\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 29, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/29\hfil On vanishing at space infinity]
{On vanishing at space infinity for a semilinear heat equation
with absorption}

\author[N. Umeda \hfil EJDE-2014/29\hfilneg]
{Noriaki Umeda}  % in alphabetical order

\address{Noriaki Umeda \newline
Graduate School of Sciences and Technology,
Meiji University, 1-1-1, Higashi-Mita, Tama-ku,
Kawasaki city, Kanagawa, 214-8571, Japan. \newline
Graduate School of Mathematical Sciences,
University of Tokyo,  3-8-1, Komaba, \newline Meguro-ku, Tokyo, 153-8914, Japan}
\email{umeda\_noriaki@cocoa.ocn.ne.jp}

\thanks{Submitted February 24, 2012. Published January 16, 2014.}
\subjclass[2000]{35K15, 35K55}
\keywords{Vanishing at space infinity; semilinear heat equation with absorption}

\begin{abstract}
 We consider  a Cauchy problem for a semilinear heat equation with absorption.
 The initial datum of the problem is bounded and its infimum is positive.
 We study solutions which do not vanish in the total space at the vanishing time;
 they vanish only at space infinity.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

We consider the semilinear heat equation (with absorption)
\begin{equation}
     u_t = \Delta u- u^{-p}, \quad x\in \mathbb{R}^d , \; t>0 \label{CP-u}
\end{equation}
supplemented by initial data
\begin{equation}
     u(x,0) = u_0 (x)>0, \quad x\in \mathbb{R}^d \label{CP-u0},
\end{equation}
with $d\ge 1$ and $p>-1$.
The function $u_0 $ is assumed to satisfy
\begin{gather}
 u_0 \text{ is bounded and continuous in } \mathbb{R}^d , \label{H1} \\
 m:=\inf_{x\in \mathbb{R}^d } u_0 (x)>0. \label{H2}
\end{gather}

In Theorem \ref{Existence-C} we prove that the Cauchy problem 
\eqref{CP-u}-\eqref{CP-u0} has a unique positive classical solution 
under the hypotheses \eqref{H1}-\eqref{H2}. However, this solution need 
not exist globally in time. For a given initial datum $u_0 $ we define
\[
	T(u_0 )=\sup \big\{ t>0;\inf_{x\in \mathbb{R}^d } u(x,t)>0 \big\} <\infty
\]
and call it the maximal existence time of the positive solution or the 
vanishing time for \eqref{CP-u}-\eqref{CP-u0}. It is clear that
\begin{equation}\label{eq:12-11111}
	\lim_{t \to T(u_0 ) } \inf_{x\in \mathbb{R}^d } u(x,t)=0.
\end{equation}
If this happens, we say that the solution vanishes at $t=T(u_0 )$.
Usually,  \emph{quenching} happens in the case $p>0$ 
(see \cite{Kawarada,Guo1,Guo2,Guo3}), while
\emph{dead-core} occurs when $-1<p<0$  
(see \cite{HV1,HV2,GS,GMW,S2}). Here
we study \emph{vanishing} in the case $p>-1$.

Let $v$ be a space independent solution of \eqref{CP-u} with an initial 
datum $m=\inf_{x\in \mathbb{R}^d } u_0 (x)$. It is easily seen that 
the solution of the problem
\begin{equation}\label{eq:GSU-ODE}
v' = -v^{-p},\quad \text{ for } t>0, \quad v(0) = m
\end{equation}
is expressed by
\begin{equation}\label{solv1}
v(t) = \{ (p+1)(T(m) -t) \}^{1/(p+1)} \quad \text{with} \quad 
T(m) = \frac{m^{p+1} }{p+1}
\end{equation}
and
\begin{equation}\label{solv2}
v(t) = \{ m^{p+1} - (p+1)t \}^{1/(p+1)} .
\end{equation}
It is immediate that $T(u_0 )\ge T(m)$ by the comparison principle 
(see Theorem \ref{comparison-C}).
Next we study the case $T(u_0 )=T(m)$ in the following theorem
(see \cite{GSU0,GSU,GU1,GU2,Gladkov,S,S2,SSU}.

\begin{theorem} \label{ThmQSI} 
Assume \eqref{H1}-\eqref{H2}. If there exists a sequence 
$\{ a_k \}_{k=1}^{\infty } \subset \mathbb{R}^d $ such that
\begin{equation}\label{inftyu01}
	u_0 (x+a_k ) \to m \quad \text{a.e. in } \mathbb{R}^d  \text{ as } 
 k\to \infty,
\end{equation}
then $T(u_0 )=T(m)$.
Moreover, if $u_0 \not\equiv m$, then the solution of 
\eqref{CP-u}-\eqref{CP-u0} does not vanish in $\mathbb{R}^d $ at $t=T(m)$. 
(It vanishes only at space infinity.)
\end{theorem}

\begin{remark}\rm
If $u_0 \not\equiv m$, then   $|a_k | \to \infty $ as $k\to \infty $.
\end{remark}

The next theorem describes the behavior of the limit of the solution 
of \eqref{CP-u}-\eqref{CP-u0} as $t\to T(m)$.
We prove that  $u(x,T(m))=\lim_{t\to\ T(m)} u(x,t)$ for every 
$x\in \mathbb{R}^d $ (see Lemma \ref{SU-Lem41}).

\begin{theorem} \label{ThmQSIB} 
Under the hypotheses as Theorem \ref{ThmQSI},
\[
\lim_{k \to \infty} u(a_k,T(m))=0.
\]
\end{theorem}

Finally, we consider the relation between the form of initial data and 
the profile of a vanishing solution at time $T(m)$ (see \cite{GSU2}).
In \cite[\S2b]{L}, for the equation
\[
	u_t =\Delta u +f(u),
\]
a subsolution and a supersolution of the form $\varphi (T(m)-t+h(x,t))$  
were constructed.
Here  we construct a subsolution and a supersolution of the form
 $\varphi(T(m)-t+g(x,t))$ where  $g(x,t)$ decays  to zero at space infinity and
\begin{equation}
	\varphi (s) =v(T(m)-s)=\{ (p+1)s \}^{1/(p+1)} ,\label{phi-def}
\end{equation}
 to estimate the profile at the vanishing time for 
\eqref{CP-u}-\eqref{CP-u0}. It is clear that
\begin{equation}
\varphi' =\varphi^{-p}, \quad \varphi (T(m))=m, \quad
\lim_{s\to 0} \varphi (s) =0.\label{phi-def2}
\end{equation}

Let  $\psi$ be a positive function satisfying the following conditions:
\begin{gather}
	 \psi (x)  \text{ is bounded and continuous in } \mathbb{R}^d, \label{A-psi-frac0} \\
 \psi (x)>0 \quad \text{for } x \in \mathbb{R}^d, \label{A-psi-frac10}
\end{gather}
 there exists a constant $C_1 >0$ such that
\begin{equation}
	\sup_{x\in \mathbb{R}^d } \Big[	\inf_{y\in B(0,1)} \Big\{
		\sup_{z\in B(y,1)}
		\frac{\psi(x)}{\psi(x+z)}
	\Big\} \Big] \le C_1, \label{A-psi-frac} 
\end{equation}
and there exist constants $a\in ( 0,1/(4T(m)) ) $ and $C_2 >0 $ such that
\begin{equation}
	\sup_{(x,y) \in \mathbb{R}^d \times \mathbb{R}^d }
		\frac{\psi(x-y)}{\psi(x)e^{a|y|^2 } } \le C_2 . \label{A-psi-frac2}
\end{equation}
Here $B(x,r)$ denotes the open ball of radius $r$ centered at $x$.

\begin{theorem} \label{ThmUmeda} 
Let the  hypotheses in Theorem  \ref{ThmQSI} hold.
If  $\psi $ satisfies \eqref{A-psi-frac0}-\eqref{A-psi-frac2}, and
\begin{equation}\label{ThU0UL}
	 C_I \psi (x)\le u_0^{p+1} (x)-m^{p+1} \le C_{II} \psi (x)
\end{equation}
for some constants $C_I>0$ and $C_{II}$,
then there exist constants
\begin{equation}
C=C(C_1 ,C_2 ,a,T(m),C_I )>0 \quad \hbox{and} \quad
C'=C'(C_1 ,C_2 ,a,  T(m) ,C_{II} )>0
\end{equation}
 such that the solution to  \eqref{CP-u}-\eqref{CP-u0} satisfies
\[
	C \psi ^{1/(p+1)} (x) \le u(x,T(m)) \le C' \psi^{1/(p+1)} (x) .
\]
\end{theorem}

\begin{remark} \rm
Theorem  \ref{ThmUmeda}, we may be restated as follows.
If
\[
	 u_0^{p+1} (x)-m^{p+1} \ge C_I \psi (x) \quad 
(\text{or } \le C_{II} \psi (x) ),
\]
for some constant $C_I>0$ (or $C_{II} >0$),
then there exists a constant $C>0$ (or $C'>0$) such that the solution 
of \eqref{CP-u}-\eqref{CP-u0} satisfies
\[
u(x,T(m)) \ge C \{ \psi (x)\}^{1/(p+1)} \quad 
(\text{or } \le C' \{ \psi (x)\}^{1/(p+1)}).
\]
\end{remark}

If $\psi $ is a positive constant, then it satisfies 
\eqref{A-psi-frac0}-\eqref{A-psi-frac2}, and the initial datum with 
this $\psi $ satisfies \eqref{ThU0UL}. However, it does not satisfy 
the hypothesis of Theorem  \ref{ThmQSI}.
In fact, the solution of \eqref{CP-u}-\eqref{CP-u0} with such an 
initial datum does not vanish at $t=T(m)$. We shall show examples 
of $\psi$ satisfying these hypothesis.

\begin{example} \rm
Let $f$ satisfy
\begin{equation}
\text{$f(r)=(r^2 +1 )^{-b/2}$,  $f(r)=e^{-br}$  or
$f(r)= \{ \log (r+e) \}^{-b}$  for $r\ge 0$ and $b>0$}.
\label{f-form}
\end{equation}
Assume that $\psi (x)$ satisfies one of the following three conditions:
\begin{enumerate}
\item $\psi (x)=f(|x|)$.

\item $\psi (x) =\Theta (x/|x|) + \{ 1-\Theta ((x/|x|)\} \tilde{f} (|x|) $,
	where $\Theta (\theta ) \in C^{\infty } (S^{d-1} )$ satisfies
 \begin{equation*}
 \Theta (\theta ) \begin{cases}
		=0,	& \theta \in S^{d-1} \cap \overline{B(\theta_0 ,r_1 )}, \\
		\in (0,1), &  \theta \in S^{d-1} \cap B(\theta_0,r_2 ) \backslash \overline{B(\theta_0 ,r_1 )} ,\\
		=1,	& \theta \in S^{d-1} \backslash B(\theta_0 ,r_2 )
		\end{cases}
 \end{equation*}
 with some direction $\theta_0 \in S^{d-1} $, some constants
 $r_1 $, $r_2 $ satisfying $0<r_1 <r_2 $ and
 \begin{equation*}
		\tilde{f} (r)= 	\begin{cases}
			1,	& r\in [0,1), \\
			f(r-1),	& r\ge 1.
		\end{cases}
 \end{equation*}
\item $\psi (x) =\inf_{i\in {\bf N} } f(\max \{ 0,|r_i| -|x-a_i| \}) $	
	with the sequence $\{ a_i \}_{i=1}^{\infty } \subset \mathbb{R}^d$
	and $\{ r_n\}_{i=1}^{\infty } \in \mathbb{R}^+ $
	satisfying $\lim_{i\to \infty } |a_i| =\infty $, $r_1<r_2< \ldots \to \infty $.
\end{enumerate}
Then $\psi (x)$  satisfies \eqref{A-psi-frac0}--\eqref{A-psi-frac2}. 
Moreover the solution of \eqref{CP-u}-\eqref{CP-u0} with the $u_0 $ 
satisfying \eqref{ThU0UL} vanishes only at space infinity at $t=T(u_0)$. 
Here $S^{d-1}$ denotes the $(d-1)$-dimensional unit sphere and  
$\overline{B}$ does the closure of $B$.
\end{example}

This article is organized as follows. In Section 2 we prove 
Theorem  \ref{ThmQSI}. Section 3 is devoted to the proof of 
Theorem  \ref{ThmQSIB}. The proof of Theorem  \ref{ThmUmeda}
will be given in Section 4. In Section 5 (Appendix A) we show the 
existence and the uniqueness for the classical solution 
\eqref{CP-u}-\eqref{CP-u0} with the initial data satisfying
 \eqref{H1}-\eqref{H2} (Theorem  \ref{Existence-C}),
and we also prove the comparison principle of the problem 
(Theorem  \ref{comparison-C}). In the last section (Appendix B),
we will show Lemma  \ref{SU-Lem41} about the existence and the
regularity for the solution at $t=T(m)$.


\section{Vanishing only at space infinity}

In this section we prove Theorem \ref{ThmQSI}. First, we show that 
$T(u_0 )=T(m)$ (see also \cite[Theorem 1]{SU}).

\begin{lemma} \label{SUProp21}
Assume \eqref{H1}-\eqref{H2}. Let $p>-1$ and $d\ge 1$. If there exist 
sequences $\{ a_k \}_{k=1}^{\infty } \subset \mathbb{R}^d $ and 
$\{ r_k \}_{k=1}^{\infty } $ satisfying $0<r_1 <r_2 < \ldots \to \infty $ 
such that
\begin{equation}\label{SU10}
	\lim_{k\to \infty } \| u_0 -m \|_{L^{\infty } (B(a_k ,r_k ) )} =0,
\end{equation}
then the solutions $u$ and $v$ of \eqref{CP-u}-\eqref{CP-u0} with initial 
data $u_0 $ and $m$ satisfy
\[
	\lim_{k\to \infty } \| u(\cdot ,t )-v(t) \|_{L^{\infty } (B(a_k ,r_k /2)) } =0
\]
for any $t\in (0,T(m))$. Moreover $T(u_0 ) =T(m)$.
\end{lemma}

\begin{proof}
Put $\tilde{u} =u-v$ and $\tilde{u}_0 =u_0 -m$. By Theorem  \ref{comparison-C}, 
  $\tilde{u} \ge 0$ for $(x,t)\in \mathbb{R}^d \times (0,T(m))$.
It is clear that $\tilde{u} $ satisfies
\begin{equation}\label{SU9}
	\begin{gathered}
	\tilde{u}_t=\Delta \tilde{u}-(u^{-p} -v^{-p} ),	\quad
 x\in \mathbb{R}^d,\; 0<t< T(m), \\
	\tilde{u}(x,0)=\tilde{u}_0 (x),	\quad x\in \mathbb{R}^d .
	\end{gathered}
\end{equation}
 From \eqref{SU10} for any $\varepsilon >0$ there exists $k_0 >0$ 
such that for any $k\ge k_0 $,
\begin{equation}\label{SU11}
	\| \tilde{u}_0 \|_{L^{\infty } (B(a_k ,r_k )) } < \varepsilon^2 .
\end{equation}
Take $t_0 \in (0,T(m))$. By the mean value theorem we have
\[
-(u^{-p} -v^{-p} )=\int_0^1 p\{ \theta u+(1-\theta )v \}^{-p-1}
 \tilde{u} d\theta  \le \max \{ 0,p \} \{ v(t_0 ) \}^{-p-1} \tilde{u}
\]
for $t\in (0,t_0 )$. Put $K=K(t_0 )=\max \{ 0,p \} (v(t_0 ))^{-p-1} $. Thus
\begin{gather*}
\tilde{u}_t\le \Delta \tilde{u}+K\tilde{u} ,\quad
 x\in \mathbb{R}^d, 0<t< t_0, \\
	\tilde{u}(x,0)=\tilde{u}_0 (x),	\quad x\in \mathbb{R}^d .
\end{gather*}
The solution of
\begin{gather*}
	\bar{u}_t=\Delta \bar{u}+K\bar{u} ,	\quad x\in \mathbb{R}^d, 0<t< t_0, \\
	\bar{u}(x,0)=\tilde{u}_0 (x),	\quad x\in \mathbb{R}^d
	\end{gather*}
is a supersolution of \eqref{SU9}. The solution $\bar{u} $ is 
\begin{align*}
	\bar{u} (x,t)	
 & =e^{Kt} \int_{\mathbb{R}^d } G(x-y,t )\tilde{u}_0 (y)dy \\
 & =e^{Kt} \Big( \int_{ \mathbb{R}^d \setminus B(x,r_k /2 ) } 
 +\int_{B(x,r_k /2 ) } \Big) G(x-y,t )\tilde{u}_0 (y)dy \\
& =I+II,
\end{align*}
where $G(x,t)$ is the Green kernel of the heat equation given by
\begin{equation}\label{G-Senbei}
	G(x,t) =\frac{1}{(4\pi t )^{d/2} } e^{-|x|^2/4t } .
\end{equation}
 From the definition of $G$ we see that
\begin{equation}
\begin{aligned}
I & \le e^{Kt_0 } \int_{ \mathbb{R}^d \setminus B(x,r_k /2 ) } G(x-y,t)
 \tilde{u}_0 (y)dy  \\
  & \le e^{Kt_0 } \| \tilde{u}_0 \|_{L^{\infty } (\mathbb{R}^d ) }
 \int_{\mathbb{R}^d \setminus B(0,r_k /2 ) } G(y,t) dy
 < \frac{\varepsilon }{2}
\end{aligned} \label{SU-01}
\end{equation}
for any $k$ large enough, where $B(x,r)$ denotes the open ball of radius
$r$ centered at $x$. Note that $y\in B(a_k ,r_k )$ whenever
$x\in B(a_k ,r_k /2 )$ and $y\in B(x,r_k /2)$. Thus by \eqref{SU11} we obtain
\begin{equation}
\begin{aligned}
II & \le e^{Kt_0 } \int_{B(x,r_k /2 ) } G(x-y,t) \tilde{u}_0 (y) dy  \\
   & \le e^{Kt_0 } \| \tilde{u_0} \|_{L^{\infty } (B(x,r_k /2) ) }
  \int_{B(x,r_k /2 )} G(x-y,t) dy  \\
   & \le e^{Kt_0 } \| \tilde{u_0} \|_{L^{\infty } (B(a_k, r_k ) ) }
  \int_{\mathbb{R}^d } G(x-y,t) dy < \frac{\varepsilon }{2}
\end{aligned} \label{SU-U2}
\end{equation}
for any $x\in B(a_k ,r_k /2 )$ and any $\varepsilon \in (0,e^{-Kt_0} /2 )$
 with $k$ large enough. We thus have
\[
\lim_{k\to \infty } \| \bar{u} (\cdot ,t) \|_{L^{\infty } (B(a_k ,r_k /2 ))} =0,
\quad \text{for } t\in(0,t_0 ).
\]
Hence, by Theorem  \ref{comparison-C},
\[
\lim_{k\to \infty } \| \tilde{u} (\cdot ,t) \|_{L^{\infty } (B(a_k ,r_k /2 ))} =0,
 \quad \text{for } t\in(0,t_0 ).
\]
Since $t_0 \in(0,T(m))$ is arbitrary,
\begin{equation}\label{SUPro21}
	\lim_{k\to \infty } \| \tilde{u} (\cdot ,t) \|_{L^{\infty }
(B(a_k ,r_k /2 ))} =0, \quad \text{for } t\in(0,T(m)).
\end{equation}
	
Next we show that $T(u_0 )=T(m)$.
Let us assume, to the contrary, that  there exists a constant $L>0 $ such that
\begin{equation}\label{8:5-shapEEE}
	\inf_{t \in (0,T(m)) } \Big[ \inf_{k\in {\bf N} }
\Big\{\operatorname{ess\,sup}_{x\in B(a_k ,r_k /2) } u(x,t)  \Big\} \Big] \ge L .
\end{equation}
Since $v_t \le 0$ and $\lim_{t\to T(m)} v(t)=0 $, there exists
$T_0 \in [0,T(m) )$ such that
\[
	v(T_0 ) \le \frac{L}{3}.
\]
 From \eqref{SUPro21} there exists a constant $k_0 \ge 0$ such that
\[
	\sup_{k\ge k_0 } \| u(\cdot ,T_0)-v(T_0) \|_{L^{\infty}(B(a_k ,r_k /2))}
\le \frac{L}{3}.
\]
 From \eqref{8:5-shapEEE}, we see that
\begin{align*}
&\sup_{k\ge k_0 } \{ \| u(\cdot ,T_0 )-v(T_0 ) \|_{L^{\infty}(B(a_k ,r_k /2))} \} \\
& =\sup_{k\ge k_0 }  \Big\{ \operatorname{ess\,sup}_{x\in B(a_k ,r_k /2) }
u(x,T_0 ) -v(T_0 ) \Big\} \\
&\ge L-\frac{L}{3}=\frac{2L}{3}>\frac{L}{3}.
\end{align*}
This is a contradiction. We thus conclude that
\begin{equation}\label{SHStar}
	\inf_{t \in [0,T(m)) } \Big[ \inf_{k\in {\bf N} }
\Big\{ \operatorname{ess\,sup}_{x\in B(a_k ,r_k /2) } u(x,t)  \Big\} \Big]=0
\end{equation}
and $T(u_0 )\le T(m)$.
By Theorem  \ref{comparison-C}, we see that $T(u_0 )\ge T(m)$.
We thus obtain $T(u_0 )=T(m)$.
\end{proof}

The next lemma shows that the solution of \eqref{CP-u}-\eqref{CP-u0} 
does not vanish in $\mathbb{R}^d $ even at the vanishing time. 
The lemma is shown by using the argument in \cite[Lemma 2.3]{MM2} 
(see also \cite{SU}).

\begin{lemma}\label{SU-Lem31} 
Let $u(x,t)$ be a solution of \eqref{CP-u}-\eqref{CP-u0} in 
$\mathbb{R}^d \times [0,T(m))$ with $m$ defined in \eqref{H2}. 
Suppose that there exist $t_0 \in (0,T(m))$, $a \in \mathbb{R}^d$, $r_0 >0$ 
and $\theta >1$ such that
\[
	u(x,t) \ge \theta \varphi(T(m) -t) \quad \text{in } |x-a|<r_0, \;
 t_0 \le t <T,
\]
where $\varphi $ is defined in \eqref{phi-def}.
Then $u$ does not vanish at $t=T(m)$ in a neighborhood of $a$.
\end{lemma}

\begin{proof}
For convenience we let $T=T(m)$.
We shall construct a suitable supersolution.
Put $\varepsilon>0$ and $\tilde{\theta } =\tilde{\theta } (\varepsilon ) 
\in (1,\theta )$ satisfy
\begin{equation}\label{T-epsilon-t0}
	\tilde{\theta } \varphi ( T-t_0 +\varepsilon/2 ) \le \theta \varphi (T-t_0).
\end{equation}
Define
\[
\omega(x,t)=\tilde{\theta } \varphi(T-t+h(r)),
\]
where $r=|x-a|$ and
\[
h(r)=\varepsilon\Big(\frac{1+\cos{\frac{\pi r}{r_0}}}{2}\Big)
=\varepsilon\Big\{ \cos \Big(\frac{\pi r}{2r_0}\Big) \Big\}^2 .
\]
Thus, from \eqref{phi-def} and \eqref{phi-def2} we have
\begin{align*}
	\omega_t-\Delta \omega +\omega^{-p}
	&=-\tilde{\theta } \varphi '  - \tilde{\theta } \varphi ' \Delta h 
- \tilde{\theta } \varphi '' |\nabla h|^2 +(\tilde{\theta } \varphi)^{-p} \\
	&=\tilde{\theta } ( -\varphi^{-p} )
\Big\{1+\Delta h +\frac{\varphi ''}{\varphi '} |\nabla h|^2 
-\tilde{\theta }^{-p-1} \Big\},
\end{align*}
where
\begin{gather*}
	\nabla h  =h_r \nabla r =h_r \frac{x-a}{r} ,\\
	\Delta h  =\operatorname{div} (\nabla h)=h_{rr} +\frac{d-1}{r} h_r .
\end{gather*}
Since $\varphi '' =-p\varphi^{-p-1} \varphi ' $, there exists $t_0 \in (0,T)$ 
such that for $t\in (t_0,T)$
\begin{equation} \label{hhhdainyu}
\begin{aligned}
	1+\Delta h + \frac{\varphi ''}{\varphi '} |\nabla h|^2 -\tilde{\theta}^{-p-1}
& = 1+ \Delta h -p\varphi^{-p-1} |\nabla h|^2 -\tilde{\theta }^{-p-1}  \\
	&\ge (1-\tilde{\theta }^{-p-1} )+ \Delta h
-\frac{p|\nabla h|^2}{(p+1)(T-t+h)}  \\
	&\ge (1-\tilde{\theta }^{-p-1} )+ \left(h_{rr} +\frac{d-1}{r}h_r\right)
-\frac{p|\nabla h|^2}{(p+1)h}.
\end{aligned}
\end{equation}
We thus conclude that
\begin{equation}\label{inequarity333}
	\begin{gathered}
		\omega_t \le \Delta \omega -\omega^{-p} , 	\quad |x-a|<r_0,t_0\le t<T, \\
		\omega (x,t_0)\le u(x,t_0), 	\quad |x-a|<r_0, \\
		\omega (x,t) \le u(x,t), 	\quad |x-a|=r_0,t_0\le t<T
	\end{gathered}
\end{equation}
for any $\varepsilon >0$ sufficient small.

By Theorem  \ref{comparison-C}, for $x \in B(a,r_0 ) $ and $t\in [t_0 ,T)$ we
have $u(x,t)\ge \omega (x,t)$. Since $\varphi$ is an increasing function, 
we obtain
\[
u(x,t) \ge \tilde{\theta } \varphi \Big( T-t+h\big(\frac{r_0}{2}\big) \Big) 
=\tilde{\theta } \varphi \big(T-t+\frac{\varepsilon}{2} \big) 
\quad \text{for } (x,t)\in B( a, r_0 /2 ) \times [t_0 ,T).
\]
 From \eqref{T-epsilon-t0}, we see that
\[
	\tilde{\theta } \varphi ( T-t +\varepsilon/2 ) \le \theta \varphi (T-t) 
\quad \text{for } t\in (t_0 , T).	
\]
Since
\[
	u(x,t) \ge \theta \varphi (T-t ) \quad \text{for } (x,t)\in 
B( a,r_0/2) \times [t_0 ,T),
\]
we have
\[
	u(x,t) \ge \tilde{\theta } \varphi ( T-t +\frac{\varepsilon}{2} ) 
\quad \text{for } (x,t)\in B( a,r_0/2) \times [t_0 ,T)
\]
and $u$ does not vanish at $t=T$ in $B ( a,r_0/2)$.
\end{proof}

Next we show that the condition on $u_0 $ in Theorem  \ref{ThmQSI}
 is equivalent to the one in Lemma \ref{SUProp21}.

\begin{lemma}\label{DDae-infty} 
Assume \eqref{H1}-\eqref{H2}.
Condition \eqref{inftyu01} is equivalent to condition \eqref{SU10}
for sequences $\{ a_k \}_{k=1}^{\infty } \subset \mathbb{R}^d $
and $\{ r_k \}_{k=1}^{\infty } $ satisfying $0<r_1 <r_2 < \ldots \to \infty $.
\end{lemma}

\begin{proof}
If \eqref{inftyu01} is assumed, since $B(0, r_k) \subset \mathbb{R}^d$ for any 
$k \in {\bf N} $, we see that
\[
	\lim_{k\ \to \infty } \| u_0 (x+a_k ) - m  \|_{L^{\infty } (B (0, r_k)) } =0,
\]
which gives  \eqref{SU10}.

Assuming  that \eqref{SU10} holds, for $x_0 \in \mathbb{R}^d $ we let  
$k_0 =k_0 (x_0 ) >0 $ be such that
$B(x_0 ,1) \subset B(0,r_k )$ for $k\ge k_0 $. Since
\[
	\lim_{k\to \infty } \| u_0 -m \|_{L^{\infty } (B(a_k ,r_k)) } =0,
\]
we have
\[
	u_0 (x+a_k ) \to m \quad \text{a.e. in $B(x_0 ,1)$  as $k\to \infty$.}
\]
Since $x_0 \in \mathbb{R}^d $ is arbitrary, we obtain \eqref{inftyu01}.
\end{proof}

Finally, we shall prove that the vanishing occurs only at space infinity 
by using Lemma \ref{SU-Lem31}.

\begin{proof}[Proof of Theorem \ref{ThmQSI}]
Lemmas  \ref{SUProp21} and  \ref{DDae-infty} yield $T(u_0 )=T(m)$. 
Let $T=T(u_0 )=T(m)$.
We need to show that for any $a \in \mathbb{R}^d$
there exist $t_0 \in (0, T )$, $r_0>0$ and $\theta > 1 $ such that
for $x\in B(a,r_0 )$ and $t\in [t_0 ,T)$
\[
	u(x,t) \ge \theta \varphi(T -t).
\]

 From the strong maximum principle (or Theorem  \ref{comparison-C2}), we obtain
\[
	u(x,t)>v(t) \quad \text{for } (x,t)\in D\times (0,T)
\]
for any compact set $D\subset \mathbb{R}^d $.
We thus may let $u_0(x)>m$ for $x\in B(a,r_0 )$ without loss of generality. 
Let $w(x,t)$ be a solution of
\begin{equation}\label{SU-w}
\begin{gathered}
	w_t =\Delta w, \quad x \in B(a ,r_0), t> 0,\\
	w(x,t)=1,  \quad x \in \partial B(a ,r_0), t\ge 0, \\
	1 \le w(x,0)\le u_0 (x)/m, \quad x \in B(a ,r_0), \\
	w(x,0) \not\equiv 1, \quad x \in B(a ,r_0).
\end{gathered}
\end{equation}
It is clear that $vw \le u$ on $\partial B(a ,r_0) \times(0,T)$ and 
$B(a ,r_0) \times \{0\}$. From \eqref{SU-w} we obtain
\[
	(vw)_t = -v^{-p} w +v \Delta w \le -(vw)^{-p} +\Delta(vw).
\]
Then for any $a\in \mathbb{R}^d$ and any $r_0>0$, $vw$ is a subsolution 
of \eqref{CP-u}-\eqref{CP-u0} in $B(a ,r_0)$.
Thus, by the strong maximum principle,
for any $(x,t) \in B(a,r_0 ) \times (0,T )$, we see that $w(x,t) >1$.
In particular, for any $\tilde{r}_0 \in(0,r_0 )$ there exist $\theta >1$ 
and $t_0 \in (0,T)$ such that
\[
	w(x,t) \ge \theta, \quad |x-a|<\tilde{r}_0, \; t_0 \le t < T.
\]
This implies
\[
	u(x,t) \ge \theta \varphi(T-t), \quad |x-a|<\tilde{r}_0, \; t_0 \le t < T
\]
 by the comparison principle. By Lemma  \ref{SU-Lem31}, $u$ does not 
vanish in a neighborhood of $a$. Since $a\in \mathbb{R}^d $ is arbitrary,
 it does not do in $\mathbb{R}^d$.
\end{proof}

\section{Behavior at vanishing time}\label{sec4}

In this section we prove Theorem  \ref{ThmQSIB}. The proof for the 
theorem uses the argument of the proof in  \cite[Theorem 3]{SU}. 
First we introduce a lemma on  the existence for the solution to 
\eqref{CP-u}-\eqref{CP-u0} at $t=T(m)$.

\begin{lemma}\label{SU-Lem41} 
Assume the same hypotheses as in Theorem  \ref{ThmQSI}.
 Then $u(x,T) =\lim_{t\to T} u(x,t)$ exists for any $x\in \mathbb{R}^d$ with 
$T=T(m)$. Moreover $u(x,T)\in C^{\infty } (\mathbb{R}^d) $.
\end{lemma}

The proof of this lemma shall be shown in Appendix B.
Now we proceed with the proof of  Theorem  \ref{ThmQSIB}.

\begin{proof}[Proof of Theorem \ref{SU-Lem31}]
It is clear in the case $u_0 \equiv m$. We should only consider the case 
$u_0 \not\equiv m$.
Let $\{ r_k \}_{k=1}^{\infty } $ be as defined in Lemma  \ref{SUProp21}.
Let   $\varepsilon>0$ be sufficiently small so that  $\varepsilon <b-m$, 
where $b=\sup_{x\in \mathbb{R}^d } u_0 (x) $.
Let 
\begin{equation}\label{2005.6.11.8}
u^{k,\varepsilon}_0(x)=
\begin{cases}
m+\varepsilon  , & |x-a_k|<r_k -1, \\
(b-m-\varepsilon )(|x-a_k|-r_k) +b , & r_k-1 \le |x-a_k|<r_k, \\
b , & |x-a_k|\ge r_k,
\end{cases}
\end{equation}
and  $u^{k,\varepsilon} $,  $v^{\varepsilon} $ be solutions of
 \eqref{CP-u}-\eqref{CP-u0} with initial data $u^{k,\varepsilon}_0 $ 
and $m+\varepsilon $. We write $T^{\varepsilon}=T(m+\varepsilon )$ 
for simplicity.

 From Lemma  \ref{SUProp21} for any $\varepsilon>0$
there exists a natural number $k_0 \in {\bf N}$ such that for any $k>k_0$,
if $x \in B(a_k ,r_k /2 ), t \in (0,T^{\varepsilon})$, then
\begin{equation}\label{2005.6.11.5}
	v^{\varepsilon}(t)+\varepsilon \ge u^{k,\varepsilon}(x,t).
\end{equation}
By the comparison principle (see Theorem  \ref{comparison-C}) 
for any $x \in \mathbb{R}^d$ and any $t \in (0,T^{\varepsilon})$,
\begin{equation}\label{2005.6.11.6}
		u^{k,\varepsilon}(x,t) \ge u(x,t).
\end{equation}
Since  $T(m)<T^\varepsilon $, by Lemma  \ref{SU-Lem41}, \eqref{2005.6.11.5} 
and \eqref{2005.6.11.6}, for any $\varepsilon >0$ there exists 
$k_0 = k_0 (\varepsilon) \in {\bf N}$ such that for any $k>k_0$
\begin{align}\label{2005.6.11.7}
		v^{\varepsilon}(T(m))+\varepsilon \ge u(a_k,T(m)).
\end{align}
Since $\varepsilon >0$ can be chosen arbitrarily small and 
$\lim_{\varepsilon \to 0} v^{\varepsilon} (T(m))=0 $, 
\eqref{2005.6.11.7} implies
\[
\lim_{k\to\infty} u(a_k,T(m)) =0.
\]
\end{proof}

\section{Profile at vanishing}

To prove Theorem  \ref{ThmUmeda}, we construct a subsolution and
a supersolution of the form $\varphi (T-t+g(x,t))$ with $T=T(m)$, as we
have explained before. This is a modification of the method
employed in \cite{L} and \cite{Shimozyo} to study blow-up profile
for a semilinear heat equation.
Let
\begin{eqnarray}
	g^{\gamma }_{\alpha ,\beta } (x,t) =g^{\gamma ,\psi }_{\alpha ,\beta } (x,t)
 =\int_{\mathbb{R}^d } G^{\gamma }_{\alpha ,\beta } (x-y,t) \psi (y) dy, 
\label{g-psi-def}
\end{eqnarray}
where
\[
	G^{\gamma }_{\alpha ,\beta } (x,t) 
=\frac{|x|^{\beta } }{(t+\gamma )^{\alpha -d/2 } } G(x,t+\gamma )
=\frac{|x|^{\beta } }{(t+\gamma )^{\alpha} } \exp 
\Big( -\frac{|x|^2}{4(t+\gamma )} \Big)
\]
with $\alpha \in \mathbb{R} $, $\beta \ge 0$, $\gamma >0$ are constants 
(see \cite{GSU2}). Note that  $g^{\gamma }_{\alpha ,\beta } $
also may be expressed as
\[
	g^{\gamma }_{\alpha ,\beta } (x,t) 
=\int_{\mathbb{R}^d } G^{\gamma }_{\alpha ,\beta } (y,t) \psi (x-y) dy.
\]
It is easily seen that:
\begin{gather}
 |\nabla g^{\gamma }_{\alpha ,0 } |
 \le \frac{ \sqrt{d} g^{\gamma }_{\alpha +1,1} }{2} , \label{gggggg1} \\
 \Delta g^{\gamma }_{\alpha ,0 }  =\frac{g^{\gamma }_{\alpha +2 ,2} }{4} 
 -\frac{d g^{\gamma }_{\alpha +1 ,0} }{2} , \label{gggggg2} \\
 \partial_t g^{\gamma }_{\alpha ,0 } =\frac{g^{\gamma }_{\alpha +2 ,2} }{4} 
 -\alpha g^{\gamma }_{\alpha +1 ,0} , \label{gggggg3}\\
\label{ggaa}
	g^{\gamma }_{\alpha ,\beta } (x,t)=\frac{g^{\gamma }_{0,\beta } }
{(t+\gamma )^{\alpha} } .
\end{gather}

Before proving Theorem  \ref{ThmUmeda} we prove the next two propositions.

\begin{proposition} \label{Prop1Umeda} 
Assume that $p>-1$. Let $\psi $ be a positive bounded continuous function 
satisfying \eqref{A-psi-frac0}-\eqref{A-psi-frac2} and
\begin{equation}\label{Lem-2-3-gamma}
	\gamma \in \Big(0, \frac{1}{4a} -T \Big).
\end{equation}
Then for any $C>0$ the function
\begin{equation}\label{prof-w}
	W(x,t)=\varphi (T-t+Cg^{\gamma }_{-\alpha ,0} (x,t))
\end{equation}
is a supersolution of \eqref{CP-u} in ${\bf R }^d \times (0,T)$
for $\alpha $ satisfies $\alpha \ge \alpha_1 $ with some constant 
$\alpha_1 =\alpha_1 (p,d,C_1 ,C_2 ,a,T ,\gamma )>0$, where $\varphi $ is 
defined in \eqref{phi-def}.
\end{proposition}

\begin{proposition} \label{Prop2Umeda} 
Assume the same hypotheses as in Proposition  \ref{Prop1Umeda}. 
Then, for each constant $C>0$, the function
\begin{equation}\label{astwwwww}
	w(x,t)=\varphi (T-t+Cg^{\gamma }_{\alpha ,0} (x,t))
\end{equation}
is a subsolution of \eqref{CP-u} in $\mathbb{R}^d \times (0,T)$ provided 
that $\alpha $ satisfies $\alpha \ge \alpha_2  $ with some constant 
$\alpha_2 =\alpha_2 (p,d,C_1 ,C_2 ,a,T ,\gamma ) >0$.
\end{proposition}

Before proving Propositions  \ref{Prop1Umeda} and  \ref{Prop2Umeda}, 
we need one lemma on estimates for $g^{\gamma }_{0 ,\beta } $.

\begin{lemma} \label{Lem3Umeda} 
Assume the same hypotheses as in Propositions  \ref{Prop1Umeda} and  
\ref{Prop2Umeda}. Then for $\beta =0$, $1$, $2$, there exist constants 
$C_3=C_3 (C_1 ,\gamma )>0$ and $C_4 =C_4(C_2 ,a ,T,\gamma ) >0$ such that
\[
	C_3 \psi (x)\le g^{\gamma }_{0 ,\beta } (x,t)\le C_4 \psi (x) \quad 
\text{in } \mathbb{R}^d \times [0,T],
\]
where $C_1 $ and $C_2 $ are in \eqref{A-psi-frac} and \eqref{A-psi-frac2}, 
respectively.
\end{lemma}

\begin{proof}
First we show $g^{\gamma }_{0,\beta } (x,t) \ge C_3 \psi (x)$ with some $C_3 >0$.
 From \eqref{A-psi-frac} we see that there exists a vector 
$q\in B(0,\min \{ 1, 2/\sqrt{d} \}) $ such that
\begin{equation}\label{A-psi-11}
	\psi (x)\le 2C_1  \inf_{z\in B(q+x,1) } \psi (z)
\end{equation}
for each $x\in \mathbb{R}^d $.
If \eqref{A-psi-11} holds, then we see that
\begin{align*}
	g^{\gamma }_{0,\beta } (x,t) 
&\ge \inf_{z\in B(q+x,1) } \psi (z) \times \int_{B(q,1) } |y|^{\beta } 
\exp \Big( -\frac{|y|^2}{4\gamma } \Big) dy \\
&\ge \psi (x) \frac{1}{2C_1 } \int_{B(q,1) } |y|^{\beta } 
 \exp \Big( -\frac{|y|^2}{4\gamma } \Big) dy.
\end{align*}
Since $|y|^{\beta} \exp(-|y|^2 /4\gamma )$ is radially symmetric and
 $|q|<2/\sqrt{d} $, we have
\begin{align*}
\int_{B(q,1)}  |y|^{\beta } \exp \Big( -\frac{|y|^2}{4\gamma } \Big) dy
&=\int_{B(\tilde{q},1)}  |y|^{\beta } \exp \Big( -\frac{|y|^2}{4\gamma } \Big) dy \\
&\ge \int_{B(0,1)\cap [0,1]^d }  |y|^{\beta } \exp 
\Big( -\frac{|y|^2}{4\gamma } \Big) dy,
\end{align*}
where $\tilde{q}=\big( \frac{|q|}{\sqrt{d} } , \frac{|q|}{\sqrt{d} },\ldots , 
\frac{|q|}{\sqrt{d} } \big)$. We thus obtain
\begin{align*}
	g^{\gamma }_{0,\beta } (x,t) 
&\ge \psi (x) \frac{1}{2C_1 } \int_{B(0,1) \cap [0,1]^d } |y|^{\beta } 
\exp \Big( -\frac{|y|^2}{4\gamma } \Big) dy \\
&\ge \psi (x) \frac{1}{2^{d+1} C_1 } \int_{B(0,1)} |y|^{\beta } 
\exp \Big( -\frac{|y|^2}{4\gamma } \Big) dy.
\end{align*}
Let
\begin{equation}
\begin{aligned}
C_3 
& =\min_{\beta =0,1,2} \frac{2^{-1-d} }{C_1 } \int_{B(0,1)} |y|^{\beta } 
 \exp \Big( -\frac{|y|^2}{4\gamma } \Big) dy \\
&=\frac{2^{-1-d}}{C_1 } \int_{B(0,1)} |y|^2 
\exp \Big( -\frac{|y|^2}{4\gamma } \Big) dy.
	\end{aligned}
	\end{equation}
Then we have
\[
	g^{\gamma }_{0,\beta } (x,t) \ge C_3 \psi (x)
\]
for any $x\in \mathbb{R}^d $.

We next prove $g^{\gamma }_{0,\beta } (x,t) \le C_4 \psi (x)$ with some 
$C_4 >0$. In addition to satisfying  \eqref{Lem-2-3-gamma}
we may assume that
\[
	\frac{1}{4(T+\gamma )} -a>0.
\]
We thus see that from \eqref{A-psi-frac2},
\[
	g^{\gamma }_{0 ,\beta } (x,t)\le C_2 \psi (x)
\int_{\mathbb{R}^d } |y|^{\beta } \exp \Big\{ -\Big( \frac{1}{4(T+\gamma )} -a 
\Big) |y|^2 \Big\} dy
\]
for $t\in[0,T]$. By putting
\[
	C_4 =\max_{\beta =0,1,2} C_2 \int_{\mathbb{R}^d } |y|^{\beta } 
\exp \Big\{ -\Big( \frac{1}{4(T+\gamma )} -a \Big) |y|^2 \Big\} dy,
\]
we obtain
$g^{\gamma }_{0 ,\beta } (x,t)\le C_4 \psi (x)$  for $t\in[0,T]$.
\end{proof}

\begin{proof}[Proof of Proposition \ref{Prop1Umeda}]
By a direct calculation we have
\begin{equation}
\begin{aligned}
	W_t -\Delta W +W^{-p} 
& =-\varphi' +C \varphi' \partial_t g^{\gamma }_{-\alpha ,0} \\
&\quad -C \varphi' \Delta g^{\gamma }_{-\alpha ,0}
 -|C \nabla g^{\gamma }_{-\alpha ,0} |^2 \varphi'' +\varphi^{-p} .
\end{aligned}
\end{equation}
 From \eqref{gggggg2}-\eqref{gggggg3} we have
\begin{equation} \label{UProp41-1}
	(\partial_t -\Delta ) g^{\gamma }_{-\alpha ,0} 
=\Big( \alpha +\frac{d}{2} \Big) g^{\gamma }_{-\alpha+1 ,0} .
\end{equation}
Since
\begin{gather}
	\varphi'  =\varphi^{-p} , \label{phi1diff} \\
	\varphi'' =-p \varphi^{-2p-1} \label{phi2diff}
\end{gather}
and
\[
	\varphi^{p+1} =(p+1)(T-t+Cg^{\gamma }_{-\alpha ,0} ) 
\ge (p+1)(Cg^{\gamma }_{-\alpha ,0} ),	
\]
we obtain
\begin{equation}\label{UProop41-2}
	|C\nabla g_{-\alpha ,0}^{\gamma } |^2 \varphi'' 
\le \frac{C|p||\nabla g_{-\alpha ,0}^{\gamma } |^2 }{(p+1)
g_{-\alpha ,0}^{\gamma} } \varphi' .
\end{equation}
Lemma  \ref{Lem3Umeda}, \eqref{gggggg1} and \eqref{ggaa} yield
\begin{equation}\label{UProp41-3}
	|C\nabla g_{-\alpha ,0}^{\gamma } |^2 \varphi'' 
\le \frac{C|p|d(g_{-\alpha +1 ,1}^{\gamma} )^2 }{4(p+1) g_{-\alpha ,0}^{\gamma } }
 \varphi' \le \frac{C|p|d C_4^2 (t+\gamma )^{\alpha -2} \psi }{4C_3 (p+1)  } 
\varphi' .
\end{equation}
 From \eqref{phi1diff}, \eqref{UProp41-1}, \eqref{UProp41-3} and Lemma  
\ref{Lem3Umeda}, we have
\begin{align*}
	W_t -\Delta W+W^{-p} 
& \ge \Big\{ C_3 (t+\gamma )^{\alpha -1} \Big( \alpha +\frac{d}{2} \Big) 
- \frac{C|p|d C_4^2 (t+\gamma )^{\alpha -2}  }{4C_3 (p+1)  } \Big\} \varphi' \\
& \ge \Big\{ C_3 \gamma \Big( \alpha +\frac{d}{2} \Big) 
 - \frac{C|p|d C_4^2 }{4C_3 (p+1)} \Big\} (t+\gamma )^{\alpha -2} \varphi' .
\end{align*}
If $\alpha $ satisfies
\begin{equation}\label{labelalpha0}
	\alpha \ge \alpha_1 \equiv \max 
\Big\{ \frac{ C C_4^2 |p|d }{4C_3^2 \gamma (p+1) } -\frac{d}{2} ,\frac{1}{2}
 \Big\} >0,
\end{equation}
then $W$ is a supersolution of \eqref{CP-u} in $\mathbb{R}^d \times (0,T)$.
\end{proof}

\begin{proof}[Proof of Proposition \ref{Prop2Umeda}]
As before, for $\varphi =\varphi (T-t+Cg^{\gamma }_{\alpha ,0} (x,t))$ we have
\begin{equation}
\begin{aligned}
	w_t -\Delta w +w^{-p}
	&=-\varphi' +C \varphi' \partial_t g_{\alpha ,0}
 -C \varphi' \Delta g^{\gamma }_{\alpha ,0}
 -|C\nabla g^{\gamma }_{\alpha ,0} |^2 \varphi'' +\varphi^{-p}  \\
	&\le \frac{C\partial_t g^{\gamma }_{\alpha ,0} }{\varphi^p }
 +\frac{C|\Delta g^{\gamma }_{\alpha ,0} |}{\varphi^p }
 +\frac{|C p \nabla g^{\gamma }_{\alpha ,0} |^2 }{\varphi^{2p+1} }
\end{aligned} \label{Ew-ineq}
\end{equation}
by \eqref{phi1diff}-\eqref{phi2diff}. It is easily seen that
\begin{equation}\label{phi-g-est}
	\varphi^{p+1} =(p+1)(T-t+Cg^{\gamma }_{\alpha ,0} )
\ge (p+1)(Cg^{\gamma }_{\alpha ,0} ).
\end{equation}
 From Lemma  \ref{Lem3Umeda}, \eqref{gggggg1}, \eqref{ggaa} and
\eqref{phi-g-est}, it follows that
\begin{equation}\label{galpha-0-ineq}
	\big|\frac{\nabla g^{\gamma }_{\alpha ,0} }{\varphi^{p+1} } \big|
\le \frac{ \sqrt{d} g^{\gamma }_{0 ,1} }{ 2(p+1)(t+\gamma )g^{\gamma }_{0, 0} }
\le \frac{C_4 \sqrt{d} }{ 2\gamma (p+1)C C_3 } .
\end{equation}
Substituting \eqref{galpha-0-ineq} for \eqref{Ew-ineq}, and using
\eqref{gggggg2}-\eqref{ggaa}, we have
\begin{align*}
&w_t -\Delta w +w^{-p}  \\
&\le \frac{C(p+1)}{ (t+\gamma )^{ \alpha +2} \varphi }
\Big[ g^{\gamma }_{0,2} +(t+\gamma )\Big\{ -2\alpha g^{\gamma }_{0,0}
+g^{\gamma }_{0,0} +\frac{C_4 \sqrt{d} |p| g^{\gamma }_{0,1} }{4C_3 (p+1) }
\Big\} \Big] \\
&\le \frac{C(p+1)\psi }{(t+\gamma )^{\alpha +2} \varphi }
\Big[ -2\alpha \gamma C_3 +C_4 \Big\{ 1+ (T+\gamma )
 \Big(1 +\frac{C_4 d|p| }{4 C_3 (p+1) } \Big) \Big\} \Big]
\end{align*}
in $\mathbb{R}^d \times [0,T]$. If $\alpha $ satisfies
\begin{equation}\label{labelalpha}
	\alpha \ge \alpha_2 \equiv \frac{C_4 }{2\gamma C_3 }
\Big\{ 1+ (T+\gamma ) \Big(1 +\frac{C_4 d|p| }{4 C_3 (p+1) } \Big) \Big\} ,
\end{equation}
then $w$ is a subsolution of \eqref{CP-u} in $\mathbb{R}^d \times (0,T)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{ThmUmeda}]

Let  $c_1 =c_1 (C_2 ,a,\gamma ,\alpha ) $ and $c_2 =c_2 (C_1 ,\gamma  ,\alpha ) $ 
be positive constants such that
\[
	g^{\gamma }_{\alpha ,0} (x,0)\le c_1 \psi (x) , \quad 
g^{\gamma }_{-\alpha , 0} (x,0) \ge c_2 \psi (x)
\]
with $\alpha > 0$ as in  Lemma  \ref{Lem3Umeda} and \eqref{ggaa}. Hence
\[
	  m^{p+1} +C_l g^{\gamma }_{\alpha ,0} (x,0) \le u_0^{p+1} (x) 
\le m^{p+1}+ C_h g^{\gamma }_{-\alpha ,0} (x,0)
\]
with $C_l =C_I /c_1 $ and $C_h =C_{II} /c_2 $.

Since $m^{p+1} =(p+1)T$ by \eqref{solv1}, we have
\begin{align*}
	w(x,0)	&=  \{ (p+1)T+C_l g^{\gamma }_{\alpha ,0} (x,0) \}^{1/(p+1)} 
 \le u_0 (x)  \\
& \le \{ (p+1) T+C_h g^{\gamma }_{-\alpha ,0} (x,0) \}^{1/(p+1)} = W(x,0),
\end{align*}
where $W$ and $w$ are defined in \eqref{prof-w} with $C=C_h /(p+1)$ and 
\eqref{astwwwww} with $C=C_l /(p+1)$.
Propositions  \ref{Prop1Umeda},  \ref{Prop2Umeda} and the comparison 
principle (see Theorem  \ref{comparison-C}) yield
\[
	w(x,t) \le u(x,t) \le W(x,t) \quad \text{in } \mathbb{R}^d \times [0,T).
\]
We thereby get
\[
\{ C_l g^{\gamma }_{\alpha ,0} (x,T) \}^{1/(p+1) } \le u (x,T) 
\le \{ \mathstrut C_h g^{\gamma }_{-\alpha ,0} (x,T) \}^{1/(p+1)} .
\]
Taking  $C=( C_l C_3 )^{1/(p+1)} $ and $C' =(C_h C_4 )^{1/(p+1) }$,
by  Lemma  \ref{Lem3Umeda}
\[
	 C\psi^{1/(p+1)} (x) \le u(x,T) \le C' \psi^{1/(p+1)} (x).
\]
Choosing
\[
	\gamma =\frac{1}{8a}-\frac{T}{2} , \quad 
\alpha =\max \{ \alpha_1 ,\alpha_2 \}
\]
with $\alpha_1$ in \eqref{labelalpha0} and $\alpha_2 $ as in 
\eqref{labelalpha}, we see that  $C$ depends only on 
$C_1 $, $C_2 $, $a$, $T$, $C_I$, and $C'$ does only on 
$C_1 $, $C_2 $, $a$, $T$, $C_{II}$.
\end{proof}

\section{Appendix A: Existence and uniqueness of the classical solution and comparison principle}

In this section we prove that Cauchy problem \eqref{CP-u}-\eqref{CP-u0} 
with conditions \eqref{H1}-\eqref{H2} has a unique positive classical solution, 
as well as a comparison principle.

First, we consider the local existence and uniqueness of the classical 
solution for the problem in time.
We know that the solution of \eqref{CP-u}-\eqref{CP-u0} satisfies the 
integral equation:
\begin{equation}\label{zeq:2-7}
	u(x,t) =e^{t \Delta } u_0 (x) -\int_0^t e^{(t-s) \Delta } u^{-p} (x,s)ds,
\end{equation}
where
\begin{equation}\label{zeq:GGGDelta}
	e^{t \Delta } \xi (x)=\int_{ \mathbb{R}^d } G(x-y,t) \xi (y)dy
\end{equation}
with $G(x,t)$ defined in \eqref{G-Senbei} and a measurable function $\xi $.
The function  $\Xi (x,t)=e^{t\Delta } \xi (x)$ is the unique solution to
\begin{gather*}
	\Xi_t =\Delta \Xi ,	 \quad x\in \mathbb{R}^d , t>0, \\
	\Xi (x,0)= \xi (x) , \quad 	x\in \mathbb{R}^d .
\end{gather*}
Now we consider the  existence in time of a local solutions to \eqref{zeq:2-7}.

\begin{lemma} \label{Existence} 
Assume that $u$ is the solution of \eqref{zeq:2-7}, where $p>-1$, $p\neq 0$, 
$u_0$ is bounded continuous, $\inf_{x\in \mathbb{R}^d} u_0 (x) =m>0$ and 
$\sup_{x\in \mathbb{R}^d } u_0 (x) =M <\infty $. Then the solution 
satisfying $u\ge m/q$ exists in $\mathbb{R}^d \times (0,T)$ with
\begin{equation}\label{p-q-value}
	T<T_{m,M} (q) \equiv 
		\begin{cases}
		\min \{ q-1, \frac{1}{p} \} ( \frac{m}{q} 
 )^{p+1} ,	& p>0, \\
		\min \{ (q-1)M^p, \frac{1}{|p|} ( \frac{m}{q} )^p  \} \frac{m}{q}  ,
 	& -1<p<0,
		\end{cases}
\end{equation}
where $q=\bar{q} =\bar{q} (m,M)$ is a positive solution of
\begin{equation}\label{zqsol}
\begin{gathered}
		q-1=\frac{1}{p},\quad p>0, \\
		(q-1)M^p =\frac{1}{|p|} ( \frac{m}{q} )^p ,	\quad -1<q<0.
\end{gathered}
\end{equation}
Moreover, the solution is a unique classical solution of 
\eqref{CP-u}-\eqref{CP-u0}.
\end{lemma}

\begin{proof}
First we show the  existence and the uniqueness of the local solution 
of \eqref{zeq:2-7} by a fixed-point theorem.
Define
\begin{equation}\label{zeq:3-1}
	\Psi (u)=e^{t \Delta } u_0 (x)-\int_0^t e^{(t-s) \Delta } u^{-p} (x,s)ds.
\end{equation}
Set
\[
	E_T =\{ u:[0,T] \to L^{\infty} (\mathbb{R}^d) ;\| u \|_{E_T} <\infty  \}
\]
with the norm
\[
	\| u \|_{E_T} =\sup_{t\in [0,T] } \|u(\cdot ,t)\|_{L^{\infty} (\mathbb{R}^d ) } .
\]
Define
\[
	B_{m/q}^M =\big\{ u\in E_T ; \inf_{(x,t)\in \mathbb{R}^d \times [0,T] } 
u(x,t) \ge m/q, \ \| u \|_{E_T } \le M \big\}.
\]
with some $q>1$. Let $u\in B_{m/q}^M $. Then, for $p>0$,
\begin{align*}
	\Psi (u)
& =e^{t \Delta } u_0 (x)-\int_0^t e^{(t-s) \Delta } u^{-p} (x,s)ds \\
& \ge m-\int_0^T \big( \frac{q}{m} \big)^p ds =m-\big( \frac{q}{m} \big)^p T,
\end{align*}
and for $-1<p<0$,
\[
	\Psi (u) \ge m-\int_0^T M^{-p} ds =m-M^{-p} T.
\]
If $T$ satisfies \eqref{p-q-value}, then $\Psi(u) >m/q$ and $\Psi$ is a
 mapping from $B_{m/q}^M$ to itself.

Rest of the proof we should show that
\[
	\| \Psi (u_1 )-\Psi (u_2 ) \|_{E_T } \le C \| u_1 -u_2 \|_{E_T }
\]
with some $C\in (0,1)$.
For $u_1 ,u_2 \in B_{m/q}^M$,
\[
	|\Psi (u_1 )-\Psi (u_2 )|  
\le \int_0^t e^{(t-s) \Delta } |(u_1 )^{-p} (x,s)-(u_2 )^{-p} (x,s)|ds.
\]
By the mean value theorem
\[
	u_1^{-p} -u_2^{-p} =-\int_0^1 p \{ \theta u_1 
+(1-\theta) u_2 \}^{-p-1} d\theta (u_1 -u_2).
\]				
Since $u_1 $, $u_2 \ge m/q$, we obtain
\[
	\| \Psi (u_1 )- \Psi (u_2 ) \|_{E_T} \le T |p| 
\big( \frac{q}{m} \big)^{p+1} \| u_1 -u_2 \|_{E_T} <C\| u_1- u_2 \|_{E_T}
\]
with $C\in (0,1)$ for
\begin{equation}\label{zeq:3-3}
	T<\frac{m^{p+1}}{|p| q^{p+1}} .
\end{equation}
We thus see that if $T$ satisfies \eqref{p-q-value} and \eqref{zeq:3-3}, then
 $\Psi$ is a contraction map in $B_{m/q}^M$ and have one fixed point 
in $B_{m/q}^M$ for $(x,t)\in \mathbb{R}^d \times (0,T)$. 
Thus \eqref{zeq:2-7} has a unique solution in $\mathbb{R}^d \times (0,T)$.
 Since $q>1$ is arbitrary, we may let $q=\bar{q} $.


To complete the proof of Lemma  \ref{Existence}, let $u(x,t)$ be the 
nonnegative and bounded solution of \eqref{zeq:2-7} that has been obtained 
in $\mathbb{R}^d \times [0,T)$ for some $T>0$. By \eqref{zeq:2-7}, $u(x,t)$
is continuous in $\mathbb{R}^d \times [0,T)$. Moreover, by considering the
 difference quotients  $\{ u (x_1 ,x_2 ,\ldots ,x_{j-1} ,x_j +h ,x_{j+1} ,
\ldots ,x_d ,t)-u (x,t) \} /h $ with $h \to 0$, one easily sees that
 $\partial u (x,t)/\partial x_j $ is locally bounded in 
$\mathbb{R}^d \times [\tau ,T) $ for $j=1,2,\ldots ,d$ and any $\tau$ 
such that $0<\tau <T$. Then, since $u\ge m/q >0$, $u^{-p} $ are locally
 H\"older continuous functions in space uniformly with respect to time. 
It then follows from the representation formula \eqref{zeq:2-7} that $u$ 
is a classical solution of \eqref{CP-u}-\eqref{CP-u0} in 
$\mathbb{R}^d \times (0,T)$ with \eqref{H1}-\eqref{H2} 
(see \cite[Chapter 1, Theorem 10]{F}).
\end{proof}

\begin{remark}\label{Existence-rem}\rm
Note that $T_{m,M} (q)$ defined in \eqref{p-q-value} has a maximum number 
at $q=\bar{q}$. For $p>0$ it is clear that $\bar{q} =(p+1)/p$. 
On the other hand, for $-1<p<0$, $\bar{q} =\lim_{n \to \infty } q_n $ 
$(<\infty)$ with the sequence $\{ q_n \}_{n=1}^{\infty } $ satisfying
\[
	q_1 =1 , \qquad q_{n+1} =1+\frac{1}{|p|} 
\big( \frac{M}{m} \big)^{-p} q_n^{-p} \quad \text{ for } n\in {\bf N} .
\]	
\end{remark}


Next, we  recall a comparison result for the solution existing 
locally in time.

\begin{theorem}\label{comparison-C} 
Let $D \subset \mathbb{R}^d $.
Assume $u_3 (x,t)$, $u_4 (x,t) \in C^{2,1} (D \times (0,T))$ satisfy the 
partial differential inequalities
\begin{gather}
	(u_3 )_t \ge  \Delta u_3 -u_3^{-p}, \label{comp1} \\
	(u_4 )_t \le  \Delta u_4 -u_4^{-p}  \label{comp2}
\end{gather}
with $u_3 (x,0) =u_{3,0} (x)$ and $u_4 (x,0)=u_{4,0} (x)$, where $u_{3,0} $ 
and $u_{4,0} $ are continuous in $D$. Assume that $u_4$ is bounded in 
$D \times [0,T') $ for any $T' <T$. Assume that $u_3 $ is bounded from below 
in $D \times [0,T') \}$ for any $T'<T$.
For $D \neq \mathbb{R}^d $, if $u_{3,0} \ge u_{4,0} $ in $D$ and $u_3 \ge u_4 $ 
on $\partial D$, then $u_3 \ge u_4 $ in $D \times (0,T)$.
On the other hand, for $D=\mathbb{R}^d $, if $u_{3,0} \ge u_{4,0} $ in 
$\mathbb{R}^d $, then $u_3 \ge u_4 $ in $\mathbb{R}^d \times (0,T)$.
\end{theorem}

\begin{proof}
The proof is based on an maximum principle for a parabolic equation 
(see \cite{PW}) and is standard (see \cite{IKO}, \cite{GGS}, \cite{ST} 
and \cite{GU3}).
\end{proof}

Next we introduce more strong result for the comparison principle.

\begin{theorem}\label{comparison-C2} 
Assume the same hypotheses as in Theorem  \ref{comparison-C} 
with $D=\mathbb{R}^d $. If $u_{3,0} \not\equiv u_{4,0} $ in 
$\mathbb{R}^d $, then $u_3 >u_4 $ in $\mathbb{R}^d \times (0,T)$.
\end{theorem}

\begin{proof}
Let $\tilde{w}=u_3 -u_4 $. From Theorem  \ref{comparison-C}, 
we see that $\tilde{w}\ge 0$. For $t\in [0,T' )$ with $T' <T$,
\[
	\tilde{w}_t = \Delta \tilde{w} -u_3^{-p} -u_4^{-p} 
		= \Delta \tilde{w} +b(x,t) \tilde{w}
\]
with $b$ defined by
\begin{equation}\label{bxtu3u4}
b(x,t)=\int_0^1 p\{ u_3 (x,t)+\theta (u_4 (x,t)-u_3 (x,t)) \}^{-p-1} d\theta .
\end{equation}
Note that $b(x,t)$ is bounded in $\mathbb{R}^d \times (0,T')$.
Put $\tilde{b} (t)=\inf_{x\in \mathbb{R}^d } b(x,t) $ and
\[
	\tilde{W} (x,t) =\exp \big\{ -\int_0^t \tilde b(s) ds \big\} \tilde{w} (x,s).
\]
Since $b(x,t)\ge \tilde{b} (t)$ and $\tilde{W} \ge 0$, we see that
\[
	\tilde{W}_t = \Delta \tilde{W} +(b(x,t)-\tilde b(t))\tilde{W}  
\ge \Delta \tilde{W} \]
and
\begin{equation}\label{0notequiv0}
	\tilde{W} (x,0)=u_0 (x) -m \; (\ge 0, \ \not\equiv 0).
\end{equation}
By the basic comparison principle and \eqref{0notequiv0}, we get
\[
	\tilde{W} (x,t)\ge e^{t\Delta } ( u_0 (x)-m ) >0 \quad\text{in } 
\mathbb{R}^d \times (0,T') ,
\]
where $e^{t\Delta } $ is defined in \eqref{zeq:GGGDelta}.
Since $T'<T$ is arbitrary, we obtain
\[
	\tilde{W} (x,t)>0 \quad \text{in } \mathbb{R}^d \times (0,T) .
\]
We thus see that
\begin{gather*}
	\tilde{w}(x,t)>0 \text{ in } \mathbb{R}^d \times (0,T) , \\
 u_3 (x,t)>u_4 (x,t) \text{ in } \mathbb{R}^d \times (0,T) .
\end{gather*}
\end{proof}

Finally, we show the existence of solutions for 
problem \eqref{CP-u}-\eqref{CP-u0} in $\mathbb{R}^d \times (0,T(m))$.

\begin{theorem}\label{Existence-C} 
Problem \eqref{CP-u}-\eqref{CP-u0} with initial data satisfying 
\eqref{H1}--\eqref{H2} has a unique classical solution
in $\mathbb{R}^d \times (0,T(m))$.
\end{theorem}

\begin{proof}
It is clear for the case $p=0$ and the case $u_0$ is a constant 
(in the case $m=M$). We consider the other case such as
\begin{gather}
	p>-1,\ p\neq 0, \label{othercase1} \\
	m<M, \label{othercase2}
\end{gather}
where $m=\inf_{x\in \mathbb{R}^d } u_0(x) $ and 
$M=\sup_{x\in \mathbb{R}^d } u_0(x) $.

 From Lemma  \ref{Existence}, \eqref{p-q-value}, \eqref{zqsol} and 
Remark  \ref{Existence-rem}, the problem has a unique classical 
solution in $\mathbb{R}^d \times (0,T_1 ]$, where
\begin{equation}\label{tee-1}
	T_1 = \begin{cases}
		(1-\varepsilon) (\tilde{q}-1) \big( \frac{m}{\tilde{q} } \big)^{p+1} ,	
 & p>0, \\
		\frac{\tilde{q} -1}{\tilde{q} } M^p m  ,	& -1<p<0 ,
		\end{cases}
\end{equation}
with
\begin{equation}\label{tilde-q}
\tilde{q} =1+\frac{1}{|p|}
\end{equation}
and some $\varepsilon \in (0,1)$.
Note that the fact $\tilde{q} \le \bar{q} $ means $T_1 < T_{m,M} (\tilde{q} )$, 
where $T_{m,M} $ and $\bar{q} $ are defined in \eqref{p-q-value} and 
\eqref{zqsol}.

Let $v_m (t)$ and $v_M (t)$ be solutions of \eqref{eq:GSU-ODE} with initial 
data $v_m (0)=m$ and $v_M (0)=M$.
Theorem  \ref{comparison-C} yields
\begin{equation}\label{comparison-1st}
	v_m (t) \le u(x,t) \le v_M (t) \quad \text{in } 
\mathbb{R}^d \times (0,T_1 ] .
\end{equation}
Let $\{ m_n \}_{n=0}^{\infty }$ and $\{ M_n \}_{n=0}^{\infty }$ be such that 
$m_0 =m$, $M_0 =M$,
\begin{equation}\label{m-both}
	m_{n+1} = \begin{cases}
		m_n  \big\{ 1-\frac{p^p (1-\varepsilon)}{(p+1)^p } \big\}^{1/(p+1)} ,	
 & p>0, \\
		m_n  \big\{ 1- \big( \frac{m_n }{M_n } \big)^{-p} \frac{1+p}{1-p } 
\big\}^{1/(p+1)} ,	& -1<p<0
		\end{cases}
\end{equation}
and
\begin{equation}\label{M-both}
	M_{n+1} = \begin{cases}
		M_n  \big\{ 1-\frac{p^p (1-\varepsilon)}{(p+1)^p } 
\big( \frac{m_n }{M_n } \big)^{p+1} \big\}^{1/(p+1)} ,	& p>0, \\
		M_n  \big\{ 1- \frac{m_n }{M_n } \frac{1+p}{1-p } \big\}^{1/(p+1)} ,	
& -1<p<0.
		\end{cases}
\end{equation}
Then, \eqref{solv2}, \eqref{tee-1} and \eqref{tilde-q} yield $v_m (T_1)=m_1$ 
and $v_M(T_1 )=M_1$. Moreover, from \eqref{comparison-1st} we obtain
 $m_1 \le u(x,T_1 ) \le M_1 $.

Next, we consider
\begin{gather*}
	(u_1 )_t=\Delta u_1  -u_1^{-p} ,\quad x\in \mathbb{R}^d, t>0, \\
	u_1 (x,0)=u(x,T_1 ),	\quad x\in \mathbb{R}^d .
\end{gather*}
By the same argument as above, we see that the problem has a unique 
classical solution in $\mathbb{R}^d \times (0,T_2 ]$, where the sequence 
$\{ T_n \}_{n=1}^{\infty }$ is defined by
\begin{equation}\label{tn-both}
	T_n = \begin{cases}
		(1-\varepsilon) (\tilde{q}-1) \big( \frac{m_{n-1} }{\tilde{q} }
 \big)^{p+1} ,	& p>0, \\
		\frac{\tilde{q} -1}{\tilde{q} } M_{n-1}^p m_{n-1}   ,	& -1<p<0
		\end{cases}
\end{equation}
with $\tilde{q}$ defined in \eqref{tilde-q}.

It is known that $u_1 (x,t) =u(x,T_1 +t)$. We thus see that 
\eqref{CP-u}-\eqref{CP-u0} has a unique classical solution in
 $\mathbb{R}^d \times (0,\tilde{T}_2 ]$ with
\[
	\tilde{T}_n =\sum_{k=1}^{n} T_k \quad \text{ for } k\in {\bf N}.
\]
Moreover we have $v_m (\tilde{T}_2)=m_2$, $v_M(\tilde{T}_2)=M_2$ and 
$m_2 \le u(x,\tilde{T}_2 ) \le M_2 $.

For $n\in {\bf N}$, by using same argument as above $n-2$ more times,
 we see that
\begin{equation}
\text{\eqref{CP-u}-\eqref{CP-u0} has a unique classical solution in }
 \mathbb{R}^d \times (0,\tilde{T}_n ] . \label{Tnunlim}
\end{equation}
Moreover we have $v_m (\tilde{T}_n )=m_n$, $v_M(\tilde{T}_n )=M_n$ and 
$m_n \le u(x,\tilde{T}_n ) \le M_n $. Note that $n\in {\bf N} $ is arbitrary 
for these results.

Finally we should show that
\begin{equation}\label{limtildetntm}
	\lim_{n \to \infty } \tilde{T}_n =T(m).
\end{equation}
Since $v_m (t)$ is decay function with respect to $t$ and $v_m (\tilde{T}_n ) =m_n >0$ for any $n\in {\bf N} $, we only should prove
\begin{equation}\label{limTm}
	\lim_{n\to \infty } v_m (\tilde{T}_n ) =\lim_{n\to \infty } m_n =0.
\end{equation}
Put $\mu_n =m_n / M_n $. From \eqref{othercase2} we get
\begin{equation}\label{othercase2lim}
	\sup_{n \in {\bf N} } \mu_n =\sup_{n\in {\bf N} } \frac{v_m (\tilde{T}_n)}{v_M (\tilde{T}_n)} \le \sup_{t\in [0, T(m)) } \frac{v_m (t)}{v_M (t)} <1.
\end{equation}
 From \eqref{m-both} and \eqref{M-both} we see that
\begin{equation}\label{mu-both}
	\mu_{n+1} = \begin{cases}
		\mu_n  \Big\{ \frac{ 1-\frac{p^p (1-\varepsilon)}{(p+1)^p } }
 { 1-\frac{p^p (1-\varepsilon)}{(p+1)^p } \mu_n^{p+1} } \Big\}^{1/(p+1)} ,	& p>0,
 \\
		\mu_n  \Big\{ \frac{ 1- \mu_n^{-p} \frac{1+p}{1-p } }
{ 1- \mu_n \frac{1+p}{1-p } } \Big\}^{1/(p+1)} ,	& -1<p<0 .
		\end{cases}
\end{equation}
Since $\{ \mu_n \}_{n=0}^{\infty }$ is a decrasing sequence with respect to $n$,
by the monotone convergence theorem and \eqref{othercase2lim}
we have $\lim_{n \to \infty } \mu_n =b$ with some $b\in[0,1)$.
We cleam that $\lim_{n\to \infty} \mu_n = 0$.
By contraries, assume that $\lim_{n\to \infty } \mu_n =b>0$.
Then from \eqref{othercase1} we obtain that $b=1$ and a 
contradiction occurs.
We thus  obtain $\lim_{n \to \infty } \mu_n =0$. 
This means \eqref{limTm} and \eqref{limtildetntm}.
Since \eqref{Tnunlim} holds for any $n\in {\bf N}$, \eqref{CP-u}-\eqref{CP-u0} 
has a unique classical solution in $\mathbb{R}^d \times (0,T(m))$.
\end{proof}


\section{Appendix B: Existence and regularity of the solution at $t=T(m)$}

In this section we prove Lemma  \ref{SU-Lem41}. Let $T=T(m)$.

\begin{proof}[Proof of Lemma \ref{SU-Lem41}] 
 From Theorem  \ref{ThmQSI}, for any $a\in \mathbb{R}^d $ and any 
$R\in(0,\sqrt{T})$, there exist constants $C_1 =C_1 (a,R)>0$ and
 $C_2 =C_2 (a,R)>C_1$ such that
\begin{equation}
	C_1 <u(x,t)<C_2 \quad \text{for } (x,t)\in Q(R) , \label{appe01}
\end{equation}
where $Q(r) =Q(r,a) =B(a,r) \times (T-r^2 ,T)$. Then there exists 
constant $C_3 =C_3 (a,R,p)$ such that
\[
	\|u\|_{L^q (Q(R))} \le C_3 ,\quad  \|u^{-p} \|_{L^q (Q(R))} \le C_3 .
\]
From \cite[Theorem 6.4.2]{K}, we have
\[
	\| u \|_{W_q^{2,1} (Q(r_1) )}\le C_4 \left( \| u^{-p} \|_{L_q (Q(R))} 
+ \| u \|_{L_q (Q(R))} \right) \le C_5
\]
with constants  $q>(d+2)/2$, $C_4 =C_4 (d,q) <\infty $,  
$C_5=C_5 (a,R,p,d,q)< \infty $, and $r_1 =R/2$.
 From \cite[II, Lemma 3.3]{LSU} we obtain
\[
	\|u\|_{C^{\alpha} (Q(r_1)) } \le C_6 (\|u\|_{W_q^{2,1} (Q(r_1 ) )} 
+\| u \|_{L_q (Q(r))} ) \le C_7
\]
with $\alpha \in (0,1)$, $C_6 =C_6 (d,q,\alpha ) <\infty$ and 
$C_7 =C_7 (a,R,p,d,q,\alpha) <\infty $. From \eqref{appe01} we see that
\[
	\|u^{-p} \|_{C^{\alpha } (Q(r_1 ) )}\le C_8
\]
with $C_8 =C_8 (a,R,p,d,q,\alpha) <\infty $. 
From \cite[Chapter 3, Theorem 5]{F} we get
\begin{equation}
	\| u \|_{C^{2+\alpha } (Q(r_2 ) )} \le C_9 (\| u \|_{C^{\alpha } (Q(r_1 ) )} 
+\|u^{-p} \|_{C^{\alpha } (Q(r_1 ) )} ) \le C_{10} \label{appe02}
\end{equation}
with $r_2 =r_1 /2 =R/4$, $C_9 =C_9 (a,R,d,\alpha )$ and 
$C_{10} =C_{10} (a,R,p,d,q,\alpha )$.


Next put $u_1=\Delta u$. Then we see that
\[
	(u_1 )_t =\Delta u_1 +pu^{-p-1} u_1 -p(p+1) u^{-p-2} |\nabla u|^2.
\]
 From \eqref{appe01} and \eqref{appe02} we have
\begin{gather*}
	\| u_1 \|_{C^{\alpha } (Q(r_2 ) )} \le C_{10},\\
\|pu^{-p-1} u_1 -p(p+1) u^{-p-2} |\nabla u|^2 \|_{C^{\alpha } (Q(r_2 ) )} 
\le C_{11}
\end{gather*}
with $C_{11} =C_{11} (a,R,p,d,q,\alpha )$. From \cite{F} again,
\[
	\|u_1 \|_{C^{2+\alpha } (Q(r_3 ) )} 
=\|u \|_{C^{4+\alpha } (Q(r_3 ) )} \le C_{12}
\]
with $r_3=r_2 /2=R/8 $ and $C_{12} =C_{12} (a,R,p,d,q,\alpha )$.

Iterating this argument $n-1$ times, we have
\[
	\|u \|_{C^{2n+2+\alpha } (Q(r_n ) )} \le C_{13}
\]
with $r_n =R/2^n $ and $C_{13} =C_{13} (a,R,p,d,q,\alpha ,n )$. 
We thus see that
\[
	\Delta^n u_t (x,t)\le C_{14} \text{ for } (x,t)\in Q(r_n )
\]
with $C_{14} =C_{14} (a,R,p,d,q,n )$. By integrating $\Delta^n u_t (a,t)$ 
from $T-r_n /2$ to $T$ with respect to $t$ and subtracting 
$\Delta^n u_t (a,T-r_n /2)$, we obtain $\Delta^n u (a,T)$. Thus we see 
that $\Delta^n u (a,T)$ exists.

Since $a\in \mathbb{R}^d $ and $n\in {\bf N} $ are arbitrary, we obtain 
$u(\cdot ,T) \in C^{\infty } (\mathbb{R}^d )$.
\end{proof}

\subsection*{Acknowledgments}
The author is grateful to anonymous referees for their valuable comments 
on this paper.

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\end{document}