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\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
\begin{equation}
\label{eqn1}
u_t=\div\left(|u|^{m-1}|\nabla u|^{p-2}\nabla u\right)\mbox{ in  }\Omega\times{\Bbb R}^+,
\end{equation}
subject to the boundary condition
\begin{equation}
\label{eqn1.1}
 u(x,t)=0\mbox{ in } \partial\Omega\times{\Bbb R}^+,
\end{equation}
and satisfying the initial condition
\begin{equation}
\label{eqn1.2} u(x,0)=u_0(x)\mbox{ in } \Omega,
\end{equation}
 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
\begin{equation}
\label{eqn2}
\div(w^{m-1}|Dw|^{p-2}Dw)=\frac{1}{3-m-p}w
\end{equation}
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,
\begin{equation}
\label{eqn3}
|Dw|\le\gamma_1(\dist(x,\partial\Omega))^{\frac{1-m}{m+p-2}}
\end{equation}
and 
\begin{equation}
\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{equation}
\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
\begin{equation}
|u(x,t)|\le \gamma_4 t^{\frac{1}{3-m-p}} w(x).
\end{equation}
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             
\begin{equation}
\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)
\end{equation}
and
\begin{equation}
\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{equation}
\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
\begin{equation}
\label{eqn8}
\vert u(x,t)\vert\leq \gamma_6 w(x) e^{\mialfa t}, 
\end{equation}
where $w(x)$ is a  solution of the equation
\begin{equation}
\label{eqn9}
\div(\vert w\vert^{m-1}\vert Dw\vert^{p-2} Dw )+\mialfa w=0 
\end{equation}  
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 
\begin{equation}
\label{eqn10}
\gamma_7e^{{\mialfa}t} w(x)\leq u(x,t)\leq \gamma_{8}
e^{\mialfa t} w(x) 
\end{equation}
and
\begin{equation}
\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}.
\end{equation}     
Moreover,  we also have
\begin{equation}
\label{eqn12}
|Du(x,t)| 
\leq\gamma_{11}\dist(x,\partial\Omega)^{2-p}e^{\mialfa t} 
\end{equation}
\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 
\begin{equation}
\label{eqn13}
{\ell}=N(m+p-3)+p.
\end{equation} 

\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
 \begin{equation}
\label{eqn14}
 u_t=\div(\vert u\vert^{m-1}\vert Du\vert^{p-2} Du )
 \end{equation}
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  $0<t<T(\mu)$ and $ \rho
\geq r>0$ we have
\begin{equation}
\label{eqn15}
\vert\Vert u(\cdot , t)\Vert\vert_{r}\leq \gamma_{12}  
\vert\Vert\mu\Vert\vert_{r} 
\end{equation}
and
\begin{equation}
\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}}}}.
\end{equation}
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
 \begin{equation}
\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},
 \end{equation}
 where $q=p-1+{{1-\epsilon }\over{Nm+1}}$.
 In particular, if $\epsilon =1$ we obtain
 \begin{equation}
\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}}}},
 \end{equation}
 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
\begin{equation}
\label{eqn21}
K(T)=T^{-{{N(m+p-3)}\over{{\ell}}}}\phi^{m+p-3}(T)+T^{-1} 
\end{equation}
where   
\begin{equation}
\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}}.
\end{equation}
Under the assumptions of Theorem 1.6, for each $t>0$ we have
\begin{equation}
\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},
\end{equation}
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
\begin{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)
\end{equation}
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
\begin{equation}
\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}
\end{equation}
where $G$ is a function defined by
\begin{equation}
\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}
\end{equation}       

We shall use of the following elementary estimates.
Suppose that  $m\leq 1$, then we have
\begin{equation}
\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} 
\end{equation}
Hence by \nref{eqn25}--\nref{eqn27} and by the definition of $K(T)$ it
follows that
\begin{equation}
\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, 
\end{equation}
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
\begin{equation}
\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}
\end{equation}
 From \nref{eqn28}, \nref{eqn29} and the definition of $K(T)$, it follows
that
\begin{equation}
\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}}}     
\end{equation}
 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
\begin{equation}
\label{eqn211}
(u-k)_+\equiv 0\text {   in   } Q_{\infty}
 \end{equation}
 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 $G$ 
\begin{equation}
\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}
\end{equation}
 Hence, as before, by \nref{eqn25}, \nref{eqn26}, \nref{eqn212} and by  
definition of $K(T)$, we get 
\begin{equation}
\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 .
\end{equation}
 Once we have \nref{eqn213} we deduce \nref{eqn23} as before.
 \end{pf} 

For $r>0$ define the function
\begin{equation}
\label{eqn214}
 \psi(t)=\sup_{\tau\in (0,t)}\Vert\vert u(\cdot ,\tau )\vert\Vert_{r}.
\end{equation}
\begin{lemma}
 For each $t>0$ 
\begin{equation}
\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}}},
\end{equation}
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 
 \begin{equation}
\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{equation}
\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
\begin{equation}
\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}
\end{equation}
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
$t>s>0$
\begin{equation}
\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}}} .
\end{equation}
A similar result holds in the case $m+p=3$; that is, for each $t>0$
\begin{equation}
\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}}}. 
\end{equation}
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
\begin{equation}
\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} 
\end{equation}
for all 
\[
0<\tau <t\leq \gamma_{45}R^p\left(\intav_{B_{R}}u(x,\tau )\,dx\right)^
{m+p-3}.
\]
\end{propo}

This proposition is usually  referred as  a lemma on ``how fast the material
can escape a given ball'' (\ncite{A-C}) and it is useful in
 studying the initial  trace. More precisely one can prove (see \ncite{DB-H}):

\begin{theorem}
Let $u $ be a non--negative solution of \nref{eqn1} in ${\Bbb R}^N\times [0,T]$ 
for some
$0< T\leq\infty$. Then, there exists a unique $\sigma$-finite Borel measure
$\mu$ on ${\Bbb R}^N$ such that
\[
\lim_{t\to 0^+}\int_{{\Bbb R}^N}u(x,t)\phi (x)dx=\int_{{\Bbb R}}\phi d\mu
\]
for all $\phi$ continuous and compactly supported in ${\Bbb R}^N$. Moreover
for each $R>0$, and  for each $0<t\leq T$
\[
R^{-N}\int_{B_{R}}d\mu\leq\gamma_{46}\left\{\left({{R^p}\over{t}}\right)^
{{{1}\over{m+p-3}}} +\left({{t}\over{R^p}}\right)^{{{N}\over{p}}}
\left[u(0,t)\right]^{{{k}\over{p}}}\right\}.
\]
\end{theorem}

\section{$L^{\infty}$-Estimates in the Large}
In the sequel, we need sharp estimates for the function
${\cal E} (t)$ defined
as the Sobolev ratio
\begin{equation}
\label{eqn31}
{\cal E} (t)={{\int_{\Omega}\vert Dv\vert^p(x,t)\,dx}\over{
\left(\int_{\Omega}v(x,t)^{{{m+2p-3}\over{m+p-2}}}dx\right)^{{{
m+p-2}\over{m+2p-3}} p}}}
\end{equation}
where $v=u^{{m+p-2}\over{p-1}}$ is a solution of  the equation
\begin{equation}
\label{eqn32}
{{\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 )  
\end{equation}
in $\Omega\times (0,\infty)$ subject to the boundary condition
\[
v(x,t)=0\text{    in   }\partial\Omega\times{\Bbb R}^+
\]
and to the initial condition
\[
v(0,x)=u_0(x)^{{{m+p-2}\over{p-1}}}.
\]
In \ncite{B-H} (see also \ncite{S-V}) the following basic fact
is proved. 

\begin{theorem}
The function ${\cal E}(t) $ is non increasing in time.
\end{theorem}

We deduce from it the following $L^{\infty}$-bound:
\begin{propo}
Assume $m+p>3$ and $u_0$ not identically equal to  $0$. Then,
for each $t>0$,
\begin{equation}
\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}}},
\end{equation}
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 
\begin{equation}
\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.
\end{equation}
Let $B_{m,p}$ the best Sobolev constant  such that
for each $w\in W^{1,p}_{0}(\Omega)$
\begin{equation}
\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}}}.
\end{equation}
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
\begin{equation}
\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}}}. 
\end{equation}
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
\begin{equation}
\label{eqn37}
\gamma_{54}e^{-\alpha_1t}\leq\Vert u(\cdot ,t)
\Vert_{\infty ,\Omega}\leq
\gamma_{55}e^{-\alpha_2t},
\end{equation}
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
\begin{equation}
\label{eqn41}
u_t-\div (\vert u\vert^{m-1} \vert Du\vert^{p-2}Du)=0
\text{  in  }
\Omega\times (s,t) 
\end{equation}
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
\begin{equation}
\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),
\end{equation}
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  
\begin{equation}
\label{eqn43}
\vert u(x,t)\vert
\leq \gamma_{60} M\left(\dist(x,{\partial\Omega} )
\right)^{{{p-1}\over{m+p-2}}}.
\end{equation}
\end{coro}
\begin{remark}
Estimates \nref{eqn43}, \nref{eqn33} and 
\nref{eqn37} imply that for each $t>1$, if $m+p>3$
\begin{equation}
\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}}}, 
\end{equation}
 and if $m+p=3$
\begin{equation}
\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{equation}
\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 $0<s<t$ let
\begin{equation}
\label{eqn46}
r(M,s,t)=r_0\min\left\{1;\left({{t-s}\over{M^{3-m-p}}}
\right)^{{{1}\over{p}}}\right\},
\end{equation}
where $r_0\leq 1$ is a positive constant depending only
upon $N,m,p$ and
$\Vert {\partial\Omega}\Vert_{C^{1,\alpha}}$.
\begin{theorem}
For each $0<s<t$ and for each  $ x\in \Omega$ such that
$\dist(x,\partial\Omega ) \leq r(M,s,t)$ we have
\begin{equation}
\label{eqn47}
u^{{{m+p-2}\over{p-1}}}\geq\gamma_{63}
(\mu (r(M,s,t)))^{{{m+p-2}\over{p-1}}}
\left({{M^{3-m-p}}\over{t-s}}\right)^{{{1}\over{p}}}
\dist(x,\partial\Omega ),
\end{equation}
where $\gamma_{63}$ depends only upon $N,m,p$ and
$\Vert{\partial\Omega}\Vert_{
C^{1,\alpha}}$. 
\end{theorem}

\begin{coro}
For every $\eta >0$, 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  
\begin{equation}
\label{eqn48}
u(x,t)\geq \gamma_{64}\mu (r_0)\left(\dist(x,\partial\Omega ) 
\right)^{{{p-1}\over{m+p-2}}},
\end{equation}
where $x\in \Omega$, $0<s<t$ and $(t-s)\leq\eta M^{3-p-m}$.
\end{coro}

\section{The case $m+p>3$ }
Denote by $x_0(t)$ a point in $\Omega$ where the 
maximum of $\vert u\vert$
is attained at time $t$. Let
\begin{equation}
\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)}}
\end{equation}
The function $\tilde u_s$ satisfies the equation
\begin{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]
\end{equation}
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 $s>1$
\begin{equation}
\label{eqn53}
\sup_{0\leq t\leq 1}
\Vert\tilde u_s (x,t)\Vert_{C^{\alpha}(\bar\Omega)}
\leq \gamma_{65} 
\end{equation}
This estimate implies
\begin{equation}
\label{eqn54}
\sup_{s\geq 1}\Vert s^{{{1}\over{m+p-3}}} u(x,s)\Vert_{C^{\alpha}
(\bar\Omega )}\leq \gamma_{66}.
\end{equation}
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
\begin{equation}
\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)}}) 
\end{equation}
 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,
\begin{equation}
\label{eqn56}
\dist(x_0(s),{\partial\Omega} )\geq \left(
{{\gamma_{67}}\over{\gamma_{68}}}\right)
 ^{{{m+p-2}\over{p-1}}}=\sigma .
 \end{equation}
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 $t\geq  t_2$
\begin{equation}
\label{eqn57}
\inf_{(x,t)\in \Omega_{r_0,t}}u(x,s)\geq \gamma_{70}t^{-{{1}\over{m+p-3}}}, 
\end{equation}
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 $t\geq 2t_2$
\begin{equation}
\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}}}.
\end{equation}
\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
\begin{equation}
\label{eqn59}
v\in C^{1,\beta } \text{ and }
\vert Dv(0,0)\vert \leq \gamma_{74}.
\end{equation}
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
\begin{equation}
\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}}}.
\end{equation}
 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 $1\leq t\leq 2$
 \begin{equation}
 \label{eqn61}
 \gamma_{76}\dist(x,{\partial\Omega} )^{p-1}\leq 
 u(x,t)\leq \gamma_{77}
 \dist(x,{\partial\Omega} )^{p-1} .
\end{equation}
 (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}

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\end{document}