\documentstyle{amsart} \newtheorem{theorem}{Theorem}[section] %(If you want theorem numbered \newtheorem{lemma}{Lemma}[section] \newtheorem{propo}{Proposition}[section] \newtheorem{coro}{Corollary}[section] \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{ack}{Acknowledgment} \renewcommand{\theack}{} %for unnumbered ack's \renewcommand{\div}{\operatorname{div}} \renewcommand{\i}{\infty} \renewcommand{\d}{\delta} \newcommand{\diam}{\operatorname{diam}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\nref}[1]{(\ref{#1})} \newcommand{\ncite}[1]{{\bf \cite{#1}}} \newcommand{\intav}{-\hskip -1.1em\int} \newcommand{\mialfa}{(p-1)^{p-1} B_p^{-p}} \begin{document} {\noindent\small {\sc Electronic Journal of Differential Equations}\newline Vol. 1994(1994), No. 02, pp. 1-17. Published March 15, 1994.\newline ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \thanks{\copyright 1994 Southwest Texas State University and University of North Texas.} \vspace{1.5cm} \title[\hfilneg EJDE--1994/02\hfil Large Time Behavior]{Large Time Behavior of Solutions to a Class of Doubly Nonlinear Parabolic Equations} \author[J.J. Manfredi \& V. Vespri\hfil EJDE--1994/02\hfilneg] {Juan J. Manfredi\\ Vincenzo Vespri} \address{Department of Mathematics \\ University of Pittsburgh\\ Pittsburgh, PA 15260} \email{manfredi+@@pitt.edu} %\author[]{Vincenzo Vespri} \address{Universit\'a di Pavia\\ Dipartimento di Matematica\\ Via Abbiategrasso 209\\ 27100 Pavia, ITALY} \email{vespri@@vmimat.mat.unimi.it} \date{} \thanks{Submitted on October 25, 1993.} \thanks{First author supported in part by NSF and by IAN (C.N.R.) Pavia.} \thanks{Second author is a member of G.N.A.F.A (C.N.R.)} \subjclass{35K65, 35K55 } \keywords{Doubly nonlinear parabolic equations, asymptotic behavior} \begin{abstract} We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation $u_t=\div\left(|u|^{m-1}|\nabla u|^{p-2}\nabla u\right)$ in a cylinder $\Omega\times{\Bbb R}^+$, with initial condition $u(x,0)=u_0(x)$ in $\Omega$ and vanishing on the parabolic boundary $\partial\Omega\times{\Bbb R}^+$. Here $\Omega$ is a bounded domain in ${\Bbb R}^N$, the exponents $m$ and $p$ satisfy $m+p\ge 3$, $p>1$, and the initial datum $u_0$ is in $L^1(\Omega)$. \end{abstract} \maketitle \section{Introduction} The objective of this article is to study the large time asymptotic behavior of weak solutions of nonlinear parabolic equations of the following type $$\label{eqn1} u_t=\div\left(|u|^{m-1}|\nabla u|^{p-2}\nabla u\right)\mbox{ in }\Omega\times{\Bbb R}^+,$$ subject to the boundary condition $$\label{eqn1.1} u(x,t)=0\mbox{ in } \partial\Omega\times{\Bbb R}^+,$$ and satisfying the initial condition $$\label{eqn1.2} u(x,0)=u_0(x)\mbox{ in } \Omega,$$ where $u\in C({\Bbb R}^+; L^2(\Omega))$ and $u^{\frac{m-1}{p-1}}\nabla u \in L^p_{\text{loc}}(\Omega\times{\Bbb R}^+)$. We always assume that $p>1$, $m+p\ge3$, $\Omega$ is a bounded domain of ${\Bbb R}^n$ and $u_0\in L^1(\Omega)$. The notion of weak solution is standard and we refer the reader to the book \ncite{DB} for more details. Equations of type \nref{eqn1} are classified as doubly nonlinear \ncite{L} or with implicit non linearity \ncite{KA}. Two well studied cases, the porous media equation and the $p$-Laplacian, belong to this larger class. In the last few years several authors have studied these kinds of equations on account of their physical and mathematical interest (see the review paper \ncite{KA}). Indeed, it seems interesting to see if and how many of the properties of the solutions of the porous media and the $p$-Laplacian equations are preserved in this more general case. Several papers are devoted to the study of the asymptotic behavior of the solutions of the porous media and the $p$-Laplacian equations. Among them we quote \ncite{AR-PE}, \ncite{AR-CR-PE}, \ncite{BE-NA-PE}, \ncite{BE-PE}, \ncite{VA2}, \ncite{KA-VA} and \ncite{F-K}. The main point of this paper is to suggest a different approach that gives better results. While in the above references, an elliptic result is used to study the asymptotic behavior, here the basic properties of the evolution equation allow for the study of the asymptotic behavior. Moreover, the elliptic result will follow as a consequence. This approach allows generalizations to a large class of equations and and does not require a non negative datum. Furthermore, we are able to prove our results under weaker regularity assumptions on both the domain $\Omega$ and the initial datum $u_0$. For instance in \ncite{AR-PE}, $\partial \Omega$ is required to be $C^{2,1/m}$ and $u_0$ to belong to $C^{1/m}(\bar\Omega)$. \par We denote by $\gamma_i$ for $i=1,2,\ldots$, positive constants depending only on the data $N$, $m$, $p$, the $L^1$ norm of $u_0$ and the $C^{1,\alpha}$ norm of $\partial\Omega$. We proceed now to state our results. \par \begin{theorem} Suppose that $m+p>3$. There exists a unique non-negative non-trivial solution of the equation $$\label{eqn2} \div(w^{m-1}|Dw|^{p-2}Dw)=\frac{1}{3-m-p}w$$ in $\Omega$, $w\in C^0(\bar\Omega)$, $w^{\frac{m-1}{p-1}}Dw\in L^p(\Omega)$ and $w(x)=0$ for $x\in\partial\Omega$. Moreover, $$\label{eqn3} |Dw|\le\gamma_1(\dist(x,\partial\Omega))^{\frac{1-m}{m+p-2}}$$ and $$\label{eqn4} \gamma_2(\dist(x,\partial\Omega))^{\frac{p-1}{m+p-2}}\le w(x)\le \gamma_3(\dist(x,\partial\Omega))^{\frac{p-1}{m+p-2}}.$$ \end{theorem} \begin{theorem} Suppose that $m+p>3$. There exists a unique solution of the evolution equation \nref{eqn1} subject to conditions \nref{eqn1.1} and \nref{eqn1.2}. Moreover, for all $t>1$ we have the bound $$|u(x,t)|\le \gamma_4 t^{\frac{1}{3-m-p}} w(x).$$ Furthermore, there exists a sequence $t_n\to\infty$ such that $$\lim_{t_n\to\infty} t_n^{\frac{1}{m+p-3}} u(x,t_n)=z(x)$$ where $z(x)\in C^0(\bar\Omega)$ is a solution of \nref{eqn2} vanishing on $\partial\Omega$, perhaps of changing sign. \end{theorem} If the initial datum $u_0$ is non negative (and not identically zero) we can be more precise. \begin{theorem} Under the above assumptions, there exist constants $t_1,t_2<1$ depending only on the data, such that for $t>\max\{t_1,t_2\}$ we have $$\label{eqn6} (t-t_1)^{{{1}\over{3-m-p}}} w(x)\leq u(x,t)\leq (t-t_2)^{{{1}\over{3-m-p}}} w(x)$$ and $$\label{eqn7} |Du(x,t|\leq \gamma_5\dist(x,\partial\Omega)^{{{1-m}\over{m+p-2}}} t^{{{1}\over{3-m-p}}}.$$ \end{theorem} In order to state the main result for the special case $m+p=3$ we denote by $B_p$ the best constant of the embedding of $W_0^{1,p}(\Omega)$ in $L^p(\Omega)$. \begin{theorem} Suppose that $m+p=3$. There exists a unique solution of the evolution equation \nref{eqn1} satisfying \nref{eqn1.1} and \nref{eqn1.2}. Moreover, for all $t >1$ we have the bound $$\label{eqn8} \vert u(x,t)\vert\leq \gamma_6 w(x) e^{\mialfa t},$$ where $w(x)$ is a solution of the equation $$\label{eqn9} \div(\vert w\vert^{m-1}\vert Dw\vert^{p-2} Dw )+\mialfa w=0$$ in $\Omega$ such that $w(x)=0$ in $\partial\Omega$, $w\in C^0(\bar\Omega)$ and $w^{{{p-2}\over{p-1}}}Dw\in L^p(\Omega)$. \end{theorem} In this case too, we obtain a better result assuming the non negativity of the initial datum. \begin{theorem} Assume $u_0\geq 0$ and not identically zero. Then, for $t>1$ we have $$\label{eqn10} \gamma_7e^{{\mialfa}t} w(x)\leq u(x,t)\leq \gamma_{8} e^{\mialfa t} w(x)$$ and $$\label{eqn11} \gamma_{9}e^{\mialfa t}\dist(x,\partial\Omega)^{p-1}\leq u(x,t)\leq \gamma_{10}e^{\mialfa t}\dist(x,\partial\Omega)^{p-1}.$$ Moreover, we also have $$\label{eqn12} |Du(x,t)| \leq\gamma_{11}\dist(x,\partial\Omega)^{2-p}e^{\mialfa t}$$ \end{theorem} \begin{remark} The above results hold for more general operators as defined, for example, in \ncite{A-DB} and \ncite{DB-H} and satisfying more general boundary conditions (\ncite{S-V}). For the sake of brevity we choose to analyze only simple operators. \end{remark} The essential tools used below will be some quantitative $L^{\infty}$-estimates, H\"older regularity results for bounded solutions (proved in \ncite{I}, \ncite{P-V} and \ncite{V}), Harnack inequalities (stated in \ncite{V2}) and the introduction of suitable comparison functions.\par Following the scheme of ideas in \ncite{DB-H} we prove the quantitative $L^{\infty}$-estimates under more general conditions than needed. More precisely, let $\mu$ be a $\sigma$-finite Borel measure in ${\Bbb R}^N$ and $r>0$. We write $$\vert\Vert \mu\Vert\vert_{r}=\sup_{\rho\geq r}\rho^{{{-{\ell}}\over{m+p-3}}} \int_{B_{\rho}}\vert d\mu\vert\,,$$ where $\vert d\mu\vert$ is the variation of $\mu$ and $$\label{eqn13} {\ell}=N(m+p-3)+p.$$ \begin{theorem} Let $m+p>3$. For every $\sigma$-finite Borel measure $\mu$ in ${\Bbb R}^N$ such that $\vert\Vert \mu\Vert\vert_{r}<+\infty$ for some $r>0$, there exists a weak solution of $$\label{eqn14} u_t=\div(\vert u\vert^{m-1}\vert Du\vert^{p-2} Du )$$ in ${\Bbb R}^N\times (0,T(\mu))$ with initial condition $$u(x,0)=\mu$$ and satisfying $u\in C((0,T(\mu)); L^2_{\text{loc}}({\Bbb R}^N))$ and $u^{{{m-1}\over{p-1}}}Du\in L^p_{\text{loc}}({\Bbb R}^N\times (0,T(\mu))).$ Here we have set $T(\mu)=+\infty$ if $\lim_{r\to\infty} \vert\Vert\mu\Vert\vert_{r}=0$ and, otherwise $$T(\mu)= c_0\lim_{r\to\infty}\vert\Vert\mu\Vert\vert_{r}^{-(m+p-3)}$$ where $c_0=c_0(N,m,p)$.\par Moreover, for all $00$ we have $$\label{eqn15} \vert\Vert u(\cdot , t)\Vert\vert_{r}\leq \gamma_{12} \vert\Vert\mu\Vert\vert_{r}$$ and $$\label{eqn16} \Vert u(\cdot,t)\Vert_{\infty,B_{\rho}}\leq \gamma_{13}t^{{{-N} \over{{\ell}}} \rho^{{{p}\over{m+p-3}}}\vert\Vert\mu\Vert\vert_{r}^{{{p} \over{\ell}}}}.$$ Furthermore, for every bounded open set $\Omega\subset {\Bbb R}^N$ and for every $\epsilon>0$ there exist constants $c_1\equiv c_1(N,m,p,\epsilon,\diam(\Omega))$ and $c_2\equiv c_2(N,m,p,\epsilon)$ such that $$\label{eqn17} \int_0^t\int_{\Omega}\vert Du^{\frac{p-1}{m+p-2}}\vert^q\, dx\,d\tau \leq c_1\vert\Vert\mu\Vert\vert_{r}^{c_4},$$ where $q=p-1+{{1-\epsilon }\over{Nm+1}}$. In particular, if $\epsilon =1$ we obtain $$\label{eqn18} \int_0^t\int_{\Omega}\vert Du\vert^{p-1}\vert u\vert^{m-1}\,dx\,d\tau \leq c_2t^{{{1}\over{{\ell}}}} \rho^{1+{{{\ell}}\over{m+p-3}}}\vert\Vert\mu\Vert \vert_{r}^{1+{{m+p-3}\over{{\ell}}}},$$ where $c_5\equiv c_5(N,m,p,\diam(\Omega ))$. \end{theorem} As proved in \ncite{DB-H} in the case of the $p$-Laplacian, these estimates are optimal. Finally we also remark that the case $m+p<3$ studied in \ncite{DB-K-V} and \ncite{S-V} behaves quite differently from the case considered in this paper since finite extinction time phenomena occur. \section{$L^{\infty}$-Estimates} To prove Theorem~1.6, we follow the ideas in section 3 of \ncite{DB-H} where we refer the reader for more details and remarks. First, note that without loss of generality we can assume $\mu_0\in L^1(\Omega )\cap L^{\infty}(\Omega)$. Actually, if one proves the quantitave estimates \nref{eqn15}--\nref{eqn18} in such a case, then by approximating $\mu_0$ with regular functions, Theorem~1.6 will follow. As in \ncite{DB-H}, we will prove the statement via several lemmas. \begin{lemma} Consider the quantity $$\label{eqn21} K(T)=T^{-{{N(m+p-3)}\over{{\ell}}}}\phi^{m+p-3}(T)+T^{-1}$$ where $$\label{eqn22} \phi (t)=\sup_{\tau\in (0,t)}\tau^{N/\ell} \sup_{\rho\geq r} \rho^{-{{p}\over{m+p-3}}}\Vert u(\cdot, \tau )\Vert_{\infty, B_{\rho}}.$$ Under the assumptions of Theorem 1.6, for each $t>0$ we have $$\label{eqn23} \Vert u(\cdot,t)\Vert_{\infty,B_{\rho}}\leq \gamma_{14} [K(t)]^{{{N+p}\over{\lambda}}} \left( \int_{{{t}\over{4}}}^t \int_{B_{2\rho}} u^p\,dx\,d\tau \right)^{p/\lambda},$$ where $\lambda =p^2+N(m+p-3)$. \end{lemma} \begin{pf} Fix $T>0$ and $\rho >0$ consider sequences $T_n={{T}\over{2}}-{{T}\over{2^{n+1}}}$, $\rho_n=\rho +{{\rho}\over{ 2^{n+1}}}$, and $\bar\rho_n ={{1}\over{2}}(\rho_n+\rho_{n+1})$ for $n=1,2,\ldots$ Set $B_n=B_{\rho_n}$, $\bar B_n=B_{\bar\rho_n}$, $Q_n=B_n\times (T_n,T)$ and $\bar Q_n=\bar B_n\times (T_{n+1},T)$ and, let $(x,t)\to \zeta_n(x,t)$ be a smooth cut off function in $Q_n$ satisfying $\zeta_n(x,t)=1$ for $(x,t)\in\bar Q_n$, $\vert D\zeta_n(x,t)\vert\leq {{2^{n+3}}\over{\rho}}$ and $0\leq {{\partial}\over{\partial t}} \zeta_n(x,t) \leq {{2^{n+2}}\over{T}}.$ Finally, for a positive number $k$ to be determined later we will consider the increasing sequences $k_n=k-{{k}\over{2^{n+1}}}$ and $\bar k_n={{1}\over{2}}(k_n+k_{n+1})$ for $n=0,1,2\ldots$\par Assume that $u$ is non negative. Setting $v=u^{{{m+p-2}\over{p-1}}}$ we see that $v$ satisfies the equation $$\label{eqn24} {{\partial}\over{\partial t}} v^{{{p-1}\over{m+p-2}}}=\left({{p-1}\over{m+p-2}}\right)^{p-1} \div (\vert Dv\vert^{p-2} Dv)$$ If $u$ changes sign we must set $v=|u|^{{m-1}\over{p-1}} u$ and the proof presented below requires only minor modifications. Multiply \nref{eqn24} by $(v-\bar k_n)_+^{q-1}\zeta_n^p$, where $q={{p(p-1)+m-1}\over{m+p-2}}$ and integrate over $Q_n$. A standard calculation gives $$\label{eqn25} \begin{split} \sup_{T_{n}\leq t\leq T}\int_{\bar B_n(t)}&G(v(x,t))\,dx + \iint_{\bar Q_{n}}\vert D(v-\bar k_n)^{{{p+q-2}\over{p}}}\vert^p\,dx\,d\tau\\ &\leq \gamma_{16}{{2^{np}}\over{\rho^p}}\iint_{Q_n} (v-\bar k_n)^{p+q-2}\, dx\,d\tau + \gamma_{18}{{2^n}\over{T}}\iint_{Q_n(t)}G(v(x,t))\,dx\,d\tau \end{split}$$ where $G$ is a function defined by $$\label{eqn26} \begin{cases} G'(s)&=(s-\bar k_n)^{q-1} s^{{{p-1}\over{m+p-2}}-1}\\ G(\bar k_n)&=0 \end{cases}$$ We shall use of the following elementary estimates. Suppose that $m\leq 1$, then we have $$\label{eqn27} \begin{split} (s-\bar k_n &)^{{{p-1}\over{m+p-2}} +q-2} \\ & \leq s^{{{p-1}\over{m+p-2}} -1}(s-\bar k_n)^{q-1} \\ &\leq 2^{ {{p-1}\over{m+p-2}}-1}(s-\bar k_n)^{{{p-1}\over{m+p-2}} +q-2}+2^{{{p-1}\over{m+p-2}} -1}k^{{{p-1}\over{m+p-2}} -1}(s-\bar k_n)^{q-1}\\ &\leq 2^{{{p-1}\over{m+p-2}} -1}(s-\bar k_n)^{{{p-1}\over{m+p-2}} +q-2}+2^{{{p-1}\over{m+p-2}} -1}4^{n({{p-1}\over{m+p-2}} -1)} (s- k_n)^{{{p-1}\over{m+p-2}} +q-2}\\ &\leq \gamma_{19} 4^{n({{p-1}\over{m+p-2}} -1)}(s- k_n)^{{{p-1}\over{m+p-2}} +q-2} \end{split}$$ Hence by \nref{eqn25}--\nref{eqn27} and by the definition of $K(T)$ it follows that $$\label{eqn28} \sup_{T_{n+1}\leq t\leq T}\int_{\bar B_n(t)}\bar w^s_ndx +\iint_{ \bar Q_n}\vert D\bar w_n\vert^p\,dx\,d\tau\leq \gamma_{20}4^{n(p+1)}K(T)\iint_{Q_{n}}w^s_n\,dx\,d\tau,$$ where $$\bar w_n=( v -\bar k_{n})_+^{{{p+q-2}\over{p}}},\quad w_n=( v -k_n)^{{{p+q-2}\over{p}}}_+$$ and $$s=\left[ q +{{1-m}\over{m+p-2}}\right]\left[ {{p}\over{p+q-2}}\right]\,.$$ By the Gagliardo-Nirenberg's inequality (see \ncite{L-S-U}, p. 62) we have $$\label{eqn29} \begin{split} & \iint_{Q_{n+1}} w_{n+1}^{p(1+{{s}\over{N}})}\,dx\,d\tau \leq \iint_{\bar Q_n}\vert \bar w_{ n+1} \zeta_n\vert^{p(1+{{s}\over{N}})}\, dx\,d\tau \\ &\leq\gamma_{21} \left\{ \iint_{\bar Q_n}\vert D\bar w_{ n}\vert^p\,dx\,d\tau + {{4^{np}}\over{\rho^p}}\iint_{\bar Q_n}\bar w^p_{ n}\,dx\,d\tau \right\} \left(\sup_{T_{n+1}\leq t\leq T}\int_{\bar B_n(t)}\bar w_{ n}^s\,dx \right)^ {{{p}\over{N}}} \end{split}$$ From \nref{eqn28}, \nref{eqn29} and the definition of $K(T)$, it follows that $$\label{eqn210} \iint_{Q_{n+1}}w^{p(1+{{s}\over{N}})}_{n+1}\,dx\,d\tau\leq \gamma_{22}\left\{4^{np} K(T)\right\}^{{{N+p}\over{N}}}\left(\iint_{Q_n} w^s_n\,dx\,d\tau \right)^{{{N+p}\over{N}}}$$ This estimate is the starting point of the iteration process described in Lemma 3.1 of \ncite{DB-H}. An application of this result gives $$\label{eqn211} (u-k)_+\equiv 0\text { in } Q_{\infty}$$ Estimate \nref{eqn23} now follows by choosing $$k=\gamma_{23}\left[K(T)\right]^{{{N+p}\over{\lambda}}}\left( \int_{{{T}\over{4}}}^T \int_{B_{2\rho}}u^p\,dx\,d\tau\right)^{{{p}\over{\lambda}}}$$ where $\lambda =p^2+N(m+p-3)$.\par Next, consider the case $m>1$. As we did before we need the following elementary estimates for $G$ $$\label{eqn212} \begin{split} \int_{k_{n+1}}^v (s-k_{n+1})^{{{p-1}\over{m+p-2}} +q-2}ds &\leq \int_{k_{n+1}}^v (s-\bar k_{ n})^{{{p-1}\over{m+p-2}} +q-2}ds \\ &\leq 4^{n(1-{{p-1}\over{m+p-2}} )} \int_{k_{n+1}}^v (s-\bar k_{n})^{q-1}s^{{{p-1}\over{m+p-2}} -1}ds \\ &\leq 4^{n(1-{{p-1}\over{m+p-2}} )} \int_{\bar k_{n}}^v (s-\bar k_{n})^{q-1}s^{{{p-1}\over{m+p-2}} -1}ds \\ &\leq 4^{n(1-{{p-1}\over{m+p-2}} )} \int_{\bar k_{n}}^v (s-\bar k_{n})^{{{p-1}\over{m+p-2}} +q-2}ds \end{split}$$ Hence, as before, by \nref{eqn25}, \nref{eqn26}, \nref{eqn212} and by definition of $K(T)$, we get $$\label{eqn213} \sup_{T_{n+1}\leq t\leq T}\int_{\bar B_{n}(t)}w^s_{ n+1}dx + \iint_{\bar Q_n}\vert Dw_{ n+1}\vert^p\,dx\,d\tau \leq \gamma_{25} 4^{n(p+1)}K(T)\iint_{Q_n}\bar w^s_{n}\,dx\,d\tau .$$ Once we have \nref{eqn213} we deduce \nref{eqn23} as before. \end{pf} For $r>0$ define the function $$\label{eqn214} \psi(t)=\sup_{\tau\in (0,t)}\Vert\vert u(\cdot ,\tau )\vert\Vert_{r}.$$ \begin{lemma} For each $t>0$ $$\label{eqn215} \phi(t)\leq \gamma_{26}\int_0^t \tau^{-{{N}\over{\ell}}(m+p-3)} \phi^{m+p-2} (\tau )\,d\tau +\gamma_{26}\psi(t)^{{{p}\over{k}}},$$ where we have set $\ell=N(m+p-3)+p$. \end{lemma} \begin{pf} Multiply \nref{eqn213} by $\rho^{-{{p}\over{m+p-3}}}\tau^{{{N}\over{\ell}}}$ to obtain \begin{aligned} \tau^{{{N}\over{\ell}}}{{\vert\Vert u(\cdot ,\tau )\Vert\vert_{\infty ,B_{2\rho}}}\over{\rho^{{{p}\over{m+p-3}} }}} &\leq \gamma_{26}\phi (t)^{{{N+p}\over{\lambda}}(m+p-3)} t^{{{p}\over{\lambda}}{{(3-m)N}\over{\ell}}} \left(\int_{{{t}\over{4}}}^{t}\int_{B_{2\rho}} \rho^{-\lambda\over m+p-3} u^p\,dx\,d\tau\right)^{{{p}\over{\lambda}}}\\ & +\gamma_{26}t^{({{N(p-1)}\over{\ell}}-1) {{p}\over{\lambda}}}\left(\int_{{{t}\over{4}}}^t \int_{B_{2\rho}}\rho^{-\lambda\over m+p-3} u^p\,dx\,d\tau\right)^{{{p}\over{\lambda}}}\\ & \equiv H^{(1)}+H^{(2)}. \end{aligned} We proceed to estimate $H^{(1)}$ and $H^{(2)}$ separately. \begin{aligned} H^{(1)}&\leq\gamma_{27}\phi(t)^{1+{{p}\over{\lambda}} (m-3)}\left(\int_{{{t}\over{4}}}^t\tau^{-{{N(m+p-3)}\over{\ell}}} (2\rho)^{-p^2\over m+p-3}\left(\tau^{{{N}\over{\ell}}} \vert\Vert u(\cdot ,\tau )\Vert\vert_{\infty ,B_{2\rho}}\right)^{p} d\tau \right)^{{{p}\over{\lambda}}}\\ &\leq \gamma_{28}\phi(t)^{1-{{p}\over{\lambda}}}\left(\int_0^t \tau^{-{{N(m+p-3)}\over{\ell}}}\phi^{m+p-2}(\tau)\, d\tau\right)^{{{p}\over{\lambda}}}\\ & \leq {{1}\over{4}}\phi(t)+\gamma_{29}\int_{0}^t \tau^{-{{N(m+p-3)}\over{\ell}}}\phi^ {m+p-2}(\tau)\,d\tau . \end{aligned} \begin{aligned} H^{(2)}& \leq\gamma_{30}\left\{{{1}\over{t}}\int_{{{t}\over{4}}}^t \tau^{{{N(p-1)}\over{\ell}}} \left({{\vert\Vert u(\cdot ,\tau )\Vert\vert_{\infty ,B_{2\rho}}}\over{(2\rho)^{{{p}\over{m+p-3}}}}}\right)^{p-1}\!\!\! (2\rho)^{{{-\ell}\over{m+p-3}}} \int_{B_{2\rho}}\!\!u(x,\tau)\,dx\,d\tau \right\}^{{{p}\over{\lambda}}}\\ &\leq\gamma_{31}\phi(t)^{{{p(p-1)}\over{\lambda}}} \left({{1}\over{t}}\int_0^t\vert\Vert u(\cdot ,\tau )\Vert\vert_{r} d\tau\right)^{{{p}\over{\lambda}}}\\ &\leq {{1}\over{4}} \phi(t)+\gamma_{32}\psi(t)^{{{p}\over{\ell}}}. \end{aligned} \end{pf} \begin{lemma} Let $\rho\geq r>0$ and let $x\to \zeta (x)$ be a piecewise smooth cut-off function in $B_{2\rho}$ such that $\zeta= 1$ on $B_{\rho}$ and $\vert D\zeta\vert\leq {{1}\over{\rho}}$. For each $t>0$ we have $$\label{eqn216} \begin{split} \int_0^t\!\!\int_{B_{2\rho}}& \!\!\vert Du\vert^{p-1}\vert u\vert^{m-1} \zeta^{p-1}dx\,d\tau \leq\gamma_{33} \rho^{1+{{\ell}\over{m+p-3}}}\left(\int_0^t\left[\tau^{{{p+1}\over{\ell}}-1} \phi(\tau)^{{{(m+p-3)(p+1)}\over{p}}}\psi(\tau)\right.\right. \\ & \left.\left. + \tau^{{{1}\over{\ell}}-1}\phi(\tau )^{{{m+p-3}\over{p}} } \psi (\tau )\right]\,d\tau \right)^{{{p-1}\over{p}}}\left(\int_0^t\tau^{{{1}\over{\ell}}-1} \phi(\tau)^{{{m+p-3}\over{p}}}\psi(\tau )\, d\tau\right)^{{{1}\over{p}}}. \end{split}$$ \end{lemma} \begin{pf} Let $v=u^{{{m+p-2}\over{p-1}}}$. By H\"older's inequality we have \begin{aligned} \int_0^t\int_{B_{2\rho}}\vert Dv\vert^{p-1}\zeta^{p-1}\,dx\,d\tau & \leq \gamma_{34} \left(\int_0^t\int_{B_{2\rho}}\tau^{{{1}\over{p}}}\vert Dv\vert^{p} v^{({{p-1}\over{p}}{{m+p-3}\over{m+p-2}}) -1}\zeta^p \,dx\,d\tau\right)^{{{p-1}\over{p}}} \\ &\times\left(\int_0^t\int_{B_{2\rho}}\tau^{({{1}\over{p}} -1)} v^{{{p-1}\over{m+p-2}} {{m+2p-3} \over{p}}}\,dx\,d\tau\right)^{{{1}\over{p}}}\\ &\equiv J_1(t)^{{{p-1}\over{p}}} J_2(t)^ {{{1}\over{p}}}. \end{aligned} The lemma will follow by estimating $J_1(t)$ and $J_2(t)$ separately. Multiply \nref{eqn24} by $\tau^{{{1}\over{p}}}v^{(p-1)({{m+p-3}\over{m+p-2}}{{1}\over{p}})} \zeta^{p-1}$ and integrate by parts to get \begin{aligned} J_1 & =\int_0^t\int_{B_{2\rho}}\tau^{{{1}\over{p}}} v^{\left({{{p-1}\over{m+p-2}} {{m+p-3}\over{p}} }-1\right)} \vert Dv\vert^p \zeta^p\,dx\,d\tau\\ &\leq\gamma_{35}\rho^{-p}\int_0^t\int_{B_{2\rho}} \tau^{{{1}\over{p}}} v^{(p-1)({{m+p-3}\over{m+p-2}}{{1}\over{p}} +1)}\,dx\,d\tau +J_2\\ & = L_1+J_2 \end{aligned} We estimate $L_1$ as follows \begin{aligned} L_1\leq \gamma_{36}\rho^{1+{{\ell}\over{m+p-3}}} \int_0^t & \tau^{{{p+1}\over{\ell}}-1} \left(\tau^{{{N}\over{\ell}}}(2\rho)^{-{{p}\over{m+p-3}}} \vert\Vert u(\cdot ,\tau )\Vert\vert_{\infty ,B_{2\rho}} \right)^{{{(m+p-3)(p+1)}\over{p}}}\\ &\times\left(\rho^{-{{\ell}\over{m+p-3}}}\int_{B_{2\rho}}u(x,\tau)\, dx\right) d\tau \\ & \leq \gamma_{37}\rho^{1+{{\ell}\over{m+p-3}}}\int_0^t \tau^{({{p+1}\over{\ell}}-1)} \phi(\tau)^{{{(m+p-3)(p+1)}\over{p}}}\psi(\tau)\,d\tau . \end{aligned} and $J_2$ by \begin{aligned} J_2\leq \gamma_{38}\rho^{1+{{\ell}\over{m+p-3}}}\int_0^t & \tau^{{{1}\over{\ell}}-1} \left( \tau^{{{N}\over{\ell}}}\rho^{-{p\over m+p-3}} \vert\Vert u(\cdot ,\tau )\Vert\vert_{\infty ,B_{2\rho} } \right)^{ {{m+p-3}\over{p}}} \\ &\times\left(\rho^{-{{\ell}\over{m+p-3}}} \int_{B_{2\rho}} u(x,\tau)\,dx\right)\,d\tau\\ & \leq\gamma_{39}\rho^{1+{{\ell}\over{m+p-3}}}\int_0^t \tau^{{{1}\over{\ell}}-1}\phi(\tau)^{{{m+p-3}\over{p}}}\psi(\tau )\, d\tau. \end{aligned} Multiply \nref{eqn14} by $\zeta^p$ and integrate in $[0,t]$ to get $$\int_{B_{\rho}} u(x,t)\, dx\leq \int_{B_{2\rho}}u_0(x)\,dx+ \gamma_{40}\rho^{-1}\int_0^t\int_{ B_{2\rho}}\vert Du\vert^{p-1}\vert u\vert^{m-1}\zeta^{p-1}\,dx\,d\tau.$$ Hence, by the previous lemma we have $$\label{eqn217} \begin{split} \psi(t)\leq\gamma_{41}\vert\Vert u_0\vert\Vert_r& +\gamma_{41}\left(\int_0^t\tau^{ {{p+1}\over{\ell}}-1}\phi(\tau)^{{{(m+p-3)(p+1)}\over{p}}}\psi(\tau) \,d\tau \right. \\ & \left. + \int_0^t\tau^{{{1}\over{\ell}}-1}\phi(\tau)^{{{m+p-3}\over{p}}} \psi(\tau)\,d\tau \right). \end{split}$$ Via an algebraic lemma (see Lemma 3.5 of \ncite{DB-H}), \nref{eqn15} and \nref{eqn16} come from \nref{eqn215} and \nref{eqn217}. Moreover \nref{eqn15}, \nref{eqn16} and \nref{eqn216} imply \nref{eqn18}. It remains only to check \nref{eqn17} to finish the proof of Theorem 1.6. We proceed as follows \begin{aligned} \int_{{{t}\over{2}}} ^t\int_{B_{\rho}}\vert Dv\vert^q &\,dx\,d\tau \leq \int_{{{t}\over{2}}}^t\int_{B_{\rho}} t^{-\beta}v^{-\alpha}\vert Dv\vert^qt^{\beta}v^{\alpha}\,dx\,d\tau\\ & \leq\left(\int_{{{t}\over{2}}}^t\int_{B_{\rho}}t^{\beta {{p}\over{q}}} v^{-\alpha {{p}\over{q}}}\vert Dv\vert^p\,dx\,d\tau\right)^{{{q}\over{p}}}\left( \int_{{{t}\over{2}}}^t\int_{B_{\rho}}t^{-\beta {{p}\over{p-q}}}v^{{{\alpha p} \over{p-q}}}\,dx\,d\tau\right)^{1-{{q}\over{p}}}\\ &\equiv (I^{(1)})^{{{q}\over{p}}} (I^{(2)})^{1-{{q}\over{p}}} \end{aligned} where $q=p-1+{{1-{\varepsilon }}\over{Nm+1}}.$ At this point \nref{eqn18} follows from the previous inequality by choosing $\alpha p=p-q$ and arguing as in \ncite{DB-H} pages 204--205. \end{pf} \begin{remark} If we consider the Cauchy problem \nref{eqn1} instead of \nref{eqn14} we get for each $t>s>0$ $$\label{eqn218} \Vert u(\cdot ,t)\Vert_{\infty ,\Omega }\leq \gamma_{42} (t-s)^{-{{N}\over{\ell}}} \left(\int_{\Omega}u(\cdot ,s)\,dx\right)^{{{p}\over{\ell}}} .$$ A similar result holds in the case $m+p=3$; that is, for each $t>0$ $$\label{eqn219} \Vert u\Vert_{L^{\infty}(\Omega\times (t+1,t+2))}\leq \gamma_{43} \left( \iint_{\Omega\times (t,t+3)}u^{{{p}\over{p-1}}}\,dx\,d\tau \right)^{{{p-1} \over{p}}}.$$ One could prove \nref{eqn215} by repeating an argument analogous to the previous one, but we prefer to show another method. As remarked by Trudinger \ncite{TR}, all the classical estimates for parabolic equations hold in such a particular case. Hence, by considering cylinders $B(R)\times [0, -R^p]$ and repeating the classical argument of the $L^{\infty}$-estimates (see, for instance, \ncite{L-S-U}) one gets \nref{eqn219}. \end{remark}\par Before concluding this section, let us state a straightforward consequence of the previous estimates (see \ncite{DB-H}): \begin{propo} Let $u\geq 0$ be a weak solution of \nref{eqn1}. Then for each $R>0$, and for each ${\epsilon } \in (0, 1]$ we have $$\label{eqn220} \begin{split} \intav_{B_{(1+{\epsilon} )R}}&u(x,t)\,dx\geq \\ & \intav_{B_{R}}u(x,\tau )\,dx \left\{ (1+\epsilon )^{-N} -{{\gamma_{44}}\over{\epsilon }}\left(\left[\intav_{B_r}u(x,\tau )\, dx \right]^{m+p-3}R^{-p} (t-\tau )\right)^{{{1}\over{\ell}}}\right\} \end{split}$$ for all $0<\tau 0, and for each 03 and u_0 not identically equal to 0. Then, for each t>0, $$\label{eqn33} \gamma_{47}t^{{{1}\over{3-m-p}}}\leq \Vert u(x,t) \Vert_{\infty ,\Omega} \leq\gamma_{48} t^{{{1}\over{3-m-p}}},$$ where \gamma_{48}\geq\gamma_{47} >0. \end{propo} \begin{pf} Without any loss of generality, we may assume u_0\in L^{\infty}(\Omega). Multiplying \nref{eqn32} by v and integrating in \Omega, we have $$\label{eqn34} \frac{p-1}{m+2p-3} {{d}\over{dt}} \int_{\Omega}v^{{{m+2p-3}\over{m+p-2}}}+ \left({{p-1}\over{m+p-2}}\right)^{p-1}\int_{\Omega} \vert Dv\vert^p=0.$$ Let B_{m,p} the best Sobolev constant such that for each w\in W^{1,p}_{0}(\Omega) $$\label{eqn35} \left(\int_{\Omega}w^{{{m+2p-3}\over{m+p-2}}}\, dx\right)^{{{m+p-2}\over{m+ 2p-3}}}\leq B_{p,m}\left(\int_{\Omega}\vert Dw\vert^p \,dx\right)^{{{1}\over{p}}}.$$ Set z(t)=\int_{\Omega}v^{{m+2p-3}\over{m+p-2}}\,dx. From \nref{eqn34} and \nref{eqn35} we get \[ \frac{p-1}{m+2p-3} {{d}\over{dt}}z+\left({{p-1}\over{m+p-2}}\right)^{p-1} B_{p,m}^{-p}z^{{{m+p-2}\over{m+2p-3}} p}\leq 0.$ Solve the differential inequality and put $\alpha ={{m+p-2}\over{m+2p-3}} p$ to get $z(t)\leq (\alpha -1)\left( \left( (p-1)^{p-2}\frac{m+2p-3}{(m+2p-2)^{p-1}}\right) B^{-p}_{p,m}t+\left( {{z(0)}\over{\alpha -1}} \right)^{1-\alpha }\right)^{{{1}\over{1-\alpha}}}.$ Hence, we obtain $$\label{eqn36} \left(\int_{\Omega}u^{{{m+2p-3}\over{p-1}}}(\cdot ,t)\, dx\right)^{{{p-1} \over{m+2p-3}}}\leq\gamma_{49}t^{{{1}\over{3-m-p}}}.$$ On applying \nref{eqn218} we get \begin{aligned} \Vert u(\cdot ,t)\Vert_{\infty ,\Omega }&\leq\gamma_{50} 2^{{{N}\over{\ell}}}t^{- {{N}\over{\ell}}}\left(\int_{\Omega}u(\cdot ,{{t}\over{2}})\, dx\right)^{{{p}\over{\ell}}}\\ &\leq\gamma_{51}2^{{{N}\over{\ell}}}t^{-{{N}\over{\ell}}} \vert\Omega\vert^{{{m+p-2}\over{ m+p-3}} {{p}\over{\ell}}}\gamma_{49}^{{{p}\over{\ell}}} t^{{{1}\over{3-m-p}} {{p}\over{\ell}}}\\ &\leq\gamma_{52}t^{{{1}\over{3-m-p}}}. \end{aligned} From Theorem~3.1 and \nref{eqn34} we have $\frac{p-1}{m+2p-3} {{d}\over{dt}}z+({\cal E} (0))^{-1}z^{{{m+p-2}\over{m+2p-3}} p} \geq 0.$ Solve the differential inequality to obtain $\gamma_{53}t^{{{1}\over{3-m-p}}}\leq \left(\int_{\Omega} u^{{{m+2p-3}\over{p-1}}} (\cdot ,t)dx \right)^{{{p-1}\over{m+2p-3}}}.$ The statement is now proved because $\left(\int_{\Omega}u^{{{m+2p-3}\over{p-1}}}(\cdot ,t)dx\right)^{{{p-1}\over{m+2p-3}}} \leq \vert \Omega\vert^{{{p-1}\over{m+2p-3}}}\Vert u(x,t) \Vert_{\infty ,\Omega}.$ \end{pf} The case $m+p=3$ follows along the same lines. \begin{propo} Suppose that $m+p=3$ and assume $u_0$ not identically equal to $0$, then for $t>1$ we have $$\label{eqn37} \gamma_{54}e^{-\alpha_1t}\leq\Vert u(\cdot ,t) \Vert_{\infty ,\Omega}\leq \gamma_{55}e^{-\alpha_2t},$$ where $\alpha_1=({\cal E}(0))^{-1}{(p-1)^{p-1}}$ and $\alpha_2=-\mialfa$. \end{propo} \begin{pf} Reasoning as above, we get $\frac{p-1}{p} {{d}\over{dt}}z+(p-1)^{p-1}B^{-p}_{p}z\leq 0 .$ Hence, $z(t)\leq e^{{-\frac{p}{p-1}\mialfa}t}z(0)$ and $\Vert u(\cdot ,t)\Vert_{\infty ,\Omega }\leq \gamma_{56} z(0)^{{{p-1}\over{p}}} e^{-\mialfa t}.$ The lower bound is obtained analogously. \end{pf} \section{ Behavior near the boundary } It is of interest here to prove estimates from above and below near $\partial\Omega$. The argument we follow is the same of \ncite{DB-K-V}, with the only difference that we need to use the super and subsolutions introduced in \ncite{S-V} instead of the ones of \ncite{DB-K-V}. For this reason, we state only the main results, leaving the easy proofs to the reader. As all the constants are stable when $m+p\to 3$, we consider only the case $m+p>3$. \subsection{Estimates from above near $\partial\Omega$} \begin{theorem} Let $u$ be a bounded solution $|u|\le M$ of $$\label{eqn41} u_t-\div (\vert u\vert^{m-1} \vert Du\vert^{p-2}Du)=0 \text{ in } \Omega\times (s,t)$$ satisfying $u\in C(s,t; L^2(\Omega))$, $u^{{{m-1}\over{p-1}}}Du\in L^p(s,t,\Omega)$ for some $s,t \in {\Bbb R}^+$ and some $M>0$. Then, for all $(x,t)\in\Omega\times (s,t)$ we have $$\label{eqn42} \vert u(x,t)\vert^{{{m+p-2}\over{p-1}}} \leq\gamma_{58}M^{{{m+p-2}\over{p-1}}} \left({{e^{\lambda}}\over{e^{\lambda}-1}}\right) ^{{{3-m-p+mp}\over{pm}}}\dist(x,\partial\Omega),$$ where $\lambda =\min\left\{ 1, {{t-s}\over{M^{3-m-p}}}\right\}$ and $\gamma_{44}$ depends only upon $N,m,p$ and $\Vert\partial\Omega\Vert_{C^{1,\alpha}}$. \end{theorem} \begin{coro} For every $\eta >0$, there exists a constant $\gamma_{59}$ (depending only upon $N,m,p$ and $\Vert {\partial\Omega}\Vert_{C^{1,\alpha}}$) such that for all $t-s\geq \eta M^{3-m-p}$ we have $$\label{eqn43} \vert u(x,t)\vert \leq \gamma_{60} M\left(\dist(x,{\partial\Omega} ) \right)^{{{p-1}\over{m+p-2}}}.$$ \end{coro} \begin{remark} Estimates \nref{eqn43}, \nref{eqn33} and \nref{eqn37} imply that for each $t>1$, if $m+p>3$ $$\label{eqn44} \vert u(x,t)\vert \leq \gamma_{61} \dist(x,{\partial\Omega} )^{{{p-1}\over{m+p-2}}} t^{{{1}\over{3-m-p}}},$$ and if $m+p=3$ $$\label{eqn45} \vert u(x,t)\vert \leq\gamma_{62} \dist(x,{\partial\Omega})^{p-1} e^{-B^{-p}_{p,m}{(p-1)^p\over{p}} t}.$$ \end{remark} \subsection{Estimates from below near $\partial\Omega$} Suppose now that $u$ is a non negative bounded solution $u\le M$ of \nref{eqn41}. For $r>0$ let $\Omega_r=\{ x\in \Omega : \dist(x,\partial\Omega ) \geq r \}$, $\Omega_{r,t}= \Omega_r \times [s,t]$ and $\mu (r)=\inf \{u(x,\tau)\colon{(x,\tau )\in \Omega_{r,\tau }}\}$. For $00$, there exist $r_0$ and $\gamma_{64}$ (depending only upon $N$, $p$, $m$, $\vert\Omega\vert$, $\eta$ and $\Vert{\partial\Omega}\Vert_{ C^{1,\alpha}}$) such that $$\label{eqn48} u(x,t)\geq \gamma_{64}\mu (r_0)\left(\dist(x,\partial\Omega ) \right)^{{{p-1}\over{m+p-2}}},$$ where $x\in \Omega$, $03$ } Denote by $x_0(t)$ a point in $\Omega$ where the maximum of $\vert u\vert$ is attained at time $t$. Let $$\label{eqn51} \tilde u_s(x,t)={{u(x,(t+s)u(x_0(s), s)^{-(m+p-3)})} \over{u(x_0(s),s)}}$$ The function $\tilde u_s$ satisfies the equation $$\label{eqn52} {{\partial}\over{\partial t}} \tilde u_s =\div (\vert\tilde u_s\vert^{m-1}\vert D\tilde u_s \vert^{p-2} D\tilde u_s ) \text{ in } \Omega\times [-1,1]$$ and it vanishes in $\partial\Omega\times [-1,1]$. By \nref{eqn33} we get that for each $s\geq 1$, $\tilde u_s$ is uniformly bounded in $\Omega\times [-1,1]$. Hence, by the regularity results of \ncite{P-V} and \ncite{V}, we have that there is $\alpha >0$ such that for all $s>1$ $$\label{eqn53} \sup_{0\leq t\leq 1} \Vert\tilde u_s (x,t)\Vert_{C^{\alpha}(\bar\Omega)} \leq \gamma_{65}$$ This estimate implies $$\label{eqn54} \sup_{s\geq 1}\Vert s^{{{1}\over{m+p-3}}} u(x,s)\Vert_{C^{\alpha} (\bar\Omega )}\leq \gamma_{66}.$$ On the other hand, since ${\cal E} (t)$ is decreasing, reasoning as in \ncite{B-H} (see also \ncite{S-V}) we have that there is a sequence of times $s_n\to\infty$ such that $u(x,s_n)s_n^{{{1}\over{m+p-3}}}\rightharpoonup w$, where $w$ solves \nref{eqn2}. Therefore, by Minty's lemma, $u(x,s_n)s_n^{{{1}\over{m+p-3}}}\to w$. Moreover by \nref{eqn33}, $w\equiv 0$ if and only if $u_0(x)\equiv 0$.\par If $u_0(x)$ is assumed to be non negative we can be more precise. Let us recall the Harnack inequality stated in \ncite{V2}. \begin{propo} Let $u\geq 0$ be a local weak solution of the equation $u_t=\div (\vert u\vert^{m-1}\vert Du\vert^{p-2} Du )$ in some cylindrical domain $\Omega_T=\Omega\times [0,T]$. Let $(x_0,t_0)\in \Omega_T$, let $B_{\rho}(x_0)$ be the ball of radius $\rho$ centered at $x_0$ and assume $u(x_0,t_0) >0$. Then, there exist two constants $c_i=c_i(N,m,p)$, $i=0,1,$ such that $$\label{eqn55} u(x_0,t_0)\leq c_0\inf_{x\in B_{\rho}(x_0)} u(x,t_0+{{c_1\rho^{p}}\over {u^{m+p-3}(x_0,t_0)}})$$ provided the box $Q_0=B_{2\rho}(x_0)\times \left\{t_0-c_1 {{\rho^{p}} \over{(u(x_0,t_0))^{m+p-3}}}, t_0+c_1{{\rho^{p}}\over{(u(x_0,t_0))^{m+p-3}}}\right\}$ is all contained in $\Omega_T$. \end{propo} Consider now the function $\tilde u_s$ defined in \nref{eqn51}. First, let us estimate the point at which the maximum is attained. By \nref{eqn33} it follows that $u(x_0(s),s)\geq\gamma_{67}t^{-{{1}\over{m+p-3}}}$. On the other hand, by \nref{eqn43} we obtain $u(x_0(s),s)\leq \gamma_{68}\dist(x_0(s), {\partial\Omega} )^{{{p-1}\over{m+p-2}}}t^{-{{1}\over{m+p-3}}}.$ Therefore, $$\label{eqn56} \dist(x_0(s),{\partial\Omega} )\geq \left( {{\gamma_{67}}\over{\gamma_{68}}}\right) ^{{{m+p-2}\over{p-1}}}=\sigma .$$ Let $s\geq 1$ and without loss of generality assume $x_0(s)=0$. Apply \nref{eqn51} at (0,0) and choose $\rho ={{\sigma }\over{2}}$ to get $\inf_{x\in B({{\sigma}\over{2}}) } \tilde u_s (x,\underline t )\geq \bar c_0,$ where $\underline t =c_1 ({{\sigma}\over{2}} )^p$. \par We may now repeat this process starting from each point $(x,t)\in \{ \vert x\vert <{{\sigma}\over{2}} \} \times \{\underline t \}$ and continue in this fashion. Let $r_0$ be the number determined in Corollary 4.2 and let $\tilde\Omega_{r_0,\tau}= \{ x\in\Omega \text{ such that } \dist(x,{\partial\Omega} )\geq r_0 \} \times [\tau ,\tau +1 ].$ The arguments indicated above prove that there are two constants $\tau$ and $\gamma_{69}$ that can be determined apriori only in terms of $N$, $m$, $p$, ${\cal E} (0)$, $\Vert{\partial\Omega}\Vert_{C^{1,\alpha }}$ and $r_0$ such that $\inf_{(x,t)\in \tilde\Omega_{r_0,\tau}}\tilde u_s (x,t) \geq\gamma_{69} >0 .$ To summarize, we have determined a constant $t_2$ such that for each $t\geq t_2$ $$\label{eqn57} \inf_{(x,t)\in \Omega_{r_0,t}}u(x,s)\geq \gamma_{70}t^{-{{1}\over{m+p-3}}},$$ where $\Omega_{r_0,t}=\{x\in\Omega \text{ such that }\dist(x,{\partial\Omega} )\geq r_0 \} \times [t, 2t].$ Therefore, from \nref{eqn48} and \nref{eqn57} we have that for each $t\geq 2t_2$ $$\label{eqn58} u(x,t)\geq\gamma_{71} t^{-{{1}\over{m+p-3}}} \left( \dist(x,{\partial\Omega} ) \right)^{{{p-1}\over{m+p-2}}}.$$ \begin{remark} Note that inequality \nref{eqn58} implies that the support of $u(x,t)$ is $\bar\Omega$ for each $t\geq 2t_2$. \end{remark} In order to get a stronger regularity result, let $(x_0,\bar t_0 )\in \Omega\times {\Bbb R}^+$ and assume $\bar t_0 >2t_2$. Denote by $\sigma$ the distance between $x_0$ and ${\partial\Omega}$ and let $R=\min (\sigma ,1)$. Consider the change of variables $x\to {2(x-x_0)\over R}, \qquad t\to {2^p(t+\bar t_0) u(x_0,\bar t_0)^{-(m+p-3)} \over R^p}, \qquad v\to {u\over u(x_0,\bar t_0)}\cdot$ The function $v$ satisfies the equation $v_t=\div (\vert v\vert^{m-1}\vert Dv\vert^{p-2}Dv) \text{ in } B(1)\times [-1,1].$ Moreover, we have $0<\gamma_{72}\leq v <\gamma_{73}$ in $B(1)\times [-1,1]$, and as above we conclude that $v$ is uniformly $\alpha$-H\"older continuous. Hence, reasoning as in \ncite{S-V}, we get that there is a constant $\beta >0$ such that $$\label{eqn59} v\in C^{1,\beta } \text{ and } \vert Dv(0,0)\vert \leq \gamma_{74}.$$ This inequality implies \nref{eqn7} in a straightforward way. Finally, reasoning as above, we obtain that there is a sequence $t_n\to\infty$ such that $t_n^{{{1}\over{m+p-3}}} u(x,t_n)\longrightarrow w$, where $w$ is a solution \nref{eqn2}. Hence \nref{eqn3} and \nref{eqn4} are direct consequences of \nref{eqn7}, \nref{eqn58} and \nref{eqn43}. Moreover \nref{eqn6} holds because it follows from \nref{eqn3} and \nref{eqn4}. Therefore, we obtain $(t-t_1)^{-{{1}\over{m+p-3}}}w(x)\leq u(x,t)\leq (t-t_2)^{-{{1}\over{m+p-3}}}w(x)$ and $$\label{eqn510} \Vert u(x,t)t^{-{{1}\over{m+p-3}}}-w(x) \Vert_{\infty ,\Omega}\leq\gamma_{75} t^{-{{1}\over{m+p-3}}}\dist(x,{\partial\Omega} )^{{{p-1}\over{m+p-2}}}.$$ Note that \nref{eqn510} implies the uniqueness of a non negative solution of \nref{eqn2}. Indeed, if there are two solutions $w$ and $z$ of \nref{eqn2} then, by \nref{eqn510}, $\Vert z(x)-w(x) \Vert_{\infty ,\Omega }\leq \gamma_{75}t^{-{{1}\over{m+p-3}}} \dist(x,{\partial\Omega} ) \text{ for all } t>2t_2.$ Therefore we must have $z(x)\equiv w(x)$. \section{The case $m+p=3$} This case is analogous to the previous one. The only difference is that we cannot deduce the existence of a solution of \nref{eqn9} because \nref{eqn37} is not as good as \nref{eqn33}. The existence of a minimizer of the functional $\left(\int_{\Omega} \vert Du\vert^p\right)^{{{1}\over{p}}}$ on the manifold $\left(\int_{\Omega} u^p\right)^{{{1}\over{p}}}=\text{ constant}$ is well known. Therefore, we can assume the existence of a positive function $w$ that satisfies \nref{eqn9}.\par Considering now a solution to the evolution problem $u_t=\div (\vert u\vert^{m-1}\vert Du\vert^{p-2} Du ) \text{ in } \Omega\times {\Bbb R}^+$ satisfying the homogeneous boundary condition $u(x,t)=0\text{ for } x\in\partial\Omega$ and the initial condition $u(x,0)=u_0(x)\text{ for } x\in\Omega.$ Reasoning as in the previous section we have that $u$ satisfies \nref{eqn3} and \nref{eqn4}. If we assume that $u_0$ is non negative, arguing as in the previous section we get that for each $1\leq t\leq 2$ $$\label{eqn61} \gamma_{76}\dist(x,{\partial\Omega} )^{p-1}\leq u(x,t)\leq \gamma_{77} \dist(x,{\partial\Omega} )^{p-1} .$$ (Note that in this case the estimate deteriorates as $t\longrightarrow +\infty$). Estimate \nref{eqn10} follows now by applying the maximum principle. \begin{ack} The authors wish to thank G.N.A.F.A. (C.N.R.) for supporting the visit of the first author to Italy. \end{ack} \begin{thebibliography}{ABCDEFGH} \bibitem[A-DB]{A-DB} D. Andreucci and E. Di Benedetto, {\em A new approach to the initial trace in non linear filtration}, Ann. Inst. Henri Poincar\e {\bf 4} (1990), 187--224. \bibitem[A-C]{A-C} D.G. Aronson and L.A. Caffarelli, {\em The initial trace of a solution of the porous medium equation}, Trans. Am. Math. Soc. {\bf 380} (1983), 351--366. \bibitem[AR-CR-PE]{AR-CR-PE}D.G. Aronson, M.G. Crandall and L.A. Peletier, {\em Stability of a degenerate non linear diffusion problem}, Nonlinear Analysis, {\bf 6} (1982), 1001--1022. \bibitem[AR-PE]{AR-PE} D.G. Aronson and L.A. Peletier, {\em Large time behavior of solutions of the porous medium equation in bounded domains}, J. Diff. Equations {\bf 39} (1981), 378--412. \bibitem[B-H]{B-H}J.G. Berryman and C.J. Holland, {\em Stability of the separable solution for fast diffusion equations}, Arch. Rat. Mech. Anal. {\bf 74} (1980), 279--288. \bibitem[BE-NA-PE]{BE-NA-PE} M. Bertsch, T. Nanbu and L.A. Peletier, {\em Decay of solutions of a non linear diffusion equation}, Nonlinear Analysis, {\bf 6} (1982), 539--554. \bibitem[BE-PE]{BE-PE}M. Bertsch and L.A. Peletier, {\em The asymptotic profile of solutions of a degenerate diffusion equation}, Arch. Rat. Mech. Anal. {\bf 91} (1985), 207--229. \bibitem[DB]{DB} E. Di Benedetto, {\em Degenerate Parabolic Equations}, Universitext, Springer Verlag 1993. \bibitem[DB-H]{DB-H}E. Di Benedetto and M.A. Herrero, {\em On the Cauchy problem and initial trace for a degenerate parabolic equation}, Trans. Am. Math. Soc. {\bf 314} (1989), 187--224. \bibitem[DB-K-V]{DB-K-V}E. Di Benedetto, Y. Kwong and V. Vespri, {\em Local space analyticity and asymptotic behavior of solutions of certain singular parabolic equations}, Indiana Univ. Math. J. {\bf 40} (1991), 741--765. \bibitem[E-VA]{E-VA} J.R. Esteban and J.L. V\'azquez, {\em Homogeneous diffusion in $\Bbb R$ with power like non linear diffusivity}, Arch. Mat. Mech. Anal. {\bf 103}(1988), 39--80. \bibitem[F-K]{F-K}A. Friedman and S. Kamin, {\em The asymptotic behavior of a gas in an $n$-dimensional porous medium}, Trans. Am. Math. Soc. {\bf 262}(1980), 551--563. \bibitem[I]{I}A.V. Ivanov, {\em Uniform H\"older estimates for weak solutions of quasilinear degenerate parabolic equations}, Lomi preprints E--10--89 (1989), 1--22. \bibitem[I-M]{I-M} A.V. Ivanov. and P.Z. Mkrtchan, {\em The weight estimate of gradient of non negative weak solutions of quasilinear parabolic equations admitting double degeneration}, Zan. Nauch. Semin. Lomi {\bf 181 } (1990), 3--23. \bibitem[KA]{KA} A.S. Kalashnikov, {\em Some problems of the qualitative theory of the non linear degenerate second order parabolic equations}, Russian Math. Surveys {\bf 42} (1987), 169--222. \bibitem[KA-VA]{KA-VA} S. Kamin and J.L. V\'azquez, {\em Fundamental solutions and asymptotic behavior for the $p$-Laplacian equation}, Rev. Mat. Iberoamericana {\bf 4}(2) (1988), 339--354. \bibitem[L-S-U]{L-S-U} O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural'tseva, {\em Linear and quasilinear equations of parabolic type}, Trans. Math. Monographs {\bf 23}, Amer. Math. Soc. 1968. \bibitem[L]{L} J.L. Lions, Quelques m\'ethodes de resolution des probl\emes aux {\em limites nonlin\'eares}, Dunod, Paris 1969. \bibitem[P-V]{P-V} M.M. Porzio and V. Vespri, {\em H\"older estimates for local solutions of some doubly non linear degenerate parabolic equations}, J. Diff. Equations {\bf 103} (1993), 146--178. \bibitem[S-V]{S-V} G. Savar\`e and V. Vespri, {\em The asymptotic profile of solutions of a class of doubly non linear equations}, Nonlinear Analysis, to appear. \bibitem[TR]{TR} N.S. Trudinger, {\em On Harnack type inequalities and their applications to quasi linear parabolic equations}, Comm. Pure. Appl. Math. {\bf 21} (1968), 205--226. \bibitem[VA]{VA} J.L. V\'azquez, {\em Two non linear diffusion equations with finite speed of propagation}, in Problems involving change of type, K. Kirchg\"assner, ed. Lecture Notes in Physics {\bf 359}, 197--206. \bibitem[VA2]{VA2} J.L. V\'azquez, {\em Asymptotic behavior and propagation properties of one dimensional flow of gas in a porous medium}, Trans. Am. Math. Soc. {\bf 277} (1983), 507--527. \bibitem[V]{V}V.Vespri, {\em On the local behavior of a certain class of doubly non linear parabolic equations}, Manuscripta Math {\bf 75} (1992), 65--80. \bibitem[V2]{V2} V. Vespri, {\em Harnack type inequalities for solutions of certain doubly non linear parabolic equations}, J. Math. Anal. Appl., to appear. \end{thebibliography} \end{document}