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\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 70, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE-2004/70\hfil Existence of solutions]
{Existence of solutions to nonlinear parabolic problems with delay} 

\author[Deliang Hsu\hfil EJDE-2004/70\hfilneg]
{Deliang Hsu} 

\address{Deliang Hsu \hfill\break
Department of Applied Mathematics \\
Shanghai Jiaotong University, 200240, Shanghai, China}
\email{hsudl@online.sh.cn}


\date{}
\thanks{Submitted November 18, 2003. Published May 11, 2004.}
\subjclass[2000]{35R10, 58F40}
\keywords{Nonlinear parabolic equation, delay, global existence, mass decay}


\begin{abstract}
  We prove the existence and uniqueness of global solutions 
  to semilinear parabolic equations with a nonlinear delay term.
  We study these problems in the whole space $\mathbb{R}^n$, 
  obtain classic solutions, and give a mass decay result of the 
  solution.
\end{abstract}

\maketitle
\newtheorem{theorem}{Theorem}[section] 
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}

\section{Introduction}

This paper is devoted to the study of existence and uniqueness of solutions
to parabolic problems with delay in unbounded domains. 
Let $\mathbb{R}^n$ be Euclidean space, $n\geq 1$ and consider the problem
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}=\Delta u+\mu | \nabla u|^p+f( t,x,u_t)
\quad  x\in \mathbb{R}^n \\
u( x,s) =\phi ( x,s)  \quad -r\leq s\leq 0,\; x\in \mathbb{R}^n
\end{gathered}  \label{a}
\end{equation}
where $f:\mathbb{R}^{+}\times \mathbb{R}^n\times C\to \mathbb{R}$ is a locally Lipschitz
functions with respect to $u_t$, $C$ denotes the phase space will be defined
in section 2, $r>0$, $\mu \neq 0$, and $u_t=u( x,t+\theta ) $,
for $-r\leq \theta \leq 0$, denotes the delay term,
$\phi ( x,s)\in C^2( \mathbb{R}^n\times [-r,0] ) $.

In the recent years many authors have been concerned with the nonlinear
partial differential equations involving delay, for the examples we refer to
the book \cite{wu} and references cited therein. Generally speaking, this
type of problems can be described as an abstract nonlinear partial
functional differential equation
\begin{gather*}
\frac{du}{dt}=A_Tu( t) +F( t,u,u_t) \\
u( \theta ) =\phi ( \theta ) ,\quad \text{for }-r\leq \theta \leq 0
\end{gather*}
and then it can be written as an integral equation
\[
u( t) =T( t) \phi ( 0) +\int_0^tT(t-s) F( s,u,u_s) ds
\]
where $T( t) $ is a strongly continuous semigroup of bounded
linear operators with $A_T$ its infinitesimal generator. Then many methods
similar to those adopted in ordinary equations can be used to study this
type of problems in abstract space, for the details we refer to \cite{wu}.
However, the problem \eqref{a} we investigate here cannot be carried on
along this line for at least in two reasons: first, the operator $\Delta $
on $\mathbb{R}^n$ is not compact, so the related semigroup bear some more complexity
than the compact cases; second, the general nonlinear term
$F(t,u,u_t) $ depend on $\nabla u$, which may cause some difficulty for
the study of our problem. Many mechanic and physical problems reduced to the
problem \eqref{a}, for the case of $f( t,x,u_t) =0$, \eqref{a}
has emerged in recent years in a number of interesting, and quite different
models, for example in the one-dimensional case and $1<p<2$, it has been
known as the KPZ-equation, serving as a physical model of growth of
surfaces, see Krug and Spohn \cite{ks1}, \cite{ks2}. In \cite{ba1} the
authors studied this problem in high dimension and proved the existence
results, and get some useful mass decay results. There also has been a
strong interest in the related equation
\begin{equation}
u_t=\Delta u+\mu | \nabla u| ^p+u^q  \label{b}
\end{equation}
where $p,q>1$, $\mu <0$, both in bounded and unbounded domains, see \cite{ba4}, \cite{wu}
and the references cited therein. In this paper, we focus on studying
problem \eqref{a} by using some estimates in parabolic equation involving
the nonlinearity $| \nabla u| ^p,$which is called the damping
nonlinear gradient term. To our knowledge there are no results about the
problem we present here, even if in the case of $\mu =0$ the solution of (%
\ref{a}) may blow up in finite time. This paper is motivated by the recent
results in \cite{ba1} and \cite{ba4}. Our main results are the following theorem.

\begin{theorem} \label{thm1}
Let $\phi \in C$. Assume that $p\geq 1$, $f$ satisfies a Lipschitz-type
condition,
\begin{equation}
| \nabla _yf( t,x,y) | \leq L| y| \label{H1}
\end{equation}
for some constant $L>0$, then there exists a unique classical solution of 
\eqref{a} in $\Omega _{T_0}$, where $T_0=\min \left\{ [2^{p+1}C_e]^{-2},\frac
12\right\} >0$, and $C_e$ will be defined later.
\end{theorem}

\begin{theorem} \label{thm2}
Let $f( t,x,u_t) =g( t,u( t-r) ),\phi \in C$,
here $g:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ satisfies the strong
Lipschitz-condition
\begin{equation}
| g_u( t,u) | \leq L  \label{H1*}
\end{equation}
for any $(t,u)\in \mathbb{R}\times \mathbb{R}$. Then \eqref{a} has a unique classical solution
$u( x,t) $ in $\mathbb{R}^n\times [0,T]$, where $T>0$ is any real number,
and for every $t\in [0,T]$, $u( \cdot ,t) \in C_b^2(\mathbb{R}^n) $.
\end{theorem}

\begin{theorem}\label{thm3}
Assume that $0\leq \phi ( 0,x) \in C_b^2(\mathbb{R}^n) \cap L^1( \mathbb{R}^n) $,
$\mu <0$ , $p<(n+2)/(n+1)$ and $f( t,x,u) \leq 0$ for any
$( t,u) \in \mathbb{R}^{+}\times C$. If $u(x,t) $ is a $C^2$ solution
of \eqref{a} in $\mathbb{R}^n\times [0,\infty )$, then it decays as
$t\to +\infty $ in the sense that
\[
\lim_{t\to+\infty} \int_{\mathbb{R}^n}u( x,t) dx=0.
\]
\end{theorem}

\section{Proof of Main Theorems}

In what follows we set 
$C=\{ u( \cdot ,t) \in C_b^2(\mathbb{R}^n),\text{ for any }t\in [-r,0]\} $ 
as the phase space with the norm
\[
\| u( x,t) \| _C=\max_{t} \{ \|u( x,t) \| _{L^\infty }
+\| \nabla u( x,t)\| _{L^\infty }\}
\]
where
\begin{gather*}
C_b^2( \mathbb{R}^n) =\{ u: u\in C^2( \mathbb{R}^n) ,u,\nabla
u,\nabla ^2u\in L^\infty ( \mathbb{R}^n)\}, \\
\Omega _t=\mathbb{R}^n\times [-r,t], \quad t\geq 0\,.
\end{gather*}

To prove our theorems, we begin by establishing the existence of the
solution for a short time. The idea of the proof is inspired by the papers of
Amour and Ben-Artzi \cite{al} and \cite{ba1}.

\begin{proof}[Proof of Theorem \ref{thm1}] Define
\[
u^0( x,t) =\begin{cases}
\phi ( x,t)  &\mbox{for }t\in [-r,0] \\
\int_{\mathbb{R}^n}G( x-y,t) \phi ( y,0) dy &\text{for } t\in (0,T_0)
\end{cases}
\]
where the heat kernel is,
\[
G( x,t) =( 4\pi t) ^{-\frac n2}e^{-\frac{|x| ^2}{4t}},\quad t>0,
\]
which satisfies,
\begin{equation}
\int_{\mathbb{R}^n}G( x,t) dx=1,\quad
\int_{\mathbb{R}^n}| \nabla G( x,t) | dx=\beta t^{-1/2},  \label{2.1}
\end{equation}
with $\beta =\int_{\mathbb{R}^n}| \nabla G( x,1) | dx$. Then we
may solve by iterations the following linear heat equation, with delay,
\begin{equation}
\begin{gathered}
\frac{\partial u^k}{\partial t}-\Delta u^k=\mu | \nabla u^{k-1}|
^p+f( t,x,u_t^{k-1}) \\
u^k( x,s) =\phi ( x,s) \quad \text{for }-r\leq s\leq 0
\end{gathered}  \label{2.2}
\end{equation}
$k=1,2,3,\dots $. By Duhamel's principle from (\ref{2.2}), we get
\begin{equation} \label{2.3}
\begin{aligned}
u^k( x,t) &=\int_{\mathbb{R}^n}G( x-y,t) \phi (
y,0) dy+\mu \int_{\mathbb{R}^n}\int_0^tG( x-y,t-s) | \nabla
u^{k-1}( y,s) | ^p  \,dy\,ds\\
&\quad +\int_{\mathbb{R}^n}\int_0^tG( x-y,t-s) f(s,x,u_s^{k-1}) \,dy\,ds   \\
u^k( x,t) &= \phi ( x,t) \quad \text{for }t\in [-r,0]
\end{aligned}
\end{equation}
\begin{equation} \label{2.4}
\begin{aligned}
\nabla u^k( x,t) &=\int_{\mathbb{R}^n}G( x-y,t) \nabla \phi
( y,0) dy+\mu \int_{\mathbb{R}^n}\int_0^t\nabla _xG( x-y,t-s)
| \nabla u^{k-1}( y,s) | ^p  \,dy\,ds\\
&\quad +\int_{\mathbb{R}^n}\int_0^t\nabla _xG( x-y,t-s) f(s,x,u_s^{k-1}) \,dy\,ds
\end{aligned}
\end{equation}
Setting $M_k( t) =\sup_{\Omega _t}| \nabla u^k( x,t) | $, and
$U_k( t)=\sup_{\Omega_t} | u^k( x,t) | $, in view of (\ref{2.1}), (\ref{2.3}),
and (\ref{2.4}), we have
\begin{gather}
M_k( t) \leq M_0( t) +\beta | \mu |
\int_0^t( t-s) ^{-1/2}M_{k-1}^p( s) ds
+L\int_0^t( t-s) ^{-1/2}U_{k-1}( s) ds \label{2.5} \\
U_k( t) \leq U_0( t) +\beta | \mu |
\int_0^tM_{k-1}^p( s) ds+L\int_0^tU_{k-1}( s) ds\,.
\label{2.6}
\end{gather}
Since $M_0( t) \leq \| \nabla \phi ( x,t)\| _C$ and
$U_k( t) \leq \| \phi ( x,t)\| _C$, it follows inductively from (\ref{2.5})
that for $t\leq T_0$,
\begin{equation}
\begin{gathered}
\| \nabla u^k( x,t) \| _{L^\infty ( \mathbb{R}^n) }
\leq 2\| \phi ( x,t) \| _C,   \\
\| u^k( x,t) \| _{L^\infty (\mathbb{R}^n) } \leq 2\| \phi ( x,t) \| _C
,\quad k=0,1,2,\dots
\end{gathered}  \label{2.7}
\end{equation}
To prove the convergence of the iterations, we need the following inequality
\begin{equation}
\big| | \nabla u^k( x,t) | ^p-| \nabla u^{k-1}( x,t) | ^p\big|
\leq C_p\| \phi ( x,t) \| _C^{p-1}| \nabla
u^k( x,t) -\nabla u^{k-1}( x,t) |  \label{2.8}
\end{equation}
which can be derived by using (\ref{2.7}) and the classical inequality: $%
a^p-b^p\leq C_p( a-b) ( a^{p-1}-b^{p-1}) $, $a>0$, $b>0$.
Now setting
\[
N_k( t) =\sup_{\Omega _t}| \nabla u^k(x,t) -\nabla u^{k-1}( x,t) | ,\quad
V_k( t) =\sup_{\Omega _t}| u^k( x,t) -u^{k-1}(x,t) | ,
\]
then from (\ref{2.3}), (\ref{2.4}) and (\ref{2.8}), we have
\begin{equation}
\begin{aligned}
&N_k( t) \\
&\leq | \mu | \sup \int_0^t\int_{\mathbb{R}^n}|
| \nabla u^{k-1}( y,s) | ^p-| \nabla
u^{k-2}( y,s) | ^p| | \nabla _xG(x-y,t-s) |  \,dy\,ds\\
&\quad +L\sup \int_0^t\int_{\mathbb{R}^n}| \nabla _xG( x-y,t-s)
| | u^k( y,s) -u^{k-1}( y,s) | \,dy\,ds \\
&\leq C_p| \mu | \| \phi ( x,t) \|
_C^{p-1}\beta \int_0^t( t-s) ^{-1/2}N_{k-1}( s) ds
+L\beta \int_0^t( t-s) ^{-1/2}V_{k-1}( s) ds\,.
\end{aligned} \label{2.9}
\end{equation}
\begin{equation}
\begin{aligned}
V_k( t) &\leq C_p| \mu | \| \phi (x,t) \| _C^{p-1}\int_0^tN_{k-1}( s)
ds+L\int_0^tV_{k-1}( s) ds  \\
&\leq C_p| \mu | \| \phi ( x,t) \|_C^{p-1}\int_0^t( t-s) ^{-1/2}N_{k-1}( s) ds
+L\int_0^t( t-s) ^{-1/2}V_{k-1}( s) ds
\end{aligned}\label{2.10}
\end{equation}
Choosing $C_e=\max \{ C_p| \mu | \| \phi (x,t) \| _C^{p-1}\beta ,L\beta 
,C_p| \mu | \|\phi ( x,t) \| _C^{p-1},L\} >0$,
from (\ref{2.9}), (\ref{2.10}), we have
\begin{equation}
N_k( t) +V_k( t) \leq C_e\int_0^t( t-s)^{-1/2}[ N_{k-1}( s) +V_{k-1}( s) ] ds
\label{2.11}
\end{equation}
then it follows inductively that (see, \cite[Chapter 3]{hb}),
\begin{equation}
N_k( t) +V_k( t) \leq C_e^kt^{\frac k2}\Gamma \big(
\frac{k+2}2\big) ^{-1}.  \label{2.12}
\end{equation}
In particular, $\sum_k( N_k( T_0) +V_k( T_0)) <\infty $ , we conclude that
$\{ \nabla u^k\}_{k=1}^\infty $ and $\{ u^k\} _{k=1}^\infty $ converge uniformly in
$\Omega _{T_0}$. Then we set
\begin{equation}
u( x,t) =\lim_{k\to\infty} u^k(x,t)  \label{2.13}
\end{equation}

Next, we prove the uniform boundedness of $\{ \nabla ^2u^k\}_{k=1}^\infty $
in $\Omega _{T_0}$, $f\in C^2$ one has
$| \nabla |\nabla f| | \leq C_n| \nabla ^2f| $, hence also
$| \nabla | \nabla f| ^p| \leq C_{p,n}| \nabla
f| ^{p-1}| \nabla ^2f| $, here $C_n$ and $C_{p,n}$ 
depend only on $n$ and $p,n$ respectively. Denoting
\[
L_k( t) =\sup_{\Omega _t}| \nabla ^2u^k(x,t) |
\]
it follows from Duhamel's principle and previous estimates that
\begin{equation}
\begin{aligned}
\nabla ^2u^k( x,t) \
&=\int_{\mathbb{R}^n}G( x-y,t) \nabla^2\phi ( y,0) dy\\
&\quad +\mu \int_{\mathbb{R}^n}\int_0^t\nabla _xG(x-y,t-s) 
\nabla | \nabla u^{k-1}( y,s) | ^p\,dy\,ds \\
&\quad +\int_{\mathbb{R}^n}\int_0^t\nabla _xG( x-y,t-s) 
\nabla f(s,x,u_s^{k-1}) \,dy\,ds
\end{aligned}  \label{2.14}
\end{equation}
then
\begin{equation}
L_k( t) \leq L_0( t) +C_{p,n}\| \nabla \phi
\| _C^{p-1}\beta | \mu | \int_0^t( t-s) ^{-1/2}L_{k-1}( s) ds 
+2L\beta \| \phi \| _Ct^{1/2}.  \label{2.15}
\end{equation}
If $\Lambda >0$ is large so that
\[
C_{p,n}\| \phi \| _C^{p-1}\beta | \mu |
\int_0^{T_0}s^{-1/2}e^{-\Lambda s}ds<\frac 12,
\]
then it follows inductively, using $L_0( t) \leq \| \nabla^2\phi \| _C$ that
\begin{equation}
L_k( t) \leq 2\| \nabla ^2\phi \| _C \label{2.16}
\end{equation}
for $t\in [ -r,T_0] $, $k=0,1,2,\dots$.

Then the same argument as in the proof of \cite[Proposition 2.4]{b.a2}
shows that $\{ \nabla ^2u^k\} _{k=1}^\infty $ is equicontinuous
in $\Omega _{T_0}$.
Using the Arzela-Ascoli theorem and the convergence of (\ref{2.13}) we
conclude from the standard regularity results of parabolic equation that the
solution of (\ref{2.13}) satisfies the \eqref{a} in classic sense in 
$\Omega_{T_0}$.

If $v( \cdot ,t) \in C^2$ is another classical solution in 
$\Omega_{T_0}$, $v( x,t) =\phi (x,t) $  for 
$-r\leq t\leq 0$, then setting $N_1( t) =\sup_{\Omega _t}| \nabla u-\nabla v|$,
 $N_2( t)=\sup_{\Omega _t}| u-v| $, we obtain as in (\ref{2.11}) that
\[
N_1( t) +N_2( t) \leq C\int_0^t( t-s)
^{-1/2}( N_1( s) +N_2( s) ) ds,
\]
for a sufficiently large constant $C$; this implies $N_1( t)
+N_2( t) \equiv 0$ and $u\equiv v$.
This completes the proof of the Lemma.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
By Theorem \ref{thm1}, there exists a  local solution to \eqref{a}. 
Hence we just need to get an estimate of $\nabla u$.
Taking differential on both side we have
\[
\frac{\partial u_j}{\partial t}-\Delta u_j-g_u( t,u( t-r)
) u_j( t-r) =\sum_{i=1}^n\psi _i( x,t) \frac{\partial u_j}{\partial x_j}
\]
where $u_j=\frac{\partial u}{\partial x_j}$, and $\psi _i( x,t)
=\mu p| \nabla u| ^{p-2}\frac{\partial u}{\partial x_j}\in
L^\infty ( \Omega _T) $. By a maximum principle of linear
parabolic equation which was proved in the appendix of \cite{ba1}, we can
obtain, for $0\leq t\leq r$,
\[
| u_j( t,x) | \leq C\big( | u_j( 0,x)| +\max_{t\in [-r,0]} | \phi _j(t,x) | \big)
\]
where $C$  depends only on $L,p,n$. By reiterating the procedure above on 
$[ r,2r]$, $[ 2r,3r]$, \dots, we conclude that, for $t\in [ 0,T] $
\[
| u_j( t,x) | \leq C_3e^{rT}\big( | u_j(0,x) | 
+\max_{t\in [-r,0]}| \phi_j( t,x) | \big) ,
\]
that is
\[
| \nabla u( t,x) | \leq C_3e^{rT}\big( | \nabla
u( 0,x) | +\max_{t\in [-r,0]}| \nabla \phi ( t,x) | \big) .
\]
So from Theorem 1, we get the existence of a global solution, and this
completes the proof.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm3}]
Let $Q( t)=\int_{\mathbb{R}^n}u( x,t) dx$. Integrating  \eqref{a} and
using the Gauss formula, we have
\begin{equation}
\frac d{dt}Q( t) \leq 0.  \label{2.22}
\end{equation}
Let $\widetilde{u}( x,t) $ be the solution of the heat equation $%
\frac{\partial \widetilde{u}}{\partial t}-\Delta \widetilde{u}=0$, having
the same initial data $\widetilde{u}( x,0) =\phi (
x,0) $. Since $\mu <0$, and $f( t,x,u_t) \leq 0$, it
follows by standard comparison principle that
\begin{equation}
0\leq u( x,t) \leq \widetilde{u}( x,t) .  \label{2.23}
\end{equation}
Integrating (\ref{a}) with respect to $x$ and $t$ yield
\[
Q( T) =Q( 0) +\mu \int_0^T\int_{\mathbb{R}^n}| \nabla
u( y,s) | ^p\,dy\,ds+\int_0^T\int_{\mathbb{R}^n}f( s,x,u_s)
\,dy\,ds
\]
by (\ref{2.22}) and $f( t,x,u) $ $\leq 0$ we can easily conclude
that
\begin{equation}
\int_0^\infty \int_{\mathbb{R}^n}| \nabla u( x,t) |^p\,dx\,dt<\infty .  \label{2.24}
\end{equation}
Fix $\varepsilon >0$. It follows from (\ref{2.24}) that there exists a
sequence $1<t_1<t_2<\dots<t_m\to +\infty $ such that
\[
\int_{\mathbb{R}^n}| \nabla u( x,t_j) | ^pdx\leq \varepsilon t_j^{-1},
\]
$j=1,2,\dots$. Using the Sobolev inequality we have
\begin{equation}
\int_{\mathbb{R}^n}u( x,t_j) ^{p^{\star }}dx\leq C_p( \varepsilon
t_j^{-1}) ^{p^{\star }/p}  \label{2.25}
\end{equation}
where $p^{\star }=np/(n-p)$. Now using the condition $p<(n+2)/(n+1)$ 
it is easy to see that one can choose $\delta >0$ such that
\begin{equation}
-\frac 1p+( \frac 12+\delta ) n( 1-\frac 1{p^{\star
}}) <0,  \label{2.26}
\end{equation}
then the H\"{o}lder inequality and (\ref{2.25}) imply
\[
\int_{| x| \leq t_j^{\frac 12+\delta }}u( x,t_j)
dx\leq C_{p,n}\varepsilon ^{\frac 1p}t_j^{-\frac 1p}t_j^{( \frac
12+\delta ) n( 1-\frac 1{p^{\star }}) },
\]
$j=1,2,\dots $, so that by (\ref{2.26}) we have
\begin{equation}
\int_{| x| \leq t_j^{\frac 12+\delta }}u( x,t_j)
dx\leq C_{p,n}\varepsilon ^{1/p}.  \label{2.27}
\end{equation}
By standard linear parabolic theory, it is easy to see that
\begin{equation}
\int_{| x| \leq t_j^{\frac 12+\delta }}\widetilde{u}(x,t_j) dx\to 0  \label{2.28}
\end{equation}
as $j\to \infty $, which in conjunction with (\ref{2.22}), (\ref
{2.23}) and (\ref{2.27}) gives
\[
\lim_{t\to\infty} \int_{\mathbb{R}^n}u( x,t) dx=0\,.
\]
This completes the proof.
\end{proof}

\begin{remark} \label{rmk1} \rm
In the paper \cite{ba4}, the authors give the decay results in the critical 
case $p=(n+2)/(n+1)$ when $f( t,u) \equiv 0$. We think the same
result will be true in the critical case, we will study this problem in a
later work.
\end{remark}

\subsection*{Acknowledgments} 
The author thanks Professor Shunian Zhang for his useful
discussion and suggestions about this paper. Professor Zhang died  two
years ago, for which the author feels much sorrow.

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