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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 131, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/131\hfil Quenching for diffusion equations]
{Quenching for singular and degenerate quasilinear diffusion equations}

\author[Y. Nie, C. Wang, Q. Zhou \hfil EJDE-2013/131\hfilneg]
{Yuanyuan Nie, Chunpeng Wang, Qian Zhou}  % in alphabetical order

\address{Yuanyuan Nie \newline
 School of Mathematics, Jilin University, Changchun
130012, China}
\email{nieyuanyuan@live.cn}

\address{Chunpeng Wang \newline
 School of Mathematics, Jilin University, Changchun
130012, China}
\email{wangcp@jlu.edu.cn}

\address{Qian Zhou \newline
 School of Mathematics, Jilin University, Changchun
130012, China}
\email{zhouqian@jlu.edu.cn}

\thanks{Submitted January 23, 2013. Published May 29, 2013.}
\subjclass[2000]{35K65, 35K67, 35B40}
\keywords{Quench; singular; degenerate}

\begin{abstract}
 This article concerns the quenching phenomenon of the solution to
 the Dirichlet problem of a singular and degenerate quasilinear diffusion
 equation. It is shown that there exists a critical length for the special
 domain in the sense that the solution exists globally in time if
 the length of the special domain is less than this number  while the
 solution quenches if the length is greater than this number.
 Furthermore, we also study the quenching properties for the quenching solution,
 including the location of the quenching points  and the blowing up of the
 derivative of the solution with respect to the time at the quenching time.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In the paper, we consider the  problem
\begin{gather}\label{a-1.1}
x^q\frac{\partial u}{\partial t}-\frac{\partial^2 u^m}{\partial x^2}=f(u^m),\quad (x,t) \in (0,a)\times(0,T),\\
\label{a-1.2}
u(0,t)=0=u(a,t),\quad t \in (0,T), \\
\label{a-1.3}
u(x,0)=0,\quad x \in (0,a),
\end{gather}
where $a>0$, $q\in\mathbb R$, $m\geq1$ and $f\in C^2([0,c^m))$ with $c>0$
satisfies
$$
f(0)>0,\quad f'(0)>0,\quad f''(s)\geq 0 \quad \text{for }0<s<c^m,
\quad\lim_{s \to c^m}f(s)=+\infty.
$$
At $x=0$, \eqref{a-1.1} is singular if $q>0$ and degenerate if $q<0$.
Furthermore, \eqref{a-1.1} is degenerate
at the points where $u=0$ in the quasilinear case $m>1$.
If $q=0$ and $m>1$, \eqref{a-1.1} is the famous porous medium equation,
which arises from many physical and biological models \cite{NDE}.
If $q>0$ and $m=1$, \eqref{a-1.1} can be used to describe
the Ockendon model for the flow in a channel of a fluid whose viscosity
is temperature-dependent
\cite{another2,another3}.

Due to the properties of $f$, the solution $u$ to
\eqref{a-1.1}--\eqref{a-1.3}
may quench at a finite time.
That is to say, there exists a finite time $T>0$ such that
\begin{align*}
\lim_{t\to T^-}\sup_{(0,a)}u(\cdot,t)=c.
\end{align*}
Quenching phenomena were introduced by  Kawarada \cite{Kawarada} in 1975
for the problem \eqref{a-1.1}--\eqref{a-1.3} in the case
$q=0$, $m=1$ and
$$
f(s)=\frac1{1-s},\quad 0\le s<1,
$$
where  Kawarada proved the existence of the critical length (which is $2\sqrt2$).
That is to say, the solution exists globally in time if
$a$ is less than the critical length, while it quenches if $a$ is greater
than the critical length.
For the quenching case,  Kawarada also showed
that $a/2$ is the quenching point
and the derivative of the solution with respect to the time blows up at
the quenching time.
Since then, there are many interesting results on quenching phenomena
for semilinear uniformly parabolic equations
(see, e.g., \cite{AW,Boni,DL,Levine1,LL,LM} and the references therein)
and for singular or degenerate semilinear parabolic equations
(see, e.g., \cite{C1,CHK,C6,C2,GH,KN} and the references therein).
Among these,  Chan and  Kong \cite{C1} considered \eqref{a-1.1}--\eqref{a-1.3}
in the semilinear case $m=1$,
where the authors showed the existence of the critical length,
the location of the quenching points and the blowing up of
the derivative of the solution with respect to the time at the quenching time.

Recently, there are few results on quenching phenomena for quasilinear
diffusion equations \cite{C3,DX,Winkler,YYJ,Zheng}.
\cite{C3} and \cite{Zheng}  showed some sufficient conditions for quenching
 solutions to the Dirichlet problems of porous medium equations.
In \cite{DX} and \cite{YYJ}, the authors considered quenching phenomena
for the one-dimensional homogeneous porous medium equation and
$p$-Laplacian equation with singular boundary flux,
respectively. It is shown that the solution quenches at the singular boundary and
the quenching rate was estimated.
 Winkler \cite{Winkler} studied the following problem
for a strongly degenerate diffusion equation with strong absorption
\begin{gather*}
\frac{\partial u}{\partial t}-u^p\frac{\partial^2 u}{\partial x^2}=-u^{-\beta}\chi_{\{u>0\}},\quad
(x,t) \in (0,a)\times(0,T), \\
u(0,t)=0=u(a,t),\quad t \in (0,T), \\
u(x,0)=u_0(x),\quad x \in (0,a),
\end{gather*}
where $a>0$, $p>1$, $-1<\beta<p-1$, $0\le u_0\in C([0,a])$ and
$\chi$ is the characteristic function.
Due to $p>1$, this equation in non-divergence form cannot be transformed
into the porous medium equation.
Winkler \cite{Winkler} ruled out
the possibility of quenching in infinite time under certain assumptions
on $p$, $\beta$ and $a$.

In this article, we study the quenching phenomenon of
the solution to the problem \eqref{a-1.1}--\eqref{a-1.3}.
The equation \eqref{a-1.1} is quasilinear in the case $m>1$.
Furthermore, there are two kinds of singularity or degeneracy in \eqref{a-1.1}:
one is the degeneracy at $u=0$ in the case $m>1$, the other
is the singularity ($q>0$) or degeneracy ($q<0$) at $x=0$.
Therefore, the classical solution to the problem \eqref{a-1.1}--\eqref{a-1.3}
may not exist and the weak solution should be considered.
By precise estimates near the parabolic boundary,
it is shown that the problem \eqref{a-1.1}--\eqref{a-1.3}
admits a continuous solution before the quenching time.
By constructing suitable super and sub solutions,
we prove the existence of the critical length.
For the quenching solution, we also study the location of the quenching points
and the blowing up of the derivative of the solution with respect to the
time at the quenching time by energy estimates and many kinds of comparison
principles. Due to the quasilinearity and the two kinds of singularity or
degeneracy in \eqref{a-1.1},
we have to overcome some technical difficulties when doing estimates,
constructing super and sub solutions, and using comparison principles.

This paper is arranged as follows. The well-posedness of the problem
\eqref{a-1.1}--\eqref{a-1.3}
is shown in $\S 2$. The existence of the critical length is proved in $\S 3$.
Subsequently, in $\S 4$ we study the quenching properties for the quenching
solution, including the location of the quenching points
and the blowing up of the derivative of the solution with respect to the
time at the quenching time.

\section{Well-posedness}

Solutions to \eqref{a-1.1}, and super and sub solutions,
are defined as follows.

\begin{definition}
A nonnegative function $u\in L^\infty((0,a)\times(0,T))$
is said to be a super (sub) solution to \eqref{a-1.1} in $(0,T)$
for some $0<T\le+\infty$, if for any $0<\tilde T<T$,
$\sup_{(0,a)\times(0,\tilde T)}u<c$ and
\begin{align*}
-\int_0^{\tilde T}\int_0^a x^{q}u\frac{\partial\varphi}{\partial t} \,dx\,dt
-\int_0^{\tilde T}\int_0^a u^m\frac{\partial^2\varphi}{\partial x^2} \,dx\,dt
\ge(\le)\int_0^{\tilde T}\int_0^af(u^m)\varphi \,dx\,dt
\end{align*}
for each $0\le\varphi\in C^2_0([0,a]\times[0,\tilde T])$.
Furthermore, $u$ is said to be a solution to \eqref{a-1.1}, if it
is both a super solution and a sub solution.
\end{definition}

The following comparison principle can be established by a duality argument.
The proof is similar to \cite[Theorem 1.3.1]{NDE}
and it is omitted.


\begin{theorem} \label{comparison}
Assume that $\hat u$ and $\check u$ are a super solution
and a sub solution to \eqref{a-1.1} in $(0,T)$ for some $0<T\le+\infty$,
respectively. Furthermore, $\hat u,\check u\in C([0,a]\times[0,T))$.
If
$$
\hat u(\cdot,0)\ge\check u(\cdot,0) \text{ in }(0,a),\quad
\hat u(0,\cdot)\ge\check u(0,\cdot),\quad
\hat u(a,\cdot)\ge\check u(a,\cdot) \text{ in }(0,T),
$$
then $\hat u\ge\check u$ in $(0,a)\times(0,T)$.
\end{theorem}

Let us turn to the  existence of local solution.

\begin{theorem} \label{existence}
The problem \eqref{a-1.1}--\eqref{a-1.3} admits uniquely a solution
$u$ in $(0,T)$ for some $T>0$.
Furthermore, $u\in C^{2,1}((0,a)\times(0,T])\cap C([0,a]\times[0,T])$,
$\frac{\partial u}{\partial t},\frac{\partial u}{\partial x}\in C^{2,1}((0,a)\times(0,T])$ and
\begin{gather*}
u(x,t)>0,\quad \frac{\partial u}{\partial t}(x,t)>0,
\quad(x,t)\in(0,a)\times(0,T),
\\
\int_0^a\Big(\frac{\partial u^m}{\partial x}(x,t)\Big)^2dx
\le 2ma\int_0^{\max_{(0,a)}u(\cdot,t)}s^{m-1}f(s^m)ds,\quad t\in(0,T).
\end{gather*}
\end{theorem}

\begin{proof}
Fix $0<c_0<c$.
For an integer $n\ge\frac{(2\text{e})^{1/m}}{c_0}$ and $k\ge1$,
the classical theory yields that the problem
\begin{gather} \label{a-2.1}
\Big(x+\frac 1k\Big)^q\frac{\partial u_{n,k}}{\partial t}
-\frac{\partial^2 u_{n,k}^m}{\partial x^2}=f(u^m_{n,k}),
\quad (x,t)\in(0,a)\times(0,T), \\
\label{a-2.2}
u_{n,k}(0,t)=u_{n,k}(a,t)=\frac1n,\quad t\in(0,T),
\\
\label{a-2.3}
u_{n,k}(x,0)=\frac1n,\quad x\in(0,a)
\end{gather}
admits a unique solution
$u_{n,k}\in C^{2,1}((0,a)\times[0,T])\cap C([0,a]\times[0,T])$
locally in time.
Set
$$
\bar u_{n,k}(x,t)=\frac{c_0}{2^{1/m}}
\Big(1-\Big(\frac xa\Big)^{3/2}+ \mathrm{e} ^{t/{T_0}-1}\Big)^{1/m},
\quad(x,t)\in[0,a]\times[0,{T_0}],
$$
where
$$
{T_0}=\frac{c_0\min\big\{\delta^q,(a+1)^q\big\}}{2 m \mathrm{e}  f(c_0^m)},\quad
\delta=\min\Big\{\frac a2,\frac{9c_0^{2m}}{64 a^{3}f^{2}(c_0^m)}\Big\}.
$$
Then
\[
\frac{c_0}{(2\text{e})^{1/m}}\le\bar u_{n,k}\le c_0,
\quad(x,t)\in[0,a]\times[0,{T_0}]
\]
and
\begin{align*}
\Big(x+\frac1k\Big)^q\frac{\partial \bar u_{n,k}}{\partial t}
-\frac{\partial^2 \bar u_{n,k}^m}{\partial x^2}
&=\frac{c_0^{m}\Big(x+\frac1k\Big)^q}{2 m\bar u^{m-1}_{n,k}{T_0}}\mathrm{e}^{t/{T_0}-1}
+\frac{3c_0^{m}}{8a^{3/2}x^{1/2}}
\\
&\ge \begin{cases}
\frac{3c_0^{m}}{8a^{3/2}\delta^{1/2}}\ge f(c_0^m),
&(x,t)\in(0,\delta)\times(0,{T_0}),
\\
\frac{c_0\min\big\{\delta^q,(a+1)^q\big\}}{2 m e {T_0}},
&(x,t)\in(\delta,a)\times(0,{T_0}).
\end{cases}
\end{align*}
Note that $\frac{c_0\min\big\{\delta^q,(a+1)^q\big\}}{2 m \mathrm{e} {T_0}}
\ge f(c_0^m)$.
Therefore, $\bar u_{n,k}$ is a supersolution to  \eqref{a-2.1}--\eqref{a-2.3}
in $(0,{T_0})$.
The classical comparison principle yields that
$u_{n,k}$ exists in $(0,{T_0})$ and
\begin{equation}
\label{a-2.4}
\frac1n\le u_{n,k}(x,t)\le c_0,\quad (x,t)\in(0,a)\times(0,{T_0}).
\end{equation}
Since $\frac{\partial u_{n,k}}{\partial t}$ solves
\begin{gather*}
\Big(x+\frac 1k\Big)^q\frac{\partial }{\partial t}\Big(\frac{\partial u_{n,k}}{\partial t}\Big)
-m\frac{\partial^2}{\partial x^2}\Big(u_{n,k}^{m-1}(x,t)\frac{\partial u_{n,k}}{\partial t}\Big)\\
-mf'(u^m_{n,k}(x,t))u^{m-1}_{n,k}(x,t)\frac{\partial u_{n,k}}{\partial t}=0,
\quad (x,t)\in(0,a)\times(0,T_0),
\\
\frac{\partial u_{n,k}}{\partial t}(0,t)=\frac{\partial u_{n,k}}{\partial t}(a,t)=0,\quad t\in(0,T_0),
\\
\frac{\partial u_{n,k}}{\partial t}(x,0)\ge0,\quad x\in(0,a),
\end{gather*}
the classical comparison principle leads to
\begin{equation}
\label{a-2.4-0}
\frac{\partial u_{n,k}}{\partial t}(x,t)\ge 0,\quad (x,t)\in(0,a)\times(0,{T_0}).
\end{equation}
Set
\[
\bar v_{n,k}(x,t)=\Big(\Big(\frac 1n\Big)^{m}+\frac1{2}f(c_0^m)x(a-x)\Big)^{1/m},
\quad(x,t)\in[0,a]\times[0,{T_0}],
\]
Then, $\bar v_{n,k}$ is a supersolution of the problem
\begin{gather*}
\Big(x+\frac 1k\Big)^q\frac{\partial v_{n,k}}{\partial t}
-\frac{\partial^2 v_{n,k}^m}{\partial x^2}=f(c_0^m),
\quad (x,t)\in(0,a)\times(0,T_0),
\\
v_{n,k}(0,t)=v_{n,k}(a,t)=\frac1n,\quad t\in(0,T_0),
\\
v_{n,k}(x,0)=\frac1n,\quad x\in(0,a).
\end{gather*}
Furthermore, \eqref{a-2.4} shows that $v_{n,k}$ is a supersolution
of the problem \eqref{a-2.1}--\eqref{a-2.3} in $(0,{T_0})$.
It follows from the classical comparison principle that
\begin{equation}
\label{a-2.4-1}
u_{n,k}(x,t)\le v_{n,k}(x,t)\le \bar v_{n,k}(x,t),
\quad(x,t)\in(0,a)\times(0,{T_0}).
\end{equation}

For a fixed $\tilde x\in(0,a)$, denote
$\delta=\frac12\min\{\tilde x,a-\tilde x\}$.
Set
\begin{gather*}
\underline w_{n,k}(x,t)=\frac 1n+(x-\tilde x+\delta)
(\tilde x+\delta-x)\min\{t,T_1\},\\
(x,t)\in[\tilde x-\delta,\tilde x+\delta]\times[0,{T_0}],
\\
\bar z_{n,k}(x,t)=\frac 1n+\Big(c_0-\frac1n\Big)
\delta^{-2}(x-\tilde x)^2+\frac t{T_2},
\quad(x,t)\in[\tilde x-\delta,\tilde x+\delta]\times[0,{T_2}],
\end{gather*}
where $0<T_1,T_2<T_0$.
Then, there exist sufficiently small $T_1,T_2>0$,
which are independent of $n$ and $k$, such that
\begin{gather*}
\Big(x+\frac 1k\Big)^q\frac{\partial \underline w_{n,k}}{\partial t}
-\frac{\partial^2 \underline w_{n,k}^m}{\partial x^2}\le f(0),
\quad(x,t)\in(\tilde x-\delta,\tilde x+\delta)\times(0,{T_0}),
\\
\Big(x+\frac 1k\Big)^q\frac{\partial \bar z_{n,k}}{\partial t}
-\frac{\partial^2 \bar z_{n,k}^m}{\partial x^2}\ge f(c_0^m),
\quad(x,t)\in(\tilde x-\delta,\tilde x+\delta)\times(0,{T_2}).
\end{gather*}
Therefore, $\underline w_{n,k}$ is a subsolution of the problem
\begin{gather}
\label{a-2.5-1}
\Big(x+\frac 1k\Big)^q\frac{\partial w_{n,k}}{\partial t}
-\frac{\partial^2w_{n,k}^m}{\partial x^2}=f(0),
\quad (x,t)\in(\tilde x-\delta,\tilde x+\delta)\times(0,{T_0}),
\\
\label{a-2.5-2}
w_{n,k}(\tilde x-\delta,t)=w_{n,k}(\tilde x+\delta,t)=\frac1n,\quad t\in(0,T_0),
\\
\label{a-2.5-3}
w_{n,k}(x,0)=\frac1n,\quad x\in(\tilde x-\delta,\tilde x+\delta),
\end{gather}
$\bar z_{n,k}$ is a supersolution to the problem
\begin{gather}
\label{a-2.6-1}
\Big(x+\frac 1k\Big)^q\frac{\partial z_{n,k}}{\partial t}
-\frac{\partial^2z_{n,k}^m}{\partial x^2}=f(c_0^m),
\quad (x,t)\in(\tilde x-\delta,\tilde x+\delta)\times(0,{T_2}),
\\
\label{a-2.6-2}
z_{n,k}(\tilde x-\delta,t)=z_{n,k}(\tilde x+\delta,t)=c_0,\quad t\in(0,T_2),
\\
\label{a-2.6-3}
z_{n,k}(x,0)=\frac1n,\quad x\in(\tilde x-\delta,\tilde x+\delta).
\end{gather}
Further, \eqref{a-2.4} shows that $u_{n,k}$ is a supersolution to the problem
\eqref{a-2.5-1}--\eqref{a-2.5-3} in $(0,{T_0})$
and a subsolution to the problem \eqref{a-2.6-1}--\eqref{a-2.6-3} in $(0,{T_2})$.
It follows from the classical comparison principle that
\begin{gather}
\label{a-2.7-1}
u_{n,k}(x,t)\ge w_{n,k}(x,t)\ge \underline w_{n,k}(x,t),
\quad(x,t)\in(\tilde x-\delta,\tilde x+\delta)\times(0,{T_0}),
\\
\label{a-2.7-2}
u_{n,k}(x,t)\le z_{n,k}(x,t)\le \bar z_{n,k}(x,t),
\quad(x,t)\in(\tilde x-\delta,\tilde x+\delta)\times(0,{T_2}).
\end{gather}
Set
$$
E_{n,k}(t)=\frac 12 \int_0^a\Big(\frac{\partial u_{n,k}^m}{\partial x}(x,t)\Big)^2dx
-m\int_0^a\int_0^{u_{n,k}(x,t)}s^{m-1}f(s^m)\,ds\,dx,
$$
for $t\in[0,T_0]$.
Integrating by parts, one gets
\begin{align*}
E'_{n,k}(t)
&=\int_0^a\frac{\partial u_{n,k}^m}{\partial x}(x,t)\frac{\partial^2 u_{n,k}^m}{\partial t\partial x}(x,t)dx
-m\int_0^a\frac{\partial u_{n,k}}{\partial t}u_{n,k}^{m-1}(x,t)f(u^m_{n,k}(x,t))dx
\\
&=\frac{\partial u_{n,k}^m}{\partial x}(a,t)\frac{\partial u_{n,k}^m}{\partial t}(a,t)
-\frac{\partial u_{n,k}^m}{\partial x}(0,t)\frac{\partial u_{n,k}^m}{\partial t}(0,t)
\\
&\quad-m\int_0^a u_{n,k}^{m-1}(x,t)\frac{\partial u_{n,k}}{\partial t}(x,t)
\Big(\frac{\partial^2 u_{n,k}^m}{\partial x^2}(x,t)+f(u^m_{n,k}(x,t))\Big)dx
\\
&=-m\int_0^a u_{n,k}^{m-1}(x,t)\Big(\frac{\partial u_{n,k}}{\partial t}(x,t)\Big)^2dx\le0,\quad
t\in(0,T_0).
\end{align*}
Therefore,
$$
E_{n,k}(t)\le E_{n,k}(0)\leq 0,\quad t\in(0,T_0),
$$
which leads to
\begin{equation} \label{www-c1}
\begin{aligned}
\int_0^a\Big(\frac{\partial u_{n,k}}{\partial x}(x,t)\Big)^2dx
&\le 2m\int_0^a\int_0^{u_{n,k}(x,t)}s^{m-1}f(s^m)\,ds\,dx\\
&\le 2ma\int_0^{\max_{(0,a)}u_{n,k}(\cdot,t)}s^{m-1}f(s^m)ds,\quad t\in(0,T_0).
\end{aligned}
\end{equation}
From the classical comparison principle and \eqref{a-2.4-0},
we have
\[
u_{n_2,k}(x,t)\le u_{n_1,k}(x,t),\quad
(x,t)\in[0,a]\times[0,{T_0}],\quad
n_2\ge n_1\ge\frac{(2\text{e})^{1/m}}{c_0},
\]
for $k\ge 1$,
and
\[
\begin{cases}
u_{n,k_2}(x,t)\le u_{n,k_1}(x,t), &\text{if }q\ge0,
\\
u_{n,k_2}(x,t)\ge u_{n,k_1}(x,t), &\text{if }q\le0,
\end{cases}
\]
for $(x,t)\in(0,a)\times(0,{T_0})$,
$n\ge (2\text{e})^{1/m}/c_0$, $k_2\ge k_1\ge1$.
Let
$$
u(x,t)=\lim_{k\to\infty}\lim_{n\to\infty}u_{n,k}(x,t),
\quad(x,t)\in[0,a]\times[0,{T_0}].
$$
Due to \eqref{a-2.4}, \eqref{a-2.4-0}, \eqref{a-2.4-1}, \eqref{a-2.7-1}
and \eqref{a-2.7-2}, the function $u$ satisfies
\begin{gather}
\label{a-2.8-1}
0\le u(x,t)\le c_0,\quad (x,t)\in(0,a)\times(0,{T_0}),
\\
\label{a-2.8-5}
u(x,\cdot)\text{ is increasing in } (0,T_0),\quad x\in(0,a),
\\
\label{a-2.8-2}
0\le u(x,t)\le\Big(\frac1{2}f(c_0^m)x(a-x)\Big)^{1/m},
\quad (x,t)\in(0,a)\times(0,{T_0}),
\\
\label{a-2.8-3}
u(x,t)\ge (x-\tilde x+\delta)(\tilde x+\delta-x)\min\{t,T_1\},
\quad(x,t)\in(\tilde x-\delta,\tilde x+\delta)\times(0,{T_0}),
\\
\label{a-2.8-4}
u(x,t)\le c_0\delta^{-2}(x-\tilde x)^2+\frac t{T_2},
\quad(x,t)\in(\tilde x-\delta,\tilde x+\delta)\times(0,{T_2}).
\end{gather}
It is not hard to show that $u$ is a solution of \eqref{a-1.1}--\eqref{a-1.3}
in $(0,T_0)$.
Furthermore, \eqref{a-2.8-3} yields that
\begin{equation} \label{ww-1}
u(x,t)>0,\quad(x,t)\in(0,a)\times(0,T_0).
\end{equation}
Therefore, $u\in C^{2,1}((0,a)\times(0,T_0])$,
which together with \eqref{a-2.8-1}--\eqref{ww-1} and $f\in C^2([0,c^m))$,
implies that
$u\in C([0,a]\times[0,T_0])$ satisfies \eqref{a-1.2} and \eqref{a-1.3},
$\frac{\partial u}{\partial t},\frac{\partial u}{\partial x}\in C^{2,1}((0,a)\times(0,T_0])$ and
\begin{align} \label{ww-2}
\frac{\partial u}{\partial t}(x,t)\ge0,\quad(x,t)\in(0,a)\times(0,T_0).
\end{align}
Noting $\frac{\partial u}{\partial t}\in C^{2,1}((0,a)\times(0,T_0])$ with \eqref{ww-2} solves
\begin{equation} \label{www-w1}
x^q\frac{\partial }{\partial t}\Big(\frac{\partial u}{\partial t}\Big)
-m\frac{\partial^2}{\partial x^2}\Big(u^{m-1}(x,t)\frac{\partial u}{\partial t}\Big)
-mf'(u^m(x,t))u^{m-1}(x,t)\frac{\partial u}{\partial t}=0,
\end{equation}
for $(x,t)\in(0,a)\times(0,T_0)$.
From the classical strong maximum principle and \eqref{ww-1} we obtain
\begin{equation} \label{www}
\frac{\partial u}{\partial t}(x,t)>0,\quad(x,t)\in(0,a)\times(0,T_0).
\end{equation}
Indeed, if \eqref{www} is wrong, then there exists
$(x_0,t_0)\in(0,a)\times(0,T_0)$ such that
$\frac{\partial u}{\partial t}(x_0,t_0)=0$. For any $0<\varepsilon<\min\{x_0,a-x_0\}$ and any
$0<\tau<t_0$,
\eqref{ww-2} shows that
$$
\frac{\partial u}{\partial t}(x_0,t_0)=0=\min_{(\varepsilon,a-\varepsilon)\times(\tau,t_0)}\frac{\partial u}{\partial t}.
$$
Since \eqref{www-w1} is uniformly parabolic in
 $(\varepsilon,a-\varepsilon)\times(\tau,t_0)$
due to \eqref{ww-1},
 from the classical strong maximum principle, we have
$$
\frac{\partial u}{\partial t}(x,t)=0, \quad (x,t)\in(\varepsilon,a-\varepsilon)\times(\tau,t_0).
$$
Then, it follows from the arbitrariness of $\varepsilon\in(0,\min\{x_0,a-x_0\})$
and $\tau\in(0,t_0)$ that
$$
\frac{\partial u}{\partial t}(x,t)=0, \quad (x,t)\in(0,a)\times(0,t_0),
$$
which contradicts \eqref{a-1.3} and \eqref{ww-1}.
Finally, \eqref{www-c1} leads to
$$
\int_0^a\Big(\frac{\partial u^m}{\partial x}(x,t)\Big)^2dx
\le 2ma\int_0^{\max_{(0,a)}u(\cdot,t)}s^{m-1}f(s^m)ds,\quad t\in(0,T_0).
$$
\end{proof}


Denote
$$
T_*=\sup\big\{T>0:\text{ the problem \eqref{a-1.1}--\eqref{a-1.3}
admits a solution in } (0,T)\big\}.
$$
We call $T_*$ the life span of the solution to
problem \eqref{a-1.1}--\eqref{a-1.3}.

\begin{remark} \label{existencere} \rm
By the standard extension process, one can show that
Problem \eqref{a-1.1}--\eqref{a-1.3} admits uniquely a solution $u$
in $(0,T_*)$.
Furthermore, $u\in C^{2,1}((0,a)\times(0,T_*))\cap C([0,a]\times[0,T_*))$,
$\frac{\partial u}{\partial t},\frac{\partial u}{\partial x}\in C^{2,1}((0,a)\times(0,T_*))$ and
\begin{gather*}
u(x,t)>0,\quad \frac{\partial u}{\partial t}(x,t)>0,
\quad(x,t)\in(0,a)\times(0,T_*),
\\
\int_0^a\Big(\frac{\partial u^m}{\partial x}(x,t)\Big)^2dx
\le 2ma\int_0^{\max_{(0,a)}u(\cdot,t)}s^{m-1}f(s^m)ds,\quad t\in(0,T_*).
\end{gather*}
\end{remark}

\section{Critical length}

Assume that $u$ is the solution to \eqref{a-1.1}--\eqref{a-1.3}
and $T_*$ is its life span.
If $T_*=+\infty$, then $u$ exists globally in time.
If $T_*<+\infty$, then $u$ must quench at a finite time, i.e.
\begin{align*}
\lim_{t\to T_*^-}\sup_{(0,a)}u(\cdot,t)=c.
\end{align*}

Let us study the relation between $T_*$ and $a$ in this section.
For convenience, we denote $u_a$ by the solution to
 \eqref{a-1.1}--\eqref{a-1.3},
and $T_*(a)$  its life span.

\begin{lemma} \label{lemma-3.1}
If $a$ is positive and sufficiently small, then $T_*(a)=+\infty$ and
\[
\sup_{(0,a)\times(0,+\infty)}u_a<c.
\]
\end{lemma}

\begin{proof}
Fix $0<c_0<c$ and $0<a\le\Big(\frac{8c_0^m}{f(c_0^m)}\Big)^{1/2}$.
Set
$$
\bar u_a(x,t)=\Big(\frac{f(c_0^m)}{2}x(a-x)\Big)^{1/m},\quad
(x,t)\in [0,a]\times[0,+\infty).
$$
Then, $\bar u_a$ satisfies
\begin{gather*}
0\le\bar u_a(x,t)\le\Big(\frac{f(c_0^m)a^2}{8}\Big)^{1/m}\le c_0,\quad
 (x,t)\in [0,a]\times[0,+\infty),\\
x^q\frac{\partial \bar u_a}{\partial t}-\frac{\partial^2\bar u_a^m}{\partial x^2}
=f(c_0^m)\ge f(\bar u_a^m), \quad(x,t)\in(0,a)\times(0,+\infty).
\end{gather*}
The comparison principle (Theorem \ref{comparison}) shows that
$$
u_a(x,t)\le\bar u_a(x,t)\le c_0,\quad(x,t)\in [0,a]\times[0,+\infty).
$$
\end{proof}

\begin{lemma}\label{lemma-3.2}
If $a>0$ is sufficiently large, then $T_*(a)<+\infty$.
\end{lemma}

\begin{proof}
Set
$$
\underline u_a(x,t)=\frac{t}{T}\Big(\frac{f(0)}4(x-a/2)(a-x)\Big)^{1/m},\quad
(x,t)\in [a/2,a]\times[0,T]
$$
with
$$
T={2}{\max\{(a/2)^q,a^q\}}\Big(\frac{a^2}{64f^{m-1}(0)}\Big)^{1/m}.
$$
Then, $\underline u_a$ satisfies
\begin{align*}
x^q\frac{\partial \underline u_a}{\partial t}-\frac{\partial^2\underline u_a^m}{\partial x^2}
&=\frac{x^q}{T}\Big(\frac{f(0)}4(x-a/2)(a-x)\Big)^{1/m}
+\frac{f(0)}{2}\Big(\frac{t}{T}\Big)^m\\
&\le f(0)\le f(\underline u^m_a),
\quad (x,t)\in(a/2,a)\times(0,T).
\end{align*}
The comparison principle (Theorem \ref{comparison}) shows that
$$
u_a(x,t)\ge\underline u_a(x,t),\quad(x,t)\in [a/2,a]\times[0,T].
$$
Particularly,
$$
u_a(3a/4,T)\ge\Big(\frac{f(0)a^2}{64}\Big)^{1/m},
$$
which yields $T_*(a)<+\infty$ if $a\ge 8 c^{m/2}f^{-1/2}(0)$.
\end{proof}

\begin{lemma} \label{lemma-3.3}
For any $0<a_1<a_2$,
$$
u_{a_1}(x,t)<u_{a_2}(x,t),\quad (x,t)\in (0,a_1)\times(0,T_*(a_2)).
$$
\end{lemma}

\begin{proof}
For any $0<a_1<a_2$, Remark \ref{existencere} shows that
$$
u_{a_2}(a_1,t)>0,\quad t\in(0,T_*(a_2)).
$$
Then, it follows from
the comparison principle (Theorem \ref{comparison}) that
$$
u_{a_1}(x,t)\le u_{a_2}(x,t),\quad (x,t)\in (0,a_1)\times(0,T_*(a_2)).
$$
Set
$$
w(x,t)=u_{a_1}(x,t)-u_{a_2}(x,t),\quad (x,t)\in [0,a_1]\times[0,T_*(a_2)).
$$
By Remark \ref{existencere}, $w\in C^{2,1}((0,a_1)\times(0,T_*(a_2)))
\cap C([0,a_1]\times[0,T_*(a_2)])$ and solves
\begin{align*}
&x^q\frac{\partial w}{\partial t}-m\frac{\partial^2}{\partial x^2}
\Big(w\int_0^1(\sigma u_{a_1}(x,t)+(1-\sigma)u_{a_2}(x,t))^{m-1}d\sigma\Big)
\\
&=mw\int_0^1f'(\sigma u^m_{a_1}(x,t)+(1-\sigma)u^m_{a_2}(x,t)) d\sigma\\
&\quad\times \int_0^1(\sigma u_{a_1}(x,t)+(1-\sigma)u_{a_2}(x,t))^{m-1}d\sigma,
\quad (x,t)\in (0,a_1)\times(0,T_*(a_2)),
\end{align*}
where $u_{a_1},u_{a_2}\in C^{2,1}((0,a_1)\times(0,T_*(a_2)))
\cap C([0,a_1]\times[0,T_*(a_2)])$ with
$$
u_{a_1}(x,t)>0,\quad u_{a_2}(x,t)>0,\quad (x,t)\in (0,a_1)\times(0,T_*(a_2)).
$$
The classical strong maximum principle (a similar discussion to \eqref{www}
in Theorem \ref{existence}) leads to
$$
w(x,t)<0,\quad (x,t)\in (0,a_1)\times(0,T_*(a_2)),
$$
i.e.
$$
u_{a_1}(x,t)<u_{a_2}(x,t),\quad (x,t)\in (0,a_1)\times(0,T_*(a_2)).
$$
\end{proof}

\begin{lemma} \label{lemma-3.3new}
There exists at most one $a>0$ such that $u_a$ quenches at the infinite time,
i.e. $T_*(a)=+\infty$ and $\sup_{(0,a)\times(0,+\infty)}u_{a}=c$.
\end{lemma}

\begin{proof} Assume that $u_{a_0}$ quenches at the infinite time for some
$a_0>0$. For $a>a_0$, set
$$
\underline u_a(x,t)=\lambda^{2/m}
u_{a_0}(\lambda^{-1}x,\lambda^{-2/m-q}t),\quad(x,t)\in [0,a]\times[0,+\infty),
\quad \lambda=\frac{a}{a_0}.
$$
Then, $\lambda>1$, and $\underline u_a$ solves
$$
x^q\frac{\partial \underline u_a}{\partial t}-\frac{\partial^2\underline u_a^m}{\partial x^2}
=f(\lambda^{-2}\underline u_a^m),
\quad(x,t)\in(0,a)\times(0,+\infty).
$$
Therefore, $\underline u_a$ is a subsolution to \eqref{a-1.1}--\eqref{a-1.3}.
Since
$$
\lim_{t\to+\infty}\sup_{(0,a)}\underline u_a(\cdot,t)=\lambda^{2/m} c>c,
$$
$u_a$ must quench at a finite time.
\end{proof}

\begin{theorem} \label{thm3.1}
There exists $a_*>0$ such that
\begin{itemize}
\item[(i)] $T_*(a)=+\infty$ and $\sup_{(0,a)\times(0,+\infty)}u_{a}<c$ if $0<a<a_*$,

\item[(ii)] $T_*(a)<+\infty$ if $a>a_*$.
\end{itemize}
\end{theorem}

\begin{proof}
Set
$$
S=\big\{a>0: T_*(a)=+\infty \text{ and }\sup_{(0,a)\times(0,+\infty)}u_a<c\big\}.
$$
By Lemmas \ref{lemma-3.1} and \ref{lemma-3.2}, this set is bounded.
Denote
$$
a_*=\sup S.
$$
By Lemma \ref{lemma-3.3}, $a\in S$ for each $0<a<a_*$.
For $a>a_*$, the definition of $S$ shows that
$T_*(a)<+\infty$ or $u_a$ quenches at the infinite time.
Let us prove that the latter case is impossible by contradiction.
Otherwise, assume that $u_{a_0}$ quenches at the infinite time
for some $a_0>a_*$.
From the definition of $S$ and Lemma \ref{lemma-3.3},
$u_{\tilde a}$ must quench at the infinite time for each $a_*<\tilde a<a_0$,
which contradicts Lemma \ref{lemma-3.3new}.
\end{proof}

\begin{remark} \rm
Using Lemma \ref{lemma-3.3}, it is not difficult to show that
$$
u_{a_*}(x,t)=\lim_{a\to a_*} u_{a}(x,t),\quad (x,t)\in (0,a_*)\times(0,T_*(a_*)).
$$
Therefore, $T_*(a_*)=+\infty$. However,
it is unknown whether $u_{a_*}$ quenches or not at the infinite time.
\end{remark}

\section{Quenching properties}

Assume that $u$ is the solution of \eqref{a-1.1}--\eqref{a-1.3}.
According to Theorem \ref{thm3.1}, $u$ quenches at a finite time
if and only if $a>a_*$.
In this section, we investigate the location of the quenching points
and the blowing up of $\frac{\partial u}{\partial t}$.

\begin{definition} \rm
Assume that the solution $u$ to  \eqref{a-1.1}--\eqref{a-1.3} quenches at
 $0<T_*<+\infty$.
A point $x\in[0,a]$ is said to be a quenching point if there exist
two sequences
$\{t_n\}_{n=1}^\infty\subset(0,T_*)$ and
$\{x_n\}_{n=1}^\infty\subset(0,a)$ such that
\[
\lim_{n\to\infty}t_n=T_*,\quad\lim_{n\to\infty}x_n=x,\quad
\lim_{n\to\infty}u(x_n,t_n)=c.
\]
\end{definition}

\begin{theorem} \label{thm3.2}
Assume that $a>a_*$. Then
\begin{itemize}
\item[(i)] there is no quenching point in $(a/2,a)$ if $q>0$,

\item[(ii)] there is no quenching point in $(0,a/2)$ if $q<0$.
\end{itemize}
\end{theorem}

\begin{proof}
We prove the case $q>0$ only; the other case can be proved similarly.
By Remark \ref{existencere},
\begin{equation} \label{www-1}
u(x,t)>0,\quad\frac{\partial u}{\partial t}(x,t)>0,\quad(x,t) \in (0,a)\times(0,T_*).
\end{equation}
Set
$$
v(x,t)=u(a-x,t),\quad (x,t)\in [0,a/2]\times[0,T_*).
$$
Then, $v$ is a solution of the equation
\begin{align} \label{www-2}
(a-x)^q\frac{\partial v}{\partial t}-\frac{\partial^2 v^m}{\partial x^2}=f(v^m),\quad(x,t)\in (0,a/2)\times(0,T_*).
\end{align}
By \eqref{www-1}, $u$ is a supersolution to \eqref{www-2}.
Similar to the proof of Lemma \ref{lemma-3.3}, one can show that
\begin{equation} \label{www-3}
u(x,t)>v(x,t),\quad(x,t)\in (0,a/2)\times(0,T_*).
\end{equation}
Set
$$
w(x,t)=u^m(x,t)-v^m(x,t),
\quad (x,t)\in [0,a/2]\times[0,T_*).
$$
Then $w$ solves
\begin{equation} \label{a-3.6}
x^q h(x,t)\frac{\partial w}{\partial t}-\frac{\partial^2 w}{\partial x^2}
+x^q \frac{\partial h}{\partial t}(x,t)w\ge g(x,t)w,
\quad (x,t)\in (0,a/2)\times(0,T_*),
\end{equation}
where
\begin{gather*}
h(x,t)=\frac{1}{m}\int_0^1(\sigma{u^m}(x,t)+(1-\sigma){v^m}(x,t))^{1/m-1} d\sigma,
\\
g(x,t)=\int_0^1f'(\sigma u^m(x,t)+(1-\sigma)v^m(x,t)) d\sigma\ge f'(0)>0,
\end{gather*}
for $(x,t)\in (0,a/2)\times(0,T_*)$.
From \eqref{www-1} and \eqref{www-3}, for $(x,t)\in (0,a/2)\times(0,T_*)$,
follows that
\begin{gather*}
\frac{\partial h}{\partial t}(x,t)=\frac{1}{m}\Big(\frac{1}{m}-1\Big)
\int_0^1\Big(\sigma \frac{\partial u^m(x,t)}{\partial t}+(1-\sigma)
\frac{\partial v^m(x,t)}{\partial t}\Big)^{1/m-1}
d\sigma<0,\\
w(x,t)>0.
\end{gather*}
Therefore, $w$ satisfies
\begin{equation}\label{a-3.7}
x^q h(x,t)\frac{\partial w}{\partial t}-\frac{\partial^2 w}{\partial x^2}\ge 0,
\quad (x,t)\in (0,a/2)\times(0,T_*).
\end{equation}
For any $0<\eta<a/4$, set
$$
\delta=\min_{(\eta,a/2-\eta)}(u^m(\cdot,T_*/2)-v^m(\cdot,T_*/2)).
$$
Let $z$ be the solution to the problem
\begin{gather}
\label{a-3.8}
x^q h(x,t)\frac{\partial z}{\partial t}-\frac{\partial^2 z}{\partial x^2}=0,\quad
(x,t)\in (\eta,a/2-\eta)\times(T_*/2,T_*),
\\
\label{a-3.9}
z(\eta,t)=z(a/2-\eta,t)=0,\quad t\in (T_*/2,T_*),
\\
\label{a-3.10}
z(x,T_*/2)=\delta\sin\Big(\frac{2\pi(x-\eta)}{a-4\eta}\Big),\quad
x\in(\eta,a/2-\eta).
\end{gather}
Since \eqref{a-3.8} is a uniformly parabolic equation in
$(\eta,a/2-\eta)\times(T_*/2,T_*)$,
 from the classical strong maximum principle it follows that
\begin{equation} \label{www-a2}
z(x,t)>0,\quad(x,t)\in(\eta,a/2-\eta)\times[T_*/2,T_*].
\end{equation}
By \eqref{a-3.7} and \eqref{www-3}, $w$ is a supersolution to
\eqref{a-3.8}--\eqref{a-3.10}.
The classical comparison principle leads to
\begin{align*}
w(x,t)\ge z(x,t),\quad(x,t)\in(\eta,a/2-\eta)\times(T_*/2,T_*);
\end{align*}
i.e.,
\begin{align*}
u^m(a-x,t)\le u^m(x,t)-z(x,t),\quad(x,t)\in(\eta,a/2-\eta)\times(T_*/2,T_*).
\end{align*}
So, there is no quenching point in $(a/2+\eta,a-\eta)$ owing to \eqref{www-a2}.
Then, (i) is proved due to the arbitrariness of $0<\eta<a/4$.
\end{proof}

\begin{theorem} \label{thm3.3}
Assume that $a> a_*$ and $M=\int_0^{c} s^{m-1}f(s^m)ds<+\infty$. Then
$$
M\ge\frac{c^{2m}}{ma^2}
$$
and
\begin{itemize}
\item[(i)] the quenching points belong to $[c^{2m}/(2Mma),a/2]$ if $q>0$,

\item[(ii)] the quenching points belong to $[c^{2m}/(2Mma),a-c^{2m}/(2Mma)]$
if $q=0$,

\item[(iii)] the quenching points belong to $[a/2,a-c^{2m}/(2Mma)]$ if $q<0$.
\end{itemize}
\end{theorem}

\begin{proof}
From Remark \ref{existencere}, one gets
\begin{equation} \label{www-5}
\int_0^a\Big(\frac{\partial u^m}{\partial x}(x,t)\Big)^2dx
\le 2mMa,\quad t\in(0,T_*).
\end{equation}
Then, it follows from \eqref{www-5} and the Schwarz inequality that
\begin{equation} \label{www-6}
\begin{aligned}
u^m(x,t)&=\int^x_0 \frac{\partial u^m}{\partial x}(y,t) dy\leq x^{1/2}
\Big(\int_0^a\Big(\frac{\partial u^m}{\partial x}(y,t)\Big)^2dy\Big)^{1/2}\\
&\le (2mMax)^{1/2},\quad (x,t)\in[0,a/2]\times[0,T_*)
\end{aligned}
\end{equation}
and
\begin{equation} \label{www-7}
\begin{aligned}
u^m(x,t)&=-\int_x^a \frac{\partial u^m}{\partial x}(y,t) dy\leq(a-x)^{1/2}
\Big(\int_0^a\Big(\frac{\partial u^m}{\partial x}(y,t)\Big)^2dy\Big)^{1/2}\\
&\le (2mMa(a-x))^{1/2},\quad (x,t)\in[a/2,a]\times[0,T_*).
\end{aligned}
\end{equation}
Since
\begin{align*}
\lim_{t\to T_*^-}\sup_{(0,a)}u(\cdot,t)=c
\end{align*}
by Theorem \ref{thm3.1},  from \eqref{www-6} and \eqref{www-7} it follows that
$$
M\ge\frac{c^{2m}}{ma^2}.
$$
Furthermore, (i)--(iii) follow from Theorem \ref{thm3.2}, \eqref{www-6}
and \eqref{www-7} directly.
\end{proof}

\begin{theorem} \label{th3.4}
Assume that $a>a_*$ and $\int_0^{c} s^{m-1}f(s^m)ds<+\infty$. Then
the solution $u$ of \eqref{a-1.1}--\eqref{a-1.3} satisfies
$$
\lim_{t\to T_*^-}\sup_{(0,a)}\frac{\partial u}{\partial t}(\cdot,t)=+\infty.
$$
\end{theorem}

\begin{proof}
From Theorem \ref{thm3.3},  there exist $0<x_1<x_2<x_3<x_4<a$ such that
\begin{gather} \label{www-b1}
\lim_{t\to T_*^-}\sup_{(x_2,x_3)}u(\cdot,t)=c,\\
\label{www-b2}
\sup_{(0,x_2)\times(0,T_*)} u<c,\quad \sup_{(x_3,a)\times(0,T_*)}u<c.
\end{gather}
Set
$$
w(x,t)=u^m(x,t),\quad (x,t)\in [x_1,x_4]\times[0,T_*).
$$
Then $w$ and $\frac{\partial w}{\partial t}$ solve
\begin{equation}\label{www-b3-0}
\frac{x^q}{m} w^{1/m-1}\frac{\partial w}{\partial t}-\frac{\partial^2 w}{\partial x^2}
=f(w),\quad (x,t)\in (x_1,x_4)\times(0,T_*)
\end{equation}
and
\begin{equation} \label{www-b3}
\begin{aligned}
&\frac{x^q}{m} w^{1/m-1}(x,t)\frac{\partial }{\partial t}\Big(\frac{\partial w}{\partial t}\Big)
+\Big(\frac{1}{m}-1\Big)\frac{x^q}{m} w^{1/m-2}(x,t)\Big(\frac{\partial w}{\partial t}\Big)^2
\\
&-\frac{\partial^2 }{\partial x^2}\Big(\frac{\partial w}{\partial t}\Big)=f'(w(x,t))\frac{\partial w}{\partial t},
\quad (x,t)\in (x_1,x_4)\times(0,T_*),
\end{aligned}
\end{equation}
respectively.
By Remark \ref{existencere},
\begin{align}
\label{www-b4}
\frac{\partial w}{\partial t}(x,t)>0,\quad(x,t) \in (0,a)\times(0,T_*).
\end{align}
Let $z$ be the solution to the problem
\begin{gather} \label{www-b5}
\frac{x^q}{m} w^{1/m-1}(x,t)\frac{\partial z}{\partial t}-\frac{\partial z^2 }{\partial x^2}=0,\quad
(x,t)\in (x_1,x_4)\times(T_*/2,T_*),
\\
\label{www-b6}
z(x_1,t)=z(x_4,t)=0,\quad t\in(T_*/2,T_*),
\\
\label{www-b7}
z(x,T_*/2)=\delta\sin\Big(\frac{\pi(x-x_1)}{x_4-x_1}\Big),\quad
x\in(x_1,x_4),
\end{gather}
where
$$
\delta=\min_{(x_1,x_4)}\frac{\partial w}{\partial t}(\cdot,T_*/2).
$$
Since \eqref{www-b5} is a uniformly parabolic equation,
one gets from the classical maximum principle that
\begin{equation}
\label{www-b8}
z(x,t)>0,\quad(x,t)\in(x_1,x_4)\times[T_*/2,T_*].
\end{equation}
By \eqref{www-b3} and \eqref{www-b4}, the function $\frac{\partial w}{\partial t}$ is a supersolution
to \eqref{www-b5}--\eqref{www-b8}.
The classical comparison principle leads to
\begin{equation}
\label{www-b9}
\frac{\partial w}{\partial t}(x,t)\ge z(x,t),\quad(x,t)\in(x_1,x_4)\times(T_*/2,T_*).
\end{equation}
Set
$$
v(x,t)=\frac{\partial w}{\partial t}(x,t)-\kappa f(w(x,t)),\quad(x,t)\in[x_2,x_3]\times[T_*/2,T_*).
$$
By \eqref{www-b2}, \eqref{www-b4}, \eqref{www-b8} and \eqref{www-b9},
there exists $\kappa>0$ such that
\begin{equation}
\label{www-b10}
v(x,t)\ge 0,\quad(x,t)\in \{x_2,x_3\}\times[T_*/2,T_*)
\cup [x_2,x_3]\times\{T_*/2\}.
\end{equation}
From \eqref{www-b3-0} and \eqref{www-b3},
$v$ solves
\begin{align*}
&\frac{x^q}{m}w^{1/m-1}(x,t)\frac{\partial v}{\partial t}-\frac{\partial^2 v}{\partial x^2}-f'(w(x,t))v
\\
&=\frac{x^q}{m}w^{1/m-1}\Big(\frac{\partial^2 w}{\partial t^2}-\kappa f'(w)\frac{\partial w}{\partial t}\Big)-
\frac{\partial^3 w}{\partial t \partial x^2}+\kappa f''(w)\Big(\frac{\partial w}{\partial x}\Big)^2
\\
&\quad
+\kappa f'(w)\frac{\partial^2 w}{\partial x^2}-f'(w)\Big(\frac{\partial w}{\partial t}-\kappa f(w)\Big)
\\
&=
\Big(1-\frac{1}{m}\Big)\frac{x^q}{m} w^{1/m-2}\Big(\frac{\partial w}{\partial t}\Big)^2
+\kappa f''(w)\Big(\frac{\partial w}{\partial x}\Big)^2
\\
&\ge 0,\quad(x,t)\in(x_2,x_3)\times(T_*/2,T_*).
\end{align*}
Then,  from the classical comparison principle with \eqref{www-b10}
it follows that
$$
v(x,t)\ge0,\quad(x,t)\in[x_2,x_3]\times[T_*/2,T_*),
$$
which, together with \eqref{www-b1}, yields
$$
\lim_{t\to T_*^-}\sup_{(x_2,x_3)}\frac{\partial u}{\partial t}(\cdot,t)=+\infty.
$$
\end{proof}

\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation of China,
the 985 Program of Jilin University and the Basic Research Foundation of
Jilin University.

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\end{document}