\documentclass{amsart} \begin{document} {\noindent\small {\em Electronic Journal of Differential Equations}, Vol.\ 1999(1999), No.~01, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)} \thanks{\copyright 1999 Southwest Texas State University and University of North Texas.} \vspace{1.5cm} \title[\hfilneg EJDE--1999/01\hfil C-infinity interfaces of solutions] {$C$-infinity interfaces of solutions for one-dimensional parabolic $p$-Laplacian equations} \author[Yoonmi Ham \& Youngsang Ko\hfil EJDE--1999/01\hfilneg] {Yoonmi Ham \& Youngsang Ko } \address{ {\sc Yoonmi Ham}\hfill\break Department of Mathematics\\ Kyonggi University \hfill\break Suwon, Kyonggi-do, 442-760, Korea } \email{ymham@@kuic.kyonggi.ac.kr} \address{{\sc Youngsang Ko}\hfill\break Department of Mathematics\\ Kyonggi University \hfill\break Suwon, Kyonggi-do, 442-760, Korea } \email{ysgo@@kuic.kyonggi.ac.kr} \date{} \thanks{Submitted November 11, 1998. Published January 5, 1999.} \subjclass{35K65} \keywords{p-Laplacian, free boundary, C-infinity regularity.} \maketitle \begin{abstract} We study the regularity of a moving interface $x = \zeta (t)$ of the solutions for the initial value problem $$ \displaylines{ u_t = \left(|u_x|^{p-2}u_x \right)_x \cr u(x,0) =u_0 (x)\,,} $$ where $u_0\in L^1({\mathbb R})$ and $p>2$. We prove that each side of the moving interface is $C^{\infty}$. \end{abstract} \newtheorem{thm}{Theorem}[section] \newtheorem{pro}[thm]{Proposition} \newcommand{\ol}{\overline} \makeatletter \numberwithin{equation}{section} \makeatother \section{\bf Introduction} We consider the Cauchy problem of the form \begin{equation}\label{eq:plap} \begin{gathered} u_t = \left(|u_x|^{p-2}u_x\right)_{x}\text{ in } S := {\mathbb R}\times (0,\infty) \\ u(x,0) =u_0 (x) \end{gathered} \end{equation} where $p > 2$. This equation has application to many physical situations, and has been studied by many authors; see for example \cite{EV88} and references therein. In the study of this equation, the velocity of propagation, $V(x,t)$, is very important, and can be obtained in terms of $u$ by writing (\ref{eq:plap}) as the conservation law $$u_t + (uV)_x = 0\,.$$ In this way we obtain $V= -v_x |v_x |^{p-2}$, where the nonlinear potential $v(x,t)$ is \begin{equation}\label{eq:defv} v = \frac{p-1}{p-2}u^{(p-2)/(p-1)}. \end{equation} By a direct computation, we realize that \begin{equation}\label{eq:eqv} v_t = (p-2)v|v_x|^{p-2}v_{xx} + |v_x|^p. \end{equation} In \cite{EV88}, it is shown that $V$ satisfies $V_x \leq \frac{1}{2(p-1)t}$ which can also be written as \begin{equation}\label{eq:V} (v_x |v_x|^{p-2})_x \geq -\frac{1}{2(p-1)t}\,. \end{equation} Without loss of generality, we assume that $u_0$ vanishes on ${\mathbb R}^{-}$ and that $u_0$ is a continuous positive function on an interval $(0,a)$ with $a>0$. Let $$P[u] =\{(x,t) \in S:u(x,t) >0\}$$ be the positivity set of a solution $u$. Then $P[u]$ is bounded from the left in the $(x,t)$-plane by the left interface curve $x=\zeta(t)$, where $$\zeta(t) = \inf\{x\in {\mathbb R} :u(x,t) >0\}\,.$$ Moreover, there is a time $t^{*}\in [0,\infty)$, called the waiting time, such that $\zeta(t) =0$ for $0 \leq t \leq t^{*}$ and $\zeta(t) <0$ for $t>t^{*}$. It is shown in \cite{EV88} that $t^*$ is finite (possibly zero) and $\zeta(t)$ is a non-increasing $C^{1}$ function on $(t^{*},\infty)$. Actually it is shown that $\zeta'(t) <0$ for every $t>t^*$, i.e., a moving interface never stops. On the other hand, D. G. Aronson and J. L. Vazquez \cite{AV87} established Theorem~\ref{thm:1} below. Let $D= \{(x,t): t>t^{*}, \zeta(t) \leq x \leq 0\}$, and let $v$ be the pressure for the solution of the porous medium equation \begin{equation}\label{eq:porous} u_t = (u^m)_{xx} \quad in\quad Q_T = {\mathbb R}\times (0,T). \end{equation} \begin{thm}\label{thm:1} $v$ is a $C^{\infty}$ function on $D$, and $\zeta(t)$ is a $C^{\infty}$ function on $(t^*, \infty)$. \end{thm} This theorem is proven by finding bounds for $v^{(k)}$ with $k\geq 2$. \medskip The purpose of this paper is to discuss the $C^{\infty}$ regularity of the moving part of the interface of the solution to (\ref{eq:plap}). To accomplish this end, we use some ideas from \cite{AV87}. \section{\bf Upper and Lower Bounds for $v_{xx}$} Let $q=(x_0, t_0)$ be a point on the left interface, so that $x_0 = \zeta(t_0)$, $v(x,t_0) = 0$ for all $x\leq \zeta(t_0)$, and $v(x,t_0) >0$ for all sufficiently small $x>\zeta(t_0)$. We assume the left interface is moving at $q$. Thus $t_0 >t^{*}$. We shall use the notation $$R_{\delta, \eta} =R_{\delta, \eta}(t_0) = \{(x,t) \in {\mathbb R}^2 :\zeta(t) 0$. \hfill $\Box$ \begin{pro}\label{pro:2} Let $q=(x_0,t_0)$ be as above. Then there exist positive constants $C_{2}$, $\delta$, and $\eta$ depending only on $p$, $q$, and $u$ such that $$v_{xx} \leq C_{2} \quad\text{in } R_{\delta,\eta/2}\,.$$ \end{pro} \noindent\textbf{Proof.} From Theorem 2 and Lemma 4.4 in \cite{EV88} we have \begin{equation}\label{eq:ev1} \zeta'(t_0)=-v_x|v_x|^{p-2}=-v_x^{p-1}=-a \end{equation} and \begin{equation}\label{eq:ev2} v_t = |v_x |^p \end{equation} on the moving part of the interface $\{x=\zeta(t), t>t^*\}$. Choose $\epsilon>0$ such that \begin{equation}\label{eq:eeps} (p-1)a-5p\epsilon\geq 4[(p-2)^2 + (p-1)^2](a+\epsilon)\epsilon. \end{equation} Then by Theorem 2 in \cite{EV88}, there exists a $\delta=\delta(\epsilon)>0$ and $\eta=\eta(\epsilon)\in(0,t_0 -t^*)$ such that $R_{\delta,\eta}\subset P[u]$, \begin{equation}\label{eq:1} (a-\epsilon)^{\frac{1}{p-1}} \zeta^{*}(t)$ in $(t_1,t_2]$. On $P[u]$, $w\equiv v_{xx}$ satisfies \begin{eqnarray*} L(w)&=&w_t-(p-2)v|v_x|^{p-2}w_{xx}-(3p-4)|v_x|^{p-2}v_x w_x \\ &&- [(p-2)^2+2(p-1)^2]|v_x|^{p-2}w^2\\ && - 3(p-2)^2 v|v_x|^{p-4}v_x ww_x - (p-2)^2(p-3)v|v_x|^{p-4}w^3\\ &=&0\,. \end{eqnarray*} We shall construct a barrier for $w$ in $R_{\delta,\eta}$ of the form $$ \phi(x,t) \equiv \frac{\alpha}{x-\zeta (t)}+\frac{\beta}{x-\zeta^{*}(t)}, $$ where $\alpha$ and $\beta$ will be decided later. By a direct computation we have \begin{eqnarray*} L(\phi) &=&\frac{\alpha}{(x-\zeta)^2}\{\zeta' - (p-2)v|v_x|^{p-2}\frac{2}{x-\zeta} + (3p-4)|v_x|^{p-2}v_x \}\\ && + \frac{\beta}{(x-\zeta^*)^2}\{\zeta^{*'} - (p-2)v|v_x|^{p-2}\frac{2}{x-\zeta^*} + (3p-4)|v_x|^{p-2}v_x \}\\ && - [(p-2)^2 + 2(p-1)^2]|v_x|^{p-2}\phi^2 + \bar{G} \end{eqnarray*} where \begin{eqnarray*} \lefteqn{\bar{G} }\\ &=& -3(p-2)^2 vv_x |v_x|^{p-4}\phi \phi_{x} - (p-2)^2 (p-3)v|v_x|^{p-4}\phi^{3}\\ &=& (p-2)^2 v|v_x|^{p-4}\phi\left(3v_x[\frac{\alpha}{(x-\zeta)^2} +\frac{\beta}{(x-\zeta^*)^2}] -(p-3)[\frac{\alpha}{x-\zeta} + \frac{\beta}{x-\zeta*}]^2\right). \end{eqnarray*} If we choose $\alpha$ and $\beta$ satisfying $$v_x \geq|p-3|\max(\alpha,\beta),$$ then $\bar{G} \geq 0$ in $R_{\delta, \eta}$. Now set $\bar{A}= \frac{\alpha}{(x-\zeta)^2}$ and $\bar{B}= \frac{\beta}{(x-\zeta^*)^2}$. Then we have \begin{eqnarray*} \lefteqn{ L(\phi) } \\ &\geq& \bar{A}\left\{\zeta' + |v_{x}|^{p-2}\{-(p-2)v\frac{2}{x-\zeta} + (3p-4)v_x -2[(p-2)^2 + 2(p-1)^2 ]\alpha\}\right\}\\ &&+ \bar{B}\left\{\zeta^{*'} + |v_{x}|^{p-2}\{-(p-2)v\frac{2}{x-\zeta^*} + (3p-4)v_x -2[(p-2)^2 + 2(p-1)^2 ]\beta\}\right\}\\ &\geq& \bar{A}\left\{(p-1)a - (5p-7)\epsilon -2[(p-2)^2 +2(p-1)^2](a+\epsilon)^{\frac{p-2}{p-1}}\alpha\right\}\\ &&+ \bar{B} \left\{(p-1)a - (5p-6)\epsilon -2[(p-2)^2 +2(p-1)^2](a+\epsilon)^{\frac{p-2}{p-1}}\beta\right\}. \end{eqnarray*} Set $$0< \alpha\leq \frac{(p-1)a -(5p-7)\epsilon} {2[(p-2)^2 + 2(p-1)^2](a+\epsilon)^{\frac{p-2}{p-1}}}=\alpha_0$$ and \begin{equation}\label{eq:beta} \beta =\frac{(p-1)a-(5p-6\epsilon)} {2[(p-2)^2 + 2(p-1)^2](a+\epsilon)^{\frac{p-2}{p-1}}}\,. \end{equation} Then from (\ref{eq:eeps}), $\beta >0$ and $L(\phi)\geq 0$ in $R_{\delta,\eta}$ for all $\alpha\in (0, \alpha_0]$ and $\beta$. Let us now compare $w$ and $\phi$ on the parabolic boundary of $R_{\delta,\eta}$. In view of (\ref{eq:ev3}) and (\ref{eq:ev4}) we have $$v_{xx}\leq\frac{\epsilon(a-\epsilon)^{\frac{1}{p-1}}}{x-\zeta} \quad\text{in } R_{\delta,\eta}$$ and in particular $$v_{xx}(\zeta(t) + \delta, t) \leq \frac{\epsilon(a-\epsilon)^{\frac{1}{p-1}}}{\delta} \text{ in } [t_1, t_2 ]\,.$$ By the Mean Value Theorem and (\ref{eq:ev5}), we have that for some $\tau \in (t_1 , t_2 )$ \begin{eqnarray*} \zeta(t) + \delta -\zeta^{*}(t) &=& \delta + (a + 2\epsilon)(t-t_{1}) + \zeta'(\tau)(t-t_1 )\\ &\leq& \delta + 3\epsilon(t-t_1)\leq \delta + 6\epsilon\eta. \end{eqnarray*} Now set $$\eta = \min\{\eta(\epsilon), \delta(\epsilon)/6\epsilon \}.$$ Since $\epsilon$ satisfies (\ref{eq:eeps}) and $\beta$ is given by (\ref{eq:beta}) it follows that $$\phi(\zeta + \delta, t) \geq \frac{\beta}{2\delta} \geq \frac{(p-1)a-(5p-6\epsilon)} {4[(p-2)^2 +2(p-1)^2](a+\epsilon)^{\frac{p-2}{p-1}}\delta} \geq\frac{(a+\epsilon)^{\frac{1}{p-1}}}{\delta}\epsilon\geq v_{xx}\,, $$ on $[t_1,t_2]$. Moreover from (\ref{eq:eps}) and (\ref{eq:beta}) \[\phi(x,t_1) \geq \frac{\beta}{x-\zeta(t_1)} >\frac{\epsilon(a-\epsilon)^{\frac{1}{p-1}}} {x-\zeta(t_{1})}>v_{xx}(x,t_{1})\text{ on } (\zeta(t_1), \zeta(t_1) +\delta]\,.\] Let $\Gamma = \{(x,t) \in {\mathbb R}^2 : x=\zeta(t), t_1 \leq t\leq t_2\}$. Clearly $\Gamma$ is a compact subset of ${\mathbb R}^2$. Fix $\alpha \in (0,\alpha_0)$. For each point $s\in \Gamma$ there is an open ball $B_s $ centered at $s$ such that $$(vv_{xx})(x,t) \leq \alpha(a-\epsilon)^{\frac{1}{p-1}}\quad\text{in } B_s \cap P[u]\,.$$ In view of (\ref{eq:ev4}) we have $$\phi(x,t) \geq \frac{\alpha}{x-\zeta} \geq v_{xx}(x,t)\quad\text{in } B_s \cap P[u]\,.$$ Since $\Gamma$ can be covered by a finite number of these balls it follows that there is a $\gamma=\gamma(\alpha) \in (0,\delta)$ such that $$\phi(x,t) \geq w(x,t)\quad\text{in } R_{\delta,\eta}.$$ Thus for every $\alpha \in (0, \alpha_0)$, $\phi$ is a barrier for $w$ in $R_{\delta,\eta}$. By the comparison principle for parabolic equations \cite{LSU} we conclude that $$v_{xx}(x,t) \leq \frac{\alpha}{x-\zeta(t)} + \frac{\beta}{x-\zeta^{*}(t) } \quad\text{in } R_{\delta,\eta}\,,$$ where $\beta$ is given by (\ref{eq:beta}) and $\alpha \in(0,\alpha_0)$ is arbitrary. Now as $\alpha$ approaches zero, we obtain $$ v_{xx}(x,t) \leq \frac{\beta}{x-\zeta^*} \leq \frac{2\beta}{\epsilon\eta} \quad\text{in } {\mathbb R}. $$ \section{\bf Bounds for $\left(\frac{\partial}{\partial{x}}\right)^{3}v$} In this section we find the estimates of the derivatives of the form $$ v^{(3)} \equiv \left(\frac{\partial}{\partial x}\right)^{3}v. $$ By a direct computation we have, \begin{eqnarray}\label{eq:**} L_{3}(v^{(3)})&=&v^{(3)}_t-(p-2)vv_{x}^{p-2}v^{(3)}_{xx} - (A+B)v_{x}^{(3)} -Cv^{(3)} -D(v^{(3)})^2\\ &&-Ev_{x}^{p-3}v_{xx}^{3} - (p-2)^{2}(p-3)(p-4)vv_{x}^{p-5}v_{xx}^{4}=0\,,\nonumber \end{eqnarray} where \begin{eqnarray*} A &=& (p-2)v_{x}^{p-1} + (p-2)^{2}vv_{x}^{p-3}v_{xx}\,,\\ B &=& (3p-4)v_{x}^{p-1} + 3(p-2)^{2}vv_{x}^{p-3}v_{xx}\,,\\ C &=& v_{xx}v_{x}^{p-2}\{(3p-4)(p-1) + 2[(p-2)^{2} \\ && + 2(p-1)^{2}] + 6(p-2)^{2}(p-3)vv_{x}^{-2}v_{xx} + 3(p-2)^2 \}\,,\\ D &=& 3(p-2)^2 vv_{x}^{p-3}\,, \\ E &=&[(p-2)^2 +2(p-1)^2 ](p-2) + (p-2)^{2}(p-3)\,. \end{eqnarray*} Suppose that $q=(x_0,t_0)$ is a point on the left interface for which (\ref{eq:ev1}) holds. Fix $\epsilon \in (0,a)$ and take $\delta_0 = \delta_0(\epsilon) >0$ and $\eta_0=\eta(\epsilon)\in (0,t_0-t^{*})$ such that $R_0\equiv R_{\delta_0,\eta_0}(t_0)\subset P[u]$ and (\ref{eq:ev3}) holds. Thus we also have (\ref{eq:ev4}) and (\ref{eq:ev5}) in $R_0$. Then by rescaling and interior estimate we have \begin{pro}\label{pro:3} There are constants $K\in {\mathbb R}^{+}$, $\delta\in (0,\delta_0)$, and $\eta \in (0,\eta_0)$ depending only on $p$, $q$, and $C_{2}$ such that $$|v^{(3)}(x,t)|\leq \frac{K}{x-\zeta(t)}\quad\text{in } R_{\delta,\eta}\,.$$ \end{pro} \noindent\textbf{Proof.} Set $$\delta = \min\{\frac{2\delta_0}{3},2s\eta_0\},\quad \eta = \eta_0-\frac{\delta}{4s}\,,$$ and define \[ R(\ol{x},\ol{t})\equiv \left\{(x,t) \in {\mathbb R}^2 : |x-\ol{x}| < \frac{\lambda}{2}, \ol{t} -\frac{\lambda}{4s} \zeta(t_0+ \eta_0)\,,&\\ &\zeta(t_0-\eta) + \delta + \frac{\lambda}{2} <\zeta(t_0 -\eta_0)\,.& \end{eqnarray*} Also observe that for each $(\ol{x},\ol{t})\in R_{\delta,\eta}$, $R(\ol{x},\ol{t})$ lies to the right of the line $x=\zeta(\ol{t}) + s(\ol{t} - t)$. Next set $x=\lambda\xi + \ol{x}$ and $t=\lambda\tau + \ol{t}$. The function $$W(\xi,\tau)\equiv v_{xx}(\lambda\xi + \ol{x},\lambda\tau + \ol{t}) =v_{xx}(x,t)$$ satisfies the equation \begin{eqnarray}\label{eq:*} W_{\tau}&=& \left\{(p-2)\frac{v}{\lambda}v_{x}^{p-2}W_{\xi} + (3p-4)v_{x}^{p-1}W\right\}_{\xi} \nonumber\\ &&+ [2(p-2)^{2}vv_{x}^{p-3}v_{xx}-(p-2)v_{x}^{p-1}]W_{\xi}\\ &&+\lambda [(p-2)^{2}(p-3)vv_{x}^{p-4}(v_{xx})^3 - (p-2)v_{x}^{p-2}(v_{xx})^2 ]\nonumber \end{eqnarray} in the region $$B\equiv \left\{(\xi,\tau)\in {\mathbb R}^{2} : |\xi|\leq \frac{1}{2}, -\frac{1}{4s}<\tau\leq 0\right\},$$ and $|W|\leq C_{2}$ in $B$. In view of (\ref{eq:ev4}) and (\ref{eq:ev5}) $$(a-\epsilon)^{\frac{1}{p-1}}\frac{x-\zeta(t)}{\lambda} \leq\frac{v(x,t)}{\lambda} \leq (a+\epsilon)^{\frac{1}{p-1}}\frac{x-\zeta(t)}{\lambda}$$ and $$\zeta(\ol{t})\leq\zeta(t)\leq\zeta(\ol{t})+s(\ol{t}-t) \leq\zeta(\ol{t})+\frac{\lambda}{4}\,.$$ Therefore, $$\frac{\lambda}{4} = \ol{x}-\frac{\lambda}{2}-\zeta(\ol{t}) -\frac{\lambda}{4} \leq x - \zeta(t) \leq \ol{x} + \frac{\lambda}{2} - \zeta(\ol{t}) = \frac{3\lambda}{2}$$ which implies $$\frac{(a-\epsilon)^{\frac{1}{p-1}}}{4}\leq \frac{v}{\lambda} \leq \frac{3(a+\epsilon)^{\frac{1}{p-1}}}{2}\,.$$ Hence by (\ref{eq:1}) equation (\ref{eq:*}) is uniformly parabolic in $B$. Moreover, it follows from Proposition \ref{pro:2} that $W$ satisfies all of the hypotheses of Theorem 5.3.1 of \cite{LSU}. Thus we conclude that there exists a constant $K=K(a,p,C_{2})>0$ such that $$\left|\frac{\partial}{\partial\xi}W(0,0)\right| \leq K ;$$ that is, $$|v^{(3)}(\ol{x},\ol{t})|\leq \frac{K}{\lambda}\,.$$ Since $(\ol{x},\ol{t})\in R_{\delta,\eta}$ is arbitrary, this proves the proposition.\hfill$\Box$\medskip We now turn to the barrier construction. If $\gamma \in (0,\delta)$ we will use the notation $$R_{\delta,\eta}^{\gamma}=R_{\delta,\eta}^{\gamma}(t_0)\equiv \{(x,t)\in {\mathbb R}^2 :\zeta(t)+\gamma\leq x\leq\zeta(t)+\delta, t_0-\eta \leq t\leq t_0 + \eta\}.$$ \begin{pro}\label{pro:4} Let $R_{\delta_{1},\eta_{1}}$ be the region constructed in the proof of Proposition \ref{pro:2} with \begin{equation}\label{eq:delta1} 0<\delta_1 < \frac{(p-1)a^{\frac{1}{p-1}}}{12(p-2)^{2}K}\,. \end{equation} For $(x,t)\in R_{\delta_{1},\eta_{1}}^{\gamma}$, let \begin{equation}\label{eq:ba2} \phi_{\gamma}(x,t) \equiv \frac{\alpha}{x-\zeta(t) -\gamma/3} + \frac{\beta}{x-\zeta^{*}(t)}\,, \end{equation} where $\zeta^{*}$ is given by (\ref{eq:z*}), and $\alpha$ and $\beta$ are positive constant less than $K/2$. Then there exist $\delta\in(0,\delta_1 )$ and $\eta\in(0,\eta_1 )$ depending only on $a$, $p$ and $C_{2}$ such that $$L_{3}(\phi_{\gamma})\geq 0 \quad\text{in } R_{\delta,\eta}^{\gamma}$$ for all $\gamma\in (0,\delta)$. \end{pro} \noindent\textbf{Proof.} Choose $\epsilon$ such that \begin{equation}\label{eq:eps} 0<\epsilon < \frac{(p-1)a}{13p-23}\,. \end{equation} There exist $\delta_{2}\in (0,\delta_{1})$ and $\eta\in (0,\eta_{1})$ such that (\ref{eq:1}), (\ref{eq:ev4}) and (\ref{eq:ev5}) hold in $R_{\delta_{2},\eta}$. Fix $\gamma\in(0,\delta_{2})$. For $(x,t) \in R_{\delta_{2},\eta}^{\gamma}$, we have \begin{eqnarray*} L_{3}(\phi_{3})&=&\frac{\alpha}{(x-\zeta-\gamma/3)^2}\left\{\zeta^{'} -\frac{2(p-2)vv_{x}^{p-2}}{x-\zeta-\gamma/3} + A+B\right\}\\ && +\frac{\alpha}{(x-\zeta^{*})^2}\left\{\zeta^{*'} -\frac{2(p-2)vv_{x}^{p-2}}{x-\zeta^{*}} + A+B\right\}\\ &&-C\phi_{3} -D(\phi_{3})^2 -Ev_{x}^{p-3}v_{xx}^{3} - (p-2)^{2}(p-3)(p-4)vv_{x}^{p-5}v_{xx}^4\, \end{eqnarray*} where $A$, $B$, $C$, $D$, and $E$ are as above. >From (\ref{eq:ev4}), together with the fact that $x-\zeta^{*}\geq x-\zeta-\gamma/3$ we have $$\frac{v}{x-\zeta^{*}}\leq \frac{v}{x-\zeta-\gamma/3} \leq (a+\epsilon)^{\frac{1}{p-1}}\frac{x-\zeta}{x-\zeta-\gamma/3} \leq (a+\epsilon)^{\frac{1}{p-1}}\frac{\gamma}{\gamma-\gamma/3} = \frac{3}{2}(a+\epsilon)^{\frac{1}{p-1}}\,.$$ >From (\ref{eq:delta1}), we have \begin{equation}\label{eq:ab} D\alpha, D\beta < 1/2DK0$ so small that $L_{3}(\phi_{3}) \geq 0$ in $R_{\delta,\eta}^{\gamma}$. \hfill $\Box$ \medskip \noindent{\bf Remark 3.1.} >From (\ref{eq:ab}) the Proposition \ref{pro:4} will be true for any $\alpha,\beta \in (0, K)$. \begin{pro} \label{pro:5} (Barrier Transformation). Let $\delta$ and $\eta$ be as in Proposition \ref{pro:4} with the additional restriction that \begin{equation}\label{eq:sepis} \eta <\frac{\delta}{6\epsilon}, \end{equation} where $\epsilon$ is as in Proposition \ref{pro:4}. Suppose that for some nonnegative constant $\beta$ \begin{equation}\label{eq:semi} v^{(3)}(x,t) \leq \frac{\alpha}{x-\zeta(t)} + \frac{\beta}{x-\zeta^{*}(t)} \quad \text{in } R_{\delta,\eta}. \end{equation} Then $v^{(3)}$ also satisfies \begin{equation}\label{eq:final} v^{(3)}(x,t) \leq \frac{2\alpha/3}{x-\zeta(t)} + \frac{\beta + 2\alpha/3}{x-\zeta^{*}(t)} \quad\text{in } R_{\delta,\eta}. \end{equation} \end{pro} \noindent\textbf{Proof.} By Remark 3.1, for any $\gamma \in (0,\delta)$ since $\beta +2\alpha/3 \leq K$ the function $$ \phi_{3}(x,t) = \frac{2\alpha/3}{x-\zeta -\gamma/3} + \frac{\beta + 2\alpha/3}{x-\zeta^{*}} $$ satisfies $L_{3}(\phi_{3}) \geq 0$ in $R_{\delta,\eta}^{\gamma}$. On the other hand, on the parabolic boundary of $R_{\delta,\eta}^{\gamma}$ we have $\phi_{3} \geq v^{(3)}$. In fact, for $t = t_{1}$ and $\zeta_{1} + \gamma \leq x \leq \zeta_1 + \delta$, with $\zeta_{1}=\zeta(t_1)$, we have $$ \phi_{3}(x,t_1) =\frac{2\alpha}{x-\zeta_1 -\gamma/3} + \frac{\beta + 2\alpha/3}{x-\zeta_1} > \frac{4\alpha/3}{x-\zeta_1} +\frac{\beta}{x-\zeta_1} > v^{(3)}(x,t_1)$$ while for $x=\zeta +\delta$ and $t_{1}\leq t \leq t_2 $ we get, in view of (\ref{eq:sepis}), \begin{eqnarray*} \phi_{3}(\zeta + \delta,t) &\geq& \frac{2\alpha/3}{\delta -\gamma/3} + \frac{\beta}{\zeta + \delta -\zeta^{*}} + \frac{2\alpha/3}{\delta + 6\epsilon\eta}\\ &\geq& \frac{2\alpha/3}{\delta} + \frac{\delta}{\zeta + \delta - \zeta^{*}} + \frac{\alpha/3}{\delta}\geq v^{(3)}(\zeta +\delta,t). \end{eqnarray*} Finally, for $x=\zeta + \gamma$, $t_1 \leq t \leq t_2 $ we have $$\phi_{3}(\zeta + \delta, t)=\frac{2\alpha/3}{\gamma-\gamma/3} + \frac{\beta + 2\alpha/3}{\zeta + \gamma - \zeta^{*}} \geq \frac{\alpha}{\gamma}+\frac{\beta }{\zeta+\gamma-\zeta^{*}} \geq v^{(3)}(\zeta+\gamma,t).$$ By the comparison principle we get $$\phi_{3} \geq v^{(3)} \quad\text{in } R_{\delta.\eta}^{\gamma}$$ for any $\gamma \in (0,\delta)$, and (\ref{eq:final}) follows by letting $\gamma \downarrow 0$. \hfill $\Box$ \begin{pro}\label{pro:6} Let $q=(x_0,t_0)$ be a point on the interface for which (\ref{eq:ev1}) holds. Then there exist constants $C_{3}$, $\delta$ and $\eta$ depending only on $p$, $q$ and $u$ such that $$\left|\left(\frac{\partial}{\partial x}\right)^{3}v\right|\leq C_{3} \quad\text{in } R_{\delta,\eta/2}.$$ \end{pro} \noindent\textbf{Proof.} By Proposition \ref{pro:3} we have, by letting $\alpha=0$, $$ v^{(3)}(x,t) \leq \frac{\beta}{x-\zeta^{*}}\leq \frac{2\beta}{\epsilon\eta} \quad\text{in } R_{\delta,\eta/2}. $$ Even though the equation (\ref{eq:**}) is not linear for $v^{(3)}$, a lower bound can be obtained in a similar way. \hfill $\Box$ \section{\bf Main Result} In this section we prove the interface is a $C^{\infty}$ function in $(t^{*},\infty)$. We follow the methods in \cite{AV87}. First we find the estimates of the derivatives of the form $$ v^{(j)} \equiv \left(\frac{\partial}{\partial x}\right)^{j}v $$ for $j\geq 4$. For the porous medium equation, we have \cite{AV87} the following equation: \begin{eqnarray*} L_{j}v^{(j)} &\equiv& v_{t}^{(j)} - (m-1)vv_{xx}^{(j)} - (2+j(m-1))v_{x}v_{x}^{(j)} -c_{mj}v_{xx}v^{(j)}\\ && - \sum_{l=3}^{j^{*}}d_{mj}^{l}v^{(l)}v^{(j+2-l)}=0 \end{eqnarray*} for $j\geq 3$ in $P[u]$, where $j^{*} = [j/2] + 1$, and the $c_{mj}$ and $d_{mj}^{l}$ are constants which depend only on their indices, but whose precise values are irrelevant. Note that $L_{j}$ is linear in $v^{(j)}$. On the other hand for the p-Laplacian equation by a direct computation we have the following equation for $j\geq 4$, \begin{eqnarray}\label{eq:j4} L_{j}v^{(j)} &=& v_{t}^{(j)}- (p-2)vv_{x}^{p-2}v_{xx}^{(j)} - ((j-2)A+B)v_{x}^{(j)} - C_{pj}v^{(j)}\\ && - F(v, v_{x}, \ldots, v^{(j-1)})=0\nonumber \end{eqnarray} where $A$ and $B$ are as before, and $C_{pj}$ involves only $v$ and derivatives of order $< j$. Note that equation (\ref{eq:j4}) is linear in $v^{(j)}$. We also follow the method in \cite{AV87}. Hence our result is \begin{pro}\label{pro:7} Let $q=(x_0,t_0)$ be a point on the interface for which (\ref{eq:ev1}) holds. For each integer $j \geq 2$ there exist constants $C_{j}$, $\delta$ and $\eta$ depending only on $p$, $j$, $q$ and $u$ such that $$\left|\left(\frac{\partial}{\partial x}\right)^{j}v\right|\leq C_{j} \quad \text{in } R_{\delta,\eta/2}.$$ \end{pro} The proof is done by induction on $j$. Suppose that $q=(x_0,t_0)$ is a point on the left interface for which (\ref{eq:ev1}) holds. Fix $\epsilon \in (0,a)$ and take $\delta_0 = \delta_0(\epsilon) >0$ and $\eta_0=\eta(\epsilon)\in (0,t_0-t^{*})$ such that $R_0\equiv R_{\delta_0,\eta_0}(t_0)\subset P[u]$ and (\ref{eq:ev3}) holds. Thus we also have (\ref{eq:ev4}) and (\ref{eq:ev5}) in $R_0$. Assume that there are constants $C_k \in {\mathbb R}^{+}$ for $k =3, \ldots, j-1$ such that \begin{equation}\label{eq:ck} |v^{(k)}| \leq C_k \quad\text{ on} R_0\quad for \quad k=3,\ldots, j-1. \end{equation} Observe that (\ref{eq:ck}) hold for $k=3$ by Proposition \ref{pro:6}. By rescaling and interior estimates, we have \begin{pro}\label{pro:8} There are constants $K\in {\mathbb R}^{+}$, $\delta\in (0,\delta_0)$, and $\eta \in (0,\eta_0)$ depending only on $p$,$q$ and $C_{k}$ for $k\in[2, j-1]$ with $j\geq 4$ such that $$|v^{(j)}(x,t)|\leq \frac{K}{x-\zeta(t)}\quad\text{in } R_{\delta,\eta}.$$ \end{pro} \noindent\textbf{Proof.} Set $$\delta = \min\{\frac{2\delta_0}{3},2s\eta_0\}\,, \quad \eta = \eta_0-\frac{\delta}{4s}\,,$$ and define \[ R(\ol{x},\ol{t})\equiv \left\{(x,t) \in {\mathbb R}^2 : |x-\ol{x}| < \frac{\lambda}{2}, \ol{t} -\frac{\lambda}{4s} \zeta(t_0+ \eta_0)\,,&\\ &\zeta(t_0-\eta) + \delta + \frac{\lambda}{2} <\zeta(t_0 -\eta_0)\,. & \end{eqnarray*} Also observe that for each $(\ol{x},\ol{t})\in R_{\delta,\eta}$, $R(\ol{x},\ol{t})$ lies to the right of the line $x=\zeta(\ol{t}) + s(\ol{t} - t)$. Next set $x=\lambda\xi + \ol{x}$ and $t=\lambda\tau + \ol{t}$. The function $$V^{(j-1)}(\xi,\tau)\equiv v^{(j-1)}(\lambda\xi + \ol{x},\lambda\tau + \ol{t}) =v^{(j-1)}(x,t)$$ satisfies the equation \begin{eqnarray}\label{eq:11} V^{(j-1)}_{\tau}&=& \left\{(p-2)\frac{v}{\lambda}v_{x}^{p-2}V^{(j-1)}_{\xi} +[(j-2)A+B]v_{x}^{p-1}V^{(j-1)}\right\}_{\xi} \nonumber\\ &&-[(p-2)v^{p-1}_{x}+(p-2)^2 vv^{p-3}_x v_{xx}+(j-2)A+B]V^{(j-1)}_{\xi}\\ &&+\lambda[C_{pj}-((j-2)A_x+B_x)]V^{(j-1)}+\lambda F(v,\ldots,v^{(j-2)} \nonumber \end{eqnarray} in the region $$B\equiv \left\{(\xi,\tau)\in {\mathbb R}^{2} : |\xi|\leq \frac{1}{2}, -\frac{1}{4s}<\tau\leq 0\right\},$$ and $|W|\leq C_{2}$ in $B$. In view of (\ref{eq:ev4}) and (\ref{eq:ev5}) $$(a-\epsilon)^{\frac{1}{p-1}}\frac{x-\zeta(t)}{\lambda} \leq\frac{v(x,t)}{\lambda} \leq (a+\epsilon)^{\frac{1}{p-1}}\frac{x-\zeta(t)}{\lambda}$$ and $$\zeta(\ol{t})\leq\zeta(t)\leq\zeta(\ol{t})+s(\ol{t}-t) \leq\zeta(\ol{t})+\frac{\lambda}{4}.$$ Therefore $$\frac{\lambda}{4} = \ol{x}-\frac{\lambda}{2}-\zeta(\ol{t}) -\frac{\lambda}{4} \leq x - \zeta(t) \leq \ol{x} + \frac{\lambda}{2} - \zeta(\ol{t}) = \frac{3\lambda}{2}$$ which implies $$\frac{(a-\epsilon)^{\frac{1}{p-1}}}{4}\leq \frac{v}{\lambda} \leq \frac{3(a+\epsilon)^{\frac{1}{p-1}}}{2}\,.$$ Hence by (\ref{eq:1}) equation (\ref{eq:*}) is uniformly parabolic in $B$. Moreover, it follows from Proposition \ref{pro:2} that $W$ satisfies all of the hypotheses of Theorem 5.3.1 of \cite{LSU}. Thus we conclude that there exists a constant $K=K(a,p,C_{1},\ldots,C_{j-1})>0$ such that $$\left|\frac{\partial}{\partial\xi}V^{(j-1)}(0,0)\right| \leq K ;$$ that is, $$|v^{(j)}(\ol{x},\ol{t})|\leq \frac{K}{\lambda}\,.$$ Since $(\ol{x},\ol{t})\in R_{\delta,\eta}$ is arbitrary, this proves the proposition.\hfill$\Box$ \medskip We now turn to the barrier construction. If $\gamma \in (0,\delta)$ we will use the notation $$R_{\delta,\eta}^{\gamma}=R_{\delta,\eta}^{\gamma}(t_0)\equiv \{(x,t)\in {\mathbb R}^2 :\zeta(t)+\gamma\leq x\leq\zeta(t)+\delta, t_0-\eta \leq t\leq t_0 + \eta\}\,.$$ \begin{pro}\label{pro:9} Let $R_{\delta_{1},\eta_{1}}$ be the region constructed in the proof of Proposition \ref{pro:2} with For $j\geq 4$ and $(x,t)\in R_{\delta_{1},\eta_{1}}^{\gamma}$, let \begin{equation}\label{eq:ba3} \phi_{j}(x,t) \equiv \frac{\alpha}{x-\zeta(t) -\gamma/3} + \frac{\beta}{x-\zeta^{*}(t)} \end{equation} where $\zeta^{*}$ is given by (\ref{eq:z*}), and $\alpha$ and $\beta$ are positive constant. Then there exist $\delta\in(0,\delta_1 )$ and $\eta\in(0,\eta_1 )$ depending only on $a$, $p$, $C_{1}, \ldots, C_{j-1}$ such that $$L_{j}(\phi_{j})\geq 0 \quad\text{in } R_{\delta,\eta}^{\gamma}$$ for all $\gamma\in (0,\delta)$. \end{pro} \noindent\textbf{Proof.} Choose $\epsilon$ such that \begin{equation}\label{eq:2eps} 0<\epsilon < \frac{a}{(j-2)(p-2)+6p-8}\,. \end{equation} There exist $\delta_{2}\in (0,\delta_{1})$ and $\eta\in (0,\eta_{1})$ such that (\ref{eq:1}), (\ref{eq:ev4}) and (\ref{eq:ev5}) hold in $R_{\delta_{2},\eta}$. Fix $\gamma\in(0,\delta_{2})$. For $(x,t) \in R_{\delta_{2},\eta}^{\gamma}$, we have \begin{eqnarray*} L_{j}(\phi_{j}) &=&\frac{\alpha}{(x-\zeta-\gamma/3 )^2}\left\{\zeta^{'} -\frac{2(p-2)vv_{x}^{p-2}}{x-\zeta-\gamma/3 } +(j-2)A+B\right\}\\ &&-\frac{\alpha}{(x-\zeta-\gamma/3 )^2}\left\{C_{pj}(x-\zeta-\gamma/3) -\frac{(x-\zeta-\gamma/3)^{2}}{\alpha}F \right\}\\ &&+ \frac{\beta}{(x-\zeta^*)^2}\left\{\zeta^{*'} -\frac{2(p-2)vv_{x}^{p-2}}{x-\zeta^{*}} +(j-2) A+B -C_{pj}(x-\zeta^*) \right\} \end{eqnarray*} where $A$, $B$, $C_{pj}$ and $F$ are as before. >From (\ref{eq:ev4}), together with the fact that $x-\zeta^{*}\geq x-\zeta-\gamma/3$ we have $$\frac{v}{x-\zeta^{*}}\leq \frac{v}{x-\zeta-\gamma/3} \leq (a+\epsilon)^{\frac{1}{p-1}}\frac{x-\zeta}{x-\zeta-\gamma/3} \leq (a+\epsilon)^{\frac{1}{p-1}}\frac{\gamma}{\gamma-\gamma/3} = \frac{3}{2}(a+\epsilon)^{\frac{1}{p-1}}\,.$$ Then from (\ref{eq:1}), (\ref{eq:ev4}) and (\ref{eq:ck}), we have \begin{eqnarray}\label{eq:***} L_{j}(\phi_{j}) &\geq& \frac{\alpha}{(x-\zeta-\gamma/3)^{2}} \left\{a-((j-2)(p-2)+6p-9)\epsilon-\delta_2 (|C_{pj}|+\frac{\delta}{\alpha}|F|\right\}\nonumber\\ &&+\frac{\beta}{(x-\zeta^{*})^2}\left\{a-((j-2)(p-2)+6p-8)- \delta_2(|C_{pj}|\right\}\nonumber \end{eqnarray} Since $\epsilon$ satisfies (\ref{eq:2eps}) we can choose $\delta=\delta_{2}(\epsilon, p, a, C_{2})>0$ so small that $L_{3}(\phi_{3}) \geq 0$ in $R_{\delta,\eta}^{\gamma}$. \hfill $\Box$ \medskip Hence as in we have the following proposition whose proof can be found in \cite{AV87}. \begin{pro}\label{pro:10} (Barrier Transformation). Let $\delta$ and $\eta$ be as in Proposition \ref{pro:9} with the additional restriction that \begin{equation}\label{eq:epis} \eta <\frac{\delta}{6\epsilon}\,, \end{equation} where $\epsilon$ is as in Proposition \ref{pro:9}. Suppose that for some nonnegative constant $\beta$ \begin{equation}\label{eq:ssemi} v^{(j)}(x,t) \leq \frac{\alpha}{x-\zeta(t)} + \frac{\beta}{x-\zeta^{*}(t)} \quad in \quad R_{\delta,\eta}\,. \end{equation} Then $v^{(j)}$ also satisfies \begin{equation}\label{eq:ffinal} v^{(j)}(x,t) \leq \frac{2\alpha/3}{x-\zeta(t)} + \frac{\beta + 2\alpha/3}{x-\zeta^{*}(t)} \quad\text{in } R_{\delta,\eta}\,. \end{equation} \end{pro} Then as in \cite{AV87}, we can prove the $C^{\infty}$ regularity of the interface. \begin{thebibliography}{9} \bibitem{AV87} D. G. Aronson and J. L. Vazquez, Eventual $C^{\infty}$-regularity and concavity for flows in one-dimensional porous media, {\it Arch. Rational Mech. Anal.} {\bf 99} (1987),no.4, 329-348. \bibitem{EV88} J. R. Esteban and J. L. Vazquez, Homogeneous diffusion in R with power-like nonlinear diffusivity, {\it Arch. Rational Mech. Anal.} {\bf 103}(1988) 39-80. \bibitem{CF80} L. A. Caffarelli and A. Friedman, Regularity of the free boundary of a gas flow in an n-dimensional porous medium, {\it Ind. Univ. Math. J.} {\bf 29} (1980), 361-381. \bibitem{LSU} O. A. Ladyzhenskaya, N.A. Solonnikov and N.N. Uraltzeva, Linear and quasilinear equations of parabolic type, {\it Trans. Math. Monographs,} {\bf 23}, Amer. Math. Soc., Providence, R. I., 1968. \end{thebibliography} \end{document}