\documentclass[twoside]{article} \usepackage{amssymb} % used for R in Real numbers \pagestyle{myheadings} \setcounter{page}{87} \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. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ Bounded solutions of nonlinear parabolic equations with time delay \thanks{ {\em 1991 Mathematics Subject Classifications:} 35K05, 35K15, 35K55, 34K05. \hfil\break\indent {\em Key words and phrases:} Nonlinear parabolic equation, piecewise constant delay, \hfil\break\indent bounded solution, initial value problem, Fisher's equation. \hfil\break\indent \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