\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 06, pp. 1--14.\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-2008/06\hfil Existence of global solutions] {Existence of global solutions for a predator-prey model with cross-diffusion} \author[S. Xu\hfil EJDE-2008/06\hfilneg] {Shenghu Xu} \address{Department of Mathematics, Northwest Normal University, Lanzhou 730070, China} \email{xuluck2001@163.com} \thanks{Submitted October 13, 2007. Published January 12, 2008.} \subjclass[2000]{35K57, 35B35, 92D25} \keywords{Cross-diffusion; global solution; gradient estimates; stability} \begin{abstract} In this article, we prove the existence of global classical solutions for a prey-predator model when the space dimension $n<10$. Under certain conditions on the coefficients of the reaction functions, the convergence of solutions is established for the system with large diffusion by constructing a Lyapunov function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} To investigate the spatial segregation under the self and cross population pressure, Shigesada, Kawasaki and Teramoto \cite{Shigesada} proposed a competition model in 1979. Then there have been established many results in the literatures; see for example \cite{Lou, Li, Shim1, Choi1, Choi2, Le, Tuoc1, Tuoc2}. For the cross-diffusion systems with prey-predator type reaction functions, there are a few results mainly on the steady-state problems with the elliptic systems, see \cite{Kuto1, Ahmed, Kuto2, Nakashima, Ryu}. In this paper, we study the following cross-diffusion system, with prey-predator type reactions, $$\begin{gathered} u_{t}-\Delta[(d_1+\alpha_{11}u+\alpha_{12}v)u]=u(a_1-b_1u-c_1v) \quad\text{in }\Omega\times[0,\infty),\\ v_{t}- \Delta[(d_2+\alpha_{21}u+\alpha_{22}v)v]=v(a_2+b_2u-c_2v) \quad\text{in }\Omega\times[0,\infty),\\ \partial_{\eta }u=\partial_{\eta}v=0 \quad\text{on } \partial{\Omega}\times[0,\infty),\\ u(x,0)=u_{0}(x)\geq0, \quad v(x,0)=v_{0}(x)\geq0 \quad\text{in }\Omega, \end{gathered}\label{e11}$$ where $\Omega\subset\mathbb{R}^{n}$ ($n\geq1$) is a bounded domain with smooth boundary $\partial\Omega$, $\eta$ is the outward unit normal vector of the boundary $\partial\Omega$, and $\partial_\eta=\partial/\partial_\eta$. $\alpha_{ij}$ are given nonnegative constants for $i, j=1,2$. And $d_i, b_i, c_i(i=1,2)$ and $a_1$ are positive constants only $a_2$ may be non-positive. In system \eqref{e11}, $u$ and $v$ are nonnegative functions which represent the population densities of the prey and predator species, respectively, $d_1$ and $d_2$ are the random diffusion rates of the two species, $\alpha_{11}$ and $\alpha_{22}$ are self-diffusion rates, and $\alpha_{12}$ and $\alpha_{21}$ are the so-called cross-diffusion rates. When $\alpha_{ij}=0$ ($i,j=1,2$), the system is the well-known Lotka-Volterra prey-predator model. For more details on the biological background, see references \cite{Shigesada, Shim2}. Local existence (in time) of solutions to \eqref{e11} was established by Amann in a series of important papers \cite{Amann1, Amann2, Amann3}. His result can be summarized as follows: \begin{theorem} \label{thmA} Suppose that $u_0, v_0$ are in $W_p^1(\Omega)$ for some $p>n$. Then \eqref{e11} has a unique non-negative smooth solution $u(x,t), v(x,t)$ in $C([0,T),W_p^1(\Omega))\bigcap C^\infty((0,T),C^\infty(\Omega))$ with maximal existence time $T$. Moreover, if the solution $(u,v)$ satisfies the estimate $$\sup_{0\leq t\leq T} \|u(.,t)\|_{W_p^1(\Omega)}<\infty \quad\text{and}\quad \sup_{0\leq t\leq T} \|v(.,t)\|_{W_p^1(\Omega)}<\infty ,$$ then $T=\infty$. \end{theorem} However, little is known about global existence of solutions to \eqref{e11}. In 2006, Shim \cite{Shim2} proved the existence of global solutions to \eqref{e11} in two cases: Case(A) $n=1$, $d_1=d_2$ and $\alpha_{11}=\alpha_{22}=0$; Case(B) $n=1$, $0<\alpha_{21}<8\alpha_{11}$ and $0<\alpha_{12}<8\alpha_{22}$. In \cite{Shim3} the author considered the case when $\alpha_{11}, \alpha_{12}, \alpha_{22}>0$ and $\alpha_{21}=0$ for the system \eqref{e11}, and established the existence of global solutions with $n=1$. We shall prove the existence of global solutions to the following system (namely, the system \eqref{e11} for $\alpha_{12}=0$) $$\begin{gathered} u_{t}-\Delta[(d_1+\alpha_{11}u)u]=u(a_1-b_1u-c_1v)\quad\text{in } \Omega\times[0,\infty),\\ v_{t}- \Delta[(d_2+\alpha_{21}u+\alpha_{22}v)v]=v(a_2+b_2u-c_2v) \quad\text{in }\Omega\times[0,\infty),\\ \partial_{\eta }u=\partial_{\eta}v=0 \quad\text{on } \partial{\Omega}\times[0,\infty),\\ u(x,0)=u_{0}(x)\geq0,\quad v(x,0)=v_{0}(x)\geq0 \quad\text{in }\Omega. \end{gathered}\label{e12}$$ This paper draws on ideas from two papers \cite{Choi2} and \cite{Tuoc2} which deal with cross-diffusion system with competition type reactions. Duo to the difference in the reaction functions. Therefore, in order to obtain the $L^p$-estimate of $v$, we have to estimate the term $uv^p$. We also obtain result on the asymptotic stability of the global solution to \eqref{e12} if the diffusion coefficients are large enough by an important Lemma 5.1 from \cite{Wang}. We summarize our results in the following theorems: \begin{theorem} \label{thm1.1} Let $\alpha_{22}>0$ and assume that $u_0\geq0, v_0\geq0$ satisfy zero Neumann boundary condition and belong to $C^{2+\lambda}(\overline{\Omega})$ for some $\lambda>0$. Then \eqref{e12} possesses a unique non-negative solution $u, v\in C^{2+\lambda,\frac{2+\lambda}{2}}(\overline{\Omega}\times[0,\infty))$ if either {\rm (i)} $\alpha_{11}=0$ or {\rm (ii)} $\alpha_{11}>0$ and $n<10$. \end{theorem} \begin{theorem} \label{thm1.2} Assume that all conditions in Theorem \ref{thm1.1} are satisfied. Assume further that \begin{gather} -\frac{a_1b_2}{b_1c_2}<\frac{a_2}{c_2}<\frac{a_1}{c_1},\label{e13}\\ 4\rho\overline{u}\overline{v}d_1d_2>m^2(\overline{v}\alpha_{21})^2. \label{e14} \end{gather} Then \eqref{e12} has the unique positive equilibrium point $(\overline{u}, \overline{v})$ which is global asymptotic stable, where $m$ is the positive constant in Lemma \ref{lem2.1} (independent of $d_1, d_2$), $\rho=(b_2c_1+2b_1c_2)b_2^{-2}$ and $$(\overline{u},\overline{v})=\Big(\frac{a_1c_2-a_2c_1}{b_1c_2+b_2c_1}, \frac{a_2b_1+a_1b_2}{b_1c_2+b_2c_1}\Big).$$ \end{theorem} The paper is organized as follows. In section 2, we collect some well known results and prove three new lemmas that are needed in section 3 and section 4. In section 3, we establish $L^r$-estimates of the solution $v$ of \eqref{e12} and in section 4 we give a proof of Theorem \ref{thm1.1}. In section 5, we give a proof of Theorem \ref{thm1.2}. \section{Preliminaries} We list here some notation. \begin{align*} &Q_T=\Omega\times[0,T),\\ &\|u\|_{L^{p,q}(Q_T)}=\Big(\int_0^T(\int_{\Omega}|u(x,t)|^p dx )^{\frac{q}{p}}dt\Big)^{1/q}, L^{p}(Q_T):= L^{p,p}(Q_T),\\ &\|u \|_{W_P^{2,1}(Q_T)}:= \|u\|_{L^p(Q_T)}+\|u_t\|_{L^p(Q_T)}+\|\nabla u\|_{L^p(Q_T)}+\|\nabla^2u\|_{L^p(Q_T)},\\ &\|u\|_{V_2(Q_T)}:= \sup_{0\leq t\leq T}\|u(.,t)\|_{L^2(\Omega)} +\|\nabla u(x,t)\|_{L^2(Q_T)}. \end{align*} Firstly, we present some useful lemmas. \begin{lemma} \label{lem2.1} Let $u, v$ be a solution of \eqref{e12} in $[0,T)$. Then $0\leq u\leq m$ and $v\geq0$ in $Q_T$, where $m=\max\{\frac{a_1}{b_1},\|u_0\|_{L^\infty(\Omega)}\}$. \end{lemma} \begin{proof} The first equation in \eqref{e12} is expressed as $$u_t=(d_1+2\alpha_{11}u)\Delta u+2\alpha_{11}\nabla u\cdot\nabla u+u(a_1-b_1u-c_1v),\label{e21}$$ and the second equation is written as $$v_t=(d_2+\alpha_{21}u+2\alpha_{22}v)\Delta v+2(\alpha_{21}\nabla u+\alpha_{22}\nabla v)\nabla v+v(\alpha_{21}\Delta u+a_2+b_2u-c_2v).\label{e22}$$ Then application of the maximum principle for \eqref{e21} and \eqref{e22} yields the nonnegative of $u$ and $v$. Applying the maximum principle to \eqref{e21} again one can also show the boundedness of $u$. \end{proof} \begin{lemma} \label{lem2.2} There exists a positive $C_1(T)$ such that \begin{gather*} \sup_{0\leq t\leq T}\|u(.,t)\|_{L^1(\Omega)}0$such that$a(x,t,\xi)\geq d$and$a_\xi(x,t,\xi)\geq 0$for all$(x,t)\in Q_T$and$\xi$in$\mathrm{R}$. \item[(ii)] There is a continuous function$M$on$\mathrm{R}$such that$a(x,t,\xi)\leq M(\xi)$for all$(x,t)\in Q_T$. \item[(iii)] For any bounded measurable function$g$on$Q_T$,$|\nabla_{x}a(.,.,g(.,.))|$is in the space$L^{2p}(Q_T)$. \end{itemize} \begin{lemma} \label{lem2.4} Assume that$w\in W_p^{2,1}(Q_T)\bigcap C^{2,1}(\overline{\Omega}\times[0,T))$is a bounded function satisfying $$w_t\leq a(x,t,w)\Delta w+f(x,t) \quad\text{in} \quad Q_T$$ with boundary condition$\frac{\partial w}{\partial\nu}\leq0$on$\partial_{Q_T}$, where$f\in L^p(Q_T)$. Then,$\nabla w$is in$L^{2p}(Q_T)$. \end{lemma} The proof of the above lemma can be found in \cite[Proposition 2.1]{Tuoc2}. \begin{lemma} \label{lem2.5} Let$q>1, \widetilde{q}=2+\frac{4q}{n(q+1)}$,$\widetilde{\beta}$in$(0,1)$and let$C_T>0$be any number which may depend on$T$. Then there is a constant$M_1$depending on$q, n, \Omega, \widetilde{\beta}$and$C_T$such that for any$g$in$C([0,T),W_2^1(\Omega))$with$(\int_\Omega|g(.,t)|^{\widetilde{\beta}}dx)^{1/\widetilde{\beta}} \leq C_T$for all$t\in[0,T]$, we have the inequality $$\|g\|_{L^{\widetilde{q}}(Q_T)}\leq M_1\big\{1 +\big(\sup_{0\leq t\leq T}\|g(.,t)\|_{L^{2q/q+1}(\Omega)}\big)^{4q/n(q+1) \widetilde{q}}\|\nabla g\|^{2/\widetilde{q}}_{L^2(Q_T)}\big\} .$$ \end{lemma} The proof of the above lemma can be found in \cite[Lemmas 2.3, 2.4]{Choi2}. \section{$L^r$-estimates for$v$} \begin{lemma} \label{lem3.1} There exists a constant$C_3(T)$such that$\|\nabla u\|_{L^4(Q_T)}\leq C_3(T)$. \end{lemma} \begin{proof} Let$\delta=\alpha_{11}/d_1$,$w_1=(1+\delta u)u$. By Lemma \ref{lem2.1},$u$is bounded. Therefore,$w_1$is also bounded. By Lemma \ref{lem2.3}, we have$w_1\in W_2^{2,1}(Q_T)$. Moreover,$w_1satisfies \begin{align*} w_{1t} &\leq d_1(1+2\delta u)\Delta w_1+a_1u(1+2\delta u)\\ &= \sqrt{d_1^2+4\delta d_1 w_1}\Delta w_1+a_1u(1+2\delta u). \end{align*} By Lemma \ref{lem2.4} withp=2$,$a(x,t,\xi)=\sqrt{d_1^2+4\delta d_1\xi}$,$f(x,t)=a_1u(x,t)(1+2\delta u(x,t))$, we obtain the desired result. \end{proof} \begin{lemma} \label{lem3.2} Let$r>2$and$p_r=\frac{2r}{r-2}$be two positive numbers. Assume that$\alpha_{22}>0$and assume also that there is a constant$M_{r,T}>0 $depending only on$r, T,\Omega$and the coefficients of \eqref{e12} such that $$\|\nabla u\|_{L^r(Q_T)}\leq M_{r,T}.$$ Then for any$q>1$, there exists a constant$C(r,q,T)>0such that \begin{aligned} &\|v(.,t)\|^q_{L^q(\Omega)}+\|\nabla(v^{q/2})\|^2_{L^2(Q_t)} +\|\nabla(v^{(q+1)/2})\|^2_{L^2(Q_t)}\\ &\leq C(r,q,T)\big(1+\|v\|^{q-1}_{L^{\frac{p_r(q-1)}{2}}(Q_t)}\big).\label{e31} \end{aligned} \end{lemma} \begin{proof} For any constantq>1$, multiplying the second equation of \eqref{e12} by$qv^{q-1}and using the integration by parts, we obtain \begin{align*} &\frac{\partial}{\partial t}\int_{\Omega}v^{q}dx \\ &= q\int_{\Omega}v^{q-1}\nabla\cdot[(d_{2}+\alpha_{21}u+2\alpha_{22}v)\nabla v+\alpha_{21}v\nabla u]dx +q\int_{\Omega}v^{q}(a_{2}+b_{2}u-c_{2}v) dx \\ &= -q(q-1)\int_{\Omega}v^{q-2}(d_{2}+\alpha_{21}u+2\alpha_{22}v)|\nabla v|^{2} dx -\alpha_{21}(q-1)\int_{\Omega}\nabla(v^{q})\cdot\nabla u\,dx \\ &\quad +q\int_{\Omega}v^{q}(a_{2}+b_{2}u-c_{2}v) dx \\ &\le -q(q-1)d_{2}\int_{\Omega}v^{q-2}|\nabla v|^{2} dx-2\alpha_{22}q(q-1)\int_{\Omega}v^{q-1}|\nabla v|^{2} dx \\ &\quad -\alpha_{21}(q-1)\int_{\Omega}\nabla(v^{q})\cdot\nabla u\,dx +q\int_{\Omega}v^{q}(a_{2}+b_{2}u-c_{2}v) dx \\ &= -\frac{4(q-1)d_{2}}{q}\int_{\Omega}|\nabla(v^{\frac{q}{2}})|^{2}dx -\frac{8\alpha_{22}q(q-1)}{(q+1)^{2}}\int_{\Omega} |\nabla(v^{\frac{q+1}{2}})|^{2}dx \\ &\quad -\alpha_{21}(q-1)\int_{\Omega}\nabla(v^{q})\cdot\nabla u\,dx +q\int_{\Omega}v^{q}(a_{2}+b_{2}u-c_{2}v) dx. %\label{e32)} \end{align*} Integrating the above inequality from0$to$t, we have \begin{aligned} & \int_{\Omega}v^{q}(x,t)dx+\frac{4(q-1)d_{2}}{q} \int_{Q_{t}}|\nabla(v^{\frac{q}{2}})|^{2}dx\,dt +\frac{8\alpha_{22}q(q-1)}{(q+1)^{2}} \int_{Q_{t}}|\nabla(v^{\frac{q+1}{2}})|^{2}\,dx\,dt \\ &\le \int_{\Omega}v^{q}(x,0)dx-\alpha_{21}(q-1)\int_{Q_t}\nabla(v^{q}) \cdot\nabla u\,dx\,dt+q\int_{Q_t}v^{q}(a_{2}+b_{2}u-c_{2}v) \,dx\,dt. \end{aligned} \label{e33} By H\"older's inequality, we have \begin{aligned} &q\int_{Q_t}v^{q}(a_{2}+b_{2}u-c_{2}v) \,dx\,dt\\ &= a_2q\int_{Q_t}v^{q} dx dt-c_2q\int_{Q_t}v^{q+1}dx dt+b_2q\int_{Q_t}uv^{q}dx dt\\ &\leq -c_2q\|v\|^{q+1}_{L^{q+1}(Q_t)}+|a_2|q|Q_T|^{\frac{1}{q+1}}\|v\|^{q}_{L^{q+1}(Q_t)} +b_2q\int_{Q_t}uv^{q}dx dt\\ &\leq -c_2q\|v\|^{q+1}_{L^{q+1}(Q_t)}+|a_2|q \big[\varepsilon\|v\|^{q+1}_{L^{q+1}(Q_t)} +\varepsilon^{-q}|Q_T|^{\frac{q}{q+1}}\big] +b_2q\int_{Q_t}uv^{q}dx dt\\ &\leq B_1+b_2q\int_{Q_t}uv^{q}dx dt, \end{aligned}\label{e34} where\varepsilon=\frac{c_2}{|a_2|}$,$B_1$depends on$T, q, |\Omega|$and the coefficients of \eqref{e12}. On the other hand, since that$\frac{1}{r}+\frac{1}{2}+\frac{1}{p_r}=1, using the H\"older's inequality and Poincar\'e inequality, we have \begin{aligned} \int_{Q_t}uv^{q}dx dt &= \int_{Q_t}u\cdot v^{\frac{q-1}{2}}\cdot v^{\frac{q+1}{2}}\,dx\,dt\\ &\leq \|v^{\frac{q-1}{2}}\|_{L^{p_r}(Q_t)}\cdot \|v^{\frac{q+1}{2}}\|_{L^{2}(Q_t)}\cdot\|u\|_{L^{r}(Q_t)}\\ &\leq C_4m\|v\|^{(q-1)/2}_{L^{\frac{p_r(q-1)}{2}}(Q_t)}\cdot\| \nabla(v^{\frac{q+1}{2}})\|_{L^{2}(Q_t)}. \end{aligned} \label{e35} The substitution \eqref{e35} into \eqref{e34} leads to \begin{aligned} q\int_{Q_t}v^{q}(a_{2}+b_{2}u-c_{2}v) dx dt &\leq B_1+C_5\|v\|^{(q-1)/2}_{L^{\frac{p_r(q-1)}{2}}(Q_t)} \cdot\|\nabla(v^{\frac{q+1}{2}})\|_{L^{2}(Q_t)}. \label{e36} \end{aligned} Since that\frac{1}{r}+\frac{1}{2}+\frac{1}{p_r}=1$and$\nabla u$is in$L^r(Q_T), using the H\"older's inequality, we have \begin{align*} \big|-\int_{Q_t}\nabla(v^{q})\cdot\nabla u\,\,dx\,dt\big| &= \frac{2q}{q+1}\big|\int_{Q_t}v^{\frac{q-1}{2}}\cdot \nabla(v^{\frac{(q+1)}{2}})\cdot\nabla u\,dx\,dt\big|\\ &\leq \frac{2q}{q+1}\|v^{\frac{q-1}{2}}\|_{L^{p_r}(Q_t)} \cdot\|\nabla(v^{\frac{q+1}{2}})\|_{L^2(Q_t)} \cdot\|\nabla u\|_{L^r(Q_t)}\\ &\leq \frac{2q}{q+1}\|v\|^{\frac{q-1}{2}}_{L^{\frac{p_r(q-1)}{2}}(Q_t)} \cdot\|\nabla(v^{\frac{q+1}{2}})\|_{L^2(Q_t)} \cdot\|\nabla u\|_{L^r(Q_t)}\\ &\leq \frac{2q}{q+1}M_{r,T}\|v\|^{\frac{q-1}{2}}_{L^{\frac{p_r(q-1)}{2}} (Q_t)}\cdot\|\nabla(v^{\frac{q+1}{2}})\|_{L^2(Q_t)}. %\label{e37} \end{align*} The substitution \eqref{e36} and the above inequality into \eqref{e33} leads to \begin{aligned} &\int_{\Omega}v^{q}(x,t)dx+\frac{4(q-1)d_{2}}{q} \int_{Q_{t}}|\nabla(v^{\frac{q}{2}})|^{2}dx\,dt +\frac{8\alpha_{22}q(q-1)}{(q+1)^{2}}\int_{Q_{t}} |\nabla(v^{\frac{q+1}{2}})|^{2}\,dx\,dt \\ &\le B_2+C_6\|v\|^{\frac{q-1}{2}}_{L^{\frac{p_r(q-1)}{2}}(Q_t)} \cdot\|\nabla(v^{\frac{q+1}{2}})\|_{L^2(Q_t)} \\ &\le B_2+\frac{C_6}{4\varepsilon}\|v\|^{q-1}_{L^{\frac{p_r(q-1)}{2}} (Q_t)}+C_6\varepsilon\|\nabla(v^{\frac{q+1}{2}})\|^2_{L^2(Q_t)}, \end{aligned}\label{e38} whereB_2>0$depending on$q, T,\Omega$coefficients of \eqref{e12} and initial datal$v_0$. For any$\varepsilon>0$, from \eqref{e38} and by choosing a sufficiently small$\varepsilon$, such that$C_6\varepsilon<\frac{8\alpha_{22}q(q-1)}{(q+1)^{2}}$, we get \eqref{e31}. This completes the proof of the lemma. \end{proof} For any number$a$, we denote$a_+=\max\{a,0\}$. \begin{proposition} \label{prop3.3} Let$\alpha_{22}>0$. \begin{itemize} \item[(i)] If$\alpha_{11}>0$, then there is a constant$C_7(T)>0$such that $$\|v\|_{V_2(Q_T)}\leq C_7(T).$$ Moreover, for any constant$r<\frac{4(n+1)}{(n-2)_+}$, there exists a positive constant$C_{T}$such that $$\|v\|_{L^r(Q_T)}\leq C_{T}.$$ \item[(ii)] If$\alpha_{11}=0$, then $$\|v\|_{L^r(Q_T)}\leq C_{T} \quad\text{for any}\quad r>1.$$ \end{itemize} \end{proposition} \begin{proof} (i) Set$w=v^{(q+1)/2}$so that$v^q=w^{2q/(q+1)}$and$v^{q+1}=w^2. Then \begin{align*} E&\equiv \sup_{0\leq t\leq T} \int_\Omega v^q(x,t)dx +\int_{Q_T}|\nabla(v^{(q+1)/2})|^2\,dx\,dt\\ &= \sup_{0\leq t\leq T} \int_\Omega w^{2q/q+1}dx +\int_{Q_T}|\nabla w|^2\,dx\,dt. \end{align*} Letr_0=4$,$p_0=\frac{2r_0}{r_0-2}$. By Lemma \ref{lem3.1}, we see that$\nabla u$is in$L^{r_0}(Q_T)$. So, from Lemma \ref{lem3.2}, we have $$E+\|\nabla(v^{\frac{q}{2}})\|^2_{L^2(Q_T)} \leq C(r_0,q,T)\Big(1+\|w\|^{\frac{2(q-1)}{q+1}}_{L^{\frac{p_0(q-1)}{q+1}}(Q_T)} \Big), \label{e39}$$ where$C(r_0,q,T)>0$depending only$T, \Omega$, initial data$u_0, v_0$and the coefficients of \eqref{e12}. Since$q>1$, if we restrict our$q$so that $$(np_0-2n-4)q\leq 2n+np_0.\label{e310}$$ Then,$\frac{p_0(q-1)}{q+1}\leq\widetilde{q}$, where$\widetilde{q}=2+\frac{4q}{n(q+1)}$. Therefore, by H\"older's inequality $$\|w\|_{L^{\frac{p_0(q-1)}{q+1}}(Q_T)}\leq C_8(q,T)\|w\|_{L^{\widetilde{q}}(Q_T)},\label{e311}$$ where$C_8(q,T)=|Q_T|^{\frac{q+1}{p_0(q-1)}-\frac{1}{\widetilde{q}}}$. Setting$\widetilde{\beta}=2/(q+1)\in(0,1)$, by Lemma \ref{lem2.2} we have $$\|w(.,t)\|_{L^{\widetilde{\beta}}(\Omega)}=\|v(.,t)\|^{\frac{1}{\widetilde{\beta}}}_{L^1(\Omega)} \leq ( C_1(T))^{\frac{1}{\widetilde{\beta}}},\quad \forall t\in[0,T).\label{e312}$$ Hence, by Lemma \ref{lem2.5} and the definition of$E$, \eqref{e312} yields $$\|w\|_{L^{p_0(q-1)/q+1}(Q_T)}\leq C_8(q,T)\|w\|_{L^{\widetilde{q}}(Q_T)}\leq C_8(q,T)M_1\{1+E^{2/n\widetilde{q}}E^{\frac{1}{\widetilde{q}}}\}. \label{e313}$$ Then \eqref{e39} together with the above inequality, we can find a constant$C_9(q,T)>0$such that $$E\leq C_9(q,T)(1+E^\mu E^\nu)\label{e314}$$ with $$\mu=\frac{4(q-1)}{n\widetilde{q}(q+1)}, \quad \nu=\frac{2(q-1)}{\widetilde{q}(q+1)}.$$ Since $\mu+\nu = \frac{2(q-1)}{\widetilde{q}(q+1)}\big[\frac{2}{n}+1\big] <\frac{1}{\widetilde{q}}\big[\frac{4q}{n(q+2)}+2\big]=1,$ it is easy to see from \eqref{e314} that$E$is bounded. Therefore, from \eqref{e313} and \eqref{e314} we get$w\in L^{\widetilde{q}}(Q_T)$which in turn implies that$v\in L^{r}(Q_T)$with$r=\frac{\widetilde{q}(q+1)}{2}$for any$q$satisfying \eqref{e310}. Now, looking at \eqref{e310}, if$n\leq2$, we have $$np_0-2n-4=2(n-2)\leq0,\label{e315}$$ then \eqref{e310} holds for all$q$. so for$n\leq2$,$v\in L^r(Q_T)$for all$r>1$. Now, suppose that$n>2, we see \eqref{e310} is equivalent to 10 and r<\frac{4(n+2)}{(n-2)_+}. Next, we consider the case \alpha_{11}=0. By H\"older's inequality, we have \begin{aligned} &q\int_{Q_t}v^{q}(a_{2}+b_{2}u-c_{2}v) \,dx\,dt \\ &= a_2q\int_{Q_t}v^{q} dx dt-c_2q\int_{Q_t}v^{q+1}dx dt+b_2q\int_{Q_t}uv^{q}dx dt\\ &\leq -c_2q\|v\|^{q+1}_{L^{q+1}(Q_t)}+|a_2|q|Q_T|^{\frac{1}{q+1}}\|v\|^{q}_{L^{q+1}(Q_t)}\\ &\quad +b_2q\|v\|^{q}_{L^{q+1}(Q_t)}\cdot\|u\|_{L^{q+1}(Q_t)}\\ &\leq -c_2q\|v\|^{q+1}_{L^{q+1}(Q_t)}+|a_2|q|Q_T|^{\frac{1}{q+1}}\|v\|^{q}_{L^{q+1}(Q_t)} +b_2qm\|v\|^{q}_{L^{q+1}(Q_t)}\\ &\leq -c_2q\|v\|^{q+1}_{L^{q+1}(Q_t)}+q \varepsilon\|v\|^{q+1}_{L^{q+1}(Q_t)}+B_3\\ &\leq B_3, \hspace{8cm}\label{e316} \end{aligned} where \varepsilon=c_2 and B_3>0 which depends only on T, q, |\Omega|, \|u_0\|_{L^{\infty}(\Omega)} and the coefficients of \eqref{e12}. We can integrate by parts once to obtain from Lemma \ref{lem2.1} and analogue of \cite[Theorem 9.1, p. 341-342]{Ladyzenskaja} for Neumann boundary condition \cite[p.351]{Ladyzenskaja} \begin{aligned} &\big|-\int_{Q_t}\nabla(v^{q})\cdot\nabla u\,\,dx\,dt\big|\\ &= \big|-\int_{Q_t}v^{q}\Delta u\,\,dx\,dt\big|\\ &\leq \|v\|^{q}_{L^{q+1}(Q_T)}\cdot\|\Delta u\|_{L^{q+1}(Q_T)}\\ &\leq C_{10} \|v\|^{q}_{L^{q+1}(Q_T)} \Big(\|u(a_1-b_1u-c_1v)\|_{L^{q+1}(Q_T)}+\|u_0\| _{W^{2-\frac{2}{q+1}}_{q+1}(\Omega)}\Big)\\ &\leq C_{11}\Big(1+\|v\|^{q+1}_{L^{q+1}(Q_T)}\Big). \end{aligned}\label{e317} The substitution of \eqref{e316} and \eqref{e317} into \eqref{e33} leads to $$\sup_{0\leq t\leq T} \|v^q(t)\|^q_{L^q(\Omega)} +\|\nabla(v^{(q+1)/2})\|_{L^2(Q_T)}^2 \leq C_{12}\big(1+\|v\|^{q+1}_{L^{q+1}(Q_T)}\big).\label{e318}$$ We introduce w=v^\frac{q+1}{2}, then \eqref{e318} leads to $$E\equiv\sup_{0\leq t\leq T} \|w(t)\|^{\frac{2q}{q+1}} _{L^{\frac{2q}{q+1}}(\Omega)} +\|\nabla w\|^2_{L^2(Q_T)} \leq C_{12}\big(1+\|w\|^{2}_{L^{2}(Q_T)}\big). \label{e319}$$ Recall that Lemma \ref{lem2.2} implies v\in L^2(Q_T), so \|w\|_{L^{\frac{4}{q+1}}(Q_T)}\leq C_{13}. Since \frac{4}{q+1}<2\leq\widetilde{q}. Then we see from H\"older's inequality $$\|w\|^{2}_{L^{2}(Q_T)}\leq\|w\|^{2(1-\lambda)}_{L^{\widetilde{q}}(Q_T)} \|w\|^{2\lambda}_{L^{\frac{4}{q+1}}(Q_T)} \leq C_{13}^{2\lambda}\|w\|^{2(1-\lambda)}_{L^{\widetilde{q}}(Q_T)},\label{e320}$$ where \lambda=(\frac{1}{2}-\frac{1}{\widetilde{q}})/(\frac{q+1}{4} -\frac{1}{\widetilde{q}}). Setting \widetilde{\beta}=2/(q+1)\in(0,1), we have \|w(.,t)\|_{L^{\widetilde{\beta}}(\Omega)} =\|v(.,t)\|^{\frac{1}{\widetilde{\beta}}}_{L^1(\Omega)} \leq C_1(T)^{\frac{1}{\widetilde{\beta}}} for all t\in[0,T) by Lemma \ref{lem2.2}. Then it follow from \eqref{e319}, \eqref{e320} and Lemma \ref{lem2.5} that $$E\leq C_{14}(1+E^{\alpha})\label{e321}$$ with \alpha=\frac{2(1-\lambda)}{\widetilde{q}}\big(\frac{2}{n}+1\big)<1. $$Thus \eqref{e321} implies$$ \sup_{0\leq t\leq T}\|w(t)\|^{\frac{2q}{q+1}}_{L^{\frac{2q}{q+1}}(\Omega)}\leq E\leq C_{15} $$with some C_{15}>0, let r=q>1, so that \sup_{0\leq t\leq T} \|v(t)\|_{L^{r}(\Omega)} \leq C_{T} and the proof is complete. \end{proof} \section{Proof of Theorem \ref{thm1.1}} The first step of the proof is to show v is in L^{r}(Q_T) for any r>1. \begin{lemma} \label{lem4.1} Let \alpha_{11}>0 and suppose that there are r_1>\max\{\frac{n+2}{2},3\} and a positive constant C_{r_1,T} such that$$ \|v\|_{L^{r_1}(Q_T)}\leq C_{r_1,T}. $$Then, v is in L^r(Q_T) for any r>1. \end{lemma} \begin{proof} The proof is almost identical to \cite[Lemma 4.1]{Tuoc2}, but for completeness we repeat it here. First, the equation for u can be written in the divergence form as $$u_t=\nabla\cdot[(d_1+2\alpha_{11}u)\nabla u]+u(a_1-b_1u-c_1v),\label{e41}$$ where d_1+2\alpha_{11}u is bounded in \overline{Q}_T by Lemma \ref{lem2.1} and u(a_1-b_1u-c_1v) is in L^{r_1} with r_1>\frac{n+2}{2}. Application of the H\"older continuity result in \cite[Theorem 10.1, p. 204]{Ladyzenskaja} to \eqref{e41} yields $$u\in C^{\beta,\frac{\beta}{2}}(\overline{Q}_T) \quad\text{with some } \beta>0. \label{e42}$$ Moreover, we have $$w_{1t}=(d_1+2\alpha_{11}u)\Delta w_1+f_1, \label{e43}$$ where w_1=(d_1+\alpha_{11}u)u is as in the proof of Lemma \ref{lem2.3}, f_1=(d_1+2\alpha_{11}u)u(a_1-b_1u-c_1v). Since u is bounded and by the assumption of this Lemma, we see that f_1 is in L^{r_1}(Q_T). From \eqref{e42}, Lemma \ref{lem2.1} and Proposition \ref{prop3.3}, applying \cite[Theorem 9.1, pp. 341-342]{Ladyzenskaja} and its remark \cite[P. 351]{Ladyzenskaja}, we have $$w_1\in W^{2,1}_{r_1}(Q_T).\label{e44}$$ This implies \nabla u=\frac{1}{d_1+2\alpha_{11}u}\nabla w_1 in L^{r_1}(Q_T). Now, following the proof of Proposition \ref{prop3.3} with r_1 instead of r_0 and p_1=\frac{2r_1}{r_1-2} instead of p_0, we see that either v is in L^r(Q_T) for any r>1 or else v is in L^{r_2}(Q_T) with$$ r_2:=\frac{(n+1)r_1}{n+2-r_1}. $$The later case happens if and only if n+2-r_1>0. If v is in L^{r_2}(Q_T), we see that f_1 is in L^{r_2}(Q_T). Therefore, applying \cite[Theorem 9.1, p. 341-342]{Ladyzenskaja} and its remark \cite[p. 351]{Ladyzenskaja} again, we have \nabla u in L^{r_2}(Q_T). Then we go back and do the same argument again. Keep doing likes this we will get a sequence of numbers $$r_{k+1}:=\frac{(n+1)r_k}{n+2-r_k}.\label{e45}$$ We stop and get the conclusion that v is in L^r(Q_T) for any r>1 when $$n+2-r_k\leq0.\label{e46}$$ Since r_1>3, from \eqref{e45} we can prove by induction that r_k>3,k=1,2,\dots . Then, we have $$\frac{r_{k+1}}{r_{k}}=\frac{n+1}{n+2-r_k}\geq\frac{n+1}{n-1}>1.\label{e47}$$ Thus, the sequence r_k is strictly increasing. Therefore, there must be some k such that \eqref{e46} holds. we stop at this k and conclude that v is in L^r(Q_T) for any r>1, namely, there is a positive constant C_{16} such that \|v\|_{L^r(Q_T)}\leq C_{16}, where C_{16}>0 depending on q, T, \Omega and the coefficients of the system \eqref{e12} but not on r. \end{proof} So, from Proposition \ref{prop3.3} and Lemma \ref{lem4.1}, we have the following lemma. \begin{lemma} \label{lem4.2} Let \alpha_{22}>0 and suppose \mathrm{(i)} \alpha_{11}=0 or \mathrm{(ii)} \alpha_{11}>0 and n<10. Then there exists M_2 such that$$ \|v\|_{L^r(Q_T)}\leq M_2 \quad\text{for any } r>1. $$Moreover, for any r>1, v is in V_2(Q_T). \end{lemma} \begin{proof}[Proof of Theorem \ref{thm1.1}] We give the proof only in case \alpha_{11}>0 because the proof for \alpha_{11}=0 is essentially the same. By Lemma \ref{lem4.2}, v is bounded in \overline{Q}_T. From \eqref{e43}, we have$$ w_{1t}=(d_1+2\alpha_{11}u)\Delta w_1+f_1, $$where f_1=(d_1+2\alpha_{11}u)u(a_1-b_1u-c_1v) is bounded in \overline{Q}_T by Lemma \ref{lem2.1} and Lemma \ref{lem4.2}, (d_1+2\alpha_{11}u)\in C^{\beta,\frac{\beta}{2}}(Q_T) by \eqref{e42}. By \cite[Theorem 9.1, p.341-342]{Ladyzenskaja}, we have$$ \|w_1\|_{W^{2,1}_r}(Q_T)u^2(\alpha_{21}\overline{v})^2.\label{e52} By the Lemma \ref{lem2.1} and Theorem \ref{thm1.1}, the condition \eqref{e14} implies \eqref{e52}. Therefore, when all conditions in Theorem \ref{thm1.2} hold, there exists positive constant\delta$depending on$b_1, b_2, c_1$and$c_2$such that $$\frac{dH(u,v)}{dt}\leq -\delta\int_{\Omega}[(u-\bar{u})^{2}+(v-\bar{v})^{2}]dx.\label{e53}$$ To obtain the uniform convergence of the solution to \eqref{e12}, we recall the following result which can be find in \cite{Wang}. \begin{lemma} \label{lem5.1} Let$a$and$b$positive constant. Assume that$\varphi,\psi\in C^1[a,+\infty)$,$\psi(t)\geq0$,$\varphi$is bounded. If$\varphi'(t)\leq-b\psi(t)$and$\psi'(t)$is bounded in$[a,+\infty)$, then$\lim_{t\to\infty}\psi(t)=0$. \end{lemma} Using integration by parts, H\"older's inequality, Lemma \ref{lem2.1}, and Lemma \ref{lem4.2}, one can easily verify that$\frac{d}{dt}\int_{\Omega}[(u-\bar{u})^{2}+(v-\bar{v})^{2}]dx$is bounded from above. Then from Lemma \ref{lem5.1} and \eqref{e53}, we have $$\|u(\cdot,t)-\overline{u}\|_{L^{\infty}(\Omega)}\to 0,\quad \|v(\cdot,t)-\overline{v}\|_{L^{\infty}(\Omega)}\to0\quad(t\to\infty).$$ Namely,$(u,v)$converges uniformly to$(\overline{u},\overline{v})$. By the fact that$H(u,v)$is decreasing for$t\geq0$, it is obvious that$(\overline{u},\overline{v})\$ is global asymptotic stable, and the proof of Theorem \ref{thm1.2} is complete. \end{proof} \subsection*{Acknowledgements} The author would like to thank professor Sheng-mao Fu for the encouragement and useful discussions, also the anonymous referee for the very careful reading of the original manuscript and the helpful suggestions. \begin{thebibliography}{99} \bibitem{Shigesada} N. Shigesada, K. Kawasaki, E. Teramoto, \emph{Spatial segregation of interacting species}, J. Theor. Biology, 79(1979), 83-99. \bibitem{Lou} Y. Lou, W. Ni, Y. Wu, \emph{On the global existence of a cross-diffusion system}, Discrete Contin. Dynam. Systems, 4(1998), 193-203. \bibitem{Li} Y. Li, C. Zhao, \emph{Global existence of solutions to a cross-diffusion system in higher dimensional domains}, Discrete Contin. Dynam. Systems, 12(2005), 193-203. \bibitem{Shim1} Seong-A Shim, \emph{Uniform boundedness and convergence of solutions to the systems with cross-diffusion dominated by self-diffusion}, Nonlinear Analysis: RWA, 4(2003), 65-86. \bibitem{Choi1} Y. S. Choi, R. Lui, Y. Yamada, \emph{Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion}, Discrete Contin. Dynam. Systems, 9(2003), 1193-1200. \bibitem{Choi2}Y. S. Choi, R. Lui, Y. Yamada, \emph{Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion}, Discrete Contin. Dynam. Systems, 10(2004), 719-730. \bibitem{Le} D. Le, T. Nguyen, \emph{Global existence for a class of triangular parabolic system on domains of arbitary dimension}, Proceedings of AMS, 7(2005), 1985-1992. \bibitem{Tuoc1} P. V. Tuoc, \emph{Global eistence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion system on domains of arbitary dimensions}, Proceedings of AMS, 135(2007), 3933-3941. \bibitem{Tuoc2} P. V. Tuoc, \emph{On global existence of solutions to a cross-diffusion system}, IMA Preprint Series 2149. \bibitem{Kuto1} K. Kuto, Y. Yamada, \emph{Multiple coexistence states for a prey-predator system with cross-diffusion}, J. Differential Equations, 197(2004), 315-348. \bibitem{Ahmed} E. Ahmed, A. S. Hegazi, A. S. Elgazzar, \emph{On persistence and stability of some biological systems with cross diffusion}, Adv. Compelex Syst, 7(2004), 65-76. \bibitem{Kuto2}K. Kuto, \emph{Stability of steady-state solutions to a prey-predator system with cross-diffusion}, J. Differential Equations, 197(2004), 293-314. \bibitem{Nakashima} K. Nakashima, Y. Yamada, \emph{Positive steady ststes for prey-predator models with cross-diffusion}, Adv. Differential Equations, 1(1996), 1099-1122. \bibitem{Ryu} K. Ryu, I. Ahn, \emph{Positive steady-ststes for two interacting species models with linear self-cross diffusions}, Discrete Contin. Dynam. Systems, 9(2003), 1049-1061. \bibitem{Amann1} H. Amann, \emph{Dynamic theory of quasilinear parabolic equations: Abstract evolution equations}, Nonlinear Analysis, 12(1988), 859-919. \bibitem{Amann2} H. Amann, \emph{Dynamic theory of quasilinear parabolic equations: Reaction-diffusion}, Diff. Int. Eqs, 3(1990), 13-75. \bibitem{Amann3} H. Amann, \emph{Dynamic theory of quasilinear parabolic equations: Global existence}, Math. Z., 202(1989), 219-250. \bibitem{Shim2} Seong-A Shim, \emph{Long-time properties of prey-predator system with cross-diffusion}, Commum. Korean Math. Soc. 21(2006), 293-320. \bibitem{Shim3} Seong-A Shim, \emph{Global existence of solutions to the prey-predator system with a singel cross-diffusion}, Bull. Korean Math. Soc. 43(2006), 443-459. \bibitem{Ladyzenskaja} O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Ural'ceva, \emph{Linear and quasilinear equations of parabolic type}, Translations of Mathematical Monographs, 23, AMS, 1968. \bibitem{Wang} M. X. Wang, \emph{Nonlinear Parabolic Equation of Parabolic Type}. Science Press, Beijing, 1993 (in Chinese). \end{thebibliography} \end{document} ?