\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Monotone method \hfil EJDE--2003/Conf/10} {EJDE--2003/Conf/10 \hfil Azmy S. Ackleh \& Keng Deng \hfil} \begin{document} \setcounter{page}{11} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent Fifth Mississippi State Conference on Differential Equations and Computational Simulations, \newline Electronic Journal of Differential Equations, Conference 10, 2003, pp 11--22. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Monotone method for nonlinear nonlocal hyperbolic problems % \thanks{ {\em Mathematics Subject Classifications:} 35A05, 35A35, 35L60, 92D99. \hfil\break\indent {\em Key words:} Nonlinear nonlocal hyperbolic IBVP, monotone approximation, \hfil\break\indent existence uniqueness. \hfil\break\indent \copyright 2003 Southwest Texas State University. \hfil\break\indent Published February 28, 2003. } } \date{} \author{Azmy S. Ackleh \& Keng Deng} \maketitle \begin{abstract} We present recent results concerning the application of the monotone method for studying existence and uniqueness of solutions to general first-order nonlinear nonlocal hyperbolic problems. The limitations of comparison principles for such nonlocal problems are discussed. To overcome these limitations, we introduce new definitions for upper and lower solutions. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \section{Introduction} This paper is concerned with the first-order hyperbolic initial-boundary value problem $$\begin{gathered} u_{t}+(g(x,t)u)_{x}=F(x,t,u,\phi(u)(x,t))\quad \text{in }D_{T},\\ g(a,t) u(a,t)=\int_{a}^{b}\beta(y,t) u(y,t)dy \quad\text{on }(0,T),\\ u(x,0)=u_{0}(x) \quad \text{in }[a,b]\,. \end{gathered} \label{1}$$ Here $D_{T}=(a,b)\times(0,T)$ for some $T>0$, $0\leq a0\ \text{on}\ [a,b)\times[0,T]$. In addition, if $b<\infty$ then $\ g(b,t)=0$, $t\in[0,T]$. Otherwise, $\lim_{x\rightarrow\infty}g(x,t)=0$ for $t\in[0,T]$. \item[(A2)] $\beta\in C_{B}(D_{T})$ is a nonnegative function. \item[(A3)] $F\in C_{B}^{1}(D_{T}\times\mathbb{R\times R})$. \item[(A4)] $u_{0}\in C_{B}^{1}(a,b)$ is nonnegative and satisfies the compatibility condition $g(a,0)u_{0}(a)=\int_{a}^{b}\beta(y,0)u_{0}(y)dy.$ \end{itemize} It is worth noting that (A4) can be considerably relaxed for certain cases (see, e.g., \cite{ad2,ad3,ad4}). The paper is organized as follows. In Section 2, we present a comparison principle and show that this principle holds for the case $F_{\phi}% (x,t,u,\phi)\geq0$. In Section 3, we discuss the case $F_{\phi}(x,t,u,\phi )\leq0$. Section 4 is devoted to the case where $F_{\phi}$ has no sign restriction. An unbounded domain (i.e., $b=\infty$) is considered in Section 5. \section{The case $F_{\phi}(x,t,u,\phi)\geq0$} In this section we assume that $g=g(x)$, $\beta=\beta(x)$, $b<\infty$, $\phi(u)(t)=\int_{a}^{b}u(y,t)dy,F_{\phi}\geq0$ and $F_{u}% +M\geq0$ for some positive constant $M$. We begin with the definition of upper and lower solutions of problem (\ref{1}). \begin{definition} \rm A function $u(x,t)$ is called an upper (a lower) solution of (\ref{1}) on $D_{T}$ if all the following hold. \begin{itemize} \item [(i)]$u\in C(D_{T})\cap L^{\infty}(D_{T})$. \item[(ii)] $u(x,0)\geq(\leq)\ u_{0}(x)$ in $[a,b]$. \item[(iii)] For every $t\in(0,T)$ and every nonnegative $\xi(x,t)\in C^{1}(\overline{D_{T}})$, \begin{align*} &\int_{a}^{b}u(x,t)\xi(x,t)dx\\ &\geq(\leq)\int_{a}^{b}u(x,0)\xi(x,0)dx+\int_{0}% ^{t}\xi(a,\tau)\int_{a}^{b}\beta(x)u(x,\tau)\,dx\,d\tau\\ &\quad +\int_{0}^{t}\int_{a}^{b}u(x,\tau)[\xi_{\tau }(x,\tau)+g(x)\xi_{x}(x,\tau)]\,dx\,d\tau\\ &\quad +\int_{0}^{t}\int_{a}^{b}\xi(x,\tau )F(x,\tau,u,\phi(u)(\tau))\,dx\,d\tau. \end{align*} \end{itemize} \end{definition} The following comparison principle is established in \cite{ad1}. To our knowledge, this result is the only comparison result for problem (\ref{1}) available in the literature. \begin{theorem} Let $u$ and $v$ be an upper solution and a lower solution of (\ref{1}), respectively. Then $u\geq v$ in $\overline{D_{T}}$. \end{theorem} Next we construct the following monotone approximation. Let ${\underline{u}% }^{0}$ and ${\overline{u}}^{0}$ be a lower solution and an upper solution of (\ref{1}), respectively. For $k=1,2,\dots$ let ${\underline{u}}^{k}$ and ${\overline{u}}^{k}$ satisfy the uncoupled systems \begin{gather*} ({\underline{u}}^{k})_{t}+(g{\underline{u}}^{k})_{x}% =F(x,t,{\underline{u}}^{k-1},\phi({\underline{u}}^{k-1}))-M({\underline{u}% }^{k}-{\underline{u}}^{k-1})\quad \text{in } D_{T},\\ g(a){\underline{u}}^{k}(a,t)=\int_{a}^{b}\beta(y){\underline{u}% }^{k}(y,t)dy \quad \text{on } (0,T),\\ {\underline{u}}^{k}(x,0)=u_{0}(x) \quad \text{in }\ [a,b] \end{gather*} and \begin{gather*} ({\overline{u}}^{k})_{t}+(g{\overline{u}}^{k})_{x}% =F(x,t,{\overline{u}}^{k-1},\phi({\overline{u}}^{k-1}))-M({\overline{u}}% ^{k}-{\overline{u}}^{k-1})\quad \text{in } D_{T},\\ g(a){\overline{u}}^{k}(a,t)=\int_{a}^{b}\beta(y){\overline{u}% }^{k}(y,t)dy \quad \text{on } (0,T),\\ {\overline{u}}^{k}(x,0)=u_{0}(x) \quad \text{in } [a,b]. \end{gather*} The functions ${\underline{u}}^{k}$ and ${\overline{u}}^{k}$ exist since they satisfy linear equations. Furthermore, it can be shown that (see \cite{ad1}) ${\underline{u}}^{0}\leq{\underline{u}}^{1}\leq\dots\leq{\underline{u}}% ^{k}\leq{\overline{u}}^{k}\leq\dots\leq{\overline{u}}^{k}\leq{\overline{u}% }^{0}\quad \text{in }\overline{D_{T}}.$ The following convergence result was established in \cite{ad1}. \begin{theorem} Let ${\underline{u}}^{0}$ and ${\overline{u}}^{0}$ be a lower solution and an upper solution of (\ref{1}), respectively, and they are continuously differentiable in $t$. Then the monotone sequences defined above converge in $L^{2}(a,b)$ to the unique solution $u(x,t)$ uniformly on $0\leq t\leq T$. Moreover, the order of convergence is linear. \end{theorem} In \cite{ad1} it was shown via a counter example that the restriction $F_{\phi}\geq0$ is necessary for establishing a comparison between upper and lower solutions. To overcome this obstacle, in the next section we define a new pair of upper and lower solutions and use this definition to establish a comparison principle. \section{The case F$_{\phi}(x,t,u,\phi)\leq0$} In this section we restrict our attention to the case $F(x,t,u,\phi )=-m(x,t,\phi)u$. We assume that $b<\infty$, $\phi(u)(t) =\int_{a}^{b}u(y,t)dy$, and $m_{\phi}\geq0$. We introduce the following definition of upper and lower solutions. \begin{definition} \rm A pair of functions $u(x,t)$ and $v(x,t)$ are called an upper solution and a lower solution of (\ref{1}) on $D_{T}$, respectively, if all the following hold. \begin{itemize} \item [(i)] $u,v\in L^{\infty}(D_{T})$. \item[(ii)] $u(x,0)\geq u_{0}(x)\geq v(x,0)$ in $[a,b]$. \item[(iii)] For every $t\in(0,T)$ and every nonnegative $\xi(x,t)\in C^{1}(\overline{D_{T}})$, % \begin{aligned} &\int_{a}^{b}u(x,t)\xi(x,t)dx\\ &\geq\int_{a}^{b}u(x,0)\xi(x,0)dx+\int_{0}^{t}% \xi(a,\tau)\int_{a}^{b}\beta(x,\tau)u(x,\tau)\,dx\,d\tau\\ &\quad+\int_{0}^{t}\int_{a}^{b}[\xi_{\tau}\left( x,\tau\right) +g(x,\tau)\xi_{x}\left( x,\tau\right) ]u(x,\tau)\,dx\,d\tau\\ &\quad-\int_{0}^{t}\int_{a}^{b}\xi(x,\tau)m(x,\tau ,\phi(v)(\tau))u(x,\tau)\,dx\,d\tau \end{aligned} \label{31}% and % \begin{aligned} &\int_{a}^{b}v(x,t)\xi(x,t)dx\\ &\leq\int_{a}^{b}v(x,0)\xi(x,0)dx+\int_{0}^{t}% \xi(a,\tau)\int_{a}^{b}\beta(x,\tau)v(x,\tau)\,dx\,d\tau\\ &\quad+\int_{0}^{t}\int_{a}^{b}[\xi_{\tau}\left( x,\tau\right) +g(x,\tau)\xi_{x}\left( x,\tau\right) ]v(x,\tau)\,dx\,d\tau\\ &\quad-\int_{0}^{t}\int_{a}^{b}\xi(x,\tau)m(x,\tau ,\phi(u)(\tau))v(x,\tau)\,dx\,d\tau. \end{aligned} \label{32}% \end{itemize} \end{definition} A function $u(x,t)$ is called a solution of (\ref{1}) on $D_{T}$ if $u$ satisfies (\ref{31}) with $\geq$'' replaced by $=$'' and $\phi (v)(\tau)$ by $\phi(u)(\tau)$. Based on this definition, the following comparison result was established in \cite{ad2}. \begin{theorem} Let $u$ and $v$ be a nonnegative upper solution and a nonnegative lower solution of (\ref{1}), respectively. Then $u\geq v$\ a.e. in $D_{T}$. \end{theorem} As a consequence, the following uniqueness result can be proved (see \cite{ad2}). \begin{theorem} Let $u(x,t)$ be a nonnegative solution of (\ref{1}) with $\phi(u)(t)\in C([0,T])$. Then $u$ is unique. \end{theorem} We now construct monotone sequences of upper and lower solutions. To this end, let ${\underline{u}}^{0}(x,t)$ and ${\overline{u}}^{0}(x,t)$ be a nonnegative lower solution and a nonnegative upper solution of (\ref{1}), respectively. We then define two sequences $\left\{ {\underline{u}}% ^{k}\right\} _{k=0}^{\infty}$ and $\left\{ {\overline{u}}^{k}\right\} _{k=0}^{\infty}$ as follows: For $k=1,2,\dots$ \begin{gather*} {\underline{u}}_{t}^{k}+(g(x,t){\underline{u}}^{k})_{x}% =-m(x,t,\phi({\overline{u}}^{k-1})){\underline{u}}^{k} \quad \text{in }D_{T}, \\ g(a,t){\underline{u}}^{k}(a,t)=\underline{B}^{k-1}(t) \quad \text{on }(0,T),\\ {\underline{u}}^{k}(x,0)=u_{0}(x) \quad \text{in }[a,b], \end{gather*} where $\underline{B}^{k-1}(t)\equiv\int_{a}^{b}\beta (y,t){\underline{u}}^{k-1}(y,t)dy$, and \begin{gather*} {\overline{u}}_{t}^{k}+(g(x,t){\overline{u}}^{k})_{x}% =-m(x,t,\phi({\underline{u}}^{k-1})){\overline{u}}^{k} \quad \text{in }D_{T},\\ g(a,t){\overline{u}}^{k}(a,t)=\overline{B}^{k-1}(t) \quad \text{on }(0,T),\\ {\overline{u}}^{k}(x,0)=u_{0}(x) \quad \text{in }[a,b], \end{gather*} where $\overline{B}^{k-1}(t)\equiv\int_{a}^{b}\beta(y,t){\overline {u}}^{k-1}(y,t)dy$. Since $\underline{B}^{k-1}$ and $\overline{B}^{k-1}$ are given functions, the existence of solutions ${\underline{u}}^{k}$ and ${\overline{u}}^{k}$ easily follows. Furthermore, we can show that these sequences satisfy ${\underline{u}}^{0}\leq{\underline{u}}^{1}\leq\cdot\cdot\cdot\leq {\underline{u}}^{k}\leq{\overline{u}}^{k}\leq\cdot\cdot\cdot\leq{\overline{u}% }^{1}\leq{\overline{u}}^{0}\ \quad\mbox{a.e. in } D_{T}.$ Upon establishing the monotonicity of our sequences, we can prove the following convergence result (see \cite{ad2}). \begin{theorem} Suppose that ${\underline{u}}^{0}(x,t)$ and ${\overline{u}}^{0}(x,t)$ are a nonnegative lower solution and a nonnegative upper solution of (\ref{1}), respectively. Then, the sequences $\left\{ {\underline{u}}^{k}\right\} _{k=0}^{\infty}$ and $\left\{ {\overline{u}}^{k}\right\} _{k=0}^{\infty}$ converge uniformly to the unique solution $u(x,t)$ of problem (\ref{1}) on $D_{T}$. Moreover, the order of convergence is linear. \end{theorem} \begin{remark} \rm In \cite{ad2} this monotone method was used to numerically solve (\ref{1}). The resutls in that paper indicate that such a scheme converges rapidly to the solution. \end{remark} \section{No restriction on the sign of F$_{\phi}(x,t,u,\phi)$} In this section we assume that $b<\infty$, $\phi(u)(t)=\int _{a}^{b}d(y)u(y,t)dy$, and that $F(x,t,u,\phi)=-m(x,t,\phi)u$ with $M+m_{\phi }\geq0$ for some positive constant $M$. Consider the following new definition of upper and lower solutions: \begin{definition} \rm A pair of functions $u(x,t)$ and $v(x,t)$ are called an upper solution and a lower solution of (\ref{1}) on $D_{T}$, respectively, if all the following hold. \begin{itemize} \item [(i)] $u,v\in L^{\infty}(D_{T})$. \item[(ii)] $u(x,0)\geq u_{0}(x)\geq v(x,0)$ a.e. in $(a,b)$. \item[(iii)] For every $t\in(0,T)$ and every nonnegative $\xi(x,t)\in C^{1}(\overline{D_{T}})$, \end{itemize} % \begin{aligned} &\int_{a}^{b}u(x,t)\xi(x,t)dx\\ &\geq\int_{a}^{b}u(x,0)\xi(x,0)dx+\int_{0}^{t}\xi (a,\tau)\int_{a}^{b}\beta(x,\tau)u(x,\tau)dx\,d\tau\\ &\quad +\int_{0}^{t}\int_{a}^{b}[\xi_{\tau}\left( x,\tau\right) +g(x,\tau)\xi_{x}\left( x,\tau\right) ]u(x,\tau)\,dx\,d\tau\\ &\quad -\int_{0}^{t}\int_{a}^{b}\xi(x,\tau)\left[ m(x,\tau ,\phi(v)(\tau))+M \phi(v)(\tau)-M \phi(u)(\tau)\right] u(x,\tau) \,dx\,d\tau \end{aligned} \label{42a} and \begin{aligned} &\int_{a}^{b}v(x,t)\xi(x,t)dx\\ &\leq\int_{a}^{b}v(x,0)\xi(x,0)dx+\int_{0}^{t}\xi (a,\tau)\int_{a}^{b}\beta(x,\tau)v(x,\tau)\,dx\,d\tau\\ &\quad +\int_{0}^{t}\int_{a}^{b}[\xi_{\tau}\left( x,\tau\right) +g(x,\tau)\xi_{x}\left( x,\tau\right) ]v(x,\tau)\,dx\,d\tau\\ &\quad -\int_{0}^{t}\int_{a}^{b}\xi(x,\tau)\left[ m(x,\tau ,\phi(u)(\tau))+M \phi(u)(\tau)-M \phi(v)(\tau)\right] v(x,\tau) \,dx\,d\tau. \end{aligned} \label{42} \end{definition} A function $u(x,t)$ is called a solution of (\ref{1}) on $D_{T}$ if $u$ satisfies (\ref{42a}) with $\geq$" replaced by =" and $\phi(v)(\tau)$ by $\phi(u)(\tau)$. Using this definition, we establish the following comparison principle \cite{ad3}. \begin{theorem} Let $u$ and $v$ be a nonnegative upper solution and a nonnegative lower solution of (\ref{1}), respectively. Then $u\geq v$ a.e. in $D_{T}$. \end{theorem} Furthermore, we prove the following uniqueness result. \begin{corollary} Let $u(x,t)$ be a nonnegative solution of (\ref{1}) with $\phi(u)(t)\in C([0,T])$. Then $u$ is unique. \end{corollary} We now construct a pair of nonnegative lower and upper solutions of (\ref{1}). Let ${\underline{u}}^{0}(x,t)=0$. Choose a constant $\gamma$ large enough such that $\max_{\overline{D_{T}}}\beta(x,t)/\min_{[0,T]}g(a,t)\leq \gamma/2.$ Fix this $\gamma$ and choose $\delta$ large enough such that $\Vert u_{0}\Vert_{\infty}\leq(\delta/2)\exp(-\gamma b).$ Now choose $\sigma$ large enough such that $\sigma\geq2M\delta\Vert\eta\Vert_{\infty}\exp(-\gamma a)/\gamma+\gamma \max_{\overline{D_{T}}}g(x,t)+\max_{\overline{D_{T}}}% |g_{x}(x,t)|.$ Let ${\overline{u}}^{0}(x,t)=\delta\exp(\sigma t)\exp(-\gamma x)$. Then it can be easily shown that ${\underline{u}}^{0}$ and ${\overline{u}}^{0}$ are a pair of lower and upper solutions of (\ref{1}) on $[a,b]\times[0,T_{0}]$ with $T_{0}=\min\{T,(\ln2)/\sigma\}$. We then define two sequences $\left\{ {\underline{u}}^{k}\right\} _{k=0}^{\infty}$ and $\left\{ {\overline{u}}^{k}\right\} _{k=0}^{\infty}$ as follows: \\ For $k=1,2,\dots$ $$\begin{gathered} {\underline{u}}_{t}^{k}+(g(x,t){\underline{u}}^{k})_{x}=-D^{k-1}% (x,t) {\underline{u}}^{k} \quad \text{in }D_{T_{0}}, \\ g(a,t){\underline{u}}^{k}(a,t)=\underline{B}^{k-1}(t) \quad \text{on }(0,T_{0}),\\ {\underline{u}}^{k}(x,0)=u_{0}(x) \quad \text{in }[a,b], \end{gathered}\label{43}%$$ where \begin{gather*} D^{k-1}(x,t)=m(x,t,\phi({\overline{u}}^{k-1}))+M \phi({\overline{u}} ^{k-1})-M \phi({\underline{u}}^{k-1}),\\ \underline{B}^{k-1}(t)\equiv\int_{a}^{b}\beta(y,t){\underline{u}}^{k-1}(y,t)dy, \end{gather*} and $$% \begin{gathered} {\overline{u}}_{t}^{k}+(g(x,t){\overline{u}}^{k})_{x}% =-E^{k-1}(x,t){\overline{u}}^{k} \quad \text{in }D_{T_{0}},\\ g(a,t){\overline{u}}^{k}(a,t)=\overline{B}^{k-1}(t) \quad \text{on }(0,T_{0}),\\ {\overline{u}}^{k}(x,0)=u_{0}(x) \quad \text{in }[a,b], \end{gathered} \label{44}%$$ where \begin{gather*} E^{k-1}(x,t)=m(x,t,\phi({\underline{u}}^{k-1}))+M \phi({\underline{u}}% ^{k-1})-M \phi({\overline{u}}^{k-1}) \\ \overline{B}^{k-1}(t)\equiv\int_{a}^{b}\beta(y,t){\overline{u}}^{k-1}(y,t)dy. \end{gather*} The existence of solutions to problems (\ref{43}) and (\ref{44}) follows from the fact that $\underline{B}^{k-1}$ and $\overline{B}^{k-1}$ are given functions. By similar reasoning, we can show that ${\underline{u}}^{k}\leq{\underline{u}% }^{k+1}\leq{\overline{u}}^{k+1}\leq{\overline{u}}^{k}$ and that ${\underline {u}}^{k+1}$ and ${\overline{u}}^{k+1}$ are also a lower solution and an upper solution of (\ref{1}), respectively. Thus by induction, we obtain two monotone sequences that satisfy ${\underline{u}}^{0}\leq{\underline{u}}^{1}\leq\dots\leq{\underline{u}}% ^{k}\leq{\overline{u}}^{k}\leq\dots\leq{\overline{u}}^{1}\leq{\overline{u}% }^{0}\quad\;\mbox{a.e. in } D_{T_{0}}.$ Hence, it follows from the monotonicity of the sequences $\left\{ {\underline{u}}^{k}\right\} _{k=0}^{\infty}$ and $\left\{ {\overline{u}}% ^{k}\right\} _{k=0}^{\infty}$ that there exist functions ${\underline{u}}$ and ${\overline{u}}$ such that ${\underline{u}}^{k}\rightarrow{\underline{u}}$ and ${\overline{u}}^{k}\rightarrow{\overline{u}}$ pointwise in $D_{T_{0}}$. It is not too difficult to argue that ${\underline{u}}={\overline{u}}$ a.e. in $D_{T_{0}}$. We denote this common limit by $u$. Upon establishing the monotonicity of our sequences, we can also prove the following convergence result. \begin{theorem} The sequences $\left\{ {\underline{u}}^{k}\right\} _{k=0}^{\infty }$ and $\left\{ {\overline{u}}^{k}\right\} _{k=0}^{\infty}$ converge uniformly along characteristic curves to a limit function $u(x,t)$. Moreover, the function $u$\ is the unique solution of problem (\ref{1}) on $[a,b]\times[0,T_{0}]$. \end{theorem} \begin{remark} \rm It is not too difficult to show that this local solution is indeed a global solution. \end{remark} \section{Unbounded Domains} This section is concerned with a special model which describes the aggregation of phytoplankton (see \cite{af}). Here $a=0$, $b=\infty$, $\phi(u)\left( x,t\right) =\frac{1}{2}\int_{0}^{x}\eta (x-y,y)u(x-y,t)u(y,t)dy-\int_{0}^{\infty}\eta(x,y)u(x,t)u(y,t)dy$ and $F(x,t,u,\phi)=\phi+f(x,t)u.$ We assume that $\eta$ and $f$ are bounded continuous functions. Let $C_{0,r}^{1}(D_{T})=\{\psi\in C^{1}(D_{T}):\exists\ x_{\psi}\in(0,\infty )\ \mathrm{such\ that}\ \psi\equiv0\ \mathrm{for}\ x\geq x_{\psi}\}$. We then introduce the following definition of coupled upper and lower solutions of problem (\ref{1}).\bigskip \begin{definition} \rm A pair of functions $u(x,t)$ and $v(x,t)$ are called an upper solution and a lower solution of (\ref{1}) on $D_{T}$, respectively, if all the following hold. \begin{itemize} \item [(i)] $u,v\in L^{\infty}((0,T);L^{1}(0,\infty))$. \item[(ii)] $u(x,0)\geq u_{0}(x)\geq v(x,0)$ a.e. in $(0,\infty)$. \item[(iii)] For every $t\in(0,T)$ and every nonnegative $\xi\in C_{0,r}^{1}(D_{T})$, % \begin{aligned} &\int_{0}^{\infty}u(x,t)\xi(x,t)dx\\ &\geq\int_{0}^{\infty}u(x,0)\xi(x,0)dx+\int_{0}^{t}% \xi(0,\tau)\int_{0}^{\infty}\beta(x,\tau)u(x,\tau)\,dx\,d\tau\\ &\quad +\int_{0}^{t}\int_{0}^{\infty}[\xi_{\tau}% (x,\tau)+g(x,\tau)\xi_{x}(x,\tau)]u(x,\tau)\,dx\,d\tau\\ &\quad +\int_{0}^{t}\int_{0}^{\infty}\xi(x,\tau )\mathcal{F(}u)(x,\tau)\,dx\,d\tau\\ &\quad -\int_{0}^{t}\int_{0}^{\infty}\xi(x,\tau)\int _{0}^{\infty}\eta(x,y)u(x,\tau)v(y,\tau)dydxd\tau \end{aligned} \label{51} and \begin{aligned} &\int_{0}^{\infty}v(x,t)\xi(x,t)dx\\ &\leq\int_{0}^{\infty}v(x,0)\xi(x,0)dx+\int_{0}^{t}% \xi(0,\tau)\int_{0}^{\infty}\beta(x,\tau)v(x,\tau)\,dx\,d\tau\\ &\quad +\int_{0}^{t}\int_{0}^{\infty}[\xi_{\tau}% (x,\tau)+g(x,\tau)\xi_{x}(x,\tau)]v(x,\tau)\,dx\,d\tau\\ &\quad +\int_{0}^{t}\int_{0}^{\infty}\xi(x,\tau )\mathcal{F(}v)(x,\tau)\,dx\,d\tau\\ &\quad -\int_{0}^{t}\int_{0}^{\infty}\xi(x,\tau)\int _{0}^{\infty}\eta(x,y)v(x,\tau)u(y,\tau)dydxd\tau, \end{aligned} \label{52} where $\mathcal{F(}w)(x,t)=\frac{1}{2}\int_{0}^{x}\eta (x-y,y)w(x-y,t)w(y,t)dy+f(x,t)w(x,t).$ \end{itemize} \end{definition} A function $u(x,t)$ is called a solution of (\ref{1}) on $D_{T}$ if $u$ satisfies (\ref{51}) with $\geq$'' replaced by $=$'' and $v(y,\tau)$ in the last integral by $u(y,\tau)$. The following comparison principle was established in \cite{ad4}. \begin{theorem} Let $u$\ and $v$\ be a nonnegative upper solution and a nonnegative lower solution of (\ref{1}), respectively. Then $u\geq v$\ a.e. in $D_{T}$. \end{theorem} \begin{corollary} Let $\underline{u}$ and $\overline{u}$ be a nonnegative lower solution and a nonnegative upper solution of (\ref{1}), respectively. If $u$ is a solution of (\ref{1}), then $\underline{u}\leq u\leq\overline{u}$\ a.e. in $D_{T}$. \end{corollary} We now construct monotone sequences of upper and lower solutions. Suppose that ${\underline{u}}^{0}(x,t)$ and ${\overline{u}}^{0}(x,t)$ are a pair of lower and upper solutions of (\ref{1}). Since $f$ and $\eta$ are bounded we can choose a positive constant $M$ such that $M-\int_{0}^{\infty}\eta (x,y)u(y,t)dy+f(x,t)\geq0$ for $(x,t)\in{\overline{D}_{T}}$ and ${\underline {u}}^{0}(x,t)\leq u(x,t)\leq{\overline{u}}^{0}(x,t)$. We then set up two sequences $\{{\underline{u}}^{k}\}_{k=0}^{\infty}$ and $\{{\overline{u}}% ^{k}\}_{k=0}^{\infty}$ by the following procedure: \\ For $k=1,2,\dots$ let ${\underline{u}}^{k}$ and ${\overline{u}}^{k}$ satisfy the systems \begin{gather*} {\underline{u}}_{t}^{k}+(g{\underline{u}}^{k})_{x}=\mathcal{F(}% {\underline{u}}^{k-1})-M({\underline{u}}^{k}-{\underline{u}}^{k-1}% )-{\underline{u}}^{k-1}\int_{0}^{\infty}\eta(x,y){\overline{u}}^{k-1}% (y,t)dy \ \mbox{in } D_{T},\\ g(0,t){\underline{u}}^{k}(0,t)=\int_{0}^{\infty}\beta (y,t){\underline{u}}^{k-1}(y,t)dy \quad \mbox{on } (0,T),\\ {\underline{u}}(x,0)=u_{0}(x) \ \mbox{in } [0,\infty) \end{gather*} and \begin{gather*} {\overline{u}}_{t}^{k}+(g{\overline{u}}^{k})_{x}=\mathcal{F(}% {\overline{u}}^{k-1})-M({\overline{u}}^{k}-{\overline{u}}^{k-1})-{\overline {u}}^{k-1}\int_{0}^{\infty}\eta(x,y){\underline{u}}^{k-1}(y,t)dy \ \mbox{in }D_{T},\\ g(0,t){\overline{u}}^{k}(0,t)=\int_{0}^{\infty}\beta (y,t){\overline{u}}^{k-1}(y,t)dy \quad \mbox{on } (0,T),\\ {\overline{u}}(x,0)=u_{0}(x) \ \mbox{in } [0,\infty). \end{gather*} By induction, we can show that the sequences satisfy ${\underline{u}}^{0}\leq{\underline{u}}^{1}\leq\dots\leq{\underline{u}}% ^{k}\leq{\overline{u}}^{k}\leq\dots\leq{\overline{u}}^{1}\leq{\overline{u}% }^{0}\quad \mbox{a.e. in }\ D_{T}.$ Then, we have the following existence-uniqueness result. \begin{theorem} Suppose that $\underline{u}^{0}(x,t)$ and $\overline {u}^{0}(x,t)$ are a nonnegative lower solution and a nonnegative upper solution of (\ref{1}), respectively. Then there exist monotone sequences $\{\underline{u}^{k}(x,t)\}$ and $\{\overline{u}^{k}(x,t)\}$ which converge to the unique solution of (\ref{1}). \end{theorem} \begin{remark} \rm As an example, for a large class of initial data such as $u_{0}(x)=O(e^{-x})$ as $x\rightarrow\infty$, we can construct a pair of nonnegative lower and upper solutions of (\ref{1}) as follows: Let ${\underline{u}}^{0}(x,t)=0$ and ${\overline{u}}^{0}(x,t)=c_{3}e^{c_{2}t}/(1+c_{1}^{2}x^{2})$ with $c_{1}% ,c_{2},c_{3}$ positive constants. First choose $c_{1}$ so large such that $\pi\max_{{\overline{D}_{1}}}\beta(x,t)/\min_{[0,1]}g(0,t)\leq c_{1}.$ Fix this $c_{1}$ and choose $c_{3}$ large enough such that $c_{3}/(1+c_{1}^{2}x^{2})\geq u_{0}(x)$ for $0\leq x<\infty$. We then determine $c_{2}$. Through a routine calculation, we find \begin{align*} \int_{0}^{x}\frac{dy}{[1+c_{1}^{2}(x-y)^{2}](1+c_{1}^{2}y^{2})} &=\frac{2} {c_{1}^{2}x}\left[ \frac{c_{1}x\tan^{-1}(c_{1}x)+\log(1+c_{1}^{2}x^{2}% )}{4+c_{1}^{2}x^{2}}\right] \\ &\leq\frac{2(1+\pi)}{c_{1}(1+c_{1}^{2}x^{2})}. \end{align*} Thus we can choose $c_{2}$ sufficiently large such that $c_{2}\geq\frac{2c_{3}}{c_{1}}(1+\pi)+\max_{{\overline{D}_{1}}}g(x,t)+\max _{{\overline{D}_{1}}}|f(x,t)-g_{x}(x,t)|.$ Then it follows that ${\overline{u}}^{0}$ is a desired upper solution of (\ref{1}) on $D_{T}$ with $T=\min\{1,\log2/c_{2}\}$. \end{remark} We now show that the solution of (\ref{1}) has the following property. \begin{theorem} For the solution $u(x,t)$ of (\ref{1}), $P(t)=\int_{0}^{\infty}u(x,t)dx$ is continuous in the existence interval. \end{theorem} Finally, we establish the existence of a global solution. \begin{theorem} The unique solution of (\ref{1}) exists for $0\leq t<\infty$. \end{theorem} \paragraph{Acknowledgements:} A. 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Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. \end{thebibliography} \noindent\textsc{Azmy S. Ackleh} (e-mail: ackleh@louisiana.edu)\\ \textsc{Keng Deng} (e-mail: deng@louisiana.edu)\\ Department of Mathematics\\ University of Louisiana at Lafayette \\ Lafayette, Louisiana 70504, USA. \end{document}