\documentclass[reqno]{amsart} \pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \usepackage{hyperref} \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 $$\begin{gathered} \frac{\partial u}{\partial t}=\Delta u+\mu | \nabla u|^p+f( t,x,u_t) \quad x\in \mathbb{R}^n \\ u( x,s) =\phi ( x,s) \quad -r\leq s\leq 0,\; x\in \mathbb{R}^n \end{gathered} \label{a}$$ 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 $11$, $\mu <0$, 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, $$| \nabla _yf( t,x,y) | \leq L| y| \label{H1}$$ 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 $$| g_u( t,u) | \leq L \label{H1*}$$ 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, $$\int_{\mathbb{R}^n}G( x,t) dx=1,\quad \int_{\mathbb{R}^n}| \nabla G( x,t) | dx=\beta t^{-1/2}, \label{2.1}$$ with $\beta =\int_{\mathbb{R}^n}| \nabla G( x,1) | dx$. Then we may solve by iterations the following linear heat equation, with delay, $$\begin{gathered} \frac{\partial u^k}{\partial t}-\Delta u^k=\mu | \nabla u^{k-1}| ^p+f( t,x,u_t^{k-1}) \\ u^k( x,s) =\phi ( x,s) \quad \text{for }-r\leq s\leq 0 \end{gathered} \label{2.2}$$ $k=1,2,3,\dots$. By Duhamel's principle from (\ref{2.2}), we get \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} \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} 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 $t\leq T_0$, $$\begin{gathered} \| \nabla u^k( x,t) \| _{L^\infty ( \mathbb{R}^n) } \leq 2\| \phi ( x,t) \| _C, \\ \| u^k( x,t) \| _{L^\infty (\mathbb{R}^n) } \leq 2\| \phi ( x,t) \| _C ,\quad k=0,1,2,\dots \end{gathered} \label{2.7}$$ To prove the convergence of the iterations, we need the following inequality $$\big| | \nabla u^k( x,t) | ^p-| \nabla u^{k-1}( x,t) | ^p\big| \leq C_p\| \phi ( x,t) \| _C^{p-1}| \nabla u^k( x,t) -\nabla u^{k-1}( x,t) | \label{2.8}$$ 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{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} \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} 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 $$N_k( t) +V_k( t) \leq C_e\int_0^t( t-s)^{-1/2}[ N_{k-1}( s) +V_{k-1}( s) ] ds \label{2.11}$$ then it follows inductively that (see, \cite[Chapter 3]{hb}), $$N_k( t) +V_k( t) \leq C_e^kt^{\frac k2}\Gamma \big( \frac{k+2}2\big) ^{-1}. \label{2.12}$$ 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 $$u( x,t) =\lim_{k\to\infty} u^k(x,t) \label{2.13}$$ 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{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} then $$L_k( t) \leq L_0( t) +C_{p,n}\| \nabla \phi \| _C^{p-1}\beta | \mu | \int_0^t( t-s) ^{-1/2}L_{k-1}( s) ds +2L\beta \| \phi \| _Ct^{1/2}. \label{2.15}$$ 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 $$L_k( t) \leq 2\| \nabla ^2\phi \| _C \label{2.16}$$ 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 $$\frac d{dt}Q( t) \leq 0. \label{2.22}$$ 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 $$0\leq u( x,t) \leq \widetilde{u}( x,t) . \label{2.23}$$ 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 $$\int_0^\infty \int_{\mathbb{R}^n}| \nabla u( x,t) |^p\,dx\,dt<\infty . \label{2.24}$$ Fix $\varepsilon >0$. It follows from (\ref{2.24}) that there exists a sequence $10$ such that $$-\frac 1p+( \frac 12+\delta ) n( 1-\frac 1{p^{\star }}) <0, \label{2.26}$$ 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 $$\int_{| x| \leq t_j^{\frac 12+\delta }}u( x,t_j) dx\leq C_{p,n}\varepsilon ^{1/p}. \label{2.27}$$ By standard linear parabolic theory, it is easy to see that $$\int_{| x| \leq t_j^{\frac 12+\delta }}\widetilde{u}(x,t_j) dx\to 0 \label{2.28}$$ 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 $f( t,u) \equiv 0$. 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