\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 53, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/53\hfil Qualitative properties of solutions]
{Qualitative properties of solutions to
semilinear  heat equations with  singular initial data}

\author[Junjie Li\hfil EJDE-2004/53\hfilneg]
{Junjie Li}

\address{Junjie Li \hfill\break
Department of Mathematicas, Yuquan Campus, Zhejiang University,
Hangzhou 310027, China}
\email{ljj@math.zju.edu.cn}


\date{}
\thanks{Submitted December 5, 2002. Published April 8, 2004.}
\thanks{Supported by the National Foundation of China and by the Scientific
 Research Foundation \hfill\break\indent
 for Returned Scholars of the Ministry of Education of China.}
\subjclass[2000]{35B05, 35K55}
\keywords{Comparison principle, extinction, shrinking of support, existence}


\begin{abstract}
 This article concerns the nonnegative solutions to the Cauchy problem
 \begin{gather*}
 u_t - \Delta u + b(x,t)|u|^{p-1}u = 0 \quad \mbox{in } \mathbb{R}^N \times (0,{\infty}),  \\
 u(x,0) = u_0(x) \quad  \mbox{in } \mathbb{R}^N \,.
 \end{gather*}
 We investigate how the comparison principle, extinction
 in finite time, instantaneous shrinking of  support,
 and existence of solutions depend on the behaviour of
 the coefficient $b(x,t)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{exm}[theorem]{Example}
\numberwithin{equation}{section}
\allowdisplaybreaks

\section{Introduction}

In this paper we investigate the qualitative properties of the
nonnegative solutions to the Cauchy problem
\begin{gather}
 Lu:=u_t -\Delta u + b(x,t)|u|^{p-1}u = 0 \quad\mbox{in }
  \mathbb{R}^N\times (0,\infty),  \label{e1.1}\\
u(x,0)=u_0(x)\quad \mbox{in }\mathbb{R}^N , \label{e1.2}
\end{gather}
 where $0<p<1$, %\begin{equation} \label{e1.3}
$b(x,t)\ge 0$, and $u_0(x)$  satisfies the hypothesis
\begin{itemize}
\item[(H1)] $u_0(x) \in C(\mathbb{R}^N\setminus\{0\})$,
$0<u_0(x)\le f(x) := f_0 +\frac {f_1}{|x|^{k_0}}$ in $\mathbb{R}^N $
($f_0\ge 0$, $f_1>0$, $k_0> 0$).
\end{itemize}
Note that for any positive number $q$, when $k_0$ is large, $
f(x)\notin L^q_{\rm loc}(\mathbb{R}^N)$. So to give a proper
definition of the solution to \eqref{e1.1}, \eqref{e1.2} is the
first thing to be consider. Moreover, due to the singularity of
the initial value, solutions to \eqref{e1.1}, \eqref{e1.2} are in
general unbounded, and the singularity at $x=0$ cannot be ``kill''
for  $t>0$ even if $b(x,t)$ possesses some kind of singularity at
$x=0$; see for instance the following example:

\begin{exm} \rm
Let $ b(x,t)=  (k_0(k_0+2-N))/(|x|^{k_0(1-p)+2})$ with $k_0+2>N$
and $u(x,0)=1/(|x|^{k_0})$. Then $u(x,t)=\frac1{|x|^{k_0}}$  is a
classical solution to \eqref{e1.1} in
$(\mathbb{R}^N\setminus\{0\}) \times (0,{\infty})$.
\end{exm}

Due to these facts, we give the definition of solution to
 \eqref{e1.1}, \eqref{e1.2} as follows.

\begin{definition} \rm \label{def1.1}
By a solution to problem \eqref{e1.1}, \eqref{e1.2} we mean  a function
$u(x,t) \in C((\mathbb{R}^N\setminus \{0\})\times [0,\infty))
\cap C^{2,1}((\mathbb{R}^N\setminus\{0\})\times (0,\infty))$
 satisfying classically \eqref{e1.1} in  $(\mathbb{R}^N\setminus\{0\})\times (0,\infty))$
with $u(x,0)=u_0(x)$  in $ (\mathbb{R}^N\setminus\{0\})$.
\end{definition}

 One sees that this definition allows solutions of \eqref{e1.1},
\eqref{e1.2} take their potential singular points only at $x=0$. Does
problem \eqref{e1.1}, \eqref{e1.2} have such a solution?
Although we do not know at the moment the precise and proper conditions
imposed on $b(x,t)$ under which problem \eqref{e1.1}, \eqref{e1.2} is global solvable in the
sense of Definition \ref{def1.1}, we  give in Section 4  a positive answer
when $b(x,t)$ satisfies certain conditions. One also believe that if
$b(x,t)$ is only nonnegative such phenomena as comparison
principle, extinction in finite time and instantaneous shrinking
of support for solutions cease to hold. Our aim in this paper
is to give suitable conditions on $b(x,t)$ under which   the above
mentioned phenomena are valid, i.e., we are mainly interested in
the following: What conditions we add on $b(x,t)$ so that
\begin{itemize}
\item[(a)] Comparison principles for subsolutions and
supersolutions of \eqref{e1.1} hold.

\item[(b)] The solution $u(x,t)$ of \eqref{e1.1} has the property of
instantaneous shrinking of the support  (the support of $u(x,t)$
is bounded for $t > 0 $ although the initial value $u_0(x) $ is
positive every where).

\item[(c)] The solution of \eqref{e1.1} becomes extinct in finite time.

\item[(d)] Problem \eqref{e1.1}, \eqref{e1.2} has a global solution.
\end{itemize}
There are many results on  (a)--(d) when initial value
$u_0(x)$ does not possess  singular points; see \cite{f1,i1,k2,l1,l3} for (a);
 \cite{e1,k1,k2,k3,l2,l3} for  (b) and (c); and
\cite{f1,k2,l1,l3} for (d). The reader can find further references therein.
However, when initial value $u_0(x)$  is subject to (H1), to
our knowledge there are few developments in these direction. As
is known to all, the comparison principle is one of the
cornerstones in dealing with phenomena (b) and (c).
We  establish, in the next section, a comparison principle when $b(x,t)$
is under some conditions. We  also state a negative result on  comparison
principles. From this negative result one can see that the
comparison principle is not valid when the singularity of $ b(x,t)$
is not very ``strong''. These results and their proofs are of
interest in themselves. The study of phenomena (b) and (c)  is the
subject of Section 3.  Section 4 is devoted to  global existence
problems.


\section{Comparison Principle}

In the sequel we use $\epsilon_0$, $R_0$  and $b_i$
$(i=0,1)$ to denote different positive constants, their values may
change from one place to the next. The statement that a constant
depends only on the data means that this constant can be
determined in terms of $N$, $p$, $f_0$, $f_1$ and $k_0$. We also use $B_r(x_0)$
to denote the ball in $\mathbb{R}^N$
of radius $r$  and centered at $x_0 $.


\begin{definition} \rm
For $M_0 \ge f_0$, $0<T_0 \le\infty$, $\mathbb{F}_{M_0}(T_0)$ is the set of all
nonnegative functions  $u(x,t)$ in
$C((\mathbb{R}^N\setminus \{0\})\times [0,T_0))\cap C^{2,1}((\mathbb{
R}^N\setminus \{0\})\times (0,T_0))$ satisfying
\begin{equation} \label{e2.1}
u(x,t) \le M_0 +\frac {f_1}{|x|^{k_0}} \quad  \text{in }
\mathbb{R}^N\setminus \{0\})\times [0,T_0) .\end{equation}
\end{definition}

\begin{theorem}[Comparison Principle] \label{thm2.1}
Assume $0<p\le 1$ , and let $u^{\pm}(x,t) \in \mathbb{F}_{M_0}(T_0)$ satisfy
\begin{equation}\label{e2.2}
\begin{gathered}
\pm L u^{\pm} \ge  0 \quad \mbox{in }  (\mathbb{R}^N\setminus\{0\})\times (0,T_0), \\
u^-(x,0) \le  u^+(x,0) \quad \mbox{in } \mathbb{R}^N\setminus\{0\} .
\end{gathered}
\end{equation}
\item[(a)]  If $k_0 <  N-2 $ and $b(x,t) \ge 0$  in
$(\mathbb{R}^N\setminus\{0\})\times [0,T_0)$, then
$$
u^-(x,t) \le u^+(x,t) \quad \mbox{in }
(\mathbb{R}^N\setminus\{0\})\times [0,T_0);$$

\item[(b)]  if $k_0 \ge N- 2$,  and for $k>k_0$, $R_0
>0$,
\begin{equation} \label{e2.3}
b(x,t) \ge \begin{cases}
\dfrac {k_0(k+2-N)f_1^{1-p}}{p|x|^{k_0(1-p)+2}} & \mbox{in }
(B_{R_0}(0)\setminus\{0\})\times [0,T_0),\\
0 &  \text{in } (\mathbb{R}^N\setminus
B_{R_0}(0))\times [0,T_0),
\end{cases}
\end{equation}
then
$u^-(x,t) \le u^+(x,t)$  in $(\mathbb{R}^N\setminus \{0\})\times [0,T_0)$.
 \end{theorem}

\begin{proof}
Suppose the contrary, then there exists a point
$(x^0,t^0) \in (\mathbb{R}^N\setminus\{0\})\times (0,T_0)$ such that
\begin{equation} \label{e2.4}
u^-(x^0,t^0) - u^+(x^0,t^0) = 2 a > 0 .
\end{equation}
Let $\theta_0 \in (0,1)$  and $\lambda_0 \ge 1$ be
fixed, and set
\begin{gather*}
v^{\pm}(x,t) = \begin{cases}
\dfrac{|x|^{k_0+\theta_0}}{(1+|x|^2)^{\frac {k_0+2\theta_0}2}}u^{\pm}(x,t),
  &  x\ne 0,\ t \in [0,T_0), \\
0, & x =0,\ t \in [0,T_0),\end{cases}
\\
w(x,t) = e^{\lambda_0(t^0-t)}(v^-(x,t)-v^+(x,t)), \quad
 E=\{(x,t)\in \mathbb{R}^N\times (0,t^0];w(x,t)>0\}.
\end{gather*}
 From the hypotheses, one easily see that
$v^{\pm}(x,t) \in C(\mathbb{R}^N\times [0,T_0))
\cap C^{2,1}((\mathbb{R}^N\setminus\{0\})\times (0,T_0))$ and
\begin{equation} \label{e2.5}
\lim_{|x|\to \infty}v^{\pm}(x,t) = 0 \quad\text{uniformly in  }
t \in [0,t^0] .\end{equation}
A straightforward calculation yields
\begin{align*}
\pm\bigl(v_t^{\pm}-\Delta v^{\pm} + 2b_i(x)v_{x_i}^{\pm}\bigr)
& =  \pm \frac {|x|^{k_0+\theta_0}}{(1+|x|^2)^{\frac {k_0+2\theta_0}2}}Lu^{\pm} \\
&\quad  \pm \bigl\{c(x)v^{\pm} + \frac {(k_0+\theta_0)(k_0+\theta_0+2-N)}{|x|^2}v^{\pm}\\
&\quad -\bigl(\frac{|x|^{k_0+\theta_0}}{(1+|x|^2)^{\frac{k_0+2\theta_0}2}}\bigr)^{1-p}b(x,t)(v^{\pm})^p\bigr\},\\
&  \ge \pm\bigl\{-\bigl(\frac{|x|^{k_0+\theta_0}}{(1+|x|^2)^{\frac{k_0+2\theta_0}2}}
\bigr)^{1-p}b(x,t)(v^{\pm})^p + c(x)v^{\pm} \\
&\quad  +\frac{(k_0+\theta_0)(k_0+\theta_0+2-N)}{|x|^2}v^{\pm}\bigr\}
\end{align*}
in $ (\mathbb{R}^N\setminus \{0\})\times (0,T_0)$,
where
\begin{gather*}
b_i(x) = \frac {(k_0+\theta_0) x_i}{|x|^2} - \frac {(k_0+2\theta_0)x_i}{1+|x|^2},\\
c(x) =\frac{(k_0+2\theta_0)(N-2k_0-2\theta_0)}{1+|x|^2}
-\frac {(k_0+2\theta_0)(2-k_0-2\theta_0)|x|^2}{(1+|x|^2)^2 }.
\end{gather*}
Therefore,
\begin{equation} \label{e2.6}
\begin{aligned}
&w_t - \Delta w + 2b_i(x)w_{x_i} \\
& \le  -\lambda_ow - \frac {p b(x,t)w}{(u^-)^{1-p}}+N(k_0+2\theta_0)w
+ \frac {(k_0+\theta_0)(k_0+\theta_0+2-N)}{|x|^2}w
\end{aligned}
\end{equation}
in $E$, where we have used the elementary inequality:
$$ a^p-b^p \ge \frac {p(a-b)}{a^{1-p}} \quad \text{for }  a > b \ge 0.
$$
Case i)  $k_0 \ge N-2 $.  In view of \eqref{e2.3}, \eqref{e2.6} and
$ u^-\in  \mathbb{F}_{M_0}(T_0)$, we first fix $ \theta_0$ and
$\bar R \in (0,R_0)$   small enough such that
\begin{equation} \label{e2.7}
\frac {pb(x,t)}{(u^-(x,t))^{1-p}} \ge
\frac{k_0(k+2-N)f_1^{1-p}}{(M_0+\frac
{f_1}{|x|^{k_0}})^{1-p}|x|^{k_0(1-p)+2}} \ge \frac
{(k_0+\theta_0)(k_0+\theta_0+2-N)}{|x|^2} \end{equation}
in
$(B_{\bar R}(0)\setminus\{0\})\times (0,t^0]$, and then take
$\lambda_0$ satisfying
\begin{equation} \label{e2.8}
\lambda_0 \ge \frac {(k_0+\theta_0)(k_0+\theta_0+2-N)}{\bar R^2}
+(k_0+2\theta_0)(N+3k_0+4\theta_0+2),
\end{equation}
to obtain
$ w_t - \Delta w + 2b_i(x)w_{x_i} \le 0$  in $E$.

\noindent Case ii)  $k_0 < N-2 $. This time we choose
$\lambda_0$  as in \eqref{e2.8} and $\theta_0$ small
enough such that $ k_0+\theta_0+2-N \le 0 $.
Then from \eqref{e2.6} we get
$$ w_t-\Delta w + 2b_i(x)w_{x_i} \le 0 \quad  \text {in }  E .
$$
Thus, we have for small $\theta_0$ and  large $\lambda_0 $,
\begin{equation} \label{e2.9}
w_t - \Delta w + 2b_i(x) w \le  0\quad \text{in }  E.
\end{equation}
For any $ t \in(0,t^o]$, if , $w(x,t)$ attains its
positive maximum at $x(t)$ (\eqref{e2.5} implies that $w(x,t)$ cannot
attain its positive maximum at infinity), then by \eqref{e2.9} we deduce
\begin{equation} \label{e2.10}
w_t(x(t),t) \le 0 .
\end{equation}
We introduce now the function
$$
W(t) = \sup_{x\in\mathbb{R}^N}\,w(x,t).$$
 From $w(x,0) \le 0$ and \eqref{e2.4},
\begin{equation} \label{e2.11}
W(t^0) \ge 2a ,\quad  W(0) \le 0 .
\end{equation}
Set
\begin{equation} \label{e2.12}
t^*=\inf\{t\in[0,t^0]; W(\tau) \ge a , \tau \in(t,t^0]\} .
\end{equation}
By \eqref{e2.11} and the continuity of $W(t)$ on $\{t\in [0,t^0];
W(t)\ge \frac a2\}$, we have
\begin{equation} \label{e2.13}
0<t^*<t^0, \quad  \text { and }  W(t^*) =a .
\end{equation}
 From the proof of \cite[Theorem 4.5]{f2}, one can easily see that
$W(t)$ is Lipschitz continuous on $[t^*,t^0]$ and
\begin{equation} \label{e2.14}
W'(t) \le w_t(x(t),t) \quad\text{a.e.  in } [t^*,t^0] ,
\end{equation}
 where $x(t)$ is a point $ \in \mathbb{R}^N\setminus\{0\}$
satisfying $w(x(t),t) = W(t)$.
Therefore, from \eqref{e2.10} and \eqref{e2.14},
$$
W'(t) \le 0 \quad\mbox{a.e. in }  [t^*,t^0] ,
$$
which implies $a=W(t^*) \ge  W(t^0) \ge 2a$, a contradiction.
\end{proof}

For the case $ k_0 \ge N-2$ we do not know at the moment in a
certain sense a precise or a sharp  condition on $b(x,t)$ under
which the comparison principle remains valid. However, we have the
following negative result.

\begin{theorem} \label{thm2.2}
Assume $k_0 \ge N-2$, $N>4$ and $0<p<1 $. Suppose that for positive constants
$\epsilon_0$, $b_0$, $b_1$,  and $R_0 $,
\begin{equation} \label{e2.15}
b_0(1+\frac 1{|x|^{N-2}})^{1-p} \le b(x,t) \le
b_1(1+\frac1{|x|^{N-2}})^{1-p}
\end{equation}
in $(B_{R_0}(0)\setminus \{0\})\times [0,T_0)$,
 and
\begin{equation} \label{e2.16}
 b(x,t) \ge \epsilon_0 \quad  \text{in }  (\mathbb{R}^N\setminus
B_{R_0}(0))\times [0,T_0).
\end{equation}
Then there exist $u^{\pm}\in \mathbb{F}_{M_0}(T_0)$
satisfying \eqref{e2.2} and  a point
$(x^0,t^0) \in (\mathbb{R}^N\setminus \{0\})\times  (0,T_0)$ such that
\begin{equation} \label{e2.17}
u^-(x^0,t^0) > u^+(x^0,t^0) ,\end{equation}
where $M_0 =f_0$ if $f_0 >0$; $M_0 > 0$ if $f_0 = 0$.
\end{theorem}

\begin{proof} If not, then we have for any pair $u^{\pm} \in \mathbb{F}_{M_0}(T_0)$
satisfying \eqref{e2.2},
\begin{equation} \label{e2.18}
u^-(x,t) \le u^+(x,t) \quad  \text{in }  (\mathbb{R}^N
\setminus \{0\})\times [0,T_0).
\end{equation}
Let $ \alpha, \beta$  and $\bar f \in (0,1)$ be
fixed, denote $a_+ = \max \{a,0\}$ and set
\begin{equation}
V^*(x) = \frac {\bar f}{\alpha^{2\omega}|x|^{N-2}}(\alpha^2-|x|^2)_+^{\omega},\quad
U^*(x,t)=\bar f(1+\frac 1{|x|^{N-2}})(1-\beta t)_+^{\sigma},
\end{equation}
where $ \omega = 2/(1-p)$,  $\sigma =  1/(1-p)$.

We may choose $\bar f$ small enough such that $ V^*$, $U^*$ are in
$\mathbb{F}_{M_0}(\infty)$. One can easily verify
\begin{equation} \label{e2.20}
\begin{aligned}
 LV^*&=(\alpha^2-|x|^2)_+^{\omega p}\bigl(\frac {\bar f}{|x|^{N-2}}\bigr)^p
 \Big[\frac {b(x,t)}{\alpha^{2\omega p}} - \frac {4\omega(\omega-1)|x|^2}
 {\alpha^{2\omega}}(\frac {\bar f}{ |x|^{N-2}})^{1-p} \\
& \quad - \frac {2\omega(N-4)}{\alpha^{2\omega}}(\frac {\bar
f}{|x|^{N-2}})^{1-p}
(\alpha^2-|x|^2)_+\Big] \\
&\le (\alpha^2-|x|^2)_+^{\omega p}(\frac {\bar
f}{|x|^{N-2}})^{1-p}\Big[\frac {b(x,t)}{\alpha^{2\omega p}} - \frac
{\bar \omega }{\alpha^{2(\omega-1)}}(\frac {\bar
f}{|x|^{N-2}})^{1-p}\big],
\end{aligned}
\end{equation}
 and
\begin{equation} \label{e2.21}
LU^* = (1-\beta t)_+^{\sigma p}\bigl[\bar f(1+\frac
1{|x|^{N-2}})]^p[b(x,t) - \beta \sigma (\bar f (1+ \frac
1{|x|^{N-2}}))^{1-p}\bigr],
\end{equation}
where $ \bar \omega = 2 \omega \min\{2\omega -2,N-4\}$.
 From  \eqref{e2.15}, \eqref{e2.16}, \eqref{e2.20} and \eqref{e2.21}, it easily
follows
\begin{gather*}
LV^* \le 0  \quad  \text{ in }  (\mathbb{R}^N\setminus\{0\})\times (0,\infty), \\
 LU^* \ge 0  \quad  \text{ in }  (\mathbb{R}^N\setminus
\{0\})\times (0,\infty),
\end{gather*}
provided that $\alpha$ and $\beta $ are small enough. Hence by
$V^*(x)\le U^*(x,0)$ in $\mathbb{R}^N \setminus \{0\}$ and \eqref{e2.18},
\begin{equation} \label{e2.22}
V^*(x) \le U^*(x,t) \quad  \text{in }
(\mathbb{R}^N\setminus\{0\})\times [0,t^*] ,
\end{equation}
where $ t^* =\min \{1,\frac {T_0}2 \}$. One can establish  step by step on
$t$ that
\begin{equation} \label{e2.23}
V^*(x) \le U^*(x,t) \quad \text{ in }
(\mathbb{R}^N\setminus \{0\})\times [0,\infty), \end{equation}
in particular,
$$
\frac {\bar f}{\alpha^{2\omega}|x|^{N-2}}(\alpha^2-|x|^2)_+^{\omega}
= V^*(x) \le U^*(x,t) =0,\quad  t> \frac 1{\beta},\ x \ne 0,$$
this yields  a contradiction.
\end{proof}

\section{Instantaneous shrinking of the support: Extinction properties}

For an arbitrary nonnegative continuous function $v(x,t)$ defined
in $ (\mathbb{R}^N\setminus\{0\})\times [0,T_0) (0 <T_0 \le  \infty )$,
we set
\begin{equation}
\xi(t;v) = \sup \{|x|; v(x,t) > 0 \}, \quad t \in [0,T_0) .
\end{equation}

\begin{definition} \label{def3.1} \rm
Assume that $0\le v(x,t) \in C((\mathbb{R}^N\setminus\{0\})\times [0,T_0))$,
and $\xi(0;v) = \infty $. We say that instantaneous shrinking of support
occurs for $v $ if there exists $\tau>0$ such that
$ \xi(t;v) < \infty$  for all $t \in (0,\tau]$.
\end{definition}

\begin{definition} \label{def3.2} \rm
Assume that
$ 0 \le v(x,t) \in C((\mathbb{R}^N \setminus \{0\})\times [0,\infty))$,
we say that extinction in finite time occurs for $ v $ if
there exists $T_0 > 0$ such that
$ v(x,t) \equiv 0$ in $(\mathbb{R}^N\setminus\{0\})\times [T_0,\infty)$.
\end{definition}

We begin with a theorem on the instantaneous shrinking of support property.

\begin{theorem} \label{thm3.1}
Assume that $0<p<1$ and  $ (H_1)$ hold, and let
$u(x,t) \in \mathbb{F}_{M_0}(T_0)$ be a solution of \eqref{e1.1}, \eqref{e1.2}.
If $b(x,t)$ satisfies, in addition to $b(x,t)\ge 0$
in  $(\mathbb{R}^N\setminus\{0\})\times [0,T_0)$,
\begin{equation}  \label{e3.2}
b(x,t) \ge h(|x|)f^{1-p}(x) \quad \text{in } (\mathbb{
R}^N\setminus B_{R_0}(0))\times [0,T_0) ,
\end{equation}
 where $R_0>0, h(r)$ is a positive non-decreasing $C^2$-function in $
[R_0,\infty)$ satisfying for some positive constant $h_0$,
\begin{equation} \label{e3.3}
\begin{gathered}
\lim_{r\to \infty}h(r) = \infty ,  \\
h'(r) + |h''(r)| \le h_0 h(r) , \quad  r \in [R_0,\infty).
\end{gathered}
\end{equation}
Then instantaneous shrinking of support occurs for $u$. Further,
$$ \xi(t;u) \le h^{-1}(\frac 1{\beta t}),\quad \forall \,  t \in (0,\tau],
$$
where $ \beta$ is a positive constant depending only on $p$;
$\tau $ is small enough, and
$h^{-1}(a) = \sup\{r;h(r) =a\}$.
\end{theorem}

\begin{proof} Denote $ \omega = \frac 2{1-p}$,  and
let $\beta \in(0,1)$  and $l_0 \ge  R_0 + 1 $
be fixed. We introduce the function
$$
U(x,t)= 2f(x)(1-\beta t h(|x|))_+^{\omega}:=2f(x)Z^{\omega}(x,t) .
$$
We wish to prove that for small $\tau >0$  and large $l_0$,
\begin{equation} \label{e3.4}
u(x,t) \le U(x,t) \quad \text{in } \{|x|\ge l_0\}\times [0,\tau].
\end{equation}
 From the definition of $h^{-1}(\cdot)$, one can see that
 $h^{-1}(\frac 1{\beta t})>l_0$ for $t\in (0,\tau]$, provided that
 $\tau$ is small
enough. Hence the assertion of the theorem easily follows from
\eqref{e3.4}.
Set
$$
Q^+ = \{ (x,t) \in(\mathbb{R}^N\setminus B_{l_0}(0))\times (0,T_0);
\,Z(x,t) >0 \}.
$$
It is obvious that $ U(x,t) \in C^{2,1}((\mathbb{
R}^N\setminus\{0\})\times [0,T_0))$, $0<Z(x,t)\le 1$ in
$Q^+$  and $LU=0$  in
$(\mathbb{R}^N\setminus B_{l_0}(0))\times (0,T_0) \setminus Q^+ $. A
straightforward calculation yields
\begin{equation} \label{e3.5}
\begin{aligned}
LU&=-2\omega \beta h(|x|)f(x)Z^{\omega -1}-\frac {2k_0(k_0+2-N)f_1}{|x|^{2+k_0}}Z^{\omega} \\
& \quad -\frac {4\omega k_0f_1}{|x|^{1+k_0}}\beta th'Z^{\omega-1}-  2\omega(\omega-1)(\beta t h')^2fZ^{\omega p} \\
& \quad + 2\beta t\omega
h''fZ^{\omega-1}+\frac{2\omega(N-1)}{|x|}\beta
th'fZ^{\omega-1}+2^pbf^pZ^{\omega p} .
\end{aligned} \end{equation}
By \eqref{e3.2}, \eqref{e3.3} and \eqref{e3.5}, we get

\begin{equation} \label{e3.6} \begin{aligned}
 LU&\ge2fZ^{\omega p}\bigl[-\omega \beta h(|x|)-\frac {k_0(k_0+2)}{l_0^2}
 -\frac {2\omega k_0 h_0}{l_0}
 - \omega(\omega-1)h_0^2-h_0\omega + \frac {h(|x|)}2 \bigr]  \\
  &\ge 0\quad \text{in } Q^+ ,
\end{aligned} \end{equation}
 provided that $\beta \ $ is small  and   $ l_0\, $ is large.
>From $ U(x,0) > u(x,0)\ \text{ on } |x| = l_0$ and the continuity
of $U$ and $u$, there exists $ \tau=\tau(l_0)>0\, $ such that
\begin{equation} \label{e3.7}
u(x,t) \le U(x,t) ,\ |x|=l_0,\ t \in [0,\tau] .
\end{equation}
 From \eqref{e3.6}, \eqref{e3.7} and $ U(x,0) >u(x,0)$ in
$\mathbb{R}^N\setminus B_{l_0}(0)$, an application of the standard
comparison principle immediately leads to the desired estimate
\eqref{e3.4}.
\end{proof}

 The next theorem demonstrates that the hypothesis
 $\lim_{r\to\infty}h(r)=\infty$ in Theorem \ref{thm3.1} is in a certain
sense sharp.

\begin{theorem} \label{thm3.2}
Assume $0<p<1$. Let $ u \in\mathbb{F}_{M_0}(T_0)$ be a solution of \eqref{e1.1},
\eqref{e1.2}  with $u_0(x) = f(x)$, and suppose that for a positive constant
$ b_1$,
\begin{equation}  \label{e3.8}
0 \le b(x,t) \le  b_1 f^{1-p}(x) \quad  \text{in }
(\mathbb{R}^N\setminus B_{R_0}(0))\times [0,T_0) .
\end{equation}
Then there exist positive constants $H$ and $\tau$ ($H$ depends
only on the data, $R_0$ and $b_1$; $\tau $ is small enough) such that
\begin{equation} \label{e3.9}
u(x,t)\ge \frac {f(x)}2 (1-Ht)_+^{\frac 1{1-p}} \quad  \text{in }
  (\mathbb{R}^N\setminus B_{R_0}(0))\times [0,\tau].
\end{equation}
 In particular, instantaneous shrinking of support does not occur for $u$.
\end{theorem}

\begin{proof}. Let $H>0$ be fixed , consider the function
$$
V(x,t)=\frac {f(x)}2(1-Ht)_+^{\sigma}:=\frac {f(x)}2 Z^{\sigma}(t)\quad
(\sigma = \frac1{1-p}) .
$$
In view of \eqref{e3.8}, a direct calculation gives
\begin{equation} \label{e3.10} \begin{aligned}
LV &= -\frac {H\sigma f}2 Z^{\sigma p} - \frac {k_0(k_0+2-N)f_1}{2|x|^{2+k_0}}
Z^{\sigma} +(\frac f2)^pbZ^{\sigma p} \\
&\le  fZ^{\sigma p}(-\frac {H\sigma}2 + \frac {k_0 N}{2R_0^2} +
\frac { b_1}{2^p}) < 0
\end{aligned} \end{equation}
in  $ (\mathbb{R}^N\setminus B_{R_0}(0))\times (0,T_0)$, provided
$H$ is large enough. Since $ u(x,0) >V(x,0)$  when $|x|=R_0$,
arguing similarly as in the proof of Theorem \ref{thm3.1}, one can
easily see the validity of \eqref{e3.9} for small $\tau >0$.
\end{proof}

We pass now to the extinction in finite time phenomenon. We distinguish two cases:
$k_0 < N-2$ and $k_0 \ge N-2$.

\begin{theorem}[Case $k_0<N-2$] \label{thm3.3}
Assume $0<p<1$ and that (H1) holds, and let  $ u\in \mathbb{F}_{M_0}(\infty)$
be a solution to \eqref{e1.1}, \eqref{e1.2}. If
\begin{equation} \label{e3.11}
b(x,t) \ge  b_0f^{1-p}(x) \quad \text{in }(\mathbb{R}^N\setminus\{0\})
\times [0,\infty)\ (b_0>0),
\end{equation}
 then extinction in finite time occurs for $u$.\end{theorem}

\begin{proof} Let $\beta$ be fixed, and consider the function
\begin{equation} \label{e3.12}
U(x,t) =f(x)(1-\beta t)_+^{\sigma}\quad (\sigma = \frac 1{1-p}).
\end{equation}
A quick calculation gives
\begin{equation} \label{e3.13}
LU =(1-\beta t)_+^{\sigma p}\bigl[-\beta \sigma f(x) - \frac
{f_1k_0(k_0+2-N)}{|x|^{k_0+2}}(1-\beta t)_+ +b(x,t)f^p(x)\bigr].
\end{equation}
 Then by the hypotheses, we have
$$
LU \ge (1-\beta t)_+^{\sigma p}f(x)(-\beta \sigma + b_0) \ge 0,
$$
provided $\beta$ is small enough. Hence by Theorem \ref{thm2.1},
$$
u(x,t) \le U(x,t) \quad \text{in } (\mathbb{R}^N\setminus\{0\})\times [0,\infty),
$$
which implies
$ u(x,t) \equiv 0$ in
$(\mathbb{R}^N\setminus\{0\})\times [\frac 1{\beta},\infty)$.
\end{proof}

\begin{theorem}[Case $k_0 \ge N-2$] \label{thm3.4}
 Assume $p$, $u_0(x)$  and $u(x,t)$ are as in Theorem \ref{thm3.3}. Suppose
that for positive constants $ k>k_0$, $R_0$  and $\epsilon_0 $,
\begin{equation} \label{e3.14}
 b(x,t)\ge \begin{cases}
\frac{k_0(k+2-N)f_1^{1-p}}{p|x|^{k_0(1-p)+2}}\ & \text{in }
 (B_{R_0}(0))\setminus\{0\})\times [0,\infty),\\
\epsilon_0\ & \text{in }  (\mathbb{R}^N\setminus B_{R_0}(0))\times
[0,\infty) .
\end{cases}
\end{equation}
Then extinction in finite time occurs for $u$.
\end{theorem}

\begin{proof} This time we introduce the function
$$
U(x,t) =(\tilde f_0 +\frac {f_1}{|x|^{k_o}})(1-\beta t)_+^{\sigma}\quad
 (\sigma=\frac 1{1-p}),$$
where
\begin{equation} \label{e3.15}
\tilde f_0 = f_0 +1+( \frac{2f_1k_0(k_0+2-N)}{\epsilon_0R_0^{k_0+2}})^{1/p}.
\end{equation}
First, by the hypotheses and \eqref{e3.15}, one can see in $ (\mathbb{
R}^N\setminus B_{R_0}(0))\times (0,\infty)$,
\begin{equation} \label{e3.16} \begin{aligned}
LU&=(1-\beta t)_+^{\sigma p}\bigl\{-\beta\sigma(\tilde f_0 +
\frac {f_1}{|x|^{k_0}})-\frac
{f_1k_0(k_0+2-N)}{|x|^{k_0+2}}(1-\beta t)_+\\
&\quad +b(x,t)(\tilde f_0+\frac {f_1}{|x|^{k_0}})^p\bigr\} \\
&\ge  (1-\beta t)_+^{\sigma p}\bigl\{-\beta\sigma(\tilde f_0
+\frac {f_1}{R_0^{k_0}})-\frac {f_1k_0(k_0+2-N)}{R_0^{k_0+2}} +
\epsilon_0 \tilde f_0^p\bigr\} \\
&\ge (1-\beta t)_+^{\sigma p}\bigl\{-\beta \sigma (\tilde f_0 +
\frac {f_1}{R_0^{k_0}})+\frac {\epsilon_0 \tilde f_0^p}2\bigr\}.
\end{aligned} \end{equation}
Next, in $(B_{R_0}(0))\times (0,\infty)$,
\begin{equation} \label{e3.17}
LU \ge (1-\beta t)_+^{\sigma p}\bigl\{-\beta \sigma (\tilde f_0
+ \frac {f_1}{|x|^{k_0}}) +(\frac1p -1)\frac
{k_0(k+2-N)f_1^{1-p}}{|x|^{k_0(1-p)+2}}(\tilde f_0+\frac
{f_1}{|x|^{k_0}})^p\bigr\}.
\end{equation}
 From \eqref{e3.16} and \eqref{e3.17}, one can easily conclude
$$ LU \ge 0 \quad \text{in } (\mathbb{R}^N\setminus \{0\})\times (0,\infty),
$$
provided $\beta$ is small enough. Thus, by comparison
\begin{equation} \label{e3.18}
u(x,t) \le U(x,t) \quad \text{in }(\mathbb{R}^N\setminus\{0\})\times [0,\infty).
\end{equation}
 The assertion of the theorem  follows immediately from \eqref{e3.18} and
the definition of $U(x,t)$.
\end{proof}

The following negative result shows that in a certain sense the
condition  $b(x,t) \ge \epsilon_0 >0$ in
$(\mathbb{R}^N\setminus B_{R_0}(0))\times [0,\infty)$ in \eqref{e3.14} is
necessary.

\begin{theorem} \label{thm3.5}
Assume $ k_0 \ge N-2$ and $\frac 12<p<1 $, and let $u$ in
$\mathbb{F}_{M_0}(\infty)$ be a solution of \eqref{e1.1}, \eqref{e1.2}
with $u_0(x) = f(x)$. If
\begin{equation} \label{e3.19}
b(x,t) \ge \frac {k_0(k+2-N)f_1^{1-p}}{p|x|^{k_0(1-p)+2}}
\quad  \text{in }(B_{R_0}(0)\setminus\{0\})\times [0,\infty),
\end{equation}
 and
\begin{equation} \label{e3.20}
0 \le b(x,t)\le g(|x|)f^{1-p}(x) \quad \text{in } (\mathbb{
R}^N\setminus B_{R_o}(0))\times [0,\infty),
\end{equation}
 where
$g(r) $ is a positive non-increasing $C^2$-function in $[R_0,\infty)$
satisfying
\begin{equation} \label{e3.21}
\begin{gathered}
\lim_{r\to\infty}g(r)=0,  \\
-\frac {g'(r)}r +|g''(r)|\le g_0g^2(r)\quad  \text{in }
[R_0,\infty)\ (g_0>0).
\end{gathered}
\end{equation}
 Then extinction in finite time does not occur for $ u$.
\end{theorem}

\begin{proof} Let $\gamma$ be large enough such that $(1-\gamma g(|x|))_+>0$
implies $|x| \ge R_0$. Consider the function
$$
V(x,t) = f(x)(1-\gamma (t+1)g(|x|))_+^{\sigma}:=f(x)Z^{\sigma}(x,t)\quad
(\sigma = \frac 1{1-p}).$$
Since
$\frac12<p<1$,
$V(x,t)\in C^{2,1}((\mathbb{R}^N\setminus\{0\})\times [0,\infty))$. Set
$$
Q^+ = \{(x,t)\in(\mathbb{R}^N\setminus\{0\})\times (0,\infty);Z(x,t)>0\}.
$$
It is obvious that $ LV=0$ in
$(\mathbb{R}^N\setminus\{0\})\times (0,\infty)\setminus Q^+$ and
$Q^+\subseteq (\mathbb{R}^N\setminus B_{R_o}(0))\times (0,\infty)$.
Now, we compute in $Q^+$,
\begin{equation} \label{e3.22} \begin{aligned}
LV & =  f(x)Z^{\sigma p}(x,t)\bigl[-\sigma \gamma g(|x|)
-\frac {f_1k_0(k_0+2-N)}{|x|^{k_0+2}f(x)}Z(x,t)
-\frac{2\sigma k_0f_1\gamma(1+t)g'(|x|)}{|x|^{k_0+1}f(x)}  \\
& \quad -\sigma(\sigma-1)(\gamma(1+t)g'(|x|))^2Z^{-1}(x,t)+\sigma(N-1)\frac{\gamma(1+t)g'(|x|)}{|x|}  \\
& \quad  +\sigma\gamma(1+t)g''(|x|)+ \frac{b(x,t)}{f^{1-p}(x)}\bigr].
\end{aligned} \end{equation}
 Then by
\eqref{e3.19}--\eqref{e3.22}, we have
\[
LV  \le  f(x)Z^{\sigma p}(x,t)[-\sigma\gamma g(|x|)+\sigma g_0(2k_0+1)g(|x|)
+ g(|x|)] \le 0,
\]
 provided $\gamma$ is large enough. Hence
from \eqref{e3.19} and \eqref{e3.20}, an application of Theorem
2.2 yields
\begin{equation} \label{e3.24}
u(x,t) \ge V(x,t) \quad \text{in } (\mathbb{
R}^N\setminus\{0\})\times [0,\infty).
\end{equation}
 From the definition of $V(x,t)$ and $\lim_{r\to\infty}g(r)=0 $,
we see that for any $T>0$, there exists a point
$x=x(T) \in \mathbb{R}^N\setminus\{0\}$ such that $ V(x,T)>0$, whence
by \eqref{e3.24}  $u(x,T)>0$.
\end{proof}

For the case $ k_0 < N-2$ we have the following result.

\begin{theorem} \label{thm3.6}
Assume  $k_0 <N-2$ and $\frac12<p<1$. Let $u(x,t) \in  \mathbb{F}_{M_0}(\infty)$ be a solution
to the problem \eqref{e1.1}, \eqref{e1.2} with $u_0(x) =f(x)$. If
\begin{equation} \label{e3.25}
0\le b(x,t) \le \frac{b_1f^{1-p}(x)}{|x|^2}\quad \mbox{in }
(\mathbb{R}^N\setminus\{0\})\times [0,\infty)\quad
(b_1>0).
\end{equation}
 Then extinction in finite time does not occur for $u$.
\end{theorem}

\begin{proof}  We introduce the function
$$
V(x,t) =f(x)(1-\frac{\gamma t}{|x|^2})_+^{\sigma}:= f(x)Z^{\sigma}(x,t)\quad
 (\sigma=\frac1{1-p}),
$$
where $ \gamma $ is a large constant to be chosen.
A simple calculation yields
\begin{align*}
&LV \\
&= -\frac{\sigma \gamma}{|x|^2}f(x)Z^{\sigma p}(x,t)
-\frac{k_0(k_0+2-N)f_1}{|x|^{k_o+2}}Z^{\sigma}(x,t)
-\frac{2\sigma(N-4)\gamma t}{|x|^4}f(x)Z^{\sigma p}(x,t)\\
&\quad-\frac{4\sigma(\sigma-1)}{|x|^6}(\gamma
t)^2f(x)Z^{\sigma-2}(x,t)+\frac{4\sigma k_0f_1}{|x|^{k_0+4}}\gamma
tZ^{\sigma p}(x,t)+b(x,t)f^p(x)Z^{\sigma p}(x,t) .
\end{align*}
Then by \eqref{e3.25}, we have for large $\gamma$
\[
LV  \le  f(x)Z^{\sigma p}(x,t)\{-\frac {\sigma \gamma}{|x|^2}
+\frac{k_o(N+4\sigma)+8\sigma}{|x|^2}+\frac{b_1}{|x|^2}\}
  \le  0 ,
\]
 and hence by comparison,
$u(x,t) \ge V(x,t)$ in $(\mathbb{R}^N\setminus\{0\})\times [0,\infty)$.
The remaining of the proof is as before.
\end{proof}

\section{Existence}

We begin with a global existence result.

\begin{theorem} \label{thm4.1}
Assume that $0 <p \le 1$ and (H1) hold, and suppose that
$b(x,t) \in C^{\alpha_0,\frac {\alpha_0}2}((\mathbb{
R}^N\setminus\{0\})\times [0,\infty))(\alpha_0 \in(0,1))$
satisfies
\begin{equation} \label{e4.1}
b(x,t) \ge \begin{cases}
\frac {k_0(k_0+2-N)_+f_1^{1-p}}{|x|^{k_0(1-p)+2}}
&\text{in }\ (B_{R_0}(0)\setminus\{0\})\times [0,\infty),\\
0 & \text{in } (\mathbb{R}^N\setminus
B_{R_0}(0))\times [0,\infty) .
\end{cases}  \end{equation}
Then there exists a solution $u(x,t) \in  \mathbb{F}_{M_0}(\infty)$
to  \eqref{e1.1}, \eqref{e1.2} for some constant $M_o$ depending only on
the data and $R_0$.
\end{theorem}

\begin{proof}. For any large $n$, let $u_n(x,t)$ be the
unique solution of the approximated problem
\begin{gather}
(u_n)_t - \Delta u_n + b_n(x,t)|u_n|^{p-1}u_n = 0  \quad \text{in }
 B_n(0)\times(0,n^2],  \\
u_n(x,t) = 0  \quad \text{on }  \partial B_n(0)\times(0,n^2] , \\
u_n(x,0) = u_{o,n}(x)\psi_n(x) \quad\mbox{in }  B_n(0).
\end{gather}
Here $\psi_n(x) $ is a smooth cutoff function in
$ B_n(0)$ satisfying $0 \le \psi_n \le 1$, $\psi_n(x) = 0$
\ on  $\partial B_n(0)$ and $\psi_n(x) = 1$
in $B_{n-1}(0)$; $u_{0,n}(x)$ and
$b_n(x,t)$ are, respectively, the smooth approximation of $u_0(x)$
and $b(x,t)$ satisfying
\begin{equation} \label{e4.5}
\begin{gathered}
\lim_{n\to\infty}u_{0,n}(x)=u_0(x) \quad \text{in } \mathbb{R}^N\setminus\{0\},\\
0\le u_{0,n}(x) \le f_0+\frac{f_1}{(|x|^2+n^{-2})^{\frac{k_0}2}}
\quad \text{in } \mathbb{R}^N ,
\end{gathered}
\end{equation}
\begin{equation}\lim_{n\to \infty}b_n(x,t)=b(x,t)\quad \text{in }
(\mathbb{R}^N\setminus\{0\})\times [0,\infty)
\end{equation}
 and
\begin{equation} \label{e4.7}
b_n(x,t) \ge \begin{cases}
\frac{k_0(k_0+2-N)_+f_1^{1-p}}{(|x|^2+n^{-2})^{\frac{k_0(1-p)+2}2}} ,&
\text{in }B_{R_o}(0)\setminus\{0\})\times [0,\infty),\\
0 & \text{in } (\mathbb{R}^N\setminus B_{R_0}(0))\times [0,\infty) .
\end{cases}
\end{equation}
 Clearly, $u_n(x,t) \ge 0$ in $B_n(0)\times [0,n^2]$.
 To estimate  the upper bound of $u_n(x,t) $, we introduce the function
\begin{equation}
U_n(x,t)=\tilde M + f_1 g(\sqrt{|x|^2+n^{-2}}) ,
\end{equation}
where $ \tilde M \ge f_0 $ is a large constant to be fixed, and
\begin{equation} \label{e4.9}
g(r)=\begin{cases}
\frac 1{r^{k_0}}, & 0<r\le r_0:=\frac{k_0+1}{2(k_0+3)}R_0 ,\\
-\frac{k_0(k_0+1)^2}{12r_0^{k_0+3}}(r-\frac{R_0}2)^3
+\frac{k_0+3}{3(k_0+1)r_0^{k_0}},&  r_0<r\le \frac {R_0}2 ,\\
\frac {k_0+3}{3(k_0+1)r_0^{k_0}},&  r>\frac{R_0}2 .
\end{cases}
\end{equation}
It is readily to verify that $ g(r) \in C^2(0,\infty)$. From the
definition of $ g(r)$, we compute
\begin{equation}
rg''(r)+(N-1)g'(r)\le \begin{cases}
\frac{k_0(k_0+2-N)_+}r g(r),&   0<r\le r_0,\\
C ,&  r_0<r\le \frac{R_0}2 ,\\
0,&  r>\frac{R_0}2 ,
\end{cases}
\end{equation}
where $C$ depends only on $N$, $k_0$, and $R_0$;
and
\begin{equation} \label{e4.11}
\begin{aligned}
LU_n & = b_n(x,t)U_n^p(x,t)-f_1\bigl\{\frac{|x|^2}{|x|^2+n^{-2}}
g''(\sqrt{|x|^2+n^{-2}}) \\
&\quad+(\frac N{\sqrt{|x|^2+n^{-2}}}-\frac{|x|^2}{(|x|^2+n^{-2})^{\frac32}})
g'(\sqrt{|x|^2+n^{-2}})\bigr\} \\
&\ge b_n(x,t)U_n^p(x,t) -\frac{f_1}{\sqrt{|x|^2+n^{-2}}}
\bigl[\sqrt{|x|^2+n^{-2}}g''(\sqrt{|x|^2+n^{-2}}) \\
&\quad+(N-1)g'(\sqrt{|x|^2+n^{-2}})\bigl] .
\end{aligned} \end{equation}
By \eqref{e4.7}--\eqref{e4.11} we have for large $\tilde M$,
$$ LU_n \ge 0 \quad \text{in } B_n(0)\times (0,n^2] ,
$$
and hence by comparison,
\begin{equation} \label{e4.12}
u_n(x,t) \le U_n(x,t) \le 2 \tilde M + \frac
{f_1}{|x|^{k_o}}\quad  \text{in }
(B_n(0)\setminus\{0\})\times [0,n^2].
\end{equation}
 From this equation,
according to the  well-known interior estimates of solutions and
their continuity module (see for instance \cite{d1,f1,l1}), it follows
that for any $l\ge 2$ and $n\ge l+4$, we have
\begin{gather} \label{e4.13}
\|u_n\|_{C^{2+\alpha_1,1+\frac{\alpha_1}2}(Q_l)}\le C_l ,\\
|u_n(x,t)-u_n(x',t')|\le\omega_l(|x-x'|+|t-t'|^{\frac12}) \label{e4.14}
\end{gather}
for all $(x,t),(x',t')\in \widehat {Q_l}$,
 where $Q_l=(B_l(0)\setminus B_{1/l}(0))\times [\frac 1{l^2},l^2]$,
$\widehat {Q_l}=(B_l(0)\setminus B_{1/l}(0))\times [0,l^2]; \alpha_1(>0)$
is independent of $n$ and $l$, $C_l$  and $\omega_l(r)$  are
independent of $n$, and $\omega_l(r)$ tends to zero as $r\downarrow 0$.

The above estimates together with Arzela's lemma and a diagonal
argument imply that  there exists a function
$u(x,t)\in \mathbb{F}_{M_0}(\infty)$ $(M_0=2\tilde M)$ such that,
after extracting a subsequence if necessary,
$$ \lim_{n\to\infty}u_n(x,t) = u(x,t)
\quad \text{in }(\mathbb{R}^N\setminus\{0\})\times [0,\infty) ,
$$
and $u(x,t)$ solves \eqref{e1.1}, \eqref{e1.2} in the classical sense.
\end{proof}

 From Theorems \ref{thm2.1} and \ref{thm4.1} we have the following statement.

\begin{corollary} \label{coro4.1}
Assume $0<p\le 1$ and (H1) hold, and suppose that
$ b(x,t)\in C^{\alpha_0,\frac{\alpha_0}2}((\mathbb{R}^N\setminus\{0\})
\times [0,\infty))$, $\alpha_0 \in(0,1)$,
satisfies: \begin{itemize}
\item[(a)] For  $k_0<N-2$,
$b(x,t) \ge 0$ in $(\mathbb{R}^N\setminus\{0\})\times [0,\infty)$

\item[(b)] For $ k_0 \ge N-2$, $k>k_0$,
$$
b(x,t) \ge \begin{cases}
\frac{k_0(k+2-N)f^{1-p}_1}{p|x|^{k_0(1-p)+2}}&
\text{in } (B_{R_0}(0)\setminus\{0\})\times [0,\infty) ,\\
0 & \text{in } (\mathbb{R}^N\setminus B_{R_0}(0))\times [0,\infty).
\end{cases}
$$
\end{itemize}
Then problem \eqref{e1.1}, \eqref{e1.2} is uniquely solvable in
$ \mathbb{F}_{M_0}(\infty)$  for some $M_0 \ge f_0$.
\end{corollary}

\begin{thebibliography}{99}

\bibitem{e1} L. C. Evans, B. F. Knerr,
\emph{Instantaneous shrinking of the support of non-negative solutions to
certain nonlinear parabolic equations and variational
inequalities}, Illinois J.Math., 23  (1979)153--166

 \bibitem{d1} E. DiBenedetto, \emph{On the local behaviour of solutions of
degenerate parabolic equations with measurable coefficients}, Ann.
Sc. NOrm. Sup.  13 (3)(1986)485--535

\bibitem{f1} A. Friedman, \emph{Partial Differential Equations of
Parabolic Type}, Prentice-Hall, Inc. (1964)

\bibitem{f2} A. Friedman, B. McLeod, \emph{Blow-up of positive
solutions of semilinear heat equations}, Indiana Univ. Math. J., 34
(1985) 425--447.
\bibitem{i1} A. M. Il'in, A. S. Kalashinikov and
O. A. Oleinik , \emph{Second order linear equations of parabolic type},
Russian Math. Surveys, 17(1962) 1--143.

\bibitem{k1} A. S. Kalashnikov, \emph{Instantaneous shrinking of the support for
solutions to certain parabolic equations and systems},
Rend. Mat. Acc. Lincei. Serie 9, vol 8(1997) 263--272.

\bibitem{k2} R. Kersner, F. Nicolosi, \emph{The nonlinear heat equation with
absorption: effects of variable coefficients}, J. Math. Anal. Appl.,
170 (1992) 551--566

\bibitem{k3} R. Kersner , A. Shishkov,
\emph{Instantaneous shrinking of the support of energy solutions},
J. Math. Anal. Appl. 198 (1996) 729--750.

\bibitem{l1} O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Ural'ceva,
\emph{Linear and Quasilinear Equations of Parabolic Type}, AMS
Providence. Rhode island (1968).

\bibitem{l2} Junjie Li, \emph{Instantaneous shrinking of the
support of solutions to certain parabolic equations with unbounded
initial data}, Nonlinear Analysis, TMA, 48(2002) 1--12.

\bibitem{l3} Junjie Li, \emph{Qualitative properties for
solutions of semilinear heat equations with strong absorption},
J. Math. Anal. Appl. 281(2003) 382--394.

\end{thebibliography}

\end{document}