\documentclass[twoside]{article}
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\markboth{Description of regional blow-up}
{ Carmen Cort\'azar, Manuel del Pino, \& Manuel Elgueta}
\begin{document}
\setcounter{page}{85}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
2001-Luminy conference on Quasilinear Elliptic and Parabolic Equations
and Systems,\newline
Electronic Journal of Differential Equations,
Conference 08, 2002, pp 85--101. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
Description of regional blow-up in a porous-medium equation
%
\thanks{ {\em Mathematics Subject Classifications:} 35B40, 35B45, 35J40.
\hfil\break\indent
{\em Key words:} Multiple-bump, pattern formation, mathematical biology,
\hfil\break\indent
singular perturbation.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Published October 21, 2002.} }
\date{}
\author{Carmen Cort\'azar, Manuel del Pino, \& Manuel Elgueta}
\maketitle
\begin{abstract}
We describe the (finite-time) blow-up phenomenon for a non-negative
solution of a porous medium equation of the form
$$
u_t = \Delta u^m + u^m
$$
in the entire space. Here $m>1$ and the initial condition is
assumed compactly supported. Blow-up takes place exactly inside a
finite number of balls with same radii and exhibiting the same
self-similar profile.
\end{abstract}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\section{ Introduction }
This paper deals with the description of the blow-up phenomenon in
the porous-medium equation in $\mathbb{R}^N$, $N\ge 1$,
\begin{equation} \label{0}
\begin{gathered}
u_t = \Delta u^m + u^m \\
u(x,0) = u_0(x) \end{gathered}
\end{equation}
where $m>1$ and $u_0(x)$ is a compactly supported, not identically
zero nonnegative function whose regularity will be specified
later. This gives rise to the interesting phenomenon of {\em
regional blow-up}, meaning this blow-up taking place only in a
compact set with nonempty interior.
The purpose of this note is to describe the blow-up in
\eqref{0} in the following sense: we show that for any
initial condition the solution $u$ develops (exactly) a finite
number of similar {\em spherical hot spots}: more precisely, there
is a finite number of disjoint balls with common radii $R^*$
outside which the solution remains uniformly bounded, while inside
each of them it develops a common self-similar radially symmetric
profile $(\bar{T}-t)^{-1/(m-1)}w_*(r)$, where $r$ is the
distance to the center of these balls and $w_*$ is a strictly
positive function.
The presence of regional blow-up in this equation was first
observed and studied in the case $N=1$ in \cite{G1}.
The elliptic problem found when searching by separation of variables
a solution of the form
$$ u(x,t) = (\bar{T}-t)^{-1/(m-1)} \theta (x)
$$
has been studied for radial symmetry in \cite{CEF,CEF1,CEF2,G}.
The result contained in this paper is a sequel of the work
\cite{CDE} where the following partial result was established: Let
$\bar{T}>0$ be the time at which blow-up occurs. Let $t_n$ be any
sequence $t_n\uparrow \bar{T}$. Then there is a subsequence of
$t_n$ which we still denote $t_n$, and a nontrivial compactly
supported solution $w(x)$ of the elliptic equation
\begin{equation} \label{est}
\Delta w^ m + w^m -{1\over m-1} w = 0 ,
\end{equation}
such that
$(\bar{T}-t_n)^{1/(m-1)} u(x, t_n )\to w(x)$ uniformly.
On the other hand, it was established in \cite{CEF1} that the
support of any finite energy solution $w$ of (\ref{est}) consists
of a finite number of disjoint balls of the same radii and that
the solution is radially symmetric inside each of them. This
radially symmetric solution $w_*(|x|)$ turns out to be unique, as
established in \cite{CEF2}. Thus $w$ can be written as
\begin{equation}
w(x) = \sum_{j=1}^k w_*(|x-x_j|) ,
\label{dd}
\end{equation}
where, if $R^*$ is the radius of the support of $w_*$,
$|x_i-x_j|\ge 2R^*$ for $i\ne j$.
Let $BU(u_0)$ be the set of {\em blow-up points} of $u$, namely
the set of points $x$ for which there are sequences $x_n \to x$
and $t_n \to \bar{T}$ such that $u(x_n , t_n) \to +\infty$. It was
also shown in \cite{CDE} that this set is compact and it is
precisely constituted by the union of the supports of all possible
limiting $w$'s. The important point unsolved in \cite{CDE} was
whether there is an {\em actual unique blow-up profile}, rather
than oscillation between different limiting configurations. The
question turns out to be rather subtle, and we answer it
affirmatively in the following result.
\begin{theorem} \label{thm1.1}
Let $u(x,t)$ be the solution of \eqref{0}, where $u_0(x)$
is compactly supported, continuous and such that
$u_0^m\in H^1(\mathbb{R}^N)$. Let $\bar{T}>0$ be the blow-up
time of this solution.
Then there are points $x_1,\ldots x_k\in \mathbb{R}^N$
such that
$$
\lim_{t\to \bar{T}} (\bar{T}-t)^{1/(m-1)} u(x,t) = \sum_{j=1}^k w_*(|x-x_j|)
$$
uniformly. Here $w_* (|x|)$ is the unique compactly
supported, radially symmetric solution of \eqref{est}.
If $R^*$ is the radius of its support then we also
have $|x_i -x_j|\ge 2R^*$ for $i\ne j$. Moreover $u(x,t)$ remains
uniformly bounded up to its blow-up time on compact subsets
of $\mathbb{R}^N \setminus \bigcup_{j=1}^k \bar B(x_j , R^*)$. In other words,
$$
BU(u_0) = \bigcup_{j=1}^k \bar B(x_j , R^*).
$$
\end{theorem}
Next we describe the proof of the above results. Let us introduce
the change of variables
\begin{equation}
v(x, t) =(\bar{T}-\tau)^{1/(m-1)} u(x,\tau )
\big|_{\tau =\bar{T}(1-e^{-t})}. \label{1}
\end{equation}
It is readily checked that $v$ is
globally defined in time and satisfies the equation
\begin{gather}
v_t = \Delta v^m + v^m -{1\over m-1} v \label{11}\\
v(x, 0) = \bar{T}^{1/(m-1)} u_0(x) . \label{2}
\end{gather}
From \cite[Proposition 4.1]{CDE}, $v$ is
a bounded function and given a sequence $t_n \to +\infty$ there is
a subsequence, which we denote in the same way, and a nontrivial,
compactly supported solution of \eqref{est} so that
$$
v(x, t_n )^m \to w(x)^m \quad \hbox{as } n\to \infty ,
$$
both in uniform and $H^1$-senses. Thus our task in establishing
Theorem \ref{thm1.1} is precisely to prove that the limit $w(x)$ is
actually the same along {\em every} sequence $t_n\to +\infty$.
Here $w(x)$ has the form \eqref{dd}, and for the sake of simplicity
we restrict ourselves in what follows of this paper only to the
case $k=1$. The proof for $k>1$ is similar if
$|x_i -x_j|> 2R^*$ for all $i\ne j$, only to the expense of some extra notation.
Instead, if
$|x_i -x_j|= 2R^*$ for some $i\ne j$, the situation is substantially
more involved. The complete proof is contained in the forthcoming
work \cite{CDE2}. It should be mentioned that conditions for
one-ball blow-up (i.e. $k=1$) are provided in \cite{CDE}. For
instance, if the support of the initial condition is contained in
a ball of radius less than $R^*$ this is the case. The one-ball
blow up can be shown to be stable, with basically the same proof
below. This means that if an initial condition $u_0$ leads to
blow-up with $k=1$ in Theorem \ref{thm1.1}, then this is the case for all
sufficiently close initial data, where the limiting ball is also
close to that associated to $u_0$. Instead, the two-ball blow-up
is not stable as the following example shows. Let us fix points
$x_1$ and $x_2$ with $|x_1 -x_2| > 2R^*$. Then the function
$$
u(x,t) = (\bar{T}_1- t)^{-1/(m-1)}w_* (|x-x_1|) +
(\bar{T}_2- t)^{-1/(m-1)}w_* (|x-x_2|)
$$
solves equation \eqref{0} for $0< t < \min\{\bar{T}_1 ,\bar{T}_2\}$.
If $\bar{T}_1 =\bar{T}_2$, then two-ball blow-up
takes place, which however disappears as soon as $\bar{T}_1$ and
$\bar{T}_2$ differ, no matter how close they are. This example
suggests that one-ball blow-up may actually hold for ``generic''
initial data.
The main feature of equation \eqref{11} is the presence of a Lyapunov
functional for it, namely
\begin{equation}
J(z) = \frac12 \int (|\nabla z^m|^2 - z^{2m})dx +{m\over m^2-1}
\int z^{m +1} dx . \label{J}
\end{equation}
In fact we have that the mapping $t \mapsto J(v(\cdot ,t))$ is
decreasing on $t>0$ and
$$
\lim_{t\to +\infty} J(v(\cdot ,t)) = J(w) .
$$
Here and in what follows the integral symbol without limits
specified means integration on the whole $\mathbb{R}^N$. The presence
of this functional implies that limit points of the trajectory
must be steady states. The problem of uniqueness of asymptotic
limits in nonlinear heat equations under the presence of a
Lyapunov functional has been analyzed in a number of works. A
general result due to L. Simon \cite{simon} shows the uniqueness
of the limit for uniformly parabolic equations in a uniform real
analytic setting on a compact manifold. Analiticity cannot be
lifted in general in this result, at least in the non-autonomous
setting, as shown in \cite{PR}. Needless to say, the compactly
supported setting we deal with makes our situation highly non
uniformly analytic.
Related uniqueness results in parabolic problems, nondegenerate
and degenerate, are contained in the works
\cite{busca,CDE1,FP,FS,FS1,GP, HP,HR,matano}. In \cite{CDE1}, a
re-normalization method based on L. Simon's ideas, used in
classifying singularities in an elliptic problem in \cite{KMPS}
was adapted to a semilinear heat equation. The general framework
of this method is what we will use here. Alternative methods for
degenerate equations of porous-medium type,
in one and higher dimensions, have been devised
in \cite{FS}, \cite{FS1}. Those techniques do not apply to the
nonlinearity of equation \eqref{2}, in particular those in
\cite{FS1}, based on analyticity, because of the presence of
compactly
supported steady states. This is explicitly commented in \cite{FS1} and
posed as an open question.
\section{Proof of the main results}
For functions $v_1$ and $v_2$ defined on $\mathbb{R}^N$ we consider
the ``distance'' between them defined as
$$
d(u_1, u_2) = \Big( \int (u_1(x)^m -u_2(x)^m)(u_1(x) -u_2(x))dx
\Big)^{1/2}.
$$
We observe that there is a constant $D>0$ for which
$$
d(u_1, u_3) \le D (\, d(u_1, u_2) \,+\, d(u_2, u_3)\,) .
$$
The theorem stated in the previous section (in the case $k=1$) will be a direct
consequence of the following result,
\begin{prop} \label{prop2.1}
There exist positive numbers $T$ and $C$ such that if $v$ is a
solution of equation \eqref{2} defined on $0\le t< +\infty$,
such that for some sequence $t_n \to +\infty$, setting
$$
v_n (x, t) \equiv v( x, t_n +t),\quad
w_n(x) = w_*(|x-x_{n}|),
$$
one has
\begin{equation}
\eta_n \equiv \sup_{t\in [0 , T]} d( v_n(t) , w_n ) \to 0 \
\hbox{as $n \to \infty$.} \label{3}
\end{equation}
Then there exists a point $\bar x_{n}$ with $|\bar x_{n} -x_{n}| \le C\eta_n
$ and
$$\sup_{t\in [0 , T]} d( v_n(t) , \bar w_n ) \le {\eta_n\over 2}
$$
for all $n$ sufficiently large, and where
$ \bar w_n(x) =w_*(|x-\bar x_{n}|)$.
\end{prop}
As a consequence, we obtain the validity of the following fact:
There exist positive numbers $T $, $\delta$, $C$ and $t^*$with the
following property:
Let $v(x,t)$ be a solution of \eqref{2}, defined in $0t^*$ is such that
$$
\eta = \sup_{t_0\le t \le t_0 +T} d(v(t) , w) < \delta .
$$
Then there exists a points $\bar x_1$ with $|\bar x_1 -x_1| \le
C\eta$ such that $$ \sup_{t_0 +T \leq t \leq t_0 +2T} d(v(t), \bar
w) \le {\eta\over 2}, $$
where $\bar w(x) =w_*(|x-\bar x_1|)$.
Let us see how Theorem \ref{thm1.1} (for $k=1$) follows from this
assertion. Let $\varepsilon$ be given and let us write $w$ as $
w(x) = w_*(|x-x_1|)$.
Let $\delta_0 < \delta$, with
$\delta$ the number predicted by Proposition \ref{prop2.1}. Assume that for
some $t_0 > t^*$ we have $ \eta_1 \equiv \sup_{t_0\le t \le t_0
+T} d(v(t) , w) \leq \delta_0 $, where $T$ and $t^*$ are the
numbers given by Proposition \ref{prop2.1}. We find then that there is a
point $x_2$ with $|x_1 - x_{2}| \le C\eta_1$ such that $\eta_2
\equiv \sup_{t_0 +T \le t \le t_0 +2T} d(v(t) , w_2) \le {\eta_1
\over 2}.$ where $ w_2(x) =
w_*(|x-x_{2}|)$. Since
$\eta_2 \le \eta_1/2 <\delta$, we can apply again Proposition \ref{prop2.1}
to find a point $x_{3}$ with now $|x_{3}-x_{2}| \le C{\eta_1\over
2}$ such that $ \eta_3 = \sup_{t\in [t_0+2T , t_0 +3T]} d(v(t) ,
w_3) \le { \eta_1\over 4} $ where $ w_3(x) = w_*(|x-x_{3}|)$.
Iterating this procedure we find a sequence $x_{j}$,
$j=1,2,\ldots$ such that $ |x_{j+1}- x_{j}| \le C{\eta\over 2^j}$
and $$ \sup_{t\in [t_0+j T , t_0 +(j+1) T]} d(v(t) , w_j) \le
{\eta_1 \over 2^j} $$ with $ w_j(x) =
w_*(|x-x_{j}|)$.
Now let $t$ be any number greater than $t_0$. Then $t\in (t_0 + jT
, t_0 + (j+1)T]$ for some $j$. Using that $w_*$ is H\"older
continuous we see that $d( w, w_j) \le C|x_j - x_1|^a$ for some
$a,C>0$. Moreover, $ |x_{j}-x_1 | \le C \sum_{l=1}^\infty \eta_1
2^{-l} = C\eta_1 ,$ hence
$$
d( v(t) , w) \le A\{(\eta_1)^2 + C^a \eta_1^a \}.
$$
We conclude that there is a $\delta >0$ such that if $\eta_1 <
\delta$, then $ d( v(t) , w) < \varepsilon$ for all $t>t_0$. Finally,
from \cite[Proposition 4.1]{CDE}, there is a sequence $s_k\to
\infty$ such that $v(x, s_k + \tau ) \to w(x)$ for some nontrivial
solution of \eqref{est}, uniformly in $x$ and for $\tau$ in bounded
intervals, in particular for $\tau \in [0,T].$ We recall that the
space support of $v$ is contained inside a ball independent of the
time variable. It follows that, given $\varepsilon >0$, there is
indeed a number $t_0 >0 $ such that $\eta_1 < \delta$. Since
$\varepsilon$ is arbitrary, we have actually established that $w$
is the unique limit point of the trajectory $v(\cdot , t)$, and
the proof of the theorem is complete. \hfill \hfill$\square$
The remaining of this paper will be devoted to the proof of
Proposition \ref{prop2.1}.
\section{Preliminaries}
Let $v_n$ be a sequence as in the statement of Proposition \ref{prop2.1}.
Then we have that $v_n(x,t)^m - w_n(x)^m \to 0 $ in $L^\infty$ and
$H^1$-senses in $\mathbb{R}^N$, uniformly locally in time $t\in
[0,\infty)$, as it follows from the results
in \cite{CDE}.
For $T>0$ fixed, which we will choose later, we use in
what follows the following notation.
\begin{equation} \phi_n (x,t) \equiv { v_n (x, t) - w_n(x) \over
\eta_n }. \label{6}\end{equation} Then $\phi_n$ satisfies the
equation
\begin{equation}
{ \partial \phi_n \over \partial t} =
m\Delta (\tilde w_n^{m-1} \phi_n) + m\tilde w_n^{m-1}
\phi_n -{1\over m-1}\phi_n \label{7}
\end{equation}
where
\begin{equation}
\tilde w_n (x,t)^{m-1} \equiv
\int_0^1 (w_n(x) +t (v_n(x, t)- w_n(x)))^{m-1}dt . \label{8}
\end{equation}
Let us observe that, by definition of the number $\eta_n$ we
have that
$$
( m\int w_n(x,t)^{m-1} \phi_n(x, t)^2 dx )^{1/2} \le 1
$$
for all $t\in [0, T]$. In the limit $w_n$ must converges (up
to subsequences) to a steady state $w$ of equation \eqref{est}.
Then $w(x) = w_*(|x-x_1|)$ for certain
$x_1\in \mathbb{R}^N$. In what follows we shall assume, with no loss
of generality that $x_1=0$
These facts suggest that on interior sets of the support $B$ of the
limit $w(x)$ of the $w_n$
we should see convergence in certain sense of $\phi_n$ to a
solution of the degenerate parabolic equation \begin{equation}
\phi_t = m\Delta (w^{m-1} \phi) +
mw^{m-1} \phi -{1\over m-1}\phi .
\label{9}\end{equation} By a {\em weak solution} of equation
\eqref{9}, we understand a function $\phi$ which is smooth in the
interior of ${\cal B} = B\times (0,\infty)$, and satisfies \eqref{9}
there, such that moreover
\begin{equation}
\int_{0}^{s} \int_{B} (m |\nabla(w^{m-1}\phi)|^2
m(w^{m-1}\phi)^2 +{1\over m-1} \phi^2 )dx dt < +\infty \quad
\hbox{for all } s>0.
\label{12}\end{equation}
\begin{lemma} \label{l1}
There is a subsequence of $\phi_n$ which we denote in the same
way, for which $\phi_n (x,t) \to \phi(x,t)$ (up to a subsequence)
in the uniform $C^{1}$-sense over compact subsets of ${\cal B}$.
Moreover, $\phi$ is a weak solution of \eqref{9}.
\end{lemma}
Before proving Lemma \ref{l1} we introduce some notation. Let us
write $ w_{n} = w_*(|x-x_{n}|) $ and consider the functions
\begin{gather}
\psi_n = {v_n^m- w_n^m\over \eta_n}. \label{psi} \\
G_n = {1\over \eta_n^2} \{ {v_n^{m+1}\over m+1} -{ w_n^{m+1}\over
m+1} - w_n^m (v_n -w_n)\}. \label{G} \\
H_n = {(m-1) (v_n^{m+1} - w_n^{m+1} ) + (m+1) (v_n w_n ^m -v_n^m
w_n) \over \eta_n^2} . \label{H}
\end{gather}
It is easily checked the existence of constants $C_1$ and $C_2$,
depending only on $m$ such that the following inequalities hold:
\begin{equation}
C_1 G_n \le {(v_n^m - w_n^m)(v_n- w_n) \over \eta_n^2 } \le C_2
G_n . \label{10}
\end{equation}
We have the validity of the following relation:
\begin{equation}
\int {\partial\over \partial t} G_n \le
\frac{1}{m^2-1}\int H_n .\label{in}
\end{equation}
In fact, let $J$
be the Lyapunov functional for \eqref{11} given by \eqref{J}. Let us
set $I =2(J(v_n ) - J( w_{n}) ).$ Then $I\ge 0$. After
integrating by parts and using the equations satisfied by $v_n$
and $w_n$ we get
$$
I = -\int v_{nt}(v_n^m - w_n^m) -
\frac{1}{m-1} \int (v_n + w_{n}) (v_n^m- w_n^m ) +
\frac{2m}{m^2-1}\int (v_n^{m+1} - w_{n}^{m+1}).
$$
Now, we have
that $\eta_n^2 \frac{\partial G_n}{\partial t} =
v_{nt}(v_n^m-w_n^m)$, so
$$
I = -\eta_n^2 \int \frac{\partial G_n}{\partial t} +\frac{1}{m-1}
\int (v_nw_n^m - v_n^m w_{n}) +\frac{1}{m+1}\int(v_n^{m+1} -
w_{n}^{m+1} ).
$$
Using the definition of $H_n$, inequality
\eqref{in} follows.
Let us observe the following consequence of \eqref{in}: since $\int
G_n(x,0)\le C$, and by definition $|H_n| \le CG_n$, then there
exist constants $a,b >0$ such that for all $n$, $t$,
\begin{equation}
\int G_n(x,t) \le be^{at} . \label{in2}\end{equation}
\paragraph{Proof of Lemma \ref{l1}.}
Let us now analyze the convergence
of $\phi_n$. We note that $\psi_n$ defined by \eqref{psi} satisfies
the equation
\begin{equation} (
\phi_n)_t = \Delta \psi_n + \psi_n
-{1\over m-1}\phi_n . \label{111}
\end{equation}
Integrating \eqref{111} against $\psi_n$, recalling that
${\partial \over \partial t} G_n = \psi_n {\partial\phi_n \over
\partial t}$,
we obtain
\begin{equation}
{\partial \over \partial t} \int G_ndx = \int {\partial
\phi_n \over \partial t} \psi_n dx =
- \int|\nabla \psi_n|^2 dx + \int \psi_n^2 -{1\over m-1} \int \phi_n
\psi_n.\label{epsi}
\end{equation}
Note that given $t>0$, there exists $C(t)>0$ such that
\begin{equation}
\int_0^t \int (| \nabla \psi_n (\cdot , s)|^2
+ | \psi_n (\cdot , s)|^2 )
ds \le C(t) \label{zzz}
\end{equation}
for all $n$. In fact, let us recall that
$$
\int G_n (s, x)dx \le b e^{as}.
$$
Since the function $v$ is bounded we see that $|\psi_n| \le C
|\phi_n|$. Hence, using \eqref{10}, we get that $\psi_n \ \phi_n <
CG_n$ and $\psi_n^2 \le C G_n$. Now integrating relation
\eqref{epsi} in time, between $s=0$ and $s=t$ and using relation
\eqref{in2} we get (\ref{zzz}).
As a consequence of the last result, the sequence $\psi_n$ can be
assumed, after passing to a subsequence, weakly convergent in
$L^2( (0,S); H^1(\mathbb{R}^N))$ for each $S>0$. Let $\psi (x,s)$ be
this limit. Recall that we are assuming $
w(x) = w_*( |x|)$ , so that the support of $w$ is the ball $B =
B(0, R^*)$. Then if we define $\phi = w^{-(m-1) }\psi$ on ${\cal
B}= B\times (0,\infty)$ then
$$
\int_0^S \int_B |\nabla (w^{m-1} \phi)|^2 + |w^{m-1}
\phi|^2 <+\infty
$$
for each $S>0$.
We recall that $\phi_n$ satisfies the equation
\begin{equation}
(\phi_n)_t = \Delta (a_n \phi_n) +a_n\phi_n -\phi_n \label{22}
\end{equation}
where $a_n = (v_n^m -w^m)/(\over v_n-w)$.
Then $a_n\to mw^{m-1}$ uniformly. Hence over compacts of ${\cal
B}$ the coefficient $a_n$ is uniformly positive and bounded. The
standard theory for quasilinear nondegenerate parabolic equations,
see \cite{LSU}, gives that this convergence is also uniform in the
$C^1$-sense over compacts of ${\cal B}$, so that $\nabla a_n$ is
also bounded there. Again the theory for nondegenerate parabolic
equations in \cite{LSU} provides uniform estimates for
$C^{1,\alpha}$ norms over compacts of ${\cal B}$, from where $C^1$
convergence follows. The fact that $\phi$ is a weak solution of
\eqref{9} is now easily checked. \hfill$\square$
Let us consider the eigenvalue problem
\begin{equation}
m\Delta (w^{m-1}\phi ) + mw^{m-1}\phi - {1\over m-1}\phi
+\lambda \phi = 0 . \label{14}\end{equation} with $\phi$ such that
$w^{m-1}\phi \in H^1(B)$ and $w^{\frac{m-1}{2}}\phi \in L^2(B)$.
Here $w(x) = w_*(|x|)$ and $B=B(0,R^*)$.
Concerning this problem, we have the validity of the following
result.
\begin{lemma}\label{l3}
There is only a finite number of negative eigenvalues (repeated
according to multiplicity)
$$
\lambda_1 \le \lambda_2 \le \ldots \le \lambda_{k } <0 ,
$$
with associated eigenfunctions $\phi_1 ,\ldots ,\phi_{k }$,
normalized so that $ \int_B \phi_i\phi_jw^{m-1} = \delta_{ij} . $
Besides, the only eigenfunctions of \eqref{14} for $\lambda =0$ are
linear combinations of the functions ${\partial w \over
\partial_{x_i}}$.
Let $\cal N$ be the finite-dimensional vector space spanned by all
eigenfunctions in $H$ associated to non-positive eigenvalues. Then
there exists a number $\lambda^* >0 $ such that
$$
\lambda^*\int_B w^{m-1}|\tilde \phi|^2 \le \int_B m (\, |\nabla
(w^{m-1}\tilde \phi)|^2 + m|w^{m-1}\tilde \phi|^2 -
\frac{w^{m-1}}{m-1}\tilde \phi^2)dx$$ for all $\tilde \phi \in H$
with $ \int w^{m-1} \tilde \phi \zeta =0 , $ for all $\zeta \in
{\cal N}$.
\end{lemma}
The proof of the above lemma, which can be found in \cite{CDE2} is based
on standard compactness arguments
in Sobolev spaces. The fact that $w^{m-1}$ vanishes quadratically
on the boundary of its support plays a role here.
We can state now the following lemma.
\begin{lemma}\label{l2}
Let $\phi(x,t)$ be the function found in Lemma \ref{l1}, then
\begin{equation}
\phi(x,t) = \sum_{j=1}^{k } D_{j}e^{-\lambda_j t}\phi_i (x)
+ \sum_{i=1}^N C_{i} {\partial w(x)\over\partial x_i}
+ \theta (x,t), \label{ll}\end{equation} where $\theta (x,t)$
converges to zero as $t\to +\infty$, exponentially uniformly
inside compact sets of the set $\cal A$
\end{lemma}
\paragraph{Proof:}
Let us consider the expansion in $B$,
$$
\phi (x,0 ) = \sum_{i=1}^{k } D_i \phi_i + \sum_{i=1}^N C_i
{\partial w\over \partial x_i} +
\theta (x)
$$
where
\begin{equation}
\int_B \theta \phi_i w^{m-1}= \int_B \theta {\partial w\over
\partial x_j}w^{m-1} =0 \label{ort}\end{equation} for all $i,j$. Now let
us consider the function
$$
\tilde \phi (x,t ) = \phi(x,t) - \sum_{i=1}^{k } D_i e^{-\lambda_i
t}\phi_i - \sum_{i=1}^N C_i {\partial w\over \partial x_i} .
$$
Let us observe that
$$
\sum C_i^2 +\sum D_i^2 \le \int_B w^{m-1}\phi^2 (x,0)dx \le 1 .
$$
Clearly $\tilde \phi (x,t)$ satisfies equation \eqref{9}.
We claim that
$$ \int_B
\tilde \phi (\cdot ,s) \phi_i w^{m-1} = \int_B \tilde \phi (\cdot
,s){\partial w\over \partial x_j}w^{m-1} =0 $$ for all $s>0$. In
fact, if we set for instance $ \varphi (s) = \int_B \tilde \phi
(\cdot ,s) \phi_i w^{m-1}$, then $ \varphi ' (s) = -\lambda _i
\varphi (s) . $ Since $ \varphi (0) = 0$, the claim follows. Now,
let us set $ \eta (s) = \int_B w^{m-1} \tilde \phi (\cdot ,s)^2 .
$ Then
$$
\eta'(s) = -2 \int_B m (|\nabla (w^{m-1}\tilde \phi)|^2 -
m(w^{m-1}\tilde \phi)^2 + {w^{m-1}\over m-1} \tilde \phi^2 )dx .
$$
Since $\tilde\phi$ satisfies the above orthogonality relations,
it follows from Lemma \ref{l3} that
$$ \eta'(s) \le -2\lambda^*
\eta (s),
$$
with $\lambda^* >0$, and hence
$$
\eta(s) \le \eta(0) e^{- 2\lambda^* s}.
$$
Also, $\eta (0) \le 1 $. Finally,
linear parabolic regularity implies that exponential decay at this
rate for $\tilde \phi$ also holds uniformly on compact subsets of
$B$. \hfill$\square$
\section{Analysis near the boundary}
In last section we have found the validity of convergence of
$\phi_n$ to $\phi$ {\em essentially} in the interior of the
support of the limiting $w$. Here we will show estimates which
provide control of $\phi_n$ near the boundary of the support of
$w$. Our main purpose is to show that the contribution of the
region near the boundary on the integral of $G_n$ is basically
negligible.
\begin{lemma}\label{corr}
Let $\varepsilon >0$ be given. Then there exist numbers $0< r_0 <
R^*$ and $s^* >0$ such that
for each given $\bar s\ge s^*$ and all $n$ sufficiently large we
have
$$
\sup_{s\in [s^*, \bar s]}\int_{
\{ |x| > r_0\}} G_n(s, x)dx <\varepsilon.
$$
\end{lemma}
The proof is divided into two steps.\\
{\bf Step 1:}
We make the following claim:
\begin{quote}
There exist numbers $A$ and $c$, depending only on $m$, with the
following property: Given $\epsilon>0 $ and $00$, we have that for
all $n$ sufficiently large,
$$
\int_{ |x|\ge r_1} G_n(x,t) dx \le
$$
\begin{equation}
A \Big[ e^{-ct} +\epsilon+ w_*^{3m-3\over 2}(r_0)
\sup_
{(s,r)\in [0,t]\times [r_0,r_1]}
\int_{|x|=r} \phi^2(x,s) d\sigma \Big] \label{car}\end{equation}
where $ r_1 = (r_0 + R^*)/2$.
\end{quote}
To prove this claim we set $ D_n= \{ x/ |x-x_{n}|\ge r_0\}$,
and define for $x \in D_n$
$$
g(x)= w_*^m( ar_0)-w_*^m( a |x-x_{n}|), \quad
a\equiv \frac{2R^*}{R^*+r_0} >1 .
$$
Let us multiply equation
\eqref{111} by $\psi_n (x,t) g(x)$ and integrate on $ D_n$. Since
$g$ vanishes on the boundary of this region, recalling that
$(G_n)_t = \psi_n (\phi_n )_t$, we find
\begin{multline}
{\partial\over
\partial t} \int_{D_n} G_n(x,t)g(x)dx \\
= B_n(t)+ \int_{D_n} {\psi_n^2\over 2 }\Delta g(x) dx +
\int_{D_n} \{ - |\nabla \psi_n |^2 + {\psi_n^2 } -{1\over m-1}
{\phi_n \psi_n } \} g(x)\,dx .\label{mm}
\end{multline}
where
$$
B_n(t) = - {a\partial w_*^m \over\partial r} (a r_0)
\int_{|x-x_{n}| =r_0} {\psi_n^2\over 2 }d\sigma.
$$
Now, $\Delta w_*^m = {w_*\over m-1} - w_*^m \ge 0 $ for $|x|\ge r_0$ if $r_0$
is sufficiently close to $R^*$, hence we assume $\Delta g(x)\le 0$
on $D_n$.
Using these observations and \eqref{mm} we get
$$ {\partial\over
\partial t} \int_{D_n} G_n(x,t)g(x)dx
\le B_n(t) +
\int_{D_n} \{ {\psi_n^2 } -{1\over m-1} {\phi_n \psi_n } \}g(x)\,dx.
$$
On the other hand, recalling the definition of $\psi_n$, estimate
\eqref{in} then reads
\begin{equation}
0\le -\int {\partial G_n \over \partial t} + \frac{1}{m-1} \phi_n
\psi_n - \frac{2}{m-1}G_n . \label{nn}
\end{equation}
We obtain then that
\begin{multline}
\int_{D_n} {\partial G_n\over \partial t} (g(x)+w_*^m(ar_0))\\
\le B_n(t)+ {1\over m-1} \int_{D_n}
{\phi_n \psi_n } (w_*^m(a r_0)- g(x))
+\int_{D_n}\{ \psi_n^2 g(x)-{{2w_*^m(a r_0)}\over{m-1}} G_n .
\}.\label{eee}
\end{multline}
Here we observe that $w_*^m(a r_0)- g(x)=0$ on $F_n= \{ a
|x-x_{n}|\ge R^* \} .$
Now, given $t^* >0$ we have that $v_n \to w$
uniformly on $[0, t^* ]\times \mathbb{R}^N$. Thus, if $r_0$ is
sufficiently close to $R^*$ we obtain that $ \psi_n^2 \le {1\over
(m-1)}G_n $ for $|x| \ge r_0$ and $00$ depending only on $m$, provided that $r_0$ is
sufficiently close to $R^*$. From these facts and \eqref{33} we see
that for given $\varepsilon >0$ and $t>0$,
$$
Y_n(t) \le Y_n (0) e^{-ct}
+\epsilon+ Cw_*^m(r_0) w_*^{3m-3\over 2}(r_0) \sup_{[0,t]} \sup_{r
\in [r_0, {{r_0+R^*}\over{2}}]} \int_{|x|=r} \phi^2 d\sigma,
$$
for all sufficiently large $n$, where we have used again that
$w_*(a r_0)0$ depending on $r_0$ such that
$$
|H_n| \le {D\over \eta_n^2} | v_n -w|^3\quad\hbox{on } B_{r_1} ,
$$
from where it follows that $\int_0^t \int_{B_{r_1 }} H_n \to 0
\mbox{ as } n \to \infty $, for each fixed $t$. Then recalling
that $|H_n| \leq CG_n$, we find that for some $C >0$ independent
of $r_0$ and all sufficiently large $n$,
$$
\int_{B_{r_1}} G_n (x,t)dx \le C [\frac{1}{m^2-1} \int_0^t
\int_{\mathbb{R}^N\setminus B_{r_1}} G_n + 1 ] .
$$
Then, passing to the limit, recalling that $G_n$ converges
uniformly in $B_{r_1}\times [0,t]$ to $mw^{m-1}\phi^2$, we get
$$
\int_{B_{r_1}} mw^{m-1} \phi^2 (x,t) dx
\le \limsup_{n\to \infty}
C( \int_0^t \int_{\mathbb{R}^N\setminus B_{r_1}}G_n + 1) .
$$
Now, from the expression \eqref{ll} for $\phi$, we obtain
$$
\phi(x,t) = \sum_{j=1}^{k } D_{j}e^{-\lambda_j t}\phi_i (x) +
O(1) .
$$
with $O(1)$ uniformly bounded in time and space inside $B_{r_1}$.
It follows that
$$
\int_{\cal A_{r_1} } w^{m-1} \phi^2 (x,t) dx = \sum_{j=1}^{k }
D_{j}^2 e^{-2\lambda_j t} +O(1) .
$$
On the other hand, we can find numbers $A$ and $c$ which depend
only on $m$ so that
\begin{multline}
\limsup_{n\to \infty} \int_{\mathbb{R}^N\setminus B_{r_1}}G_n(x,s)dx\\
\le A\{ e^{-cs} + w_*^{3m-3\over 2}(r_0)
\sup\{ \int_{|x|=r} \phi^2 (x,s)d\sigma :
s\in [0,t],\ r\in [r_0,r_1], \ 1\le i \le k\}.
\label{bb}\end{multline}
It can also be proved that if $\phi_i$
is an eigenfunction corresponding to a negative eigenvalue of
(\ref{14}), then $|\phi_i (x)| \le Cw(|x|)^{3-m\over 2}$, hence
\begin{multline}
w^{3m-3\over 2}(r_0) \sup\{ \int_{|x|=r} \phi^2 d\sigma
: s\in [0,t],\ r\in [r_0,r_1] ,\, 1\le i \le k\} \\
\le C w(r_0)^{m+3\over 2}
\sum_{j=1}^{k } D_{j}^2 e^{-2\lambda_j t} +O(1) ,
\label{ooo}\end{multline}
where $C$ is independent of $r_0$.
Combining the above relations, we get then that for certain
constant $C$
$$
\sum_{j=1}^{k } D_{j}^2 e^{-2\lambda_j t} \le
C w(r_0)^{m+3\over 2}\sum_{j=1}^{k } D_{j}^2 e^{-2\lambda_j t} +O(1),
$$
where $C$ is independent of $r_0$. Since $w(r_0)$ may be chosen
arbitrarily small, we obtain a contradiction from this last
relation for all $t$ sufficiently large if any of the $D_{j}$'s
was not zero. This completes the second step.
The result of the lemma follows now easily from combining steps
one and two. \hfill$\square$
\section{Proof of Proposition \ref{prop2.1}}
We define $\bar x_{in} = \eta_n (C_{1},\ldots , C_{N})$. Then
$ |\bar x_{n}| \le C \eta_n $ with $C=C(m)$. Let us write $ \bar
w_n(x)= w_*^m(x + \bar x_{n} ). $ We want to
estimate the quantity
$$ I_n(s) = \int (v_n(x,s)^m - \bar
w_n(x)^m) (v_n(x,s) -\bar w_n(x))dx
$$
Let us consider $r \in [0,R^*]$, to be determined later, and set
$B_r = \{ |x| < r \}$.
Then
\begin{eqnarray*}
I_n(s) &=& \int_{B_\delta} (v_n^m - \bar w_n^m) (v_n - \bar w_n)dx
+\int_{\mathbb{R}^N\setminus B_\delta} (v_n^m - \bar w_n^m) (v_n -
\bar w_n)dx \\
&=& I_n^1(s) + I_n^2 (s).
\end{eqnarray*}
We have
$$
I_n^1(s) \le C[ \int_{ B_r} (v_n^m - w_n^m) (v_n -
w_n)dx + \int_{B_r} ( \bar w_n^m - w_n^m) (\bar w_n - w_n)dx ].
$$
Now, from \ref{10}, we get
$$ \int_{B_r} (v_n^m - w_n^m) (v_n -
w_n)dx \le C\eta_n^2 \int_{ B_r} G_n(x,s)dx ,
$$
again with $C=C(m)$. Corollary \ref{corr} then implies that if $r$
is chosen
close enough to $R*$, depending on $m$, and $s\ge s^*$, with $s^*$
also depending only on $m$ then for all $n$ sufficiently large
$$
C \int_{ B_r} G_n(x,s)dx < {1\over 8} .
$$
Also,
$$
C( \bar w_n^m - w_n^m) (\bar w_n - w_n) \leq K \eta_n^2
$$
with $K$ depending on $m$ and $k$ only. Therefore, taking $r$ closer to
$R^*$, if necessary, we get
$$ C\int_{B_r} ( \bar w_n^m - w_n^m)
(\bar w_n - w_n)dx \leq \frac{\eta_n^2}{8}
$$
if $n$ is large enough. Putting these two estimates together we see
that if we choose $T \ge s^*$, then
$$ I_n^1(s) \le {\eta_n^2 \over 4}
$$
for all $s\in [T,2T]$ provided that $n$ is sufficiently large.
On the other hand, we recall that
$$
v_n(x,s) = w_n(x)+\eta_n \phi_n (x,s),
$$
so that we can write
$$
v_n(x,s) = w_n(x) +\eta_n \sum_{i=1}^N C_{i} {\partial w_*\over
\partial x_i} (x) + \eta_n \theta (x,s)+ (\phi_n(x,s)-\phi (x,s)) ,
$$
where $\theta $ decays exponentially in compact sets of $\cal A$.
Now, since $r$ has been already fixed it follows that $\bar w_n(x)
= w_*(x-x_{n} + \bar x_{n})$ and
$$
\lim_{n \to \infty} \eta_n^{-2}\int_{\mathbb{R}^N \setminus B_r}
(v_n^m - \bar w_n^m)(v_n - \bar w_n)dx
= m\int_{\mathbb{R}^N \setminus
B_r}w^{m-1}(x)\theta^2 (x,s)\,dx
$$
uniformly on $s$ on compact
subsets of $(0,\infty )$. Since $\theta (x,s)$ decays
exponentially we have that there are positive numbers $A$ and $a$,
depending only on $m$, such that $$I_n^2 (s) \leq \eta_n^2 A
e^{-at}.$$ Consequently we have $$I_n (s) \leq \eta_n^2
(\frac{1}{4} + Ae^{-aT})$$ for all $s \in [T,2T]$. Making $T$
larger if necessary (depending only on $m$) we obtain that the
quantity between brackets is less than $1/2$. This concludes the
proof of the Proposition. \hfill$\square$
\paragraph{Acknowledgements}
C. Cort\'azar and M. Elgueta
were supported by grant 1000625 from Fondecyt.
M. del Pino was supported by grant 8000010 from Lineas
Complementarias Fondecyt and by grant from Fondap Matematicas Aplicadas.
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\noindent\textsc{Carmen Cort\'azar} (e-mail: ccortaza@mat.puc.cl)\\
\textsc{Manuel Elgueta} (e-mail: melgueta@mat.puc.cl)\\[3pt]
Departamento de Matem\'aticas\\
Pontificia Universidad Cat\'olica de Chile,\\
Casilla 306, Correo 22, Santiago, Chile \smallskip
\noindent\textsc{Manuel del Pino} \\
Departamento de Ingenier\'{\i}a Matem\'atica and CMM \\
Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile\\
e-mail: delpino@dim.uchile.cl
\end{document}