\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 85, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2014/85\hfil Comparable almost periodic reaction-diffusion systems] {Convergence in comparable almost periodic reaction-diffusion systems with Dirichlet boundary conditions} \author[F. Cao, Y. Fu\hfil EJDE-2014/85\hfilneg] {Feng Cao, Yelai Fu} % in alphabetical order \address{Feng Cao \newline Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China} \email{fcao@nuaa.edu.cn} \address{Yelai Fu \newline Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China} \email{fuyelai@126.com} \thanks{Submitted November 14, 2013. Published April 2, 2014.} \subjclass[2000]{37B55, 37L15, 35B15, 35K57} \keywords{Reaction-diffusion systems; asymptotic behavior; uniform stability; \hfill\break\indent skew-product semiflows} \begin{abstract} In this article, we study the asymptotic dynamics in nonmonotone comparable almost periodic reaction-diffusion systems with Dirichlet boundary condition, which are comparable with uniformly stable strongly order-preserving system. By appealing to the theory of skew-product semiflows, we obtain the asymptotic almost periodicity of uniformly stable solutions to the comparable reaction-diffusion system. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \newcommand\norm[1]{\|#1\|} \section{Introduction} In the previous 50 years or so, many concepts from dynamical systems have been applied to the study of partial differential equations (see \cite{chen,chen2,chen3,chow,chow2,Hale,Hen,ShenYi,S}, etc.). In this paper, we shall study the long-term behaviour of the solutions of some non-autonomous comparable reaction-diffusion equations. We consider the almost periodic reaction-diffusion system with Dirichlet boundary condition: \begin{equation}\label{1.1} \begin{gathered} \frac{\partial v_i}{\partial t}= d_i(t)\Delta v_i +F_i(t,v_1,\dots,v_n),\quad x\in \Omega,\; t>0,\\ v_i(t,x)=\,0, \quad x\in \partial\Omega,\; t>0, \\ v_i(0,x)=v_{0,i}(x), \quad x\in \bar{\Omega},\; 1\leq i\leq n, \end{gathered} \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary. $d=(d_1(\cdot),\dots,d_n(\cdot))\in C(\mathbb {R},\mathbb {R}^n)$ is assumed to be an almost periodic vector-valued function bounded below by a positive real vector. The nonlinearity $F=(F_1,\dots,F_n): \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n$ is $C^1$-admissible and uniformly almost periodic in $t$, and $F$ points into $\mathbb {R}_+^n$ along the boundary of $\mathbb {R}_+^n$: $F_i(t,v)\geq 0$ whenever $v\in \mathbb {R}_+^n$ with $v_i=0$ and $t\in \mathbb{R}^+$. However, $F$ has no monotonicity properties. To study the properties of the solutions of such a non-monotone equation, an effective approach is to exhibit and utilize certain comparison techniques (see \cite{ConSm,Bro1,Bro2,Sm}). As pointed out in \cite[Section 4]{Smi3}, the comparison technique involves monotone systems in a natural way: the original non-monotone systems are comparable with certain monotone ones. Thus, we assume that there exists a function $f:\mathbb{R}\times \mathbb{R}^n_+\to \mathbb{R}^n$ with $f(t,v)\geq F(t,v)$ (or $f(t,v)\leq F(t,v)$), $\forall(t,v)\in \mathbb{R}\times \mathbb{R}_+^n$. Also, we assume that $f$ satisfies (H1)--(H4) in section 2. Then we get a strongly order-preserving system (see section 2 for details): \begin{equation}\label{1.2} \begin{gathered} \frac{\partial u_i}{\partial t}= d_i(t)\Delta u_i +f_i(t,u_1,\dots,u_n),\quad x\in \Omega,\; t>0,\\ u_i(t,x)=\,0, \quad x\in \partial\Omega,\; t>0, \\ u_i(0,x)=u_{0,i}(x), \quad x\in \bar{\Omega},\; 1\leq i\leq n. \end{gathered} \end{equation} We want to know whether such a non-monotone system \eqref{1.1} inherits certain asymptotic behaviour from its strongly order-preserving partner \eqref{1.2}. Note that a unified framework to study nonautonomous equations is based on the so-called skew-product semiflows (see \cite{Sell,ShenYi}). Since even the strongly monotone (which is a stronger notion than strongly order-preserving) skew-product semiflows can possess very complicated chaotic attractors (see \cite{ShenYi}), we hence assume that the strongly order-preserving partner is `uniformly stable', and to establish the asymptotic 1-cover property of the corresponding strongly order-preserving skew-product semiflow. As far as we know, there are only a few works on the related topics. Jiang \cite{Jiang3} proved the global convergence of the comparable discrete-time or continuous-time system provided that all the equilibria of its monotone partner form a totally ordered curve. Recently, Cao, Gyllenberg and Wang\cite{CGW} established the asymptotic 1-cover property of the comparable skew-product semiflows, whose partner systems are eventually strongly monotone and uniformly stable. Here we emphasize that for reaction-diffusion system with Dirichlet boundary condition, the cone $X_+$ has empty interior in the state space $X=\Pi_1^n C_0(\bar{\Omega})$ (see section 2 for details). Thus, the skew-product semiflow generated by its partner is only strongly order-preserving, but not eventually strongly monotone (see \cite[Chapter 6]{HirSmi}). So we have to find another way to get the corresponding asymptotic dynamics for Dirichlet problem. Motivated by \cite{JZH}, to obtain the asymptotic behavior of solutions to comparable almost periodic reaction-diffusion system \eqref{1.1}, we first prove that every precompact trajectory of the strongly order-preserving system \eqref{1.2} is asymptotic to a 1-cover of the base flow (see Proposition \ref{prop3.3}). Based on this, for the uniformly stable and strongly order-preserving skew-product semiflow generated by \eqref{1.2}, we can get the topological structure of the set of the union of all 1-covers similarly as \cite{CGW} (see Lemma \ref{lem4.1}). With such tools, we are able to establish the 1-covering property of uniformly stable omega-limit sets of comparable skew-product semiflow (see Proposition \ref{prop4.3}), and thus obtain the asymptotic almost periodicity of uniformly stable solutions to system \eqref{1.1}. This article is organized as follows. In section 2, we present some basic definitions and our main result. In Section 3 we prove the main result. \section{Preliminaries and statement of the main result} A subset $S$ of $\mathbb{R}$ is said to be {\it relatively dense} if there exists $l>0$ such that every interval of length $l$ intersects $S$. A function $f$, defined and continuous on $\mathbb{R}$, is {\it almost periodic} if, for any $\varepsilon>0$, the set $T(f,\varepsilon)=\{s\in \mathbb{R}:|f(t+s)-f(t)|<\varepsilon,\, \forall t\in \mathbb{R}\}$ is relatively dense. A continuous function $f : \mathbb{R}\times \mathbb{R}^m \mapsto \mathbb{R}^n$ is said to be {\it admissible} if, for every compact subset $K \subset \mathbb{R}^m$, $f$ is bounded and uniformly continuous on $\mathbb{R}\times K$. Besides, if $f$ is of class $C^r (r \geq 1)$ in $x \in \mathbb{R}^m$, and $f$ and all its partial derivatives with respect to $x$ up to order $r$ are admissible, then we say that $f$ is $C^r$-{\it admissible}. A function $f \in C(\mathbb{R}\times \mathbb{R}^m,\mathbb{R}^n)$ is {\it uniformly almost periodic in} $t$, if $f$ is both admissible and almost periodic in $t\in \mathbb{R}$. Let $f \in C(\mathbb{R}\times \mathbb{R}^m,\mathbb{R}^n)$ be uniformly almost periodic, one can define the Fourier series of $f$ (see \cite{ShenYi,Ve}), and the {\it frequency module} $\mathcal{M}(f)$ of $f$ as the smallest Abelian group containing a Fourier spectrum. Let $f,g\in C(\mathbb{R}\times \mathbb{R}^m,\mathbb{R}^n)$ be two uniformly almost periodic functions in $t$. One has $\mathcal{M}(f)=\mathcal{M}(g)$ if and only if the flow $(H(g),\mathbb{R})$ is isomorphic to the flow $(H(f),\mathbb{R})$ (see, \cite{Fi} or \cite[Section 1.3.4]{ShenYi}). Here $H(f)={\rm cl}\{f\cdot\tau:\tau\in \mathbb{R}\}$ is called the {\it hull of $f$}, where $f\cdot\tau(t,\cdot)=f(t+\tau,\cdot)$ and the closure is taken under the compact open topology. Let $(Y,d_Y)$ be a compact metric space with metric $d_Y$. A \emph{continuous flow} $\sigma: \mathbb{R}\times Y \to Y$, $(t,y) \to \sigma{(t,y)}=\sigma_t(y)=y\cdot t$ is called \emph{minimal} if $Y$ has no other nonempty compact invariant subset but itself. Here a subset $Y_1 \subset Y$ is \emph{invariant} if $\sigma_{t}(Y_1) = Y_1$ for every $t \in \mathbb{R}$. Consider the almost periodic reaction-diffusion system with Dirichlet boundary condition \begin{equation}\label{IBVP-sys} \begin{gathered} \frac{\partial v_i}{\partial t} = d_i(t)\Delta v_i +F_i(t,v_1,\dots,v_n),\quad x\in \Omega, \; t>0,\\ v_i(t,x)=\,0, \quad x\in \partial\Omega,\; t>0,\\ v_i(0,x)=v_{0,i}(x), \quad x\in \bar{\Omega},\; 1\leq i\leq n, \end{gathered} \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary. $\Delta$ is the Laplacian operator on $\mathbb{R}^n$. Let $d=(d_1(\cdot),\dots,d_n(\cdot))\in C(\mathbb{R},\mathbb{R}^n)$ be an almost periodic vector-valued function and for some $d_0 > 0$, $d_i(t) \geq d_0$, for all $t \in \mathbb{R}$, $1 \leq i \leq n$. The nonlinearity $F=(F_1,\dots,F_n): \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n$ is $C^1$-admissible and uniformly almost periodic in $t$. Let $v=(v_1,\dots,v_n)$, we also assume that \begin{itemize} \item[(I1)] $F_i(t,v)\geq 0$ whenever $v\in \mathbb{R}_+^n$ with $v_i=0$ and $t\in \mathbb{R}^+$. \end{itemize} Denote $X=\Pi_1^n C_0(\bar{\Omega})$ ($C_0(\bar{\Omega}):=\{\phi\in C(\bar{\Omega},\mathbb{R}):\phi|_{\partial\Omega}=0\}$) and the standard cone $X_+=\{u\in X:u(x)\in \mathbb{R}_+^n,x\in\bar{\Omega}\}$. Then the cone $X_+$ induces an \emph{ordering} on $X$ via $x_{1}\leq x_{2}$ if $x_{2}- x_{1}\in X_+$. We write $x_{1}< x_{2}$ if $x_{2}- x_{1}\in X_+\setminus \{0\}$. Let $x\in X$ and a subset $U\subset X$. We write $x<_r U$ if $x<_r u$ for all $u\in U$. Given two subsets $A$, $B\subset X$, we write $A<_r B$ if $a<_r b$ holds for each choice of $a\in A$, $b\in B$. Here $<_r$ represents $\leq$ or $<$. $x>_r U$ is similarly defined. Obviously, every compact subset in $X$ has both a greatest lower bound and a least upper bound. Let $H(d,F)$ be the hull of the function $(d,F)$. Then the time translation $(\mu,G)\cdot t$ of $(\mu,G)\in H(d,F)$ induces a compact and minimal flow on $H(d,F)$ (see \cite{Sell} or \cite{ShenYi}). By the standard theory of reaction-diffusion systems (see \cite[Chapter 6]{HirSmi}), it follows that for every $v_0\in X_+$ and $(\mu,G)\in H(d,F)$, the system \begin{equation}\label{IBVP-sys-g} \begin{gathered} \frac{\partial v_i}{\partial t}= \mu_i(t)\Delta v_i +G_i(t,v),\quad x\in \Omega,\; t>0,\\ v_i(t,x)=0, \quad x\in \partial\Omega, \; t>0,\\ v(0,x)=v_{0}(x), \quad x\in \bar{\Omega},\; 1\leq i\leq n \end{gathered} \end{equation} admits a (locally) unique regular solution $v(t,\cdot,v_0;\mu,G)$ in $X_+$. This solution also continuously depends on $(\mu,G)\in H(d,F)$ and $v_0\in X_+$ (see \cite{Hen}). Thus, \eqref{IBVP-sys-g} induces a (local) skew-product semiflow $\Gamma$ on $X_+\times H(d,F)$ with $$ \Gamma_t(v_0,(\mu,G))=(v(t,\cdot,v_0;\mu,G),(\mu,G)\cdot t),\quad \forall (v_0,(\mu,G))\in X_+\times H(d,F), \,t\geq 0. $$ Now we assume that there exists a function $f\in C^1(\mathbb{R}\times\mathbb{R}^n_+,\mathbb{R}^n)$, which is $C^1$-admissible and uniformly almost periodic in $t$, satisfying \begin{itemize} \item[(H1)] $ f(t,v)\geq F(t,v)$ for all $(t,v)\in \mathbb{R}\times \mathbb{R}_+^n$. with its frequency module $\mathcal{M}(f)=\mathcal{M}(F)$ (thus $H(d,f)\cong H(d,F)$); \item[(H2)] $f_i(t,0)=0 \,(1\leq i\leq n)$; \item[(H3)] $\frac{\partial f_i}{\partial x_j}(t,x)\geq 0$ for all $1\leq i\neq j\leq n$, and there is a $\delta>0$ such that if two nonempty subsets $I,J$ of $\{1,2,\dots,n\}$ form a partition of $\{1,2,\dots,n\}$, then for any $(t,x)\in \mathbb{R}\times\mathbb{R}^n_+$, there exist $i\in I$, $j\in J$ such that $|\frac{\partial f_i}{\partial x_j}(t,x)|\geq \delta>0$; \item[(H4)] Every nonnegative solution of ordinary differential system $\dot{u} = g(t,u), g \in H(f)$, is bounded. \end{itemize} It is easy to see that, for any $(\mu,G)\in H(d,F)$, there exists a $(\mu,g)\in H(d,f)$ such that $$ g(t,v)\geq G(t,v) \textnormal{ for all } (t,v)\in \mathbb{R}\times \mathbb{R}_+^n. $$ Denote $Y=H(d,f)$. Then we can consider the reaction-diffusion system \begin{equation}\label{IBVP-sys-g-1} \begin{gathered} \frac{\partial u_i}{\partial t}= \mu_i(t)\Delta u_i +g_i(t,u),\quad x\in \Omega, \; t>0,\\ u_i(t,x)=0, \quad x\in \partial\Omega, \; t>0,\\ u(0,x)=u_{0}(x)\in X_+, \quad x\in \bar{\Omega},\; 1\leq i\leq n, \end{gathered} \end{equation} which induces the global skew-product semiflow \begin{equation}\label{equ6} \Pi_t:X_+\times Y\to X_+\times Y;\quad (u_0,y=(\mu,g))\mapsto (u(t,\cdot,u_0,y),y\cdot t),~ t\in \mathbb{R}^+, \end{equation} where $u(t,\cdot,u_0,y)$ is the unique regular global solution of \eqref{IBVP-sys-g-1} in $X_+$. Without any confusion, we also write $u(t,\cdot,u_0,y)$ as $u(t,u_0,y)$. Clearly, by the comparison principle and (H4), the forward orbit $O^+(x,y)= \{\Pi_{t}(x,y) : t\geq 0\}$ of any $(x,y)\in X_+\times Y$ is precompact. Thus the omega-limit set of $(x,y)$, defined by $\omega(x,y)=\{(\hat{x},\hat{y}) \in X_+\times Y : \Pi_{t_{n}}(x,y)\to (\hat{x},\hat{y}) (n\to\infty) \text{ for some sequence } t_{n}\to \infty \}$, is a nonempty, compact and invariant subset in $X_+\times Y$. A forward orbit $O^+(x_0,y_0)$ of $\Pi_t$ is said to be {\it uniformly stable} if for every $\varepsilon>0$ there is a $\delta=\delta(\varepsilon)>0$, called the {\it modulus of uniform stability}, such that for every $x\in X_+$, if $s\geq 0$ and $\norm{u(s,x_0,y_0)-u(s,x,y_0)}\leq \delta(\varepsilon)$ then $$ \norm{u(t+s,x_0,y_0)-u(t+s,x,y_0)}<\varepsilon \textnormal{ for each }t\geq 0. $$ Here we assume that every forward orbit of $\Pi_t$ in \eqref{equ6} is uniformly stable, which can be guaranteed by the existence of invariant functional. Let $P:X_+\times Y \to Y$ be the natural projection. A compact positively invariant set $K\subset X_+\times Y$ is called a {\it $1$-cover} of $Y$ if $P^{-1}(y)\cap K$ contains a unique element for every $y\in Y$. If we write the 1-cover $K=\{(c(y),y):y\in Y\}$, then $c:Y\to X$ is continuous with $\Pi_t(c(y),y)=(c(y\cdot t),y\cdot t)$, $\forall t\geq0$. For the sake of brevity, we hereafter also write $c(\cdot)$ as a {\it $1$-cover} of $Y$. For skew-product semiflows, we always use the order relation on each fiber $P^{-1}(y)$, and write $(x_1,y)\leq (<)\, (x_2,y)$ if $x_1\leq x_2$ ($x_10$ such that, whenever $(x_1,y)<(x_2,y)$ there exist open subsets $U$, $V$ of $X_+$ with $x_1\in U$, $x_2\in V$ satisfying $$ \Pi_{t}(U,y) < \Pi_{t}(V,y) \quad \text{for all } t\geq t_0. $$ $\Pi_t$ is called {\it fiber-compact} if there exists a $\bar{t}>0$ such that, for any $y\in Y$ and bounded subset $B\subset X$, $\Pi_t(B,y)$ has compact closure in $P^{-1}(y\cdot t)$ for every $t>\bar{t}$. Then according to (H3), \cite[Chapter 6]{HirSmi} and \cite[Section 6]{JZH}, one can obtain that $\Pi_t$ in \eqref{equ6} is strongly order-preserving and fibre-compact. By (H1), similarly as the proof of Lemma 5.2 in \cite{CGW}, we can get that $\Gamma_t$ is upper-comparable with respect to $\Pi_t$ in the sense that if $\Gamma_t(x_1,y)\leq\Pi_t(x_2,y)$ whenever $(x_1,y),(x_2,y)\in X_+\times Y$ with $(x_1,y)\leq(x_2,y)$. Now we are in a position to state our main result. \begin{theorem}\label{thm2.1} Any uniformly stable $L^\infty$-bounded solution of \eqref{IBVP-sys} is asymptotic to an almost periodic solution. \end{theorem} \begin{remark} \rm We note that for reaction-diffusion system with Dirichlet boundary condition \eqref{IBVP-sys}, the cone $X_+$ has empty interior in the state space $X=\Pi_1^n C_0(\bar{\Omega})$. Thus, the skew-product semiflow generated by its monotone partner \eqref{IBVP-sys-g-1} is only strongly order-preserving, but not eventually strongly monotone. Consequently, the results in \cite{CGW} can't be used to study the asymptotic behavior of the solutions to system \eqref{IBVP-sys}. \end{remark} \section{Proof of Theorem \ref{thm2.1}} To obtain the asymptotic almost periodicity of solutions to system \eqref{IBVP-sys}, we first investigate the asymptotic behavior of its strongly order-preserving partner. Motivated by \cite{JZH}, we establish the 1-cover property of omega limit sets for the strongly order-preserving and uniformly stable skew-product semiflows $\Pi_t$. The following result is adopted from \cite[P. 19]{RJGR} or \cite[P. 29]{ShenYi}, see also \cite[P. 634]{NOS}. \begin{theorem}\label{thm3.1} Let $\Theta_t$ be a skew-product semiflow on $X_+\times Y$. If a forward orbit $O^+_\Theta(x_0,y_0)$ of $\Theta_t$ is precompact and uniformly stable, then its omega-limit set $\omega_\Theta(x_0,y_0)$ admits a flow extension which is minimal. \end{theorem} Now fix $(x_0,y_0)\in X_+\times Y$ and let $K=\omega(x_0,y_0)$ be its omega-limit set with respect to $\Pi_t$. For any given $y\in Y$, we define $$ (p(y),y) = \text{g.l.b. of }K \cap P^{-1}(y) .$$ Then from \cite[Proposition 3.1]{JZH}, it follows that $\omega(p(y),y)$ is $1$-cover of $Y$. Denote $\{(p_{\ast}(y),y)\}=\omega(p(y),y)\cap P^{-1}(y) $, by \cite[Proposition 3.2]{JZH} one has \begin{equation}\label{3.0} u(t,p_\ast(y),y)=p_\ast(y\cdot t)\quad \text{ for any }y\in Y \text{ and } t\in\mathbb{R}. \end{equation} So we can denote the 1-cover $\omega(p(y),y)$ by $p_\ast(\cdot)$. \begin{lemma}\label{lem3.2} Assume that there exists a point $(z,y)\in K$ such that $p_{\ast}(y) < z$. Then for any $t\in \mathbb{R}$, there exist a neighborhood $U$ of $p_{\ast}(y)$ and a neighborhood $V$ of $z$ such that $$ u(t,U,y) < u(t,V,y). $$ \end{lemma} \begin{proof} By the minimality of $K$, for any $t\in \mathbb{R}$, there is $\tau_n \to +\infty$ such that $\tau_n + t \geq 0$ and $$ \Pi_{\tau_n} \circ \Pi_t (z,y) \to \Pi_t (z,y), \quad\text{as } n \to \infty. $$ Note that the monotonicity implies that $$ \Pi_{\tau_n} \circ \Pi_t (p_{\ast}(y),y) \leq \Pi_{\tau_n} \circ \Pi_t (z,y). $$ Letting $n \to \infty$, we then get $\Pi_t(p_{\ast}(y),y) \leq \Pi_t(z,y)$, thus, \begin{equation}\label{3.1} u(t,p_{\ast}(y),y) \leq u(t,z,y),\quad \forall t \in \mathbb{R}. \end{equation} Suppose that the conclusion of the lemma does not hold. Then we claim that there exists $r_0\in \mathbb{R}$ such that \begin{equation}\label{3.2} u(t,p_{\ast}(y),y) = u(t,z,y),\quad \forall t \leq r_0. \end{equation} Otherwise. By \eqref{3.1}, one has that for any $r \in \mathbb{R}$, there exists some $\bar{t}\leq r$ such that $$ u(\bar{t},p_{\ast}(y),y) < u( \bar{t},z,y). $$ Since $\Pi_t$ is strongly order-preserving, it follows that there exist a neighborhood $\bar{U}$ of $u(\bar{t},p_{\ast}(y),y)$ and a neighborhood $\bar{V}$ of $u(\bar{t},z,y)$ such that $$ u(r - \bar{t} + t_0,\bar{U},y\cdot \bar{t}) < u(r - \bar{t} + t_0,\bar{V},y\cdot \bar{t}). $$ Note that by the continuity of $\Pi_t$, there exist a neighborhood $\hat{U}$ of $p_{\ast}(y)$ with $u(\bar{t},\hat{U},y)\subset \bar{U}$, and a neighborhood $\hat{V}$ of $z$ with $u(\bar{t},\hat{V},y)\subset \bar{V}$. So we have \begin{equation*} u(r - \bar{t} + t_0,u(\bar{t},\hat{U},y),y\cdot \bar{t}) < u(r - \bar{t} + t_0,u(\bar{t},\hat{V},y),y\cdot \bar{t}). \end{equation*} Thus, \begin{equation*} u(r + t_0,\hat{U},y) < u(r + t_0,\hat{V},y). \end{equation*} Since $r$ is arbitrary, the conclusion of the lemma holds. A contradiction. So we proved the claim. By the minimality of $K$, we obtain that $\alpha(z,y) = K$. Hence, $(z,y)\in \alpha(z,y)$. Then it follows that there exists a sequence $\tau_n \to -\infty$ such that $\tau_n \leq r_0$ and $\Pi_{\tau_n} (z,y) \to (z,y)$. Thus the 1-cover property of $\omega(p_{\ast}(y),y)$ and \eqref{3.0} imply that $\Pi_{\tau_n} (p_{\ast}(y),y) \to (p_{\ast}(y),y)$. By \eqref{3.2}, one has $$u(\tau_n,p_{\ast}(y),y) = u(\tau_n,z,y).$$ By letting $n \to +\infty$, we get $$(p_{\ast}(y),y)=(z,y).$$ A contradiction to the assumption. This completes the proof. \end{proof} The following Proposition shows the 1-cover property of omega limit sets for $\Pi_t$. \begin{proposition}\label{prop3.3} For any $(x_0,y_0)\in{X_+\times Y}$, $\omega(x_0,y_0)$ is a 1-cover of $Y$. \end{proposition} \begin{proof} Now fix $(x_0,y_0)\in{X_+\times Y}$ and set $K = \omega(x_0,y_0)$. For any $y\in Y$, by \cite[Proposition 3.1]{JZH}, we have $(p_\ast(y),y)\leq K\cap P^{-1}(y)$. We claim that $\{(p_\ast(y),y)\}= K\cap P^{-1}(y)$ for all $y\in Y$. Suppose not. Then there exist some $y\in Y$ and a point $(\hat{z},y)\in K$ such that $p_{\ast}(y) < \hat{z}$. By the minimality of $K$, we get that \begin{equation*} p_{\ast}(y) < z,\quad \forall(z,y)\in K\cap P^{-1}(y). \end{equation*} Then it follows from Lemma \ref{lem3.2} that there exist a neighborhood $U_z$ of $p_{\ast}(y)$ and a neighborhood $V_z$ of $z$ such that \begin{equation}\label{3.4} U_z < V_z. \end{equation} Since $\{V_{z} : (z,y) \in K \cap {P^{-1}(y)}\}$ is an open cover of $K \cap {P^{-1}(y)}$, we can find a finite subcover, denoted by $\{V_1,V_2, \dots, V_n\}$. Note that by \eqref{3.4} there exist neighborhoods $U_i,~i=1,2,\dots,n$ of $p_{\ast}(y)$ such that $$ U_1 < V_1, \quad U_2 < V_2,\;\dots,\; U_n < V_n. $$ Therefore, $\cap_{i=1}^{n}{U_i} < \cup_{i=1}^{n}{V_i}$. Since $K \cap {P^{-1}(y)}\subset \cup_{i=1}^{n}{V_i}$, we have \begin{equation*} \cap_{i=1}^{n}{U_i} < K \cap {P^{-1}(y)}. \end{equation*} So we can take an $\epsilon_0 > 0$ such that \begin{equation}\label{3.5} B^+(p_{\ast}(y),\epsilon_0) < K \cap {P^{-1}(y)}, \end{equation} where $B^+(p_{\ast}(y),\epsilon_0) = \{ x \in X_+ : x \geq p_{\ast}(y),~\norm{x - p_{\ast}(y)} \leq \epsilon_0\}$. By the uniform stability of $\Pi_t(p_{\ast}(y),y)$, there exists $\delta_0 = \delta_0(\epsilon_0)\leq \epsilon_0$ such that \begin{equation*} \norm{ u - p_{\ast}(y) } \leq \epsilon_0,~\forall (u,y) \in \omega(x,y)\cap{P^{-1}(y)} \end{equation*} whenever $\norm{x - p_{\ast}(y)} \leq \delta_0$. Combing with \eqref{3.5}, we get $$ (p_{\ast}(y),y) \leq \omega(x,y) \cap{P^{-1}(y)} < K \cap {P^{-1}(y)} $$ for any $x\in B^+(p_{\ast}(y),\delta_0) $. Since $\omega(x,y)$ is minimal, using \cite[Proposition 3.1(3)]{JZH}, we obtain \begin{equation}\label{3.6} \omega(x,y) = \omega(p(y),y)=p_\ast(\cdot),\quad\forall x \in B^+(p_{\ast}(y),\delta_0). \end{equation} Set $$ L = \{\tau \in [0,1]: x_\tau = p_{\ast}(y) + \tau(\hat{z} - p_{\ast}(y)), \omega(x_\tau,y) = p_\ast(\cdot) \}. $$ By \eqref{3.6}, there exists a $\bar{\tau}> 0$ such that $[0,\bar{\tau}] \subset L$. It is easy to see that $L$ is an interval. Now we show that $L$ is closed, that is, $L = [0,\tau_0]$ with $0 < \tau_0=\sup\{\tau:\tau\in L\} <1$. Note that $\Pi_t(x_{\tau_0},y)$ is uniformly stable. Let $\delta(\epsilon)$ be the modulus of uniform stability for $\epsilon>0$. Thus, we take $\tau\in[0,\tau_0)$ with $\norm{x_\tau-x_{\tau_0}}<\delta(\epsilon)$ and we get $$ \norm{u(t,x_\tau,y)-u(t,x_{\tau_0},y)}<\epsilon,\quad \forall t\geq0. $$ Since $\omega(x_\tau,y)=p_\ast(\cdot)$, there is a $\hat{t}$ such that $$ \norm{u(t,x_\tau,y)-p_\ast(y\cdot t)}<\epsilon,\quad \forall t\geq \hat{t}. $$ Then, we deduce that $$ \norm{u(t,x_{\tau_0},y)-p_\ast(y\cdot t)}<2\epsilon,~~\forall t\geq \hat{t}, $$ and hence $\omega(x_{\tau_0},y)=p_\ast(\cdot)$. So $L$ is closed. Then by a similar argument in the proof of \cite[Theorem 4.1]{JZH}, we can get a contradiction. Indeed, since $L = [0,\tau_0]$ with $0 < \tau_0 <1$, for any $\tau \in (\tau_0,1)$ we have $(p_{\ast}(y),y) \notin \omega(x_\tau,y)$. For $\epsilon_0$ defined in \eqref{3.5}, by the uniform stability of the orbit, we get \begin{equation}\label{3.7} \norm{ u(t,x_\tau,y) - u(t,x_{\tau_0},y) } < \epsilon_0, \quad\forall t \geq 0 \end{equation} whenever $0 < \tau - \tau_0 \ll 1 $. Let $\{t_n\}$ be such that $\Pi_{t_n}(x_{\tau_0},y) \to (p_{\ast}(y),y) $. Choosing a subsequence if necessary, we may assume that $\Pi_{t_n}(x_{\tau},y) \to (\tilde {x},y)$ for $0<\tau-\tau_0\ll 1$. By \eqref{3.7}, we obtain $\norm{ \tilde {x} - p_{\ast}(y) } \leq \epsilon_0$. Thus, from the monotonicity, $\tilde {x} \in B^+(p_{\ast}(y),\epsilon_0)$. So by \eqref{3.5}, $\tilde {x} < K\cap {P^{-1}(y)}$. Using \cite[Proposition 3.1 (3)]{JZH} again, we get $\omega(\tilde {x},y) = \omega(p(y),y)=p_\ast(\cdot)$. Then the minimality of $\omega(x_{\tau},y)$ implies that $\omega(x_{\tau},y) = \omega(\tilde {x},y) = p_\ast(\cdot)$, which is a contradiction to the definition of $\tau_0$. Thus, $K \cap {P^{-1}(y)} = \{(p_{\ast}(y),y)\}$ for all $y\in Y$. The minimality deduces that $K$ is a 1-cover of $Y$. \end{proof} Denote $$ A = \cup_{c(\cdot) \textnormal{ is a 1-cover for $\Pi_t$}} c(\cdot) $$ of all 1-covers of $Y$ for $\Pi_t$. For each $y \in Y$, set $A(y) = A\cap P^{-1}(y)$. Based on Proposition \ref{prop3.3}, we obtain the following result. \begin{lemma} \label{lem4.1} $A$ is totally ordered with respect to `$<$', and for each $y \in Y$, $A(y)$ is homeomorphic to a closed interval in $\mathbb{R}$. \end{lemma} The proof of the above lemma is similar to that of \cite[Theorem 3.1]{CGW}, therefore it is omitted. For any $(x_0,y_0)\in X_+\times Y$, denote the forward orbit and the omega-limit set for $\Gamma_t$ by $O^+_{\Gamma}(x_0,y_0)$ and $\omega_\Gamma(x_0,y_0)$, respectively. Now we will prove the $1$-cover property for the uniformly stable $\omega$-limit sets of the comparable skew-product semiflow $\Gamma_t$. \begin{proposition}\label{prop4.3} Assume that for point $(x_0,y_0)\in X_+\times Y$, $O^+_{\Gamma}(x_0,y_0)$ is uniformly stable. Let $\hat{K}=\omega_\Gamma(x_0,y_0)$. For any $y \in Y$, if there exists some $ (b(y),y) \in A(y)$ such that $\hat{K} \cap P^{-1}(y) \geq (b(y),y)$, then $\hat{K}$ is a 1-cover of $Y$ for $\Gamma_t$. \end{proposition} \begin{proof} Let $C_\Pi=\{c(\cdot): c(\cdot) \textnormal{ is a $1$-cover for }\Pi_t\}$. Then by a similar argument in the proof of \cite[Theorem 4.3]{CGW}, using Lemma \ref{lem4.1} we can define a nonempty totally ordered set $\mathcal{C}\subset C_\Pi$, for which \begin{equation*} \mathcal{C}=\{c(\cdot)\in C_\Pi :(c(y),y) \geq \hat{K}\cap P^{-1}(y) \,\text{ for all } \,y\in Y\}, \end{equation*} and the greatest lower bound $\inf\mathcal{C}\in \mathcal{C}$ exists. Denote $q(\cdot)=\inf\mathcal{C}$. Now we assert that $\hat{K}$ is a 1-cover of $Y$ for $\Gamma_t$, satisfying \begin{equation*} \hat{K} \cap P^{-1}(y) = (q(y),y),\quad \forall y \in Y. \end{equation*} Otherwise, there exist a $y_1 \in Y$ and some $(c,y_1) \in \hat{K}\cap P^{-1}(y_1)$ such that \begin{equation*} (q(y_1),y_1) > (c,y_1). \end{equation*} According to our assumption, we have $$ (q(y_1),y_1) > (c,y_1) \geq (b(y_1),y_1). $$ Then by \cite[Lemma 3.4]{CGW}, there is a strictly order-preserving continuous path \begin{equation}\label{4.1} J: [0,1] \to A(y_1) \quad \text{with }J(0) = (b(y_1),y_1)\text{ and }J(1) = (q(y_1),y_1). \end{equation} Since $(q(y_1),y_1) > (c,y_1)$, by the strongly order-preserving property of $\Pi_t$ and the comparability of $\Gamma_t$ with respect to $\Pi_t$, we have that there exists a neighborhood $U$ of $q(y_1)$ such that \begin{equation*} \Pi_{t_1}(U,y_1) > \Pi_{t_1}(c,y_1) \geq \Gamma_{t_1}(c,y_1)=(v(t_1,c,y_1),y_1\cdot t_1) \end{equation*} for some $t_1 > t_0$. Denote $\bar{c}=v(t_1,c,y_1)$ and $y_2=y_1\cdot t_1$. Then $(\bar{c},y_2)\in \hat{K}$ and \begin{equation}\label{4.2} (u(t_1,U,y_1),y_2)>(\bar{c},y_2). \end{equation} Note that $U$ is a neighborhood of $q(y_1)$. Then due to \eqref{4.1} we can find a point $q_1(y_1) \in U\cap A(y_1)$ with $q_1(y_1) (q_1(y_2),y_2) > (\bar{c},y_2). \end{equation*} Since $O^+_{\Gamma}(x_0,y_0)$ is uniformly stable, by Theorem \ref{thm3.1} $\hat{K}$ admits a flow extension which is minimal. Thus for any $t \in \mathbb{R}$, there is $t_n \to +\infty$ such that $t_n + t \geq 0$ and $$ \Gamma_{t_n} \circ \Gamma_t(\bar{c},y_2) \to \Gamma_t(\bar{c},y_2),~n \to \infty. $$ Then the monotonicity and the comparability of $\Gamma_t$ with respect to $\Pi_t$ imply that $$ \Pi_{t_n} \circ \Pi_t(q_1(y_2),y_2) \geq \Pi_{t_n} \circ \Pi_t(\bar{c},y_2) \geq \Gamma_{t_n} \circ \Gamma_t(\bar{c},y_2). $$ By letting $n \to \infty$ in the above, we get $\Pi_t(q_1(y_2),y_2) \geq \Gamma_t(\bar{c},y_2)$, thus, \begin{equation}\label{4.3} u(t,q_1(y_2),y_2) \geq v(t,\bar{c},y_2),~\forall t \in \mathbb{R}. \end{equation} Note that $O^+_{\Pi}(q_1(y_2),y_2)$ is uniformly stable, by Theorem \ref{thm3.1} we obtain \begin{equation}\label{4.4} u(t,q_1(y),y)=q_1(y\cdot t)\quad \text{for any }y\in Y \text{ and } t\in\mathbb{R}. \end{equation} So combining \eqref{4.3}, \eqref{4.4} and the comparability of $\Gamma_t$ with respect to $\Pi_t$, similarly as the proof of Lemma \ref{lem3.2}, we can get that for any $t \in \mathbb{R}$, there exist a neighborhood $U_t$ of $q_1(y_2)$ and a neighborhood $V_t$ of $\bar{c}$ such that \begin{equation*} u(t,U_t,y_2) > v(t,V_t,y_2). \end{equation*} In particular, for $t = 0$, there exist a neighborhood $U_0$ of $q_1(y_2)$ and a neighborhood $V_0$ of $\bar{c}$ such that \begin{equation}\label{4.5} (U_0,y_2) > (V_0,y_2). \end{equation} Recall that $\hat{K}$ is the omega-limit set of $(x_0,y_0)$ for $\Gamma_t$, there exists some sequence $t_n \to +\infty$ such that $\Gamma_{t_n}(x_0,y_0) \to (\bar{c},y_2) \in \hat{K}$, as $n \to \infty$. Also, since $q_1(\cdot)$ is a 1-cover for $\Pi_t$, we get $\Pi_{t_n}(q_1(y_0),y_0) \to (q_1(y_2),y_2)$, as $n \to \infty$. So by \eqref{4.5} there exists $N > 1$ such that \begin{equation}\label{20} \Pi_{t_N}(q_1(y_0),y_0) > \Gamma_{t_N}(x_0,y_0). \end{equation} Then by a similar argument in the proof of \cite[Theorem 4.3]{CGW}, we can get that \begin{equation*} (q_1(y),y) \geq \hat{K} \cap P^{-1}(y) \quad\text{for all }y \in Y. \end{equation*} For the sake of completeness, we include a detailed proof here. As a matter of fact, by the monotonicity of $\Pi_t$ and the comparability of $\Gamma_t$ with respect to $\Pi_t$, it follows from \eqref{20} that \begin{equation}\label{4.6} \Pi_{t+t_N}(q_1(y_0),y_0) \geq \Pi_t\Gamma_{t_N}(x_0,y_0) \geq \Gamma_{t+t_N}(x_0,y_0),\quad \forall t \geq 0. \end{equation} For any $(x,y) \in \hat{K}$, there exists $s_n \to +\infty$ such that $\Gamma_{s_n}(x_0,y_0) \to (x,y)$, as $n \to \infty$. Let $t = s_n - t_N$ in \eqref{4.6} for all $n$ sufficiently large. Then we get $\Pi_{s_n}(q_1(y_0),y_0) \geq \Gamma_{s_n}(x_0,y_0)$. Letting $n \to +\infty$, one has $(q_1(y),y) \geq (x,y)$. By the arbitrariness of $(x,y) \in \hat{K}$, we get $(q_1(y),y) \geq \hat{K} \cap P^{-1}(y)$ for all $y \in Y$. This contradicts the definition of $q(\cdot)$. So we have proved the assertion, and $\hat{K}$ is a 1-cover of $Y$ for $\Gamma_t$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2.1}] Let $v(t,\cdot,v_0;d,F)$ be an $L^\infty$-bounded solution of \eqref{IBVP-sys} in $X_+$. Then from the study in \cite{Hen} and a priori estimates for parabolic equations, it follows that $v$ is a globally defined classical solution in $X_+$, and $\{v(t,\cdot,v_0;d,F):t\geq \tau\}$ is precompact in $X_+$ for some $\tau>0$. So $\hat{K}:=\omega_\Gamma(v_0,(d,F))$ is a nonempty compact set in $X_+\times H(d,F)$. Since $0(\cdot)\in C_\Pi$ by (H2), $$ \hat{K}\cap P^{-1}(y)\geq (0,y)\in A(y),\quad \forall y\in Y. $$ If $v(t,\cdot,v_0;d,F)$ is uniformly stable, then by Proposition \ref{prop4.3} we get that $\hat{K}$ is a $1$-cover of $\Omega$ for $\Gamma_t$, and thus the uniformly stable $L^\infty$-bounded solution $v(t,\cdot,v_0;d,F)$ is asymptotic to an almost periodic solution. \end{proof} \subsection*{Acknowledgments} Feng Cao was supported by grants 11201226 from the NSF of China, 20123218120032 from SRFDP, and NS2012001 from the Fundamental Research Funds for the Central Universities. \begin{thebibliography}{00} \bibitem{Bro1} P. N. Brown; Decay to uniform states in ecological interactions, {\it SIAM J. Appl. Math.}, \textbf{38} (1980), 22-37. \bibitem{Bro2} P. N. Brown; Decay to uniform states in food webs, {\it SIAM J. Appl. Math.}, \textbf{46} (1986), 376-392. \bibitem{CGW} F. Cao, M. Gyllenberg, Y. 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