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\markboth{\hfil Bounded solutions of nonlinear parabolic equations \hfil}%
{\hfil Hushang Poorkarimi \& Joseph Wiener \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc 15th Annual Conference of Applied Mathematics, Univ. of Central Oklahoma},
\newline
Electronic Journal of Differential Equations, Conference~02, 1999, pp. 87--91.
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ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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Bounded solutions of nonlinear parabolic equations with time delay
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35K05, 35K15, 35K55, 34K05.
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{\em Key words and phrases:} Nonlinear parabolic equation, piecewise constant delay,
\hfil\break\indent
bounded solution, initial value problem, Fisher's equation.
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\copyright 1999 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Published December 9, 1999.} }
\date{}
\author{Hushang Poorkarimi \& Joseph Wiener}
\maketitle
\begin{abstract}
The existence and uniqueness of a bounded solution is established for
a nonlinear parabolic equation with piecewise continuous time delay. This
problem may be considered as a generalization of Fisher's equation which has
applications in certain ecological studies.
\end{abstract}
\newtheorem{theorem}{Theorem}
\section{Introduction}
In this paper, we are interested in finding sufficient conditions for the
existence of a unique bounded solution to a nonlinear parabolic equation
with time delay. The initial-value problem under investigation is the
following
\begin{eqnarray}
&u_{t}( x,t) =a^{2}u_{xx}( x,t) +f( u(x,t) , u( x,\left[ t\right] ) ) ,& \\
&u( x,0) =\varphi ( x) ,\quad -\infty