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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. 
\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 <x<\infty & 
\end{eqnarray}
in the domain $\mathcal{D}=( -\infty ,\infty ) \times (
0,\infty ) $, where $\left[ t\right] $ denotes the greatest-integer
function. In this case, the delay function is constant on unit intervals and
has jump discontinuities at their endpoints. Within each of these intervals
the dynamical system is described by a PDE without delay. Continuity of a
solution at the endpoints of two consecutive intervals leads to a difference
equation of an integer argument for the values of the solution at the
endpoints. Therefore, equations with piecewise constant delay describe
hybrid (continuous and discrete) dynamical systems and combine the
properties of both differential and difference equations. They include, as
particular cases, loaded and impulsive equations of control theory. This
note continues our earlier research on bounded solutions of nonlinear
hyperbolic equations with time delay [1-4].

\paragraph{Definition.}
A function $u( x,t) $ is called a solution of the initial-value
problem (1)-(2) if it satisfies the following conditions:
\begin{description}
\item{(i)} $u( x,t) $ is continuous in $\mathcal{D}=( -\infty,\infty ) 
\times ( 0,\infty ) $. 

\item{(ii)} $u_{t}$ and $u_{xx}$ exist and are continuous in $\mathcal{D}$, with
the possible exception of the points $( x,\left[ t\right] ) $
where one-sided derivatives exist.

\item{(iii)} $u( x,t) $ satisfies (1) in $\mathcal{D}$, with
the possible exception of the points $( x,\left[ t\right] ) $,
and initial conditions (2).
\end{description}

\section{Existence and uniqueness theorem}

The method of proof is based on reducing equations (1), (2) to an integral
equation in two variables and the use of successive approximations. 

\begin{theorem}
Assume the following hypotheses:
\begin{description}
\item{(i)} The function $\varphi ( x) $ is twice continuously
differentiable and bounded on ${\mathbb R}$.

\item{(ii)} The function $f:$ ${\mathbb R}${}$\times {\mathbb R}\rightarrow {\mathbb R}$ is
continuous and bounded in $\mathcal{D}$, and satisfies the Lipschitz
condition $\left| f( u,w) -f( v,w) \right| \leq
L\left| u-v\right| $ uniformly with respect to $w$, where $L$ is a positive
constant and $u,v\in ( -\infty ,\infty ) $.
\end{description}
Then there exists a unique solution to problem (1), (2) which is bounded in $
\mathcal{D}$.
\end{theorem}

\paragraph{Proof.} On the interval $0\leq t<1$, equation
(1) becomes 
\begin{equation}
u_{t}( x,t) =a^{2}u_{xx}( x,t) +f( u(
x,t) ,\varphi ( x) )\,.
\end{equation}
Using integration by parts twice for equation (1) we obtain, after some
computations, the following integral equation for the solution of problem
(1), (2): 
\begin{eqnarray}
u( x,t)  &=&\frac{1}{\sqrt{4a^{2}\pi t}}\int_{-\infty }^{\infty
}e^{-( x-\xi ) ^{2}/4a^{2}t}\varphi ( \xi )\,d\xi  \\
&&+\frac{1}{\sqrt{2\pi }}\int_{0}^{t}\frac{1}{\sqrt{t-\tau }}( \frac{1}{
\sqrt{4a^{2}\pi t}}\int_{-\infty }^{\infty }e^{-( x-\xi )
^{2}/4a^{2}t}f( u(\xi ,\tau ),\varphi (\xi ))\,d\xi )\,d\tau\, .\nonumber
\end{eqnarray}
According to the method of successive approximations, put 
\[ 
u_{0}(x,t) =\frac{1}{\sqrt{4a^{2}\pi t}}\int_{-\infty }^{\infty
}e^{-( x-\xi ) ^{2}/4a^{2}t}\varphi ( \xi )\,d\xi\,.
\] 
Since $\left| \varphi ( x) \right| \leq M$ for $x\in (
-\infty ,\infty ) $, the substitution $( x-\xi ) /2a\sqrt{t}
=\alpha $ implies that 
\begin{equation}
\left| u_{0}( x,t) \right| \leq \frac{1}{2a\sqrt{\pi t}}
\int_{-\infty }^{\infty }e^{-\alpha ^{2}}2a\sqrt{t} \left| \varphi ( \xi
) \right| \,d\alpha = \left| \varphi ( x) \right| \leq M\,,
\end{equation}
and from equation (4) one obtains the estimates 
\begin{eqnarray}
\left| u_{1}-u_{0}\right|  &\leq &\frac{1}{\sqrt{2\pi }}\int_{0}^{t}\frac{1}{
\sqrt{t-\tau }}( \int_{-\infty }^{\infty }e^{-( x-\xi )
^{2}/4a^{2}( t-\tau ) }\left| f( u_{0},\varphi (\xi ))
\right|\,d\xi )\,d\tau   \nonumber \\
&\leq &\frac{1}{\sqrt{2\pi }}\int_{0}^{t}\frac{1}{\sqrt{t-\tau }}(
\int_{-\infty }^{\infty }e^{-\beta ^{2}}\cdot \overline{M}\cdot 2a\sqrt{
t-\tau }\,d\beta )\,d\tau   \nonumber \\
&=&\sqrt{2}a\overline{M}t,\quad 0\leq t<1 \,,
\end{eqnarray}
where $\left| f( u_{0}( x,t) ,\varphi ( x)
) \right| \leq \overline{M}\ $in $\mathcal{D}$ and $( x-\xi
) /2a\sqrt{t-\tau }=\beta $, with some constants $M$ and $\overline{M}$
. Therefore, from equation (4) and inequality (6), it follows that 
\begin{equation}
\left| u_{1}\right| \leq M+\sqrt{2}a\overline{M}t,\quad 0\leq t<1\,.
\end{equation}
Furthermore, we use equation (4) for the second approximation 
\[ 
u_{2}( x,t) =u_{0}( x,t) +\frac{1}{\sqrt{2\pi }}
\int_{0}^{t}\frac{1}{\sqrt{t-\tau }}( \int_{-\infty }^{\infty
}e^{-( x-\xi ) ^{2}/4a^{2}( t-\tau ) }f(
u_{1},\varphi )\,d\xi )\,d\tau 
\] 
and obtain the estimate 
\[ 
\left| u_{2}-u_{1}\right| \leq \frac{1}{\sqrt{2\pi }}\int_{0}^{t}(
\int_{-\infty }^{\infty }e^{-\gamma ^{2}}L\left| u_{1}-u_{0}\right| \cdot 2a
\sqrt{t-\tau }\,d\gamma )\,d\tau \leq 2L\overline{M}\frac{a^{2}t^{2}}{2!
},
\] 
which implies 
\begin{equation}
\left| u_{2}\right| \leq M+2\frac{\overline{M}}{L}\frac{( Lat)
^{2}}{2!},\quad 0\leq t<1\,.
\end{equation}
In the same fashion,
\begin{eqnarray*}
\left| u_{3}-u_{2}\right|  &\leq &\frac{1}{\sqrt{2\pi }}\int_{0}^{t}\frac{1}{
\sqrt{t-\tau }}( \int_{-\infty }^{\infty }e^{-( x-\xi )
^{2}/4a^{2}t}\bigg| f( u_{2}(\xi ,\tau ),\varphi (\xi ))  \\
&& \hspace{4cm} -f(u_{1}(\xi ,\tau ),\varphi (\xi )) \bigg|\,d\xi )\,d\tau  \\
&\leq &\frac{1}{\sqrt{2\pi }}\int_{0}^{t}\frac{1}{\sqrt{t-\tau }}(
\int_{-\infty }^{\infty }e^{-( x-\xi ) ^{2}/4a^{2}t}L\left|
u_{2}-u_{1}\right|\,d\xi )\,d\tau  \\
&\leq &\frac{1}{\sqrt{2}}\cdot 2a\cdot 2L^{2}\overline{M}a^{2}\frac{t^{3}}{3!
}\,,
\end{eqnarray*}
or 
\begin{equation}
\left| u_{3}-u_{2}\right| \leq 2^{3/2}\overline{M}L^{2}\frac{(
at) ^{3}}{3!}\,.
\end{equation}
Similarly, 
\[ 
\left| u_{4}-u_{3}\right| \leq 2^{2}\overline{M}L^{3}\frac{( at)
^{4}}{4!}\,.
\] 
In general, this procedure leads to the estimate 
\begin{equation}
\left| u_{n}( x,t) -u_{n-1}( x,t) \right| \leq 2^{n/2}
\frac{\overline{M}}{L}\frac{( Lat) ^{n}}{n!},\quad n=1,2,\ldots 
\end{equation}
and since 
\[ 
u( x,t) =u_{0}( x,t) +\sum_{i=1}^{\infty }(
u_{i+1}( x,t) -u_{i}( x,t) ) ,
\]
it follows 
\[ 
\left| u( x,t) \right| \leq M+\sum_{i=1}^{\infty }\left|
u_{i+1}-u_{i}\right|\,.
\] 
With the account of inequality (10), 
\[ 
\left| u( x,t) \right| \leq M+\frac{\overline{M}}{L}e^{\sqrt{2}
aLt},\quad 0\leq t<1\,,
\] 
which proves the existence of a bounded solution for (1), (2) in the domain $
0\leq t\leq 1$, $-\infty <x<\infty $. Now, for the solution of (1), (2) on
the interval $1\leq t<2$, we note that $0\leq t-1<1$ and replacing $t$ with $
t-1$, arrive at the estimates 
\begin{eqnarray*}
\left| u_{0}\right|  &\leq &\frac{1}{\sqrt{4a^{2}\pi ( t-1) }}
\int_{-\infty }^{\infty }e^{-( x-\xi ) ^{2}/4a^{2}(
t-1) }\left| \varphi _{1}( \xi ) \right| \,d\xi \leq M_{1},
\\
\left| u_{1}-u_{0}\right|  &\leq &\frac{1}{\sqrt{2\pi }}\int_{0}^{t-1}\frac{1
}{\sqrt{t-1-\tau }}( \int_{-\infty }^{\infty }e^{-( x-\xi )
^{2}/4a^{2}( t-1) }\left| f( u_{0},\varphi _{1})
\right| \,d\xi )\,d\tau  \\
&=&\sqrt{2}a\overline{M_{1}}( t-1) , \\
\left| u_{1}\right|  &\leq &M_{1}+\sqrt{2}a\overline{M_{1}}( t-1)
,\quad 1\leq t<2\,,
\end{eqnarray*}
with some constants $M_{1}$ and $\overline{M_{1}}$, where 
\[ 
\varphi _{1}( x) =u(x,1),\quad \left| \varphi _{1}( x)
\right| \leq M_{1},\quad \mbox{and}\quad \left| f( u_{0}(
x,t) ,\varphi _{1}( x) ) \right| \leq \overline{M_{1}}\,.
\] 
On the interval $1\leq t<2$, we have the inequalities 
\begin{equation}
\left| u_{2}-u_{1}\right| \leq 2L\overline{M_{1}}\frac{a^{2}(
t-1) ^{2}}{2!}\,,
\end{equation}
and
\[ 
\left| u_{2}\right| \leq M_{1}+2\frac{\overline{M_{1}}}{L}\frac{
L^{2}a^{2}( t-1) ^{2}}{2!},\quad 1\leq t<2\,.
\] 

\smallskip \noindent Continuation of the above procedure yields the general
estimates 
\begin{equation}
\left| u_{n}( x,t) -u_{n-1}( x,t) \right| \leq
2^{n/2}L^{n-1}\overline{M_{1}}\frac{( a( t-1) ) ^{n}}{n!},\quad 1\leq t<2\,.
\end{equation}
These inequalities imply that 
\[ 
\left| u( x,t) \right| \leq M_{1}+\frac{\overline{M_{1}}}{L}e^{
\sqrt{2}aL( t-1) },\quad 1\leq t<2\,,
\] 
which proves the existence of a bounded solution to problem (1), (2) in the
domain $1\leq t<2$, $-\infty <x<\infty $. In the next step, we obtain 
\[ 
\left| u( x,t) \right| \leq M_{2}+\frac{\overline{M_{2}}}{L}e^{
\sqrt{2}aL( t-2) },\quad 2\leq t<3\,,
\] 
and, in general, 
\[ 
\left| u( x,t) \right| \leq M_{n}+\frac{\overline{M_{n}}}{L}e^{
\sqrt{2}aL( t-n) },\quad n\leq t<n+1\,,
\] 
where the variants $M_{n}$ and $\overline{M_{n}}$ are bounded. Therefore,
the function $u( x,t) $ constructed is a solution of problem (1),
(2) which is bounded in $\mathcal{D}$. Uniqueness of this solution is a
simple consequence of the Lipschitz condition.

\paragraph{Remark 1.}
The method of successive approximations also enables one to prove, under
certain assumptions, the existence of a unique bounded solution for the
differential equation with constant delay 
\[ 
u_{t}( x,t) =a^{2}u_{xx}( x,t) +f( u(
x,t) ,u( x,t-h) ) ,
\] 
satisfying the initial condition 
\[ 
u( x,t) =\varphi ( x,t) ,\quad \quad ( -h\leq
t\leq 0,-\infty <x<\infty ) .
\] 
We may also generalize problem (1), (2) to include equations of the form 
\[ 
u_{t}( x,t) =a^{2}u_{xx}( x,t) +f( u(
x,t) ,u(x,\left[ t\right] ),u_{x}( x,\left[ t\right] )
)\,.
\] 


\paragraph{Remark 2.}
Equation (1) may be considered as a generalization of Fisher's equation
\[ 
\frac{\partial u}{\partial t}=k\frac{\partial ^{2}u}{\partial x^{2}}+f(u),
\] 
the discrete and continuous versions of which have been used as models for
population dynamics and the propagation of a favored gene in a population.
The discrete model might be more appropriate in certain situations, for
example, population dispersal in a patchy environment. Equation (1) is 
also a semi-discretization of a continuous nonlinear diffusion equation.

\begin{thebibliography}{9}

\bibitem{1}  Chi, H., Poorkarimi, H., Wiener, J., and Shah, S. M.,
``On the exponential growth of solutions to non-linear hyperbolic
equations,'' Internat. Joun. Math. Math. Sci., \textbf{12}(1989), 539-546.

\bibitem{2}  Poorkarimi, H. and Wiener, J., ``Bounded solutions of
non-linear hyperbolic equations with delay,'' Proceedings of the VII
International Conference on Non-Linear Analysis, V. Lakshmikantham, Ed.,
1986, 471-478.

\bibitem{3}  Shah, S. M., Poorkarimi, H., and Wiener, J., ``Bounded
solutions of retarded nonlinear hyperbolic equations,'' Bull. Allahabad
Math. Soc. \textbf{1} (1986), 1-14.

\bibitem{4}  Wiener, J., Generalized Solutions of Functional
Differential Equations, (World Scientific, Singapore), 1993.
\end{thebibliography}
\bigskip

\noindent{\sc Hushang Poorkarimi} (e-mail: poorkar@panam.edu) \\  
{\sc Joseph Wiener} (e-mail: jwiener@panam.edu) \\
Department of Mathematics\\
University of Texas - Pan American \\
Edinburg, TX 78539 USA \\
Tel: 956-381-3534.

\end{document}