\documentclass[twoside]{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} \pagestyle{myheadings} \markboth{\hfil Periodic traveling waves \hfil EJDE--1999/26} {EJDE--1999/26\hfil Peter Bates \& Fengxin Chen \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 1999}(1999), No.~26, pp. 1--19. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Periodic traveling waves for a nonlocal integro-differential model \thanks{ {\em 1991 Mathematics Subject Classifications:} 35K55, 35Q99. \hfil\break\indent {\em Key words and phrases:} nonlocal phase transition, periodic traveling waves, stability. \hfil\break\indent \copyright 1999 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted June 15, 1999. Published August 19, 1999.} } \date{} % \author{Peter Bates \& Fengxin Chen} \maketitle \begin{abstract} We establish the existence, uniqueness and stability of periodic traveling wave solutions to an intrego-differential model for phase transitions. \end{abstract} \newtheorem{Def}{Definition}[section] \theoremstyle{plain} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{rem}{Remark}[section] \numberwithin{equation}{section} \renewcommand{\theequation}{\thesection.\arabic{equation}} \section{Introduction} In this paper, we are concerned with the following integro-differential model for phase transitions \begin{equation}\label{1.1.1} u_{t} - D u_{xx}-d (J*u -u)-f(u,t)=0, \end{equation} where $x \in {\mathbb R}$ and $D, d $ are nonnegative constants with $D + d \neq 0$; $J* u(x,t)=\int_{{\mathbb R}^n} J(x-y)u(y,t)dy$ is the convolution of $J$ and $u(x,t)$; $J\in C^1({\mathbb R})\bigcap L^1(\mathbb R) $; $f(u,\cdot\,)$ is $T-$periodic, i.e., $f(u,t+T)=f(u,t)$ for all $u,t \in {\mathbb R}$; and $f(\cdot,t)$ is bistable. Other conditions on $J$ and $f$ are specified below. A typical example of $f$ is the cubic potential function $f=\rho (1-u^2)(2u-\gamma (t))$, where $\rho >0$ is a constant, $\gamma (t)$ is $T-$periodic and $0<\gamma(t)<2$. When $d =0$, equation (\ref{1.1.1}) is the classical Allen-Cahn equation \cite{kn:fm1} for which the results are known. Therefore, we will assume $d>0$ throughout. Equation (\ref{1.1.1}) can be considered as a nonlocal version of the Allen-Cahn equation which incorporates spatial long range interaction. When $d=0$ and $f(u,\cdot\,) = f(u)$ is independent of $t$, the traveling wave solution of the form $u(x,t)= U(x-ct)$ is studied in \cite{kn:fm1} and \cite{kn:fm2}(see also their references). The nonlocal autonomous case is studied in \cite{kn:bfrw}. X. Chen \cite{kn:chen} applied a ``squeezing'' technique, due to a strong comparison principle, to study the existence, uniqueness and stability of traveling wave solutions for a variety of autonomous nonlocal evolution equations, which includes the Allen-Cahn reaction-diffusion equation, neural networks, the continuum Ising model, and a thalamic model. When $f(u,t)$ is $T$-periodic, periodic traveling wave solutions of the bistable reaction diffusion are studied in \cite{kn:abc}. In this paper, we will establish similar results to those in \cite{kn:abc} but for the more general equation (\ref{1.1.1}). We assume in this paper that \begin{enumerate} \item[{\bf H1)}] $f \in C^{2,1}({\mathbb R}\times {\mathbb R})$ is periodic in $t$ with period $T$, i.e., there is a $T>0$ such that $f(u,t)=f(u,t+T)$ for all $u,t \in {\mathbb R}$. \item[{\bf H2)}] The period map $P(\alpha):= w(\alpha, T)$, where $w(\alpha,t)$ is the solution to \begin{align}\label{1.1.2} w_t=f(w,t), \ \ \ \mbox{ for all } \ t\in {\mathbb R}, \ \ w(\alpha,0) = \alpha, \end{align} has exactly three fixed points $\alpha ^-$, $\alpha ^0$, $\alpha ^+$, satisfying $\alpha ^-< \alpha ^0<\alpha ^+$. In addition, they are non-degenerate and $\alpha^\pm $ are stable and $\alpha^0 $ is unstable, that is, \begin{equation}\label{1.1.3} \frac{d}{d\alpha}P(\alpha ^\pm) <1 < \frac{d}{d\alpha}P(\alpha ^0). \end{equation} \item[{\bf H3)}] $J(x)\in C^1({\mathbb R})$ is nonnegative, $\int_{\mathbb R} J(x) \,dx =1$, and $ \int_{\mathbb R} |J'(x)|\,dx <\infty$. In the case $D = 0$, we need the following additional condition, \item[{\bf H4)}] \begin{equation}\label{1.1.4} \sup\{f_u(u,t): u\in [W^-(t),W^+(t)], t\in [0,T]\}0$, for all $x\in R_1$ and $ t\in (\tau, t_0]$. \end{lem} \begin{proof} We may assume $\tau =0 $. We take $d>0$ since the result is standard for $d=0$. By the assumption that $u(x,t)\in C_{unif}({\mathbb R} \times [0,t_0])$, $\inf_{x\in {\mathbb R}}u(x,t)$ is continuous on $[0, t_0]$. If the conclusion of the lemma is not true, there exist constants $\epsilon >0$ and $T_0>0$ such that $ u(x,t)>-\epsilon e^{2Kt}$, for all $x\in {\mathbb R}$ and $ 0 0,\nonumber \end{eqnarray*} by the choice of $K$, which is a contradiction. Therefore $u(x,t)\ge 0$ for all $x \in {\mathbb R}$ and $t\in(0,t_0]$. Suppose $u(x,t)$ is not identically zero on $R_1\times (0, t_0]$ and there is a point $(x_2,t_2)\in R_1\times (0, t_0] $ such that $ u(x,t)$ achieves the minimum 0. By a similar argument to the above we deduce that $(J*u-u)(x_2,t_2) =0$. Therefore $u\equiv 0$, which is a contradiction. That completes the proof. %% \end{proof} \vspace{.1in} Now we are ready to state the uniqueness theorem. \begin{thm}\label{theorem1.2.1} Suppose (H1), (H2) and (H3) hold. Then problem (\ref{1.1.6})- (\ref{1.1.8}) admits at most one smooth solution. \end{thm} \begin{proof} The proof is similar to that in \cite{kn:abc}. Let $(U, c)$ and $(\overline{U}, \overline{c})$ be any two solutions of (\ref{1.1.6})-(\ref{1.1.8}) with $c\ge \overline{c}$. We prove $U=\overline{U}$ and $c=\overline c$, we divide the proof into six steps. 1. By periodicity of $U(\xi,t)$ and the comparison principle, we have $$w(-M^-,kT+t)\le U(\xi,kT+t) = U(\xi,t)\le w(M^+,kT+t),$$ where $M^{\pm}=\sup_{\xi \in {\mathbb R}}{\pm U(\xi,0)}$. Letting $ k\rightarrow \infty$ gives $W^-(t)\le U(\xi,t)\le W^+(t)$. By Lemma \ref{lemma1.2.1}, we have \begin{equation}\label{1.2.3} W^-(t)0$ and $a^\pm(T) < 1$. Moreover, there exist two constants $C_1$ and $C_2$ such that \begin{equation}\label{1.2.4} C_2a^-(t)\le a^+(t)e^{\nu t} \le C_1a^-(t). \end{equation} For $\eta>0$ and $t\in[0,T]$, let $I^{\pm}_\eta(t):=[W^\pm(t)-\eta,W^\pm(t)+\eta]$ and define \begin{displaymath} \delta_0= \frac{\sup\{\eta\colon |f_u(u,t)-f_u(W^\pm(t),t)|\le \nu^\pm/\,4,\mbox{ for } t\in [0,T], u\in I^\pm_\eta(t)\}}{2\|a^+(\cdot)\|_{C^0([0,T])} + 2\|a^-(\cdot)\|_{C^0([0,T])}} \end{displaymath} and let $\zeta(\xi)$ be a smooth function such that $0\le \zeta(\xi)\le 1$, $\zeta(\xi)= 0$ for $ \xi \le -2$, and $\zeta(\xi)= 1$ for $\xi \ge 2$. Let $a(\xi,t) = e^{\nu t}a^+(t)\zeta(\xi) + a^-(t)(1-\zeta(\xi))$. Define \begin{multline*} \xi_0=\inf\bigl\{\hat \xi\ge 2\colon |d(J*\zeta-\zeta)(\pm \xi) (a^+(t)e^{\nu t}- a^-(t))|\le \frac {\nu^\pm}{4} \min\{a^+(t)e^{\nu t}, a^-(t)\},\\ \text{and } |U(\pm\xi,t) - W^\pm (t)|<\delta_0/\,2, \forall\ \xi\ge\hat\xi,\ t\in[0,T]\bigr\}. \end{multline*} This $\xi_0$ is well defined since $\lim_{\xi\to \infty}U(\pm \xi,t)= W^\pm(t)$ uniformly in $t$ and\\ $\lim _{x\to \infty}(J*\zeta -\zeta) (\pm x) =0$. For each $\delta \in (0,\delta_0/2]$, define $U_\delta(\xi,t) = U(\xi,t) + \delta a(\xi,t)$. Then, on $(\xi_0,+ \infty)$, \begin{eqnarray} L^c U_\delta(\xi,t)&:=&U_{\delta t} - cU_{\delta \xi} -D U_{\delta \xi\xi} -d(J*U_{\delta }-U_{\delta }) - f(U_{\delta },t) \nonumber\\ &=& f(U, t) -f(U+ \delta a^+(t),t) + [\nu^+/\,2 + f_u(W^+(t),t) +\nu]\delta a^+(t)e^{\nu t} \nonumber\\ &\mbox{}& - \delta d[a^+(t)e^{\nu t}- a^-(t)](J*\zeta-\zeta) \nonumber\\ &=&\delta a^+e^{\nu t}[\nu^+/2 +\nu+ f_u(W^+(t),t) -\int _0^1f_u(U+ \delta \theta a^+(t),t) d\theta] \nonumber\\ &\mbox{}& - \delta d[a^+(t)e^{\nu t}- a^-(t)](J*\zeta-\zeta)\nonumber\\ &\ge& (\nu^+/\,4 +\nu) \delta a^+(t)e^{\nu t} -d \delta[a^+(t)e^{\nu t}- a^-(t)](J*\zeta-\zeta) \nonumber\\ &\ge&0. \end{eqnarray} where we have used the fact that $a(\xi,t)= a^+(t)e^{\nu t}$ on $(\xi_0,+\infty)$ and the definitions of $\xi_0$ and $\delta_0$. Similarly, we have $ L^c U_\delta(\xi,t)\ge0$, on $(-\infty,-\xi_0)$. That is, $U_\delta(\xi,t)$ is a super solution on $((-\infty,-\xi_0)\bigcup (\xi_0,+\infty))\times\mathbb R^+ $. 3. Since $\lim_{\xi\to \infty}\overline{U}(\pm\xi,t) = W^\pm(t)$ uniformly in $t$, by (\ref{1.2.3}), there exists a large constant $\hat z_0$ such that $$ \overline U(\xi-z +(c-\overline c)t,t)\le\begin{cases}U(\xi,t),\hfill &\mbox{if $\xi \in [-\xi_0,\xi_0]$;}\cr U(\xi,t)+\delta_0, &\mbox{if $\xi \notin [-\xi_0,\xi_0]$;}\end{cases} $$ for all $ t\in[0,T]$ and $z\ge \hat z_0$. Define $\hat\delta := \inf\{\delta >0 \,:\, \overline U(\xi-z,0) \le U(\xi,0) +\delta, \mbox{ for all }\ z\ge \hat z_0,\xi \in {\mathbb R}\}$. Obviously, $\hat\delta\le \delta_0$. We claim that $\hat \delta =0$. In fact, for $z>z_0$, $L^c\overline U(\xi-z+(c-\overline c)t,t)=0$. And on $[-\xi_0,\xi_0] \times (0,T]$, \begin{eqnarray} U_{\hat\delta}(\xi,t)& =& U(\xi,t) + \hat\delta a(\xi,t) \ge U(\xi,t) \ge\overline U(\xi -z+(c-\overline c)t,t), \end{eqnarray} and \begin{eqnarray} U_{\hat\delta}(\xi,0)&=& U(\xi,0) + \hat\delta a(\xi,0) = U(\xi,0) + \hat\delta \ge \overline U(\xi -z,0) \end{eqnarray} for all $\xi\in {\mathbb R}$. By Lemma \ref{lemma1.2.1}, we have $$ \overline U(\xi-z +(c-\overline c)t,t) \le U_{\hat \delta}(\xi,t)$$ for all $ z\ge \hat z_0$, $(\xi,t) \in {\mathbb R}\times (0,T]$. Since $z\ge \hat z_0$ is arbitrary, we have $$\overline U(\xi-z,T)\le U_{\hat\delta}(\xi,T)$$ for all $ z\ge \hat z_0$. By the periodicity of $U(\xi,\cdot)$, we have $$\overline U(\xi-z,T) \le U(\xi,0)+ {\hat\delta}a(\xi,T)$$ for all $ z\ge \hat z_0$, $\xi\in {\mathbb R}$. Therefore, $$\overline U(\xi-z,0)\le U(\xi,0) + {\hat\delta}\max\{a^+(T)e^{\nu T},a^-(T)\}$$ for all $z\ge \hat z_0$, and $\xi\in {\mathbb R}$. This contradicts the definition of $\hat{\delta}$ since $a^\pm(T)<1$. Therefore, $$ \overline U(\xi-z,0)\le U(\xi,0)$$ for all $ \xi\in {\mathbb R}$ and $ z\ge \hat z_0$. 4. By the comparison principle (Lemma \ref{lemma1.2.1}), $\overline U(\xi -z +(c-\overline c)t,t)\le U(\xi,t)$, for all $ \xi\in {\mathbb R}$, $ t\ge 0 $, and $z\ge \hat z_0$. Therefore by periodicity, $\overline U((c-\overline c)kT-z,0)\le U(0,kT)= \alpha^0$. Letting $k\to \infty$, we deduce that $c= \overline c$ since $\overline U((c-\overline c)kT-z,0)\to \alpha^+$ if $c>\overline c$. 5. Define $z_0 =\inf\{\hat z_0\ :\, \overline U(\xi-z,0)\le U(\xi,0),\mbox{ for all } \xi\in {\mathbb R}, z\ge\hat z_0\}$. Similar to the proof in step 3, we can show that $\overline U(\xi-z_0,0) = U(\xi,0)$ for $ \xi\in {\mathbb R}$. 6. We prove that $z_0=0$. If not, $\overline U(\xi-z,0) < U(\xi,0)$ for all $\xi\in {\mathbb R}, z>z_0$. By the comparison principle and periodicity, $$U(\xi-z+z_0,0) = \overline U(\xi-z_0 -z+z_0,0) = \overline U(\xi-z) < U(\xi, 0),$$ since $\overline U(\xi-z_0,0)=U(\xi,0)$. Therefore $U(\xi,0)$ is strictly increasing. Since $U(z_0,0) =\overline U(0,0) =\alpha_0 =U(0,0)$, we deduce that $z_0 =0$. That completes the proof. \end{proof} \begin{cor}\label{corollary1.2.1} Under the conditions of Theorem \ref{theorem1.2.1}, any smooth solution to (\ref{1.1.6})-(\ref{1.1.8}) is strictly increasing. \end{cor} \section{Existence of Periodic Traveling Waves} In this section, we are going to establish the existence of the periodic traveling wave solution to (\ref{1.1.6})-(\ref{1.1.8}) by a homotopy argument. Assume $(U_0,c_0)$ is the unique solution of the following problem, corresponding to the parameter $\theta=\theta_0 \le 1$, \begin{eqnarray} &\mbox{} &U_t-cU_\xi -[1-\theta (1-D)] U_{\xi \xi} -\theta d(J*U - U) -f(U,t)=0, \label{1.3.1}\\ &\mbox{}&U(\pm\infty,t)=\lim_{\xi\rightarrow\pm\infty} U(\xi,t) = w(\alpha^\pm,t), \mbox{ uniformly in } t\in {\mathbb R}, \label{1.3.2}\\ &\mbox{}&U(\cdot,T)=U(\cdot,0),\ U(0,0)=\alpha ^0, \label{1.3.3} \end{eqnarray} satisfying $U_{0\xi}>0$, $U_{0\xi}(\xi,t) \to 0$ uniformly in $t$ as $ \xi \to \pm \infty$. Let $$X_0 =\{v\ :\ v\in C_{unif}( {\mathbb R}\times {\mathbb R}), v(\cdot,t+T)=v(\cdot,t)\ \hbox{and}\ \lim_{x\to\infty} v(\pm x,t)=0, \forall \ t\in\mathbb R\}.$$ and $L=L(U_0,c_0,\theta_0)$ be the linearization of the operator in (\ref{1.3.1})-(\ref{1.3.3}) defined by $$ D(L) = X_2:= \{ v\in X_0\ :\ v_{\xi\xi}, v_\xi, v_t \in X_0\},$$ \begin{equation} \label{1.3.4} L v= v_t - [1-\theta_0 (1-D)] v_{\xi\xi} -\theta_0 d(J*v-v) -c_0v_\xi-f_u(U_0,t)v \end{equation} for $ v\in D(L)$. We first establish some lemmas. \begin{lem}\label{lamma1.3.1} $L$ has $0$ as a simple eigenvalue. \end{lem} \begin{proof} Clearly, $p=U_{0\xi}$ is an eigenfunction corresponding to the eigenvalue $0$. We only need to prove the simplicity. Suppose $\phi(\xi,t) \in X_0$ is another eigenfunction with eigenvalue $0$. We prove that $\phi=z p$, for some constant $z\in {\mathbb R}$. Let $\nu^\pm$ be defined as in Section 2. Without loss of generality we assume $\nu^+ \ge \nu^-$. Let $\nu=(\nu^+ -\nu^-)/\,2$ be as in Section 2. Suppose $\zeta(\xi) $ is a smooth function such that $\zeta(\xi) \equiv 0$, for $\xi<0$; $\zeta(\xi)\equiv 1$, for $\xi>4$; and $0 \le \zeta(\xi) \le 1$, $0\le \zeta'(\xi)\le 1$, and $|\zeta''(\xi)|\le1$, for all $\xi\in {\mathbb R}$. Define \begin{eqnarray}\label{1.3.5} &A(\xi,t) = \zeta(\xi) a^+(t) e^{\nu t} + ( 1-\zeta(\xi))a^-(t), &\\ \label{1.3.6} &B(t)=\int _0^t \max\{a^+(\tau) e^{\nu \tau}, a^-(\tau)\}\,d\tau,&\\ \label{1.3.7} &K=\frac{\nu^+ - \nu^- /2 + 1 +(D+d) +2c_0 + 2 \|f_u\| }{\min _{(\xi,t) \in [-\xi_0,\xi_0]\times [0,T]} U_{0\xi}(\xi,t)},& \end{eqnarray} where $\|f_u\| =\sup\{|f_u(u,t)|\,:\, u\in [W^-(t),W^+(t)], \ t\in [0,T]\}$ and $\xi_0$ is a large constant to be chosen later. Let $\Psi(\xi,t) =K B(t)U_{0\xi}(\xi,t) + A(\xi,t)$, then $\Psi(\xi,0) = 1$. We claim that \begin{equation}\label{1.3.8} L \Psi(\xi,t) = KB_tU_{0\xi}(\xi,t) +L A(\xi,t)\ge 0. \end{equation} We divide the proof by considering three intervals $(-\infty,-\xi_0)$, $[-\xi_0,\xi_0]$, and $(\xi_0,\infty)$. We assume $\xi_0>4$. On $(\xi_0,\infty)$, $A(\xi,t) = a^+(t)e^{\nu t}$, therefore \begin{multline*} LA(\xi,t) = [ \nu^+/\,2 + f_u(W^+(t),t) + \nu - f_u(U_0(\xi,t),t)] a^+(t)e^{\nu t}\\ -\theta_0 d(J*\zeta-\zeta)[a^+(t)e^{\nu t} -a^-(t)]. \end{multline*} Notice that $(J*\zeta - \zeta)(\xi) \to 0$, and $U_0(\xi,t) \to W^+(t)$ as $\xi \to \infty$. We deduce, by (\ref{1.2.4}), that we can choose $\xi_0$ large enough such that $$L A(\xi,t)\ge 0,\qquad\hbox{on } \ (\xi_0,\infty) \times {\mathbb R}^+.$$ Similarly we have \begin{eqnarray*} L A(\xi,t) &=& [\nu^-/\,2 + f_u(W^-(t),t) -f_u(U_0(\xi,t),t)]a^-(t) \\ && -\theta_0 d [J*\zeta -\zeta][a^+(t)e^{\nu t} - a^-(t)] \quad\mbox{on $(-\infty,-\xi_0)$.} \end{eqnarray*} Therefore there exists $\xi_0>>1$ such that $$ L A(\xi,t) \ge 0, \qquad \hbox{on} \ (-\infty,-\xi_0) \times {\mathbb R}^+.$$ We fix $\xi_0$ large enough such that $L A(\xi,t) \ge 0$, on $((-\infty,-\xi_0)\bigcup (\xi_0,\infty))\times {\mathbb R}^+$. On $[-\xi_0, \xi_0]$, \begin{eqnarray*} \lefteqn{|L A(\xi,t)|} \\ & = &|A_t -[1-\theta_0 (1-D)]A_{\xi\xi}-\theta_0 d(J*A -A) - c_0A_\xi -f_u(U_0(\xi,t),t)A(\xi,t)| \\ &\le&\max\{a^+(t)e^{\nu t}, a^-(t)\}\{\nu^+- \nu^-/2 +[1-\theta_0 (1-D)] +\theta_0 d+2c_0 + 2\|f_u\|\} \end{eqnarray*} Therefore $L\Psi(\xi,t)\ge 0$, on $[-\xi_0,\xi_0]$ by (\ref{1.3.8}) and the choice of $K$ in (\ref{1.3.7}). By the comparison principle, we have $$\phi(\xi,t) \le \Psi(\xi,t)\|\phi(\xi,0)\|_{\infty}.$$ Letting $t=kT$ and letting $k\to \infty$, we have $$|\phi(\xi,0)| \le KB(\infty)\|\phi(\xi,0)\|_{\infty}U_{0\xi}(\xi,0),$$ where $B(\infty) = \lim_{t\to\infty}B(t)$. The limit exists since $a^\pm(t) \to 0$ exponentially and (\ref{1.2.4}) holds. Let $z_* := \sup\{z \ :\ \phi(\xi,0)\ge zU_{0\xi}(\xi,0),\mbox{ for all } \xi\in {\mathbb R} \}$. We claim that $\phi(\xi,0) = z_*U_{0\xi}(\xi,0)$, for all $\xi\in {\mathbb R}$. If not, there exists a point $\xi_0$ such that $\phi(\xi_0,0)>z_*U_{0\xi}(\xi_0,0)$. Then by the comparison principle, $\phi(\xi,T)>z_*U_{0\xi}(\xi,T)$. Replacing $\phi$ by $\phi -z_* U_{0\xi}$, we can assume $z_*=0$. So, $\phi(\xi,0)>0$, for all $\xi\in {\mathbb R}$. Choose $\overline \xi$ such that $KB(\infty)\sup_{|\xi|\ge\overline \xi}U_{0\xi}(\xi,0)<1/\,4$ and choose $\epsilon$ such that $\phi(\xi,0)>\epsilon U_{0\xi}(\xi,0)$, on $[-\overline\xi,\overline\xi]$. Then $$\phi(\xi,0) -\epsilon U_{0\xi}(\xi,0)\ge -\epsilon\sup_{|\xi|\ge\overline \xi}U_{0\xi}(\xi,0),$$ and therefore, $$\phi(\xi,t) -\epsilon U_{0\xi}(\xi,t) \ge -\epsilon\Psi(\xi,t)\sup_{|\xi|\ge\overline \xi}U_{0\xi}(\xi,0).$$ Letting $t=kT$ and letting $k\to\infty$, we have $$\phi(\xi,0) -\epsilon U_{0\xi}(\xi,0)\ge- \frac{1}{4}\epsilon U_{0\xi}(\xi,0),$$ which contradicts the definition of $z_*$, and completes the proof. \end{proof} \vspace{.1in} Since $J*u-u $ is a bounded operator on $X_0$, we know that $0$ is an isolated eigenvalue of $L$ for $\theta_0<1$. Now consider the adjoint operator $L^*=L^*(U_0,c_0,\theta_0)$ of $L$. Since the comparison principle holds for $L$, we know that $0$ is an isolated eigenvalue for $L^*$ with a positive eigenfunction (see Section 11.4 and theorem 9.11 in \cite{kn:kls}). We denote by $\phi^*(x,t)$ the positive eigenfunction of $L^*$ corresponding to the eigenvalue $0$. \begin{lem}\label{lemma1.3.2} With $\theta_0$, $U_0$, and $c_0$ as above with $\theta_0<1$, there exists $\eta>0$ such that for each $\theta \in[\theta_0,\theta_0+\eta)$, (\ref{1.3.1})- (\ref{1.3.3}) has a solution $(U(\theta,\xi,t),c(\theta))$. \end{lem} \begin{proof} Consider the operator $G: (X_2\times {\mathbb R})\times {\mathbb R} \rightarrow X_0\times {\mathbb R}$ defined by \begin{align*} G(w,\theta) = &((U_0 +v)_t - [1-\theta (1-D)](U_0+ v)_{\xi\xi} -\theta d(J*(U_0 + v) -(U_0+v))\\ & -(c_0 +c)(U_0 + v)_\xi -f(U_0+v,t), v(0,0)) \end{align*} for $w=(v,c)\in X_2\times {\mathbb R}$. Then $G$ is of class $C^1$, $G(0,\theta_0) = (0,0)$ and $$ \frac{\partial G}{\partial w}(0,\theta_0) =\left[\matrix L &U_{0\xi}\cr \delta & 0\cr \endmatrix\right].$$ where $\delta$ is the $\delta$-function. We show that $ \frac{\partial G}{\partial w}(0,\theta_0)$ is invertible. Consider the equation on $X_0\times {\mathbb R}$: $$ \frac{\partial G}{\partial w}(0,\theta_0)\left[\matrix v\cr c \cr \endmatrix\right] =\left[\matrix h\cr b \cr \endmatrix\right], \qquad\mbox{for }\left[\matrix h\cr b \cr \endmatrix\right] \in X_0\times {\mathbb R},$$ i.e., \begin{eqnarray} L v + cU_{0\xi} &=&h, \label{1.3.9}\\ v(0,0) &=&b. \label{1.3.10} \end{eqnarray} By the Fredholm Alternative, (\ref{1.3.9}) is solvable if and only if $h-cU_{0\xi} \perp \phi^*$, i.e., \begin{equation}\label{1.3.11} \int_0^T\int_{\mathbb R} [h\phi^* -c U_{0\xi}\phi^*]\,dxdt=0. \end{equation} Since $U_{0\xi}>0$ and $\phi^* >0$, $c$ is uniquely determined by (\ref{1.3.11}). After we determine c, the solution $v$ of (\ref{1.3.9}) is determined up to a term $k U_{0\xi}$, where $k$ is a constant. Then (\ref{1.3.10}) determines $k$ uniquely. Therefore $ \frac{\partial G}{\partial w}(0,\theta_0)$ is invertible. The lemma now follows from the Implicit Function Theorem. % \end{proof} \begin{lem}\label{lemma1.3.3} Suppose that for $\theta\in [0,\overline\theta)$, where $\overline \theta\le 1$, there exists a solution $(U(\theta,\xi,t),c(\theta))$ of (\ref{1.3.1})- (\ref{1.3.3}). Then $\|U(\theta,\cdot,\cdot)\|_{L^\infty(\mathbb R\times [0,T])}$, $\|U_\xi(\theta,\cdot,\cdot)\|_{L^\infty(\mathbb R\times [0,T])}$ and $\|U_t(\theta,\cdot,\cdot)\|_{L^\infty(\mathbb R\times [0,T])}$ are uniformly bounded for $\theta\in[0,\overline\theta)$. \end{lem} \begin{proof} For the case $\overline\theta <1$, the conclusion of the lemma follows from classical parabolic estimates. Therefore we take $\overline\theta=1$, and prove the lemma for $\theta$ near $1$. We only prove the uniform boundedness of $U_\xi(\theta,\xi,t)$; all others are similar. Let $v(\theta,\xi,t):= U_\xi(\theta,\xi,t)$ and $M=\sup_{\xi, t\in {\mathbb R}}|J'*U(\xi,t)|$. Then $v(\theta,\xi,t)$ satisfies $$v_t-[1-\theta (1-D)]v_{\xi\xi} + \theta d v -c(\theta) v_\xi - f_u(U(\theta,\xi,t),t) v =\theta d J'*U.$$ Define $l(\theta):= \theta d -\sup\{f_u(u,t)\ :\ u\in [W^-(t), W^+(t)],\, t\in [0,T]\}$. For $\theta\in [0,1)$ such that $l(\theta)>0$, we have, by the comparison principle for parabolic equations, $$v(\theta,\xi,t)\le e^{-l(\theta)t}\sup_{\xi\in {\mathbb R}}|v(\theta,\xi,0)| + (1- e^{-l(\theta)t} )M/\,l(\theta).$$ By periodicity, we deduce that $v(\theta,\xi,t)$ is uniformly bounded for $\theta\in [0,1)$ with $l(\theta)>0$. % \end{proof} \begin{lem}\label{lemma1.3.4} Suppose that there is a sequence $\theta_j$ such that $\lim_{\theta_j\to \overline\theta}U(\theta_j,\xi,t)=U(\overline \theta,\xi,t)$ uniformly with respect to $(\xi,t)\in {\mathbb R}\times [0,T]$ for some function $U(\overline \theta,\xi,t)$. Then $\{c(\theta_j)\}$ is bounded. \end{lem} \begin{proof} First we prove the following statement. Suppose $(\overline V,\overline C)$ satisfies, for some $\overline \xi>0$, \begin{align} &\overline {V_t} -[1-\theta (1-D)]\overline {V}_{\xi\xi}-\theta d(J*\overline V-\overline V) -\overline C\, \overline {V_\xi} -f(\overline V,t)\le 0, \nonumber \\ &\qquad\hbox{in}\ (-\infty,\overline \xi)\times (0,T],\\ &\overline V(-\infty,t)\overline C$, then $U(\theta,\xi,t)$ satisfies \begin{eqnarray} L^{\overline C}U(\theta,\xi,t)&:=& U_t -(1-\theta)U_{\xi\xi}-\theta(J*U -U) -\overline C U_\xi -f(U,t)\nonumber\\ &=& (c(\theta)-\overline C) U_\xi >0.\nonumber \end{eqnarray} Let $m_0 = \inf\{m\ :\ U(\theta,\xi,0)>\overline V(\xi-m,0),\mbox{ for } \xi\in {\mathbb R}\}$. Then by assumption, $m_0$ is well defined and $m_0\ge0$. Moreover, there exists a point $\xi_0\in(-\infty,\overline \xi)$ such that $U(\theta,\xi_0,0)=\overline V(\xi_0-m_0,0$). Applying the strong comparison principle on $(-\infty,\overline \xi)\times [0,T]$, we get $U(\theta,\xi,t)> \overline V(\xi-m_0,t)$, for all $\xi\in {\mathbb R}$, $t\in [0,T]$. This is a contradiction since $U(\theta,\xi_0,T)=U(\theta,\xi_0,0) = \overline V(\xi_0-m_0,0)\le \overline V(\xi_0 - m_0,T)$, and the claim is proved. We denote $\theta_j$ by $\theta$. Let $\zeta(s) =[1+\tanh(s/\,2)]/\,2$, $W_1(t) = w(\alpha ^+ -\epsilon,t)$ and $W_2(t) = w(\alpha ^- -\epsilon,t)$, where $\epsilon$ is a small constant to be chosen. Let $\overline V(\xi,t) = W_1(t)\zeta(\xi+\xi_0) +W_2(t)(1-\zeta(\xi+\xi_0))$, where $\xi_0 $ is a constant such that $\zeta(\xi_0) = \frac{\alpha^0-\alpha^-+\epsilon }{\alpha^+ -\alpha^-}$. Since $W_i(T) >W_i(0),\ \hbox{for}\ i=1,2$, we have $\overline V(\cdot,T)\ge\overline V(\cdot,0)$. Moreover, $\overline V_\xi >0$, $\overline V(\infty,0)=\alpha^+-\epsilon$, and $\overline V(-\infty,0)=\alpha^--\epsilon$. Since $\lim_{\theta \to \overline \theta}U(\theta,\xi,t) =U(\overline\theta,\xi,t)$ uniformly and $U(\theta,+\infty,t) = W^+(t)$, we can choose $\overline \xi$ sufficiently large such that $U(\theta,\xi,t)>\overline V(\xi,t)$, for $(\xi,t)\in [\overline \xi, \infty)\times [0,T]$. For $\xi<\overline\xi$, \begin{align*} L^{\overline C}(\overline V)=&\overline V_t -[1-\theta (1-D)]\overline V_{\xi\xi}-\theta d (J*\overline V-\overline V) -f(\overline V,t) -\overline C V_\xi \nonumber \\ =&-\zeta(1-\zeta)(W_1-W_2)[\overline C +(1-\theta (1-D))(1-2\zeta)]\\ &-\theta d(W_1-W_2)(J*\zeta-\zeta) +[\zeta f(W_1,t) + (1-\zeta)f(W_2,t)\\& - f(W_1\zeta + W_2(1-\zeta),t)] \nonumber \\ =&-\zeta(1-\zeta)(W_1-W_2)[\overline C + (1-\theta (1-D))(1-2 \zeta) \nonumber \\ \mbox{}& -(W_1-W_2) f_{uu}(\sigma,t)/\,2 -\theta d/\,(1-\zeta)]-\theta d(W_1-W_2)(J*\zeta-\zeta),\nonumber \end{align*} where we use the Taylor's expansion $$\zeta f(W_1,t) + (1-\zeta)f(W_2,t) - f(W_1\zeta + W_2(1-\zeta),t) = \zeta(1-\zeta)(W_1-W_2)^2 f_{uu}(\sigma,t)$$ for some $\sigma\in [W_2,W_1]$. If we choose $\overline C = 1 + D + \frac{1}{2}\sup\{(W^+(t)-W^-(t) +2)|f_{uu}(u,t)|\ :\ u\in[W^-(t) -1,W^+(t) +1], t\in [0,T]\} + \sup_{\xi\le \overline \xi}{d}/\,(1-\zeta(\xi))$, then $L^{\overline C}(\overline V)<0$ for $\xi<\overline\xi$. Therefore $c(\theta)\le \overline C$ by our earlier observation. We can get a lower bound estimate similarly. \end{proof} \vspace{.1in} We are ready to obtain a solution to (\ref{1.3.1})-(\ref{1.3.3}). \begin{thm}\label{theorem1.3.1} Under the conditions of Theorem \ref{theorem1.1.1}, there exists a solution $(U(\theta,\xi,t),c(\theta))$ to (\ref{1.3.1})-(\ref{1.3.3}) for all $\theta\in [0,1]$. \end{thm} \begin{proof} By the result in \cite{kn:abc}, there exists a solution $(U_0,c_0$) to (\ref{1.3.1})-(\ref{1.3.3}) corresponding to $ \theta=0$, such that $U_{0\xi}>0$ and $\lim_{\xi\to\infty}U_{0\xi} =0$ uniformly with respect to $t$. By Lemma \ref{lemma1.3.2}, there exists an interval $[0,\overline \eta)$ such that for all $\theta\in [0,\overline \eta)$ system (\ref{1.3.1}) - (\ref{1.3.3}) has a solution $(U(\theta,\xi,t),c(\theta))$ with the required properties. Suppose $[0,\eta)$ is the maximal interval such that (\ref{1.3.1})-(\ref{1.3.3}) admits a solution for each $\theta \in [0,\eta)$. Then we claim that $\eta = 1$ and (\ref{1.3.1})-(\ref{1.3.3}) admits a solution for each $\theta \in [0,1]$. By Lemma \ref{lemma1.3.3} and Helly's theorem, we can choose a subsequence ${\theta_j}$ such that $\lim_{j\to\infty}\theta_j=\eta$, and $\lim_{j\to\infty} U(\theta_j,\xi,t)$ exists uniformly for all $\xi\in {\mathbb R}$ and each rational $t$. By Lemma \ref{lemma1.3.3} again, $\|U_t(\theta,\xi,t)\|$ is uniformly bounded for all $\theta \in [0,\eta)$. Therefore there exists a uniformly continuous function $U(\eta,\xi,t)$ such that $\lim_{j\to\infty} U(\theta_j,\xi,t) =U(\eta,\xi,t)$ uniformly for all $(\xi,t)\in {\mathbb R}\times [0,T]$. Moreover, by choosing a subsequence if necessary, the derivatives of $U(\theta_j,\xi,t)$ converge to the corresponding derivatives of $U(\eta,\xi,t)$ uniformly on any compact set of ${\mathbb R}\times [0,T]$. Therefore by Lemma \ref{lemma1.3.4}, we can choose a subsequence of $\{\theta_j\}$ (we label it the same) such that $c(\theta_j)\to c(\eta)$. Therefore $(U(\eta,\xi,t),c(\eta))$ is a solution to (\ref{1.3.1})-(\ref{1.3.3}) corresponding to parameter $\eta$, with the same properties as $(U_0,c_0)$. Therefore, either $\eta=1$ , or we can extend the existence interval to $[0,\eta +\epsilon) $ for some $\epsilon>0$, which would contradict the maximality of $\eta$. Therefore, for all $\theta\in[0,1]$, (\ref{1.3.1})-(\ref{1.3.3}) has a solution. \end{proof} \section{Stability of the Periodic Traveling Waves} In this section, we study the stability and asymptotic stability of the periodic traveling wave solutions $U(x-ct,t)$ obtained in Section 3. We denote by u(x,t;g) the solution to the initial value problem \begin{eqnarray} &\mbox{}&u_t - D u_{xx} -d(J*u-u) -f(u,t) =0,\ \mbox{in } {\mathbb R}\times (0,\infty),\label{1.4.1}\\ &\mbox{}&u(x,0) = g(x),\ \mbox{on } {\mathbb R},\label{1.4.2} \end{eqnarray} where $g(\cdot)\in L^\infty({\mathbb R})$. For the existence and uniqueness of (\ref{1.4.1}) and (\ref{1.4.2}), we have \begin{lem}\label{lemma1.4.1} For any $g(\cdot) \in L^\infty({\mathbb R})$ , there exists a unique solution $u(x,t;g)\in C^1([0,\infty),L^\infty({\mathbb R}))$ of (\ref{1.4.1}) and (\ref{1.4.2}). Moreover, $u(\cdot,t;g)$ is continuous from $[0,\infty)\times C_{\mbox{unif}}({\mathbb R})$ to $C_{\mbox{unif}}({\mathbb R})$. \end{lem} \begin{proof} The case $D \neq 0$ follows from standard parabolic theory. We only need to consider the case where $D =0 $. Write (\ref{1.4.1}) and (\ref{1.4.2}) as \begin{equation}\label{1.4.3} u(x,t) = g(x) + \int^t_0 (d(J*u-u) +f(u,t))\, dt. \end{equation} Then the local existence and uniqueness follow from the contraction mapping theorem in the usual way. Let $M = \sup_{x\in {\mathbb R}}|g(x)|$. Then $w(\pm M,t)$ are super- and sub-solutions of (\ref{1.4.1}) respectively. By the comparison principle, \begin{eqnarray} w(-M,t)\le u(x,t;g) \le w(M,t) \nonumber \end{eqnarray} for $t>0$. Global existence follows since $w(\pm M,t) $ are bounded. The continuous dependence can be easily proved using (\ref{1.4.3}). \end{proof} \vspace{.1in} We claim that the asymptotic behavior of the solutions to (\ref{1.4.1}) and (\ref{1.4.2}) is governed by the periodic traveling wave solution $U(x-ct,t)$. We have the following result: \begin{thm}\label{theorem1.4.1} (1) (Uniform Stability) For any $\epsilon >0$, there is a $\delta>0$ such that for any $g\in C_{\mbox{unif}}({\mathbb R})$ with $\|g(\cdot) - U(\cdot,0)\| < \delta$, one has \begin{equation}\label{1.4.4} \|u(\cdot,t;g) - U(\cdot -ct,t)\| < \epsilon \end{equation} for all $t>0$. (2). (Asymptotic Stability) For any $g \in C_{\mbox{unif}} ({\mathbb R})$ satisfying \begin{equation}\label{1.4.5} \liminf_{x\to \infty} g(x) > W^0(0),\ \ \ \limsup_{x\to - \infty} g(x) < W^0(0), \end{equation} where $ W^0(t)= w(\alpha^0,t)$ and $w(\alpha^0,t)$ is the solution of (\ref{1.1.2}). Then there is $\xi_0 \in {\mathbb R}$ such that \begin{equation}\label{1.4.6} \|u(\cdot, t; g) - U(\cdot - ct+ \xi_0,t)\| \to 0 \end{equation} exponentially as $t \to \infty$. \end{thm} In order to prove the theorem we need the following lemmas. The first lemma use the monotonicity of $U(\cdot,t)$ to construct super- and sub- solutions. \begin{lem}\label{lemma1.4.2} There exist $\beta_1 >0,\delta_1>0 $ and $ \sigma_1>0$ such that, for any $\delta\in (0,\delta_1)$, $\tau\in {\mathbb R}^+$ and $\xi_0 \in {\mathbb R}$, $v^\pm (x,t)$ are super- and sub- solutions of (\ref{1.4.1}), respectively, on $[\tau, \infty]$, where \begin{equation}\label{1.4.7} v^\pm (x,t) = U(x-c(t-\tau) +\xi_0 \pm \sigma_1\delta (1-e^{-\beta _1 (t-\tau)}),t) \pm \delta e^{- \beta_1 (t-\tau)} \end{equation} for $x\in {\mathbb R} $ and $t\in [\tau,\infty)$. \end{lem} \begin{proof} The proof of the lemma is similar to that of Lemma 2.2 in \cite{kn:chen}. We omit it. \end{proof} The next lemma is an analog of the strong comparison principle of parabolic equations. This is the key lemma to apply the ``squeezing'' technique employed in \cite{kn:chen} to prove the stability. \begin{lem}\label{lemma1.4.3} There is a positive function $\eta(\cdot,t)$ satisfying $0\le \eta(\cdot,t)\le 1$ for $t\in[0,T]$ such that $\eta(\cdot,t)$ is non-increasing and for any super-solution $u_1(x,t)$ and sub-solution $u_2(x,t)$ of (\ref{1.4.1}) on ${\mathbb R}^+$ satisfying $u_1(x,\tau)\ge u_2(x,\tau)$ for all $x\in {\mathbb R}$ and for some $\tau\in {\mathbb R}$, and $|u_i(x,t)| \le K_0 =\sup_{t\in {\mathbb R}}\{|W^-(t)|+1,|W^+(t)|+1\}$ for all $x\in {\mathbb R}$ and $t\ge \tau$, the following holds \begin{equation}\label{1.4.8} u_1(x,t) -u_2(x,t) \ge \eta(M,t-\tau)\int_z^{z+1}[u_1(y,\tau) -u_2(y,\tau)] dy \end{equation} for all $x\in {\mathbb R}$ with $|x-z|\le M$ and $t\ge \tau$. \end{lem} For the proof of this lemma, we refer the reader to a similar result in \cite{kn:chen}. To prove the stability, we first need to show that, for given initial data as in (\ref{1.4.5}), the solution with this initial data first forms a vague front of periodic traveling waves as the system evolves. In order to prove that, we need to construct various super- and sub- solutions. \begin{lem}\label{lemma1.4.4} Let $\zeta(s) =\frac{1}{2}( 1+ \tanh \frac{s}{2})$. For any given $T_0>0$ and $m_\pm\in {\mathbb R}$ with $m_-W^0(0) \mbox{ and } \limsup_{x\to - \infty} g(x) 0$, there are constants $H>0$ and $T_0>0$ such that \begin{equation}\label{1.4.16} U(x-H,T_0)-\delta \le u(x,T_0;g) \le U(x+ H , T_0) + \delta. \end{equation} \end{lem} \begin{proof} Without loss of generality, we assume, for some $0<\delta_0<1$, that \begin{align*} W^-(0)-\delta_0\le g(x)\le W^+(0) + \delta_0. \end{align*} By assumption (H2), for $\delta<<1 $, there is a $T_0>0$ such that \begin{eqnarray}\label{1.4.17} W^+(T_0)-\delta/\,4 < w(m_+,T_0) < W^+(T_0) +\delta/\,4 \end{eqnarray} for $m_+ = W^0(0) + \delta_0$ or $m_+ = W^+(0) +2 \delta_0$, and that \begin{eqnarray}\label{1.4.18} W^-(T_0)-\delta/\,4 < w(m_-,T_0) < W^-(T_0) +\delta/\,4 \end{eqnarray} for $m_- = W^0(0) - \delta_0$ or $m_- = W^-(0) -2 \delta_0$, where $w(m_{\pm},t)$ are as in Lemma \ref{lemma1.4.5}. For $T_0$ fixed as above, by Lemma \ref{lemma1.4.4}, there are $K>0$, $C>0$ and $\epsilon_0>0$ such that, for all $0<\epsilon \le \epsilon_0$, \begin{align} v^+(x,t) =& w(W^+(0) +2\delta_0,t) \zeta(\epsilon(x+h) +Ct) \nonumber\\&+ w(W^0(0) -\delta_0,t) (1-\zeta(\epsilon(x+h) +Ct)) +\rho(\epsilon) e^{Kt}\nonumber \end{align} and \begin{align} v^-(x,t) =& w(W^0(0) +\delta_0,t) \zeta(\epsilon(x-h) -Ct) \nonumber\\&+ w(W^-(0) -2\delta_0,t) (1-\zeta(\epsilon(x-h) -Ct)) -\rho(\epsilon) e^{Kt}\nonumber \end{align} are super- and sub-solutions on ${\mathbb R}\times [0,T_0]$, respectively. Fix $\epsilon <\epsilon_0$ small enough such that \begin{equation}\label{1.4.19} \rho(\epsilon) e^{KT_0}<\delta/\,4. \end{equation} By (\ref{1.4.15}), there is an $h$ large enough such that \begin{equation}\label{1.4.20} v^-(x,0) \le g(x) \le v^+(x,0). \end {equation} Hence, by the comparison principle, \begin{equation}\label{1.4.21} v^-(x,t) \le u(x,t;g) \le v^+(x,t) \end {equation} for all $x\in {\mathbb R}$ and $t\in[0,T_0]$. Since $\lim_{x\to\infty}U(x-CT_0,T_0) = W^\pm(T_0)$, by (\ref{1.4.16})- (\ref{1.4.21}), there exists $H$ large enough such that \begin{equation}\label{1.4.22} U(x-H,T_0)-\delta \le v^-(x,T_0)\le u(x,T_0;g) \le v^+(x,T_0)\le U(x+ H , T_0) + \delta \end{equation} for all $x\in {\mathbb R}$. This completes the proof. \end{proof} \vspace{.1in} The following lemma is the ``squeezing technique'' employed in \cite{kn:chen}. \begin{lem}\label{lemma1.4.6} There exists $\epsilon^*>0$ such that if $u(x,t)$ is a solution of (\ref{1.4.1}), and if for some $\tau \in {\mathbb R}^+$, $\xi\in {\mathbb R}$, $\delta \in (0,\frac {\delta_1}{2})$, and $h>0$, one has \begin{equation}\label{1.4.23} U(x-c\tau + \xi,\tau) -\delta \le u(x,\tau)\le U(x-c\tau+\xi + h,\tau) + \delta \end{equation} for all $x\in {\mathbb R}$, then for every $t\ge\tau +1$, there exist $\hat{\xi}(t)$, $\hat{\delta}(t)\ge 0$ and $\hat{h}(t)\ge 0$ satisfying \begin{eqnarray} \hat{\xi}(t)&\in& [\xi-\sigma_1\delta, \xi +h+\sigma_1 \delta],\\ 0\le\hat{\delta}(t)&\le&e^{-\beta_1(t-\tau-1)}[\delta + \epsilon^*\min\{h,1\}], \\ 0\le \hat{h}(t) &\le&[h-\sigma_1 \epsilon^*\min\{h,1\}] + 2\sigma_1 \delta, \end{eqnarray} such that (\ref{1.4.23}) holds with $ \tau$, $\xi$, $\delta$ and $h$ being replaced by $ t$ $(\ge \tau +1)$, $\hat{\xi}(t)$, $\hat{\delta}(t)$ and $\hat{h}(t)$ respectively, where $\beta_1$, $\delta_1$, and $\sigma_1$ are as in Lemma \ref{lemma1.4.2}. \end{lem} \begin{proof} The proof is similar to that of Lemma 3.3 in \cite{kn:chen}. In the proof of that lemma, the only properties used are given by Lemma \ref{lemma1.4.2} and Lemma \ref{lemma1.4.3}. For details, see \cite{kn:chen} and \cite{kn:shen}. % \end{proof} \vspace{.1in} {\bf Proof of Theorem \ref{theorem1.4.1}. } (1). Let $\epsilon >0$ be given. Since $U(\cdot,\cdot)$ is uniformly continuous on ${\mathbb R}\times [0,T]$, there is a constant $k_0>0$ such that \begin{equation}\label{1.4.27} |U(x+k,t) - U(x,t)|<\epsilon/2, \end{equation} for all $x\in {\mathbb R}$, $t\in [0,T]$ and all $k$ with $|k|\le k_0$. Let $\beta_1$, $\delta_1$ and $\sigma_1$ be given as in Lemma \ref{lemma1.4.2}. Choose $\delta>0$ such that $\delta <\min\{\delta_1, \epsilon/\,2, k_0/\,\sigma_1\}$. Then for any $g\in C_{unif}({\mathbb R})$ satisfying $\|g(\cdot) - U(\cdot,0)\| < \delta$, by Lemma \ref{lemma1.4.2}, we have \begin{align*} U(x-ct - \sigma_1\delta (1-e^{-\beta _1 t}),t) - \delta e^{- \beta_1 t} \le& u(x,t;g)\\ \le& U(x-ct + \sigma_1\delta (1-e^{-\beta _1 t}),t) + \delta e^{- \beta_1 t} \end{align*} for $x\in {\mathbb R} $ and $t\in [0,\infty)$. By (\ref{1.4.27}) and the choice of $\delta$, we have \begin{eqnarray} \|u(\cdot,t;g) - U(\cdot -ct,t)\| < \epsilon, \nonumber \end{eqnarray} for all $t>0$. (2). Let $\epsilon^*$ be given as in Lemma \ref{lemma1.4.6} and $\beta_1$, $\delta_1$ and $\sigma_1$ be given as in Lemma \ref{lemma1.4.2}. Let $\overline{\delta} = \min\{\delta_1/\,2, \epsilon^*/\,4\}$, and $\overline{\gamma} = \sigma_1\epsilon^* -2\sigma_1\overline{\delta}$. Let $t_0$ be chosen such that $e^{-\beta_1(t_0 -1)}(\overline{\delta} + \epsilon^*)\le (1-\overline{\gamma})\overline{\delta}$. By Lemma \ref{lemma1.4.5}, there are $\xi_0\in {\mathbb R}$ , $h>0$ and $T_0>0$ such that \begin{equation}\label{1.4.28} U(x-cT_0+\xi_0,T_0)-\overline{\delta} \le u(x,T_0;g) \le U(x-cT_0+ \xi_0 +h , T_0) + \overline{\delta} \end{equation} for all $x\in {\mathbb R}$. First, we may assume $0< h \le 1$. In fact, if $h>1$, we can choose integer $N>0$ such that $0\le h-N \overline{\gamma}\le1$. Applying Lemma \ref{lemma1.4.6} repeatedly, we conclude that \begin{align} U(x -c(kt_0 +T_0)+\xi_k,&kt_0 +T_0) -\overline{\delta}_k \le u(x,kt_0 +T_0;g)\nonumber\\ \le& U(x- c(kt_0 +T_0)+\xi_k +h_k,kt_0 +T_0) +\overline{\delta}_k \label{1.4.29} \end{align} for all $x\in {\mathbb R}$, where $\xi_k\in[\xi_{k-1} -\sigma_{k-1}\overline{\delta}_{k-1}, \xi_{k-1} +\sigma_{k-1}\overline{\delta}_{k-1} + h_{k-1}]$, $\overline{\delta}_{k}\le (1-\overline{\gamma})^k \overline{\delta}$, $h_k\le h_{k-1}-\overline{\gamma}$, and $\delta_0=\overline{\delta}$. Therefore (\ref{1.4.28}) holds with $\xi_0$, $\overline{\delta}$, $h$, and $T_0$ being replaced by $\xi_N$, $\overline{\delta}_N$, $h_N$, and $T_N =Nt_0 +T_0$, respectively. Now we assume $h\le 1$ and (\ref{1.4.28}) holds. Define $T_k=kt_0$, $\overline{\delta}_{k}= (1-\overline{\gamma})^k \overline{\delta}$, and $h_k= h_{k-1}-\overline{\gamma}$. Then we can show by induction that (\ref{1.4.29}) still holds. Define $\delta(t) =\overline{\delta}_k$, $\xi(t) = \xi_k -\sigma_1\overline{\delta}_k$, and $h(t) = h_k +2\sigma_1\overline{\delta}_k$, for $t\in [T_k +T_0, T_{k+1}+T_0]$ and $k=0,1,\dots$. Then, by Lemma \ref{lemma1.4.2}, \begin{eqnarray} U(x-ct +\xi(t),t) -\delta(t)\le u(x,t;g)\le U(x-ct +\xi(t) + h(t),t) + \delta(t) \end{eqnarray} for $x\in {\mathbb R}$ and $t\ge T_0$. Note that $\delta(t)\to 0$, $h(t)\to 0$ and $\xi(t) \to\xi(\infty)$ exponentially as $t\to \infty$. 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Capuzzo Dolcetta and P. L. Lions. Lecture Notes in Mathematics, 1660. Fondazione C.I.M.E.. [C.I.M.E. Foundation] Springer-Verlag, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 1997. \bibitem{kn:shen} W. Shen, \newblock Traveling waves in time almost periodic structures governed by bistable nonlinearities I. Stability and uniqueness, \newblock to appear in {\em J. Diff. Eqns.} \bibitem{kn:shen1} W. Shen, \newblock Traveling waves in time almost periodic structures governed by bistable nonlinearities II. Existence, \newblock to appear in {\em J. Diff. Eqns.} \end{thebibliography} \bigskip \noindent {\sc Peter Bates} \\ Department of Mathematics\\ Brigham Young University\\ Provo, UT 84602. USA\\ E-mail address: peter@math.byu.edu \medskip \noindent {\sc Fengxin Chen} \\ Division of Mathematics and Statistics\\ University of Texas at San Antonio\\ 6900 North Loop 1604 West \\ San Antonio, TX 78249. USA \\ E-mail address: feng@math.utsa.edu \end{document}