\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 55, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/55\hfil Gierer-Meinhardt system with three equations] {Existence of global solutions for a Gierer-Meinhardt system with three equations} \author[S. Abdelmalek, H. Louafi, A. Youkana\hfil EJDE-2012/55\hfilneg] {Salem Abdelmalek, Hichem Louafi, Amar Youkana} % in alphabetical order \address{Salem Abdelmalek \newline Department of Mathematics, College of Arts and Sciences, Yanbu Taibah University, Saudi Arabia. \newline Department of Mathematics, University of Tebessa 12002 Algeria} \email{sallllm@gmail.com} \address{Hichem Louafi \newline Faculty of Economics and Management Science, University of Batna, 5000 Algeria} \email{hichemlouafi@gmail.com} \address{Amar Youkana \newline Department of Mathematics, University of Batna, 5000 Algeria} \email{youkana\_amar@yahoo.fr} \thanks{Submitted March 22, 2012. Published April 5, 2012.} \subjclass[2000]{35K57, 92C15} \keywords{Gierer-Meinhardt system; Lyapunov functional; activator-inhibitor} \begin{abstract} This articles shows the existence of global solutions for a Gierer-Meinhardt model of three substances described by reaction-diffusion equations with fractional reactions. Our technique is based on a suitable Lyapunov functional. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In recent years, systems of Reaction-Diffusion equations have received a great deal of attention, motivated by their widespread occurrence in modeling chemical and biological phenomena. Among these systems, the Gierer-Meinhardt is an important one. Meinhardt, Koch and Bernasconi \cite{Meinhardt3} proposed activator-inhibitor models (an example is given in section 4) to describe a theory of biological pattern formation in plants (\emph{Phyllotaxis}). We consider a reaction-diffusion system with three components: \begin{equation} \begin{gathered} u_{t}-a_1\Delta u=f(u, v, w)=\sigma-b_1u+\frac{u^{p_1}}{v^{q_1}(w^{r_1}+c)}\\ v_{t} -a_2\Delta v=g(u, v, w)=-b_2v+\frac{u^{p_2}}{v^{q_2}w^{r_2}}\\ w_{t} -a_3\Delta w=h(u, v, w)=-b_3w+\frac{u^{p_3}}{v^{q_3}w^{r_3}} \end{gathered} \label{1.1} \end{equation} with $x\in\Omega$, $t>0$, and with Neummann boundary conditions \begin{equation} \frac{\partial u}{\partial\eta}=\frac{\partial v}{\partial\eta}=\frac{\partial w}{\partial\eta}=0\quad\text{on }\partial\Omega\times\{ t>0\} , \label{1.2} \end{equation} and initial data \begin{equation} \begin{gathered} u(0,x)=\varphi_1(x)>0\\ v(0,x)=\varphi_2(x)>0\\ w(0,x)=\varphi_3(x)>0 \end{gathered}\label{1.3} \end{equation} on $\Omega$, and $\varphi_i\in C(\overline{\Omega})$ for all $i=1, 2, 3$. Here $\Omega$ is an open bounded domain of class $C^{1}$ in $\mathbb{R}^{N}$, with boundary $\partial\Omega$; $\partial / \partial\eta$ denotes the outward normal derivative on $\partial\Omega$. We use the following assumptions: $a_i, b_i, p_i, q_i, r_i$ are nonnegative for $i=1, 2, 3$, with $\sigma>0$, $c\geq0$: \begin{equation} 02\max\big\{ 1,\frac{b_2+b_3}{b_1}\big\}, \quad \frac{1}{\beta}>2A_{12}^2, \label{1.6} \\ (\frac{1}{2\beta}-A_{12}^2) (\frac{1}{2\gamma} -A_{13}^2) >(\frac{\alpha-1}{\alpha}A_{23}-A_{12} A_{13}) ^2. \label{1.7} \end{gather} The main result of the paper reads as follows. \begin{theorem} \label{thm1} Suppose that the functions $f,g$ and $h$ satisfy condition \eqref{1.4}. Let $(u(t,\cdot), v(t, \cdot),w(t, \cdot)) $ be a solution of \eqref{1.1}-\eqref{1.3} and let \begin{equation} L(t)=\int_{\Omega}\frac{u^{\alpha}(t,x) }{v^{\beta}( t,x) w^{\gamma}(t,x) } \, dx. \label{1.8} \end{equation} Then the functional $L$ is uniformly bounded on the interval $[0,T^{\ast }], T^{\ast}0$, then $u, v, w$ are uniformly bounded in $\overline{\Omega}\times[0, \infty) $. \end{corollary} \section{Preliminaries} The usual norms in spaces $L^p(\Omega)$, $L^{\infty}(\Omega)$ and $C(\overline{\Omega})$ are denoted respectively by: \begin{equation} \label{2.1} \begin{gathered} \| u\|_p^p=\frac{1}{| \Omega| }\int_{\Omega}| u(x)| ^pdx, \\ \| u \|_{\infty}= \operatorname{ess\,sup}_{x\in\Omega} | u(x)| , \\ \| u\|_{C (\overline{\Omega})}=\max_{x\in\overline{\Omega}} | u(x)| . \end{gathered} \end{equation} In 1972, following an ingenious idea of Turing \cite{Turing}, Gierer and Meinhardt \cite{Gierer} proposed a mathematical model for pattern formations of spatial tissue structures of hydra in morphogenesis, a biological phenomenon discovered by Trembley in 1744 \cite{Abraham}. It is a system of reaction-diffusion equations of the form \begin{equation} \begin{gathered} u_t -a_1\Delta u=\sigma-\mu u+\frac{u^p}{v^q}\\ v_t -a_2\Delta v=-\nu v+\frac{u^{r}}{v^s} \end{gathered} \label{2.2} \end{equation} for $x\in\Omega$ and $t>0$, with Neummann boundary conditions \begin{equation} \frac{\partial u}{\partial\eta}= \frac{\partial v}{\partial\eta }=0,\quad \; x\in\partial\Omega, t>0, \label{2.3} \end{equation} and initial conditions \begin{equation} u(x, 0) =\varphi_1(x)>0, \quad v(x, 0) =\varphi_2(x)>0, \quad x\in\Omega, \label{2.4} \end{equation} where $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary $\partial\Omega$, $a_1, a_2>0$, $\mu, \nu, \sigma>0$, the indexes $p, q, r$ and $s$ are non negative with $ p>1$. Existence of solutions in $(0,\infty) $ is proved by Rothe in 1984 \cite{Rothe} in a particular situation when $p=2$, $q=1$, $r=2$, $s=0$ and $N=3$. Rothe's method cannot be applied (at least directly) to the general case. Wu and Li \cite{Wu} obtained the same results for \eqref{2.2}-\eqref{2.4} so long as $u, v^{-1}$ and $\sigma$ are suitably small. Mingde, Shaohua and Yuchun \cite{Mingde} show that solutions of this problem are bounded all the time for any initial values if \begin{equation} \frac{p-1}{r}<\frac{q}{s+1}, \quad \frac{p-1}{r}<1. \label{2.5b} \end{equation} Masuda and Takahashi \cite{Masuda} considered a more general system for $(u, v)$, \begin{equation} \begin{gathered} u_t -a_1\Delta u=\sigma_1(x) -\mu u+\rho_1(x,u) \frac{u^p}{v^q}, \\ v_t -a_2\Delta v=\sigma_2(x) -\nu v+\rho_2(x,u) \frac{u^{r}}{v^s}, \end{gathered} \label{2.6} \end{equation} with $\sigma_1, \sigma_2\in C^{1}(\overline{\Omega})$, $\sigma_1\geq0,\sigma_2\geq0$, $\rho_1,\rho_2\in C^{1}(\overline{\Omega}\times\overline{\mathbb{R}}_{+}^2) \cap L^{\infty}(\overline{\Omega}\times\overline{ \mathbb{R}}_{+}^2) $ satisfying $\rho_1\geq0,\rho_2>0$ and $p, q, r, s$ are nonnegative constants satisfying \eqref{2.5b}. Obviously, system \eqref{2.4} is a special case of system \eqref{2.6}. In 1987, Masuda and Takahashi \cite{Masuda} extended the result to $\frac{p-1}{r}<\frac{2}{N+2}$ under the sole condition $\sigma_1>0$. In 2006, Jiang \cite{jiang}, under the conditions \eqref{2.5b}, $\varphi_1,\varphi_2\in W^{2,l}(\Omega)$, $l>\max\{ N,2\}$, $\frac {\partial\varphi_1}{\partial\eta}=\frac{\partial\varphi_2}{\partial\eta}=0$ on $\partial\Omega$ and $\varphi_1\geq0,\varphi_2>0$ in $\overline{\Omega}$, showed that \eqref{2.6} has a unique nonnegative global solution $(u, v)$ satisfying \eqref{2.3}-\eqref{2.4}. It is well-known that to prove existence of global solutions to \eqref{1.1}--\eqref{1.3}, it suffices to derive a uniform estimate of $\| f(u, v, w) \|_p$, $\| g(u, v, w) \|_p$ and $\| h(u, v, w) \|_p$ on $[0;T_{\rm max})$ in the space $L^p(\Omega)$ for some $p>N/2$ (see Henry \cite{Henry}). Our aim is to construct a polynomial Lyapunov functional allowing us to obtain $L^p-$ bounds on $u, v$ and $w$ that lead to global existence. Since the functions $f, g$ and $h$ are continuously differentiable on $\mathbb{R}_{+}^{3}$, then for any initial data in $C(\overline{\Omega})$, it is easy to check directly their Lipschitz continuity on bounded subsets of the domain of a fractional power of the operator \begin{equation} \mathcal{A}=-\begin{pmatrix} a_1\Delta & 0 & 0\\ 0 & a_2\Delta & 0\\ 0 & 0 & a_3\Delta \end{pmatrix}. \label{3.1} \end{equation} Under these assumptions, the following local existence result is well known (see Henry \cite{Henry}). \begin{proposition} \label{prop1} System \eqref{1.1}-\eqref{1.3} admits a local unique, classical solution $(u, v, w)$ on $(0,T_{\rm max})\times\Omega$. If $T_{\rm max}<\infty$ then \begin{equation} \lim_{t\nearrow T_{\rm max}} (\| u(t,.) \|_{\infty}+\|v(t,.) \|_{\infty}+\| w(t,.) \|_{\infty}) =\infty. \label{3.2} \end{equation} \end{proposition} \section{Proofs of main results} For the proof of Theorem \ref{thm1}, we need some preparatory Lemmas. \begin{lemma} \label{lem1} Assume that $p$, $q$, $r$, $s$, $m$, and $n$ satisfy \[ \frac{p-1}{r}<\min(\frac{q}{s+1},\frac{m}{n},1) . \] For all $h$, $l$, $\alpha$, $\beta$, $\gamma>0$, there exist $C=C(h, l, \alpha, \beta, \gamma) >0$ and $\theta=\theta(\alpha)\in(0,1) $, such that \begin{equation} \alpha\frac{x^{p-1+\alpha}}{y^{q+\beta}z^{m+\gamma}}\leq\beta\frac {x^{r+\alpha}}{y^{s+1+\beta}z^{n+\gamma}}+C(\frac{x^{\alpha}}{y^{\beta }z^{\gamma}}) ^{\theta},\quad x\geq0,\; y\geq h,\; z\geq l. \label{4.1} \end{equation} \end{lemma} \begin{proof} For all $x\geq0, y\geq h, z\geq l$, from the inequality \eqref{4.1}, we have \begin{equation} \alpha\frac{x^{p-1}}{y^qz^{m}}\leq\beta\frac{x^{r}}{y^{s+1}z^{n}} +C(\frac{x^{\alpha}}{y^{\beta}z^{\gamma}}) ^{\theta-1}. \label{6.1} \end{equation} We can write \[ \alpha\frac{x^{p-1}}{y^qz^{m}}=\alpha\beta^{-(p-1)/r}( \beta\frac{x^{r}}{y^{s+1}z^{n}}) ^{\frac{p-1}{r}}y^{\frac{( s+1) (p-1) }{r}-q}z^{\frac{n(p-1) }{r}-m}. \] For each $\epsilon$ such that $0<\epsilon<\min(\frac{q}{s+1},\frac{m}{n},1) -\frac{p-1}{r}$, we have \[ \alpha\frac{x^{p-1}}{y^qz^{m}}=\alpha\beta^{-(p-1)/r}( \beta\frac{x^{r}}{y^{s+1}z^{n}}) ^{\frac{p-1}{r}+\epsilon} (\beta\frac{x^{r}}{y^{s+1}z^{n}}) ^{-\epsilon}v^{\frac{( s+1) (p-1) }{r}-q}z^{\frac{n(p-1) }{r}-m}. \] Then \begin{equation} \begin{split} \alpha\frac{x^{p-1}}{y^qz^{m}} & =\alpha(\beta) ^{-\frac{p-1}{r}-\epsilon}(\beta\frac{x^{r}}{y^{s+1}z^{n}}) ^{\frac{p-1}{r}+\epsilon}(\frac{1}{x^{\alpha}}) ^{\frac {r\epsilon}{\alpha}}(y) ^{\frac{(s+1) ( p-1) }{r}-q+\epsilon(s+1) }\\ &\quad\times z^{\frac{n(p-1) }{r}-m+\epsilon n} \\ & \leq\alpha(\beta) ^{-\frac{p-1}{r}-\epsilon}( \beta\frac{x^{r}}{y^{s+1}z^{n}}) ^{\frac{p-1}{r}+\epsilon}( \frac{1}{x^{\alpha}}) ^{\frac{r\epsilon}{\alpha}}(h) ^{\frac{(s+1) (p-1) }{r}-q+\epsilon( s+1) }\\ &\quad\times l^{\frac{n(p-1) }{r}-m+\epsilon n} \\ & \leq\alpha(\beta) ^{-\frac{p-1}{r}-\epsilon}( \beta\frac{x^{r}}{y^{s+1}z^{n}}) ^{\frac{p-1}{r}+\epsilon}( \frac{1}{x^{\alpha}}) ^{\frac{r\epsilon}{\alpha}}(h) ^{\frac{(s+1) (p-1) }{r}-q+\epsilon( s+1) }\\ &\quad \times l^{\frac{n(p-1) }{r}-m+\epsilon n}(\frac{y}{h}) ^{\frac{\beta r\epsilon}{\alpha}}(\frac{z}{l}) ^{\frac{\gamma r\epsilon}{\alpha}} \\ & \leq C_1(\beta\frac{x^{r}}{y^{s+1}z^{n}}) ^{\frac{p-1} {r}+\epsilon}(\frac{y^{\beta}z^{\gamma}}{x^{\alpha}}) ^{r\epsilon/\alpha}, \end{split} \label{6.2} \end{equation} where \[ C_1=\alpha(\beta) ^{-\frac{p-1}{r}-\epsilon}h^{^{\frac {(s+1)(p-1)}{r}-q+\epsilon(s+1)-\frac{\beta r\epsilon}{\alpha}}} l^{\frac{(n)(p-1)}{r}-m+\epsilon n-\frac{\gamma r\epsilon}{\alpha}}. \] Using Young's inequality for \eqref{6.2} by taking \[ C=C_1^{1+\frac {p-1+r\epsilon}{r-(p-1) -r\epsilon}}, \quad \theta=1-\frac {r\epsilon}{\alpha(1-\frac{p-1}{r}-\epsilon) }, \] where $\epsilon$ is sufficiently small, we obtain inequality \eqref{6.1}. \end{proof} \begin{lemma} \label{lem2} Let $T>0$ and $f=f(t)$ be a non-negative integrable function on $[0, T)$. Let $0<\theta<1$ and $W=W(t)$ be a positive function on $[0, T)$ satisfying the differential inequality \[ \frac{dW}{dt}\leq-W(t) +f(t)W^{\theta}(t), \quad 0\leq t0,\\ v(t,x)\geq e^{-b_2t}\min(\varphi_2(x))>0,\\ w(t,x)\geq e^{-b_3t}\min(\varphi_3(x))>0. \end{gathered} \label{4.2} \end{equation} \end{lemma} The proof of the above lemma follows immediately from the maximum principle, and it is omitted. \begin{proof} [Proof of Theorem \ref{thm1}] Differentiating $L(t) $ with respect to $t$ yields \begin{align*} L'(t) & =\int_{\Omega}\frac{d}{dt}( \frac{u^{\alpha}}{v^{\beta}w^{\gamma}}) dx\\ & =\int_{\Omega}\big(\alpha\frac{u^{\alpha-1}}{v^{\beta}w^{\gamma} }\partial_{t}u-\beta\frac{u^{\alpha}}{v^{\beta+1}w^{\gamma}}\partial _{t}v-\gamma\frac{u^{\alpha}}{v^{\beta}w^{\gamma+1}}\partial_{t}w\big) dx. \end{align*} Replacing $\partial_{t}u$, $\partial_{t}v$\ and $\partial_{t}w$ by their values in \eqref{1.1}, we obtain \[ L'(t) =I+J, \] where $I$ contains the Laplacian terms and $J$ contains the other terms, \[ I =a_1\alpha\int_{\Omega}\frac{u^{\alpha-1}}{v^{\beta}w^{\gamma}} \, \Delta udx-a_2\beta\int_{\Omega}\frac{u^{\alpha}}{v^{\beta+1}w^{\gamma}} \, \Delta vdx-a_3\gamma\int_{\Omega}\frac{u^{\alpha}}{v^{\beta}w^{\gamma+1}} \, \Delta w \, dx, \] and \begin{align*} J & =(-b_1\alpha+b_2\beta+b_3\gamma) L(t) +\alpha\int_{\Omega}\frac{u^{p_1+\alpha-1}}{v^{q_1 + \beta}w_3^{\gamma }(w^{r_1}+c)} \, dx \\ &\quad - \beta\int_{\Omega}\frac{u^{p_2+\alpha}}{v^{q_2+\beta +1}w^{r_2+\gamma}}dx-\gamma\int_{\Omega}\frac{u^{p_3+\alpha}} {v^{q_3+\beta}w^{r_3+\gamma+1}} \, dx +\sigma\alpha\int_{\Omega}\frac{u^{\alpha-1}}{v^{\beta}w^{\gamma}} \, dx. \end{align*} \subsection*{Estimation of $I$} Using Green's formula, we obtain \begin{align*} I & =\int_{\Omega}(-a_1\alpha(\alpha-1) \frac{u^{\alpha-2}}{v^{\beta}w^{\gamma}}| \nabla u| ^2+a_1\alpha \beta\frac{u^{\alpha-1}}{v^{\beta+1}w^{\gamma}}\nabla u\nabla v+a_1 \alpha\gamma\frac{u^{\alpha-1}}{v^{\beta}w^{\gamma+1}}\nabla u\nabla w\\ &\quad +a_2\beta\alpha\frac{u^{\alpha-1}}{v^{\beta+1}w^{\gamma}}\nabla u\nabla v-a_2\beta(\beta+1) \frac{u^{\alpha}}{v^{\beta+2}w^{\gamma} }| \nabla v| ^2-a_2\beta\gamma\frac{u^{\alpha} }{v^{\beta+1}w^{\gamma+1}}\nabla v\nabla w\\ &\quad +a_3\gamma\alpha\frac{u^{\alpha-1}}{v^{\beta}w^{\gamma+1}}\nabla u\nabla w-a_3\gamma\beta\frac{u^{\alpha}}{v^{\beta+1}w^{\gamma+1}}\nabla v\nabla w-a_3\gamma(\gamma+1) \frac{u^{\alpha}}{v^{\beta}w^{\gamma+2} }| \nabla w| ^2)dx\\ & =-\int_{\Omega}[\frac{u^{\alpha-2}}{v^{\beta+2}w^{\gamma+2}}( QT) \cdot T] \, dx, \end{align*} where \[ Q=\begin{pmatrix} a_1\alpha(\alpha-1) & -\alpha\beta\frac{a_1+a_2}{2} & -\alpha\gamma\frac{a_1+a_3}{2}\\ -\alpha\beta\frac{a_1+a_2}{2} & a_2\beta(\beta+1) & \beta\gamma\frac{a_2+a_3}{2}\\ -\alpha\gamma\frac{a_1+a_3}{2} & \beta\gamma\frac{a_2+a_3}{2} & a_3\gamma(\gamma+1) \end{pmatrix}. \] The matrix $Q$ is positive definite if, and only if, all its principal successive determinants $\Delta_1, \Delta_2, \Delta_3$ are positive. To see this, we have: $\Delta_1=a_1\alpha(\alpha-1) >0$ by \eqref{1.6}. Note that $$ \Delta_2=\left|\begin{matrix} a_1\alpha(\alpha-1) & -\alpha\beta\frac{a_1+a_2}{2}\\ -\alpha\beta\frac{a_1+a_2}{2} & a_2\beta(\beta+1) \end{matrix} \right| =\alpha^2 \beta^2 a_1 a_2\big(\frac{\alpha-1}{\alpha} \frac{\beta+1}{\beta}-A_{12}^2\big) $$ which is positive by \eqref{1.6}. Using \cite[Theorem 1]{abdelmalek1}, we obtain \begin{align*} (\alpha-1) \Delta_3 &=(\alpha-1) | Q|\\ &=\alpha(\alpha\gamma\beta) ^2a_1 a_2 a_3 \Big(\big(\frac{\alpha-1}{\alpha }\frac{\beta+1}{\beta}-A_{12}^2\big) \big(\frac{\alpha-1}{\alpha} \frac{\gamma+1}{\gamma}-A_{13}^2\big) \\ &\quad -\big(\frac{\alpha-1}{\alpha}A_{23}-A_{12}A_{13}\big) ^2\Big). \end{align*} Then using \eqref{1.6}-\eqref{1.7}, we obtain $\Delta_3>0$. Consequently, $I\leq 0$ for all $(t, x)\in [0, T^{\ast}] \times\Omega$. \subsection*{Estimation of $J$} According to the maximum principle, there exists $C_0$ depending on $ \| \varphi_1 \|_{\infty}$, $\| \varphi_2 \|_{\infty}$, $\|\varphi_3\|_{\infty} $ such that $v, w\geq C_0>0$. We then have \[ \frac{u^{\alpha-1}}{v^{\beta}w^{\gamma}} =(\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}) ^{(\alpha-1)/\alpha} (\frac{1}{v})^{\beta/\alpha}(\frac{1}{w}) ^{\gamma/\alpha} \leq(\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}) ^{\frac{\alpha -1}{\alpha}}(\frac{1}{C_0}) ^{(\beta+\gamma)/\alpha}. \] So \[ \frac{u^{\alpha-1}}{v^{\beta}w^{\gamma}} \leq C_2(\frac{u^{\alpha}}{v^{\beta}w^{\gamma}})^{(\alpha-1)/\alpha} \quad \text{where }C_2=(\frac{1}{C_0}) ^{(\beta+\gamma)/\alpha}. \] Using Lemma \ref{lem1}, for all $(t, x) \in [0, T^{\ast}] \times\Omega$, we obtain \begin{equation} \alpha\frac{u^{p_1+\alpha-1}}{v^{q_1+\beta}w^{\gamma}(w^{r_1}+c)} \leq\alpha\frac{u^{p_1+\alpha-1}}{v^{q_1+\beta}w^{\gamma+r_1}}\leq \beta\frac{u^{p_2+\alpha}}{v^{q_2+\beta+1}w^{r_2+\gamma}} +C\big(\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}\big) ^{\theta}, \label{4.3} \end{equation} or \begin{equation} \alpha\frac{u^{p_1+\alpha-1}}{v^{q_1+\beta}w^{\gamma+r_1}}\leq \gamma\frac{u^{p_3+\alpha}}{w^{r_3+1+\gamma}v^{q_3+\beta}} C\big(\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}\big) ^{\theta}. \label{4.4} \end{equation} We have \begin{align*} J & =(-b_1\alpha+b_2\beta+b_3\gamma) L(t) +\alpha\int_{\Omega}\frac{u^{p_1+\alpha-1}}{v^{q_1+\beta}w^{\gamma}(w^{r_1}+c) } dx\\ &\quad -\beta\int_{\Omega}\frac{u^{p_2+\alpha}}{v^{q_2+\beta+1}w^{r_2+\gamma}} dx - \gamma\int_{\Omega}\frac{u^{p_3 +\alpha}}{v^{q_3+\beta}w^{r_3+\gamma+1}} \, dx +\sigma\alpha\int_{\Omega}\frac{u^{\alpha-1}}{v^{\beta}w^{\gamma}} \, dx. \end{align*} Using \eqref{4.4}, \[ J\leq(-b_1\alpha+b_2\beta+b_3\gamma) L(t) +\int_{\Omega}C(\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}) ^{\theta}dx+\alpha\sigma\int_{\Omega}C_2(\frac{u^{\alpha}}{v^{\beta }w^{\gamma}}) ^{(\alpha-1)/\alpha} dx\,. \] Applying H\"{o}lder's inequality, for all $t \in [0, T^{\ast}]$, we obtain \[ \int_{\Omega}C(\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}) ^{\theta }dx\leq\Big(\int_{\Omega}(\frac{u^{\alpha}}{v^{\beta}w^{\gamma} }) dx\Big) ^{\theta}\Big(\int_{\Omega}C^{\frac{1}{1-\theta} }dx\Big) ^{1-\theta}. \] Then \[ \int_{\Omega}C\big(\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}\big) ^{\theta }dx\leq C_3L^{\theta}(t),\quad \text{where }C_3=C| \Omega| ^{1-\theta}. \] We have \[ \int_{\Omega}C_2\big(\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}\big) ^{(\alpha-1)/\alpha} dx \leq\Big(\int_{\Omega}(\frac{u^{\alpha} }{v^{\beta}w^{\gamma}}) dx\Big) ^{(\alpha-1)/\alpha} \Big(\int_{\Omega}(C_2) ^{\alpha}dx\big) ^{1/\alpha}. \] Whereupon \[ \int_{\Omega}C_2(\frac{u^{\alpha}}{v^{\beta}w^{\gamma}}) ^{(\alpha-1)/\alpha} dx\leq C_{4}L^{(\alpha-1)/\alpha} ( t) \quad \text{where } C_{4}=C_2| \Omega| ^{1/\alpha}. \] We have \[ J\leq(-b_1\alpha+b_2\beta+b_3\gamma) L(t) +C_3L^{\theta}(t) +\alpha\sigma C_{4}L^{\frac{\alpha-1} {\alpha}}(t). \] Whereupon \[ J\leq(-b_1\alpha+b_2\beta+b_3\gamma) L(t) +C_{5}(L^{\theta}(t) +\alpha\sigma L^{\frac{\alpha -1}{\alpha}}(t) ). \] Thus under conditions \eqref{1.6} and \eqref{1.7}, we obtain the differential inequality \[ L'(t)\leq(-b_1\alpha+b_2\beta+b_3\gamma) L( t) +C_{5}(L^{\theta}(t) +\alpha\sigma L^{(\alpha-1)/\alpha} (t) ). \] Since $-b_1\alpha+b_2\beta+b_3\gamma<0$, we obtain \begin{equation} L'(t) \leq C_{5}L^{\theta}(t) +C_{5} \alpha\sigma L^{(\alpha-1)/\alpha} (t). \label{4.5} \end{equation} Using Lemma \ref{lem2}, we deduce that $L(t)$ is bounded on $(0,T_{\rm max})$; i.e, $L(t)\leq\gamma_1$, where $\gamma_1$ dependents on the $L^{\infty}$-norm of $\varphi_1, \varphi_2$ and $\varphi_3$. \end{proof} \begin{proof}[Proof of Corollary \ref{coro1}] Since $L(t)$ is bounded on $(0,T_{\rm max})$ and the functions $$ \frac{u^{p_1}}{v^{q_1}(w^{r_1}+c)}, \quad \frac{u^{p_2}}{v^{q_2}w^{r_2}}, \quad \frac{u^{p_3}}{v^{q_3}w^{r_3}} $$ are in $L^{\infty}((0, T_{\rm max}), L^{m}(\Omega))$ for all $m>\frac{N}{2}$, as a consequence of the arguments in Henry \cite{Henry} and Haraux and Kirane \cite{Haraux}, we conclude that the solution of the system \eqref{1.1}-\eqref{1.7} is global and uniformly bounded on $\Omega\times(0, +\infty)$. \end{proof} \section{Example} In this section we present a particular activator-inhibitor model that illustrates the applicability of Theorem \ref{thm1} and Corollary \ref{coro1}. We assume that all reactions take place in bounded region $\Omega$ with smooth boundary $\partial\Omega$. \begin{example} \label{examp1} \rm The model proposed by Meinhrdt, Koch and Bernasconi \cite{Meinhardt3} to describe a theory of biological pattern formation in plants (Phyllotaxis) is \begin{equation} \label{4.1b} \begin{gathered} \frac{\partial u}{\partial t} - a_1 \frac{\partial^2 u}{\partial x^2} =-b_1 u +\frac{a^2}{v(w+k_u)} + \sigma, \\ \frac{\partial v}{\partial t} - a_2 \frac{\partial^2 v}{\partial x^2} =-b_2 v +u^2, \\ \frac{\partial w}{\partial t} - a_3 \frac{\partial^2 w}{\partial x^2} =-b_3 w +u, \end{gathered} \end{equation} for $x\in\Omega$ and $t>0$, where $u,v,w$ are the concentrations of the three substances; called activator ($u$) and inhibitors ($v$ and $w$). \end{example} We claim that \eqref{4.1b} with boundary conditions \eqref{1.2} and non-negative uniformly bounded initial data \eqref{1.3} has a global solution. 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