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\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{equation}
\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}
\end{equation}
Here $D_{T}=(a,b)\times(0,T)$ for some $T>0$, $0\leq
a<b\leq\infty$, and $\phi$ is a function of $u$. Problem (\ref{1})
often arises in applications. For example, the well-known
size-structured model, where $F(x,t,u,\phi )=-m\left(
x,t,\phi\right)  u$ and $\phi(u)(t)=\int_{a}^{b}d(y)u(y,t)dy$,
fits under the class of problems given in (\ref{1}). For the
size-structured model $g$, $m$ and $\beta$ denote the individuals
growth, mortality and reproduction rates, respectively, and $\phi$
denotes a population weight.

There are three common methods used in the literature to prove the existence
and uniqueness of solutions to certain cases of (\ref{1}). One is the
semigroup of operators theoretical approach. This approach is very elegant but
has been applied only to special cases where the parameters are
time-independent, i.e., $g=g(x)$ and $\beta=\beta(x)$. The idea is to write
the PDE as an abstract evolution equation of the form $du/dt=\mathcal{A}%
u+\mathcal{F}(u),\ u(0)=u_{0}$ and show that $\mathcal{A}$ is the
infinitesimal generator of a $C_{0}$-semigroup of operators, and
hence establish the existence and uniqueness of solutions under
some regularity conditions on the mapping $\mathcal{F}(u)$ (see,
e.g.,\cite{ackleh,af,bkw}). The second approach is based on the
classical characteristics method \cite{cs1,jiali}. This approach
is, in general, feasible and relatively easy to apply provided the
function $F(x,t,u,\phi) $ is linear in $u$. In such a case one can
obtain an implicit representation of the solution and use this
representation to transform the PDE\ into an equivalent system of
nonlinear integral equations. Then the contraction mapping
principle is applied to establish the result. The third approach
to study the well-posedness of solutions is via the finite
difference approximation technique used for the classical
conservation laws (see, e.g., \cite{crandall,smoller}). The
crucial step in this technique is to show that the developed
finite difference approximation has a bounded total variation.
Then, through the compact imbedding of the space of functions of
bounded variation in $L^{1}(a,b)$ one can extract a convergent
subsequence and show that the limit is indeed a solution
\cite{akb,afer,ai}. Application of such a technique results not
only in the existence of solutions but also in a numerical scheme
that can be used to investigate the solution quantitatively.

The goal of this paper is to present recent results on the employment of the
monotone method for investigating the existence and uniqueness of solutions of
(\ref{1}). The idea behind such a method is to replace the actual solution in
all the nonlinear and nonlocal terms with some previous guess for the
solution, then solve the resulting linear model to obtain a new guess for the
solution. Iteration of such a procedure yields the solution of the original
problem upon passage to the limit. A novelty of such a technique when applied
to (\ref{1}) is that an explicit solution representation for each of these
iterates is obtained, and hence an efficient numerical scheme can be developed
(see \cite{ad2,ad5}). The key step is a comparison principle between
consecutive guesses.

To carry out our program, let $C_{B}(\Omega)$ denote the space of continuous
and uniformly bounded functions on $\Omega$. The following assumptions will be
imposed on our parameters throughout the paper:

\begin{itemize}
\item [(A1)]$g\in C_{B}^{1}(D_{T}),\ g>0\ \text{on}\ [a,b)\times[0,T]$.
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{equation}%
\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}%
\end{equation}
and
\begin{equation}%
\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{equation}
\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{equation}%
\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}
\end{equation}
and
\begin{equation}
\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{equation}
\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{equation}
\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}%
\end{equation}
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{equation}%
\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}%
\end{equation}
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{equation}%
\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}
\end{equation}
and
\begin{equation}
\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}
\end{equation}
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. S. Ackleh was
partially supported by grant DMS-0211453 from the
National Science Foundation.  K. Deng was partially supported by
grant DMS-0211412 from the National Science Foundation.

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\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}