\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 97, pp. 1--14.\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 2003 Texas State University-San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/97\hfil Heat equation and shrinking] {The heat equation and the shrinking} \author[Masaki Kawagishi \& Takesi Yamanaka\hfil EJDE--2003/97\hfilneg] {Masaki Kawagishi \& Takesi Yamanaka} % in alphabetical order \address{Masaki Kawagishi \newline College of Science and Technology, Nihon University\\ Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan} \email{masaki@suruga.ge.cst.nihon-u.ac.jp} \address{Takesi Yamanaka \newline College of Science and Technology, Nihon University\\ Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan} \email{yamanaka@math.cst.nihon-u.ac.jp} \date{} \thanks{Submitted April 1, 2003. Published September 17, 2003.} \subjclass[2000]{35K05, 35K55, 35R10, 49K25} \keywords{Partial differential equation, heat equation, shrinking, delay, Gevrey} \begin{abstract} This article concerns the Cauchy problem for the partial differential equation $$\partial_1 u(t,x)-a\partial_2^2 u(t,x) =f(t,x,\partial_2^p u(\mu(t)t,x),\partial_2^q u(t,\nu(t)x))\,.$$ Here $t$ and $x$ are real variables, $p$ and $q$ are positive integers greater than 1, and the {\it shrinking factors} $\mu(t)$, $\nu(t)$ are positive-valued functions such that their suprema are less than 1. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{cor}[thm]{Corollary} \newcommand{\normb}{\hspace{1pt}\rule[-2pt]{.9pt}{10pt}\hspace{1pt}} \section{Introduction} The effect of {\it shrinkings} placed on the independent variables has been investigated in \cite{kawa1,kawa2,kawa3, kawa-yama1}. All of the results obtained so far can be said to be on the same line as the Cauchy-Kovalevskaja theorem. This means that the results obtained are independent of the type of differential equations, such as parabolic or hyperbolic. Therefore, there are possibilities for obtaining some new results in the study of the effect of the shrinking by taking into account the type of differential equation. The present note is our first attempt to pursue such possibilities. To be a bit more exact, we consider here the problem of introducing shrinking factors into the one dimensional heat equation $$\partial_1 u(t,x)-a \partial_2^2 u(t,x)=f(t,x). \label{a.10}$$ In this equation $u(t,x)$ is the unknown real-valued function with $(t,x)\in\mathbb{R}^2$, $\partial_i$ denotes partial differentiation with respect to the $i$th variable, $f(t,x)$ is a given function of $(t,x)$, and $a$ a positive constant. Next, we recall a well-known result about the solution of \eqref{a.10} satisfying the initial condition $$u(0,x)=\varphi(x). \label{a.20}$$ When $\varphi(x)$ and $f(t,x)$ are continuous and bounded, if $f(t,x)$ satisfies the H\"{o}lder condition with respect to $x$, then a solution to \eqref{a.10}-\eqref{a.20}, (on the domain $t\ge 0, -\infty0, \end{cases} \end{multline} where $G(t,x)=\frac{1}{2\sqrt{\pi at}}e^{-x^2/(4at)}\quad (t>0,\;-\infty\tau, \\ g(\tau,x)& \text{if } t=\tau \end{cases} \label{p1.3} is continuous for (t,\tau,x)\in\mathbb{R}^3 with 0\le\tau\le t\tau,\\ \partial_2^k g(\tau,x)& \text{if }t=\tau. \end{cases} \label{p1.13} \end{prop} \section{Gevrey functions} As stated in \S1, our purpose in the present note is to solve the Cauchy problem \eqref{a.60}-\eqref{int.4}. To be more exact, we assume that the function f(t,x,v,w) in \eqref{a.60} is a Gevrey function of (x,v,w) and seek a solution u(t,x) that is a Gevrey function of x. In this section, we shall recall the definition of Gevrey function and state some fundamental properties of Gevrey functions. \subsection*{Gevrey functions of one variable} We denote by \mathbb{Z}_+ the set of all non-negative integers. Let I be a real interval and \lambda a constant greater than 1. Let \lambda be a fix constant greater than 1. If for a C^\infty function w:I\to\mathbb{R} there are positive constants C,M such that \[ |w^{(k)}(x)|\le CM^k(k!)^\lambda$ holds for all$x\in I$and all$k\in \mathbb{Z}_+$, then$w$is called a Gevrey function on$I$of order$\lambda$. It is easy to see that a$C^\infty$function$w:I\to\mathbb{R}$is a Gevrey function of order$\lambda$if and only if there are positive constants$C',L$such that $|w^{(k)}(x)|\le 2^{-5}C'L^k(k!)^\lambda(1+k)^{-2}$ holds for all$x\in I$and all$k\in \mathbb{Z}_+$. So we write, according to Yamanaka \cite{yama1}, $\Gamma_\lambda(k)=2^{-5}(k!)^\lambda (1+k)^{-2}$ for$k=0,1,2,\ldots$and define \begin{equation*} |w|_{L}=\sup\Big\{\frac{|w^{(k)}(x)|}{L^k \Gamma_\lambda(k)} : x\in I,\;k\in \mathbb{Z}_+\Big\} \end{equation*} for each$C^\infty$function$w:I\to\mathbb{R}$. We denote by$\gamma_L(I)$the family of all$C^\infty$functions$w:I\to\mathbb{R}$such that$|w|_L<\infty$. Besides this family, we need another type of Gevrey family. For a$C^\infty$function$w:I\to\mathbb{R}$, we write $$\normb w \normb=\sup_{x\in I} |w(x)|,\ \ \|w\|_{L}=\max\{2^6 \normb w\normb,2^3L^{-1}|w'|_L\} \label{2.1t}$$ and define $\mathcal{G}_L(I)=\{w\in C^\infty(I,\mathbb{R}): \|w\|_{L}<\infty\}.$ Between the two types of Gevrey families$\gamma_{L}(I)$and$\mathcal{G}_{L}(I)$there is the following relation which proof can be found in \cite[Proposition 2.1]{kawa-yama1} and in \cite[Lemma 5.2]{yama2}. \begin{prop}\label{prop2.1t} If$00$, we write \begin{equation*} {\mathcal H}(T)=\{v\in\widetilde{\mathcal{G}}_{h}([0,T),\mathbb{R}): \sup_{00$, we set \begin{equation*} {\mathcal F}(T,K)=\{ v\in{\mathcal H}(T):{\mathbb N}_{{\mathcal H}(T)}[v]\le K\}, \end{equation*} which is a closed convex subset of the Banach space ${\mathcal H}(T)$. We now want to show that there is a $T\le T_0$ such that, if $v\in{\mathcal F}(T,C_f)$, then the integral $$w(t,x):=\int_0^t d\tau \int_{-\infty}^\infty G(t-\tau,x-\xi) f(\tau,\xi,\partial_2^p v(\mu(\tau)\tau,\xi),\partial_2^q v(\tau,\nu(\tau) \xi))d\xi \label{4.110}$$ is well-defined for all $(t,x)\in [0,T)\times\mathbb{R}$ and the function $w:[0,T)\times\mathbb{R}\to\mathbb{R}$ defined by \eqref{4.110} is again in ${\mathcal F}(T,C_f)$. In order for the integral on the right-hand side of \eqref{4.110} to be well-defined it is necessary that the values $\partial_2^p v(\mu(\tau)\tau,\xi)$ and $\partial_2^q v(\tau,\nu(\tau) \xi)$ can be substituted for $y$ and $z$ in $f(\tau,\xi,y,z)$, respectively. It is enough for this that the inequality \max\{|\partial_2^p v(\mu(\tau)\tau,\xi)|, |\partial_2^q v(\tau,\nu(\tau) \xi)|\}0$such that$T\le T_2$and$E(T)+F(T)\le (2C_f)^{-1}$. So we can define $$T^*=\max\{T:T\le T_2\ \ \text{and}\ \ E(T)+F(T)\le (2C_f)^{-1}\}. \label{4.290}$$ By \eqref{4.270}, \eqref{4.283} and \eqref{4.290}, if$v_0$and$v_1$are in${\mathcal F}(T^*,C_f)$, then the inequality $$\|\partial_2^p(v_1-v_0)(\mu(\tau)\tau,\cdot)\|_{h(\tau)} +\|\partial_2^q (v_1-v_0)(\tau,\nu(\tau)\cdot)\|_{h(\tau)} \le \frac{1}{2C_f} {\mathbb N}_{{\mathcal H}(T^*)}[v_1-v_0] \label{4.300}$$ holds for$0\le\tau\le T^*$. By \eqref{4.240} and \eqref{4.300} we obtain $$\|\delta_{(v_1,v_0)}(\tau,\cdot)\|_{h(\tau)} \le \frac{1}{2} {\mathbb N}_{{\mathcal H}(T^*)}[v_1-v_0]. \label{4.310}$$ Now, our aim is to estimate the difference$\Phi_{T^*}(v_1)-\Phi_{T^*}(v_0)$. Using the notation$\delta_{(v_1,v_0)}$, this difference is expressed as $$\label{4.320} \Phi_{T^*}(v_1)(t,x)-\Phi_{T^*}(v_0)(t,x) =\int_0^t d\tau \int_{-\infty}^\infty G(t-\tau,x-\xi) \delta_{(v_1,v_0)}(\tau,\xi)d\xi$$ By \eqref{4.320}, \eqref{4.310} and Proposition \ref{prop3.8}, we know that \begin{equation*} {\mathbb N}_{{\mathcal H}(T^*)}[\Phi_{T^*}(v_1)-\Phi_{T^*}(v_0)]\le \frac{1}{2}{\mathbb N}_{{\mathcal H}(T^*)}[v_1-v_0]. \end{equation*} This implies that the mapping$\Phi_{T^*}$from the closed subset${\mathcal F}(T^*,C_f)$of the Banach space${\mathcal H}(T^*)$into itself is a contraction. Therefore, there is one and only one element$v$of${\mathcal F}(T^*,C_f)$such that \begin{equation*} v=\Phi_{T^*}(v). \end{equation*} This element$v\in {\mathcal F}(T^*,C_f)$is a solution of the integral equation \eqref{a.70} and, accordingly, a solution of the Cauchy problem \eqref{a.60}-\eqref{int.4}. Since$v$is in${\mathcal F}(T^*,C_f)$, it belongs to the family$\widetilde{\mathcal{G}}_{h}([0,T^*),\mathbb{R})$and satisfies the inequality \eqref{p1.21}. Further, since$v$is a solution of the differential equation \eqref{a.60}, it satisfies the equality $$\partial_1 v(t,x)=a \partial_2^2 v(t,x)+f(t,x,\partial_2^p v(\mu(t)t,x),\partial_2^q v(t,\nu(t) x)). \label{4.331}$$ From this equality we see that$\partial_1 v(t,x)$is infinitely differentiable in$x$, because so is the right-hand side of \eqref{4.331}. This completes the proof of the fact that the Cauchy problem \eqref{a.60}-\eqref{int.4} has a solution$u\in \widetilde{\mathcal{G}}_{h}([0,T^*),\mathbb{R})${\small$\bigcap$}$C^{(1,\infty)}([0,T^*),\mathbb{R})$such that the inequality \eqref{p1.21} holds. The fact that the Cauchy problem has only one such solution is easily confirmed. In fact assume that$v\in\widetilde{\mathcal{G}}_{h}([0,T^*),\mathbb{R})${\small$\bigcap$}$C^{(1,\infty)}([0,T^*),\mathbb{R})$is a solution of the Cauchy problem \eqref{a.60}-\eqref{int.4} and the inequality \eqref{p1.21} holds. Then$v$is in the set${\mathcal F}(T^*,C_f)$and satisfies the integral equation \eqref{a.70}. This means that$v$is the unique fixed point of the contraction$\Phi_{T^*}: {\mathcal F}(T^*,C_f)\to{\mathcal F}(T^*,C_f)\$. \end{proof} \begin{thebibliography}{00} \frenchspacing \bibitem{kawa1} M. Kawagishi, {\em Cauchy-Kovalevskaja-Nagumo type theorems for PDEs with shrinkings}, Proc. Japan Acad., {\bf 75}, Ser. A (1999), 184-187. \bibitem{kawa2} M. Kawagishi,{\em A generalized Cauchy-Kovalevskaja-Nagumo theorem with shrinkings}, Scientiae Mathematicae Japonicae, {\bf 54}, No. 1 (2001), 39-50 \bibitem{kawa3} M. 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