constant independent of t, for some $p>n/2$. The key is to establish the uniform boundedness of $\| v\|_{p}$ on $]0,T^{*}[$, taking into account assumptions (H2) and (H3). When $p\geq 2$, we put $$\label{e3.3} \begin{gathered} \Gamma(p)=\frac{p\Gamma+1}{p-1}\,,\quad \Gamma=(a-d)^{2}[1+\frac{1}{4ad}]\,,\\ l=\frac{2\beta \rho}{\Gamma(p)\sigma}\,,\quad \omega= [\frac{S^{2}}{4adR^{2}}+\frac{p}{R^{2}}+\mu](p-1) \end{gathered}$$ where $\rho >0$, $$S=\frac{\rho}{\Gamma(p)l}, \quad R=\frac{(a-d)\rho}{\Gamma(p)(l+\rho K)}.$$ Using these definitions and notation we can state the following key proposition \begin{proposition} \label{prop3.1} Assume that $p\geq 2$ and let $$\label{e3.4} G_{N}(t)= N \int_{\Omega}u dx+\int_{\Omega}(v+\omega)^{p} \exp(-\frac{1}{\Gamma(p)}\ln(\Gamma(p)[l+\rho(K-u)]))dx,$$ where $(u,v)$ is the solution of \eqref{e1.1}--\eqref{e1.4} on $]0,T^{*}[$. Then under the assumptions (H3) and \eqref{e1.5} there exist two positive constants $N$ and $s$ such that $$\label{e3.5} \frac{dG_{N}}{dt} \leq -(p-1)\sigma G_{N}+s.$$ \end{proposition} The proof of the above proposition requires some lemmas. \begin{lemma} \label{lem3.1} If $(u,v)$ be a solution of $\eqref{e1.1}-\eqref{e1.4}$ then $$\label{e3.6} \int_{\Omega}f(u,v)dx\leq\beta|\Omega|-\frac{d}{dt}\int_{\Omega}u(t,x)dx.$$ \end{lemma} \begin{proof} We integrate both sides of \eqref{e1.1}, $$f(u,v)=\beta -\alpha u -\frac{d}{dt}u(t,x)$$ satisfied by $u$, which is positive and then we find \eqref{e3.6}. \end{proof} \begin{lemma} \label{lem3.2} Assume that $p\geq2$, then under the assumptions {\rm (H1)--(H3)}, and \eqref{e1.5} there exists $N_1$, such that $$\label{e3.7} [p (g(\xi,\tau)-\phi(\tau))(\tau+\omega)^{p-1} - \theta f(\xi,\tau) (\tau+\omega)^{p}]\leq N_1f(\xi,\tau)$$ for all $0\leq \xi \leq K$ and $\tau \geq 0$, $\theta > 0$. \end{lemma} \begin{proof} from the assumption (H3) and \eqref{e1.5}, we conclude that there exists $\tau_{0}>0$, such that for all $0\leq\xi\leq K,\tau\geq\tau_{0}$, one finds $$[p\frac{\psi(\tau)}{\tau+\omega}-\theta ](\tau+\omega)^{p}f(\xi,\tau)\leq0$$ now if $\tau$ is in the compact interval $[0,\tau_{0}]$, then the continuous function $$\chi(\xi,\tau)=p\psi(\tau)(\tau+\omega)^{p-1}-\theta (\tau+\omega)^{p}$$ is bounded. \end{proof} \begin{lemma} \label{lem3.3} For all $\tau\geq 0$ and $\omega \geq 1$, we have $$\label{e3.8} \frac{\beta \rho}{\Gamma(p)l} (\tau+\omega)^{p} - \sigma p (\tau+\omega)^{p-1}\tau+p\phi(\tau)(\tau+\omega)^{p-1}\leq -(p-1)\sigma (\tau+\omega)^{p}+ M_1$$ where $M_1$ is positive constant. \end{lemma} \begin{proof} Let us put \begin{align*} \Pi=&\frac{\beta \rho}{\Gamma(p)l} (\tau+\omega)^{p} - \sigma p (\tau+\omega)^{p-1}\tau \\ \leq &[2\frac{\beta \rho}{\Gamma(p)l}- \sigma p] \tau(\tau+\omega)^{p-1}+[2\frac{\beta \rho}{\Gamma(p)l} \frac{\omega}{\tau+\omega}-\frac{\beta \rho}{\Gamma(p)l}] (\tau+\omega)^{p} \end{align*} since $l=\frac{2\beta \rho}{\Gamma(p)\sigma}$, then \begin{equation*} \Pi+p\phi(\tau)(\tau+\omega)^{p-1}\leq -(p-1)\sigma (\tau+\omega)^{p} +[\frac{p (\sigma \omega+\phi(\tau))}{\sigma(\tau+\omega)} -\frac{1}{2}] \sigma (\tau+\omega)^{p} \end{equation*} using \eqref{e1.5} and Lemma \ref{lem3.2}, we can establish Proposition \ref{prop3.1}. \end{proof} \begin{proof}[Proof of Proposition \ref{prop3.1}] Let $$h(u)=-\frac{1}{\Gamma(p)}\ln(\Gamma(p)[l+\rho(K-u)])$$ so that $$G_{N}(t)= N \int_{\Omega}u dx+ L(t)$$ where $$L(t)=\int_{\Omega}e^{h(u)} (v+\omega)^{p}dx.$$ Differentiating $L$ with respect to $t$ and using Greens formula, we obtain that $$L'(t)=I+J$$ where \begin{align*} I=&- \int_{\Omega}[a(h''(u)+h'^{2}(u)) (v+\omega)^{p}+pc h'(u) (v+\omega)^{p-1}] e^{h(u)}\nabla u^{2} dx\\ &-\int_{\Omega} [p(a+d)h'(u) (v+\omega)^{p-1}+p(p-1) c (v+\omega)^{p-2}] e^{h(u)} \nabla u\nabla v dx\\ &-\int_{\Omega} p(p-1)d (v+\omega)^{p-2} e^{h(u)} \nabla v^{2} dx \end{align*} and \begin{align*} J=&\int_{\Omega}\beta h'(u)(v+\omega)^{p} e^{h(u)} dx\\ &+\int_{\Omega} [p g(u,v) (v+\omega)^{p-1}-h'(u) f(u,v) (v+\omega)^{p}] e^{h(u)}dx\\ &-\int_{\Omega} \alpha h'(u) u (v+\omega)^{p} e^{h(u)}dx - \int_{\Omega} \sigma p (v+\omega)^{p-1}v e^{h(u)}dx. \end{align*} We see that $I$ involves a quadratic form with respect to $\nabla u,\nabla v$. \begin{align*} D=&[a(h''(u)+h'^{2}(u)) (v+\omega)^{p}+pch'(u)(v+\omega)^{p-1}] \nabla u^{2} \\ &+[p(a+d)h'(u)(v+\omega)^{p-1}+p(p-1)c(v+\omega)^{p-2}] \nabla u\nabla v \\ &+p(p-1)d(v+\omega)^{p-2} \nabla v^{2}\\ \end{align*} which is nonnegative if \begin{align*} \delta=&[p(a+d)h'(u)(v+\omega)^{p-1}+p(p-1)(v+\omega)^{p-2}]^{2}\\ &-4p(p-1)d(v+\omega)^{p-2}[a(h''(u)+h'^{2}(u))(v+\omega)^{p}\\ &+pch'(u)(v+\omega)^{p-1}]\leq 0. \end{align*} Indeed, \begin{align*} \delta=&p^{2}(a+d)^{2}h'^{2}(u)(v+\omega)^{2p-2}+2(a+d)cp^{2}(p-1)h'(u)) (v+\omega)^{2p-3}\\ &+c^{2}p^{2}(p-1)^{2}(v+\omega)^{2p-4}-4p(p-1)ad(h''(u)+h'^{2}(u)) (v+\omega)^{2p-2}\\ &-4cdp^{2}(p-1)h'(u)(v+\omega)^{2p-3} \end{align*} and from \eqref{e3.3} we have that $v+\omega \geq 1$. It follows that \begin{align*} \delta \leq &[p^{2}(a+d)^{2}h'^{2}(u)-4p(p-1)ad (h''(u)+h'^{2}(u))] (v+\omega)^{2p-2}\\ &+[2(a+d)c p^{2}(p-1)h'(u) +c^{2} p^{2}(p-1)^{2}\\ &-4cd p^{2}(p-1) h'(u)] (v+\omega)^{2p-3}. \end{align*} Let us try $$T=p^{2}(a+d)^{2}h'^{2}(u)-4p(p-1)ad (h''(u)+h'^{2}(u))$$ the choice of $h(u)$ implies $$\label{e3.9} \frac{\rho}{\Gamma(p)[l+\rho K]}\leq h'(u)\leq\frac{\rho}{\Gamma(p)l},$$ and, consequently, $$T=-\frac{\rho^{2}4ad(a-d)^{2}p^{2}}{[\Gamma(p)(l+\rho (K-u))]^{2}} \leq -4adp^{2} R^{2}\leq 0.$$ In addition \begin{align*} \delta\leq &[2(a+d)c p^{2}(p-1)h'(u)+c^{2} p^{2}(p-1)^{2} -4cd p^{2}(p-1) h'(u)] (v+\omega)^{2p-3}\\ &+T (v+\omega)(v+\omega)^{2p-3}\\ \leq &[(p-1)c^{2}+2(a-d)c\frac{\rho}{\Gamma(p)l} -\frac{4ad R^{2}\omega}{(p-1)}] p^{2}(p-1)(v+\omega)^{2p-3} +T v(v+\omega)^{2p-3}. \end{align*} If we replace $w$ by its value \eqref{e3.3} and use the parabolic condition $c^{2}-4ad < 0$, we deduce that \begin{align*} \delta \leq&[p(c^{2}-4ad)-(c-S)^{2}-4ad R^{2}\mu] p^{2}(p-1)(v+\omega)^{2p-3}\\ &+T v(v+\omega)^{2p-3}\leq 0, \end{align*} that is, $I \leq 0$. We can control the second term by observing that \begin{align*} J\leq&\int_{\Omega}[\beta h'(u) (v+\omega)^{p} - \sigma p (v+\omega)^{p-1}v+p\phi(v)(v+\omega)^{p-1}]e^{h(u)} dx\\ &+\int_{\Omega} [p (g(u,v)-\phi(v)) (v+\omega)^{p-1}-h'(u) f(u,v) (v+\omega)^{p}] e^{h(u)}dx. \end{align*} Using \eqref{e3.9}, \begin{align*} J\leq&\int_{\Omega}[\frac{\beta \rho}{\Gamma(p)l} (v+\omega)^{p} - \sigma p (v+\omega)^{p-1}v+p\phi(v)(v+\omega)^{p-1}]e^{h(u)} dx\\ &+\int_{\Omega} [p (g(u,v)-\phi(v)) (v+\omega)^{p-1} - \frac{\rho}{\Gamma(p)[l+\rho k]} f(u,v) (v+\omega)^{p}] e^{h(u)}dx. \end{align*} Applying Lemmas \ref{lem3.2} and \ref{lem3.3}, one finds that $J\leq-(p-1)\sigma\int_{\Omega}(v+\omega)^{p}e^{h(u)} dx +M_1\int_{\Omega}e^{h(u)} dx+N_1\int_{\Omega}f(u,v)e^{h(u)} dx.$ In addition we see that $$h(u)\leq -\frac{1}{\Gamma(p)}\ln\frac{2\beta\rho}{\sigma},$$ and, consequently, $J\leq -(p-1)\sigma L +M_1|\Omega|e^{-\frac{1}{\Gamma(p)} \ln\frac{2\beta\rho}{\sigma}}+N_1e^{-\frac{1}{\Gamma(p)} \ln\frac{2\beta\rho}{\sigma}}\int_{\Omega}f(u,v)dx.$ Letting $$M=M_1|\Omega|e^{-\frac{1}{\Gamma(p)}\ln\frac{2\beta\rho}{\sigma}}, \quad N=N_1e^{-\frac{1}{\Gamma(p)}\ln\frac{2\beta\rho}{\sigma}}$$ and using Lemma \ref{lem3.1} we conclude that \begin{align*} J\leq&-(p-1)\sigma L +M+N[\beta |\Omega|-\frac{d}{dt} \int_{\Omega} u(t,x) dx]\\ \leq &-(p-1)\sigma G_{N}+(p-1)\sigma N\int_{\Omega} u dx+ M +N \beta |\Omega|- N\frac{d}{dt}\int_{\Omega} u(t,x) dx\\ \leq &-(p-1)\sigma G_{N}+|\Omega|N[(p-1)K\sigma + \beta] + M - N\frac{d}{dt}\int_{\Omega} u(t,x) dx. \end{align*} It follows that $$\frac{dG_{N}}{dt} \leq -(p-1)\sigma G_{N}+s$$ where $s=|\Omega|N[(p-1)K\sigma + \beta] + M$. \end{proof} We can now establish the main result of this manuscript. \begin{theorem} \label{thm3.1} Under the assumptions {\rm (H1)-(H3)} and \eqref{e1.5}, the solutions of \eqref{e1.1}--\eqref{e1.4} are global and uniformly bounded in $[0,+\infty[$. \end{theorem} \begin{proof} Multiplying \eqref{e3.5} by $e^{(p-1)\sigma t}$ and integrating, implies the existence of a positive constant $C>0$ independent of $t$ such that $$G_{N}(t)\leq C.$$ Since $$e^{h(u)}\geq e^{-\frac{1}{\Gamma(p)}\ln(\Gamma(p)[l+\rho k)])},$$ for all $p\geq 2$, we have $\int_{\Omega}(v+\omega)^{p} dx\quad \leq e^{\frac{1}{\Gamma(p)}\ln(\Gamma(p)[l+\rho k))}G_{N}(t) \leq C e^{\frac{1}{\Gamma(p)}\ln \Gamma(p)[l+\rho k]}= C(p).$ Consequently, $$\int_{\Omega}(v+\mu)^{p} dx \leq C(p),\quad \int_{\Omega}v^{p} dx \leq C(p).$$ Now we chose $p> n/2$ and we search for a bound for $\| \Upsilon(u,v)\|_{p}$. We put $$A_1= \max_{0\leq \tau\leq \tau_{0}} \psi(\tau),\quad A_2=\max_{0\leq \xi\leq K}\varphi(\xi),\quad A_3 = \max_{0\leq \tau\leq \tau_{0}} \phi(\tau)$$ where $\tau_{0}=max(\tau_1,\tau_2)$ such that $$\tau\geq \tau_1 \quad \Rightarrow \quad \psi(\tau)\leq \tau, \quad \tau\geq \tau_2 \quad \Rightarrow \quad \phi(\tau)\leq \tau.$$ Using {\rm (H1)--(H3)} implies $$g(u,v)\leq\psi(v) f(u,v)+\phi(v) \leq \psi(v) \varphi(u) (\mu+v)^{r}+\phi(v) \leq A_2 \psi(v) (\mu+v)^{r}+\phi(v).$$ Since $0\leq u\leq K$, we have \begin{equation*} \int_{\Omega} g(u,v)^{p} dx\leq \int_{\Omega} [A_2 \psi(v) (\mu+v)^{r} +\phi(v)]^{p}dx. \end{equation*} Now we use the inequality $$(x+y)^{q}\leq 2^{q-1}(x^{q}+y^{q})$$ all $x,y\geq0$ and $q\geq1$, to obtain the following sequence of estimates \begin{align*} &\int_{\Omega} g(u,v)^{p} dx\\ &\leq \int_{\Omega}2^{p-1}[A_2^{p} \psi(v)^{p} (\mu+v)^{rp}+\phi(v)^{p}]dx\\ &\leq 2^{p-1} \Big[A_2^{p}\Big( \int_{v\leq \tau_{0}} A_1^{p} (\mu+\tau_{0})^{rp} dx+\int_{v\geq \tau_{0}} v^{p} (\mu+v)^{rp} dx\Big) + |\Omega| A_3^{p}+\int_{v\geq \tau_{0}} v^{p}dx\Big] \\ &\leq 2^{p-1} \Big[\Big(A_2 A_1(\mu+\tau_{0})^{r}\Big)^{p}|\Omega| + A_2^{p}\int_{v\geq \tau_{0}} (\mu+v)^{(r+1)p} dx + |\Omega| A_3^{p}+\int_{v\geq \tau_{0}} v^{p}dx\Big]\\ &\leq 2^{p-1} [(A_2 A_1(\mu+\tau_{0})^{r})^{p}|\Omega|+ A_2^{p}C((r+1)p) + |\Omega| A_3^{p}+C(p)] =E_{g}^{p}\\ \end{align*} and $\int_{\Omega} f(u,v)^{p} dx\leq A_2^{p} C(rp)=E_{f}^{p}.$ We conclude that \begin{align*} \|\Upsilon(u,v)\|_{p} & \leq \|g(u,v)\|_{p} +\frac{c}{a-d}[\|f(u,v)\|_{p}+\alpha\|u\|_{p} +\beta |\Omega|] +\sigma \|v\|_{p} \\ &\leq E_{g}+\frac{c}{a-d}[E_{f}+(\alpha K+\beta)|\Omega|] +\sigma \sqrt[p]{C(p)}. \end{align*} We conclude that the unique solution of \eqref{e1.1}--\eqref{e1.4} is globally and uniformly bounded in $[0,+\infty[\times\Omega$. \end{proof} \subsection*{Acknowledgments} The author wants to thank Professors A. 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