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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 208, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/208\hfil Quenching for a semilinear diffusion]
{Quenching of a semilinear diffusion equation with convection and reaction}

\author[Q. Zhou, Y. Nie, X. Zhou, W. Guo \hfil EJDE-2015/208\hfilneg]
{Qian Zhou, Yuanyuan Nie, Xu Zhou, Wei Guo}

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

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

\address{Xu Zhou \newline
College of Computer Science and Technology,
Jilin University, Changchun 130012, China}
\email{zhouxu0001@163.com}

\address{Wei Guo \newline
School of Mathematics and Statistics,
Beihua University, Jilin 132013, China}
\email{guoweijilin@163.com}

\thanks{Submitted April 16, 2015. Published August 10, 2015.}
\subjclass[2010]{35K20, 35B40}
\keywords{Quenching; critical length}

\begin{abstract}
 This article concerns the quenching phenomenon of the solution to the
 Dirichlet problem of a semilinear diffusion equation with convection
 and reaction. It is shown that there exists a critical length for the
 spatial interval in the sense that the solution exists globally in
 time if the length of the spatial interval is less than this number
 while the  solution quenches if the length is greater than this number.
 For the solution quenching at a finite time,
 we study the location of the quenching points and the blowing up of
 the derivative of the solution with respect to the time.
\end{abstract}

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

\section{Introduction}

In this article, we consider the  problem
\begin{gather}
\label{a-1.1}
\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}+b(x)
\frac{\partial u}{\partial x}=f(u),\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 $a>0$, $b\in C^1([0,+\infty))\cap L^\infty([0,+\infty))$
and $f\in C^1([0,c))$ with $c>0$ satisfies
\begin{equation} \label{aa0}
f(0)>0,\quad f'(s)>0 \quad \text{for }0<s<c,
\quad\lim_{s \to c^{-}}f(s)=+\infty.
\end{equation}
By the properties of $f$, the solution $u$ to the problem
\eqref{a-1.1}--\eqref{a-1.3}
may quench, i.e., there exists a time $0<T_*\leq+\infty$ such that
\[
\sup_{(0,a)}u(\cdot,t)<c\quad \text{for each }0<t<T_*\quad	\text{and}\quad
\lim_{t\to T_*^-}\sup_{(0,a)}u(\cdot,t)=c.
\]
It is called that $u$ quenches at a finite time if $T_*<+\infty$,
while $u$ quenches at the infinite time if $T_*=+\infty$.

Quenching phenomena were introduced by  Kawarada \cite{Kawarada} in 1975
for the problem \eqref{a-1.1}--\eqref{a-1.3} in the case that
$b\equiv0$ and $f(s)={(1-s)^{-1}}\,(0\le s<1)$,
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.
However, it was unknown what happens when $a$ is equal to the critical
length in \cite{Kawarada}.
For the special case that $f(s)={(c-s)^{-\beta}}\,(0\le s<c,\beta>0)$,
Levine (\cite{Levine2}) in 1989 proved that the solution can not quench
in infinite time
by finding the explicit form of the minimum steady-state soluton.
Since \cite{Kawarada}, there are many interesting results on quenching phenomena
for semilinear uniformly parabolic equations
(see, e.g., \cite{AW,Boni,DL,Levine1,LL,LM}),
singular or degenerate semilinear parabolic equations
(see, e.g., \cite{C1,CHK,C6,C2,GH})
and quasilinear diffusion equations (\cite{DX,NW,Winkler,YYJ}).

In this article, we study the quenching phenomenon of
the solution to  \eqref{a-1.1}--\eqref{a-1.3}.
Since there is a convection term in \eqref{a-1.1}, it can describe more
diffusion phenomena.
By constructing suitable super and sub solutions,
we prove the existence of the critical length.
For the solution quenching at a finite time, 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 super and sub solutions.
Due to the existence of the convection term in \eqref{a-1.1},
we have to overcome some technical difficulties when doing estimates and
constructing super and sub solutions.

This paper is arranged as follows. The existence of the critical length is
proved in $\S 2$. Subsequently, in $\S 3$ 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{Critical length}

Thanks to the classical theory on parabolic equations,
 problem \eqref{a-1.1}--\eqref{a-1.3} is well-posed locally in time.
Denote
\begin{align*}
T_*=\sup\Big\{&T>0: \text{problem \eqref{a-1.1}--\eqref{a-1.3} admits a solution }
\\
&u\in C^{2,1}((0,a)\times(0,T))\cap C([0,a]\times[0,T]) \text{ and }
\sup_{(0,a)\times(0,T)}u<c\Big\}.
\end{align*}
$T_*$ is called the life span of the solution to \eqref{a-1.1}--\eqref{a-1.3}.

\begin{proposition} \label{existencere}
Problem \eqref{a-1.1}--\eqref{a-1.3} admits uniquely a solution $u$
in $(0,T_*)$. Furthermore,
$u\in C^{2,1}((0,a)\times(0,T_*))\cap C([0,a]\times[0,T_*))$ and satisfies
$u>0$ and $\frac{\partial u}{\partial t}>0$ in $(0,a)\times(0,T_*)$.
\end{proposition}

\begin{proof}
Clearly, the existence and uniqueness follow from the local well-posedness
and a standard extension process.
Set
\[
v(x,t)=\frac{\partial u}{\partial t}(x,t),\quad(x,t)\in[0,a]\times[0,T_*).
\]
Then $v$ solves
\begin{gather*}
\frac{\partial v}{\partial t}-\frac{\partial^2 v}{\partial x^2}
+b(x)\frac{\partial v}{\partial x}=f'(u(x,t))v,\quad (x,t) \in (0,a)\times(0,T_*),
\\
v(0,t)=0=v(a,t),\quad t \in (0,T_*),\\
v(x,0)=f(0),\quad x \in (0,a).
\end{gather*}
The strong maximal principles for $u$ and $v$ show that $u>0$ and $v>0$
in $(0,a)\times(0,T_*)$.
\end{proof}

If $T_*=+\infty$, then $u$ exists globally in time.
If $T_*<+\infty$, then $u$ must quench at a finite time.
Let us study the relation between $T_*$ and $a$ below.
For convenience, we denote $u_a$ to be the solution to
 \eqref{a-1.1}--\eqref{a-1.3}
and $T_*(a)$ to be its life span.

\begin{lemma} \label{lemma-3.1}
If $a>0$ is 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
\[
0<a\le\min\Big\{\Big(\frac{4c_0}{f(c_0)}\Big)^{1/2},
\frac{1}{\|b\|_{L^\infty([0,+\infty))}+1}\Big\}.
\]
Set
$$
\bar u_a(x,t)={f(c_0)}x(a-x),\quad (x,t)\in [0,a]\times[0,+\infty).
$$
Then, $\bar u_a$ satisfies
\begin{gather*}
0\le\bar u_a(x,t)\le\frac{1}{4}f(c_0)a^2\le c_0,\quad
(x,t)\in [0,a]\times[0,+\infty),
\\
\frac{\partial \bar u_a}{\partial t}
-\frac{\partial^2\bar u_a}{\partial x^2}+b(x)\frac{\partial \bar u_a}{\partial x}
=2{f(c_0)}+{f(c_0)}b(x)(a-2x) \ge f(c_0)\ge f(\bar u_a),
\\
(x,t)\in(0,a)\times(0,+\infty).
\end{gather*}
The comparison principle shows that
$u_a\le\bar u_a\le c_0$ 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
$$
\underline u_a(x,t)=\frac{t}{4T}{f(0)}x(a-x),\quad(x,t)\in [0,a]\times[0,T]
$$
with $T=\max\big\{\frac1{4}a^{2},{a(\|b\|_{L^\infty([0,+\infty))}+1)}\big\}$.
Then, $\underline u_a$ satisfies
\begin{align*}
\frac{\partial \underline u_a}{\partial t}-\frac{\partial^2\underline u_a}{\partial x^2}
 +b(x)\frac{\partial \underline u_a}{\partial x}
&=\frac{1}{4T}{f(0)}x(a-x)+\frac{t}{2T}{f(0)}+\frac{t}{4T}{f(0)}b(x)(a-2x)
\\
&\leq f(0)\le f(\underline u_a),\quad(x,t)\in(0,a)\times(0,T).
\end{align*}
The comparison principle shows
$u_a\ge\underline u_a$ in $(0,a)\times(0,T)$.
Particularly, $u_a(a/2,T)\ge\frac1{16}{f(0)a^2}$,
which yields $T_*(a)<+\infty$ if $a\ge 4\sqrt{c}/\sqrt{f(0)}$.
\end{proof}

\begin{lemma} \label{lemma-3.3}
For any $0<a_1<a_2$, we have
$u_{a_1}<u_{a_2}$ in $(0,a_1)\times(0,T_*(a_2))$ and
$\frac{\partial u_{a_1}}{\partial x}(0,\cdot)
<\frac{\partial u_{a_2}}{\partial x}(0,\cdot)$ in $(0,T_*(a_2))$.
\end{lemma}

\begin{proof}
Proposition \ref{existencere} shows $T_*(a_1)\ge T_*(a_2)$ and
$u_{a_2}(a_1,t)>0$ for each $t\in(0,T_*(a_2))$.
Set
$$
w(x,t)=u_{a_1}(x,t)-u_{a_2}(x,t),\quad (x,t)\in [0,a_1]\times[0,T_*(a_2)).
$$
Then $w$ solves
\begin{gather*}
\frac{\partial w}{\partial t}-\frac{\partial^2 w}{\partial x^2}+b(x)
\frac{\partial w}{\partial x}=h(x,t)w,\quad
(x,t) \in (0,a_1)\times(0,T_*(a_2)),
\\
w(0,t)=0,\quad w(a_1,t)=u_{a_2}(a_1,t)>0,\quad t \in (0,T_*(a_2)),
\\
w(x,0)=0,\quad x \in (0,a_1),
\end{gather*}
where
\[
h(x,t)=\int_0^1f'(\sigma u_{a_1}(x,t)+(1-\sigma)u_{a_2}(x,t)) d\sigma,
\quad (x,t)\in (0,a_1)\times(0,T_*(a_2)).
\]
The strong maximum principle leads to
$w<0$ in $(0,a_1)\times(0,T_*(a_2))$,
and thus the Hopf Lemma yields $\frac{\partial w}{\partial x}(0,\cdot)<0$ in $(0,T_*(a_2))$.
\end{proof}


\begin{lemma}\label{lemma-3.3new}
There exists at most one $a>0$ such that $u_a$ quenches at the infinite time.
\end{lemma}

\begin{proof} Assume that $u_{a_0}$ quenches at the infinite time for
some $a_0>0$. For each $a>a_0$, let us show that $u_{a}$ quenches at a
finite time by contradiction.
Otherwise, Lemma \ref{lemma-3.3} shows that $u_{a}$ must quench at the
infinite time. Proposition \ref{existencere} and Lemma \ref{lemma-3.3} yield
\begin{gather}
\label{aa2}
u_a(a_0,t)>u_a(a_0,1)>0,\quad  t\in(1,+\infty),
\\
\label{aa3}
u_a(x,1)>u_{a_0}(x,1),\quad x\in(0,a_0)
	\text{ and }\frac{\partial u_a}{\partial x}(0,t)>\frac{\partial u_{a_0}}{\partial x}(0,t),\quad t\in(1,+\infty).
\end{gather}
Let
$$
\underline u_a(x,t)=u_{a_0}(x,t)+\delta\int_0^x	
\exp \Big\{\int_0^y b(s)ds\Big\}dy,\quad (x,t)\in [0,a_0]\times[1,+\infty).
$$
By \eqref{aa2} and \eqref{aa3}, there exists $\delta>0$ such that
\begin{gather} \label{aa4}
u_a(a_0,t)>\underline u_a,\quad  t\in(1,+\infty), \quad
u_a(x,1)>\underline u_a(x,1),\quad x\in(0,a_0).
\end{gather}
Note that $\underline u_a$ satisfies
\begin{gather} \label{aa5}
\frac{\partial \underline u_a}{\partial t}-\frac{\partial^2 \underline u_a}{\partial x^2}+b(x)\
pd {\underline u_a}x=
f(u_{a_0})<f({\underline u_a}),\quad (x,t)\in(0,a_0)\times(1,+\infty).
\end{gather}
Owing to \eqref{aa4} and \eqref{aa5}, the comparison principle gives
$$
u_a(x,t)\ge\underline u_a(x,t)=u_{a_0}(x,t)+\delta\int_0^x
 \exp \Big\{\int_0^y b(s)ds\Big\}dy,\quad (x,t)\in [0,a_0]\times[1,+\infty),
$$
which contradicts that both $u_{a_0}$ and $u_a$ quench at the infinite time.
\end{proof}


\begin{theorem}\label{th3.1}
Assume that $f\in C^1([0,c))$ satisfies \eqref{aa0}.
Then 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
$$
S=\big\{a>0: T_*(a)=+\infty \text{ and }\sup_{(0,a)\times(0,+\infty)}u_a<c\big\}.
$$
From Lemmas \ref{lemma-3.1} and \ref{lemma-3.2}, $S$ is a bounded set.
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} \label{remark1} \rm
$T_*(a_*)=+\infty$. However,
it is unknown whether $u_{a_*}$ quenches or not at the infinite time.
\end{remark}

\begin{remark} \label{remark2} \rm
To consider the classical solution to  problem \eqref{a-1.1}--\eqref{a-1.3},
we need $b\in C^1([0,+\infty))\cap L^\infty([0,+\infty))$.
While $b\in L^\infty([0,+\infty))$, we can investigate the weak solution
to \eqref{a-1.1}--\eqref{a-1.3},
and it is not hard to show that Lemmas \ref{lemma-3.1}--\ref{lemma-3.3new}
also hold.
Therefore, Theorem \ref{th3.1} still holds if $b\in L^\infty([0,+\infty))$.
\end{remark}

\section{Quenching properties}

\begin{definition} \label{def3.1}
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{th3.3}
Assume that $f\in C^1([0,c))$ satisfies \eqref{aa0} and
$M=\int_0^{c}f(s)ds<+\infty$.
Let $u$ be the solution to \eqref{a-1.1}--\eqref{a-1.3} quenching
at a finite time $T_*$.
Then the quenching points belong to $[\delta,a-\delta]$
with
\[
\delta=\frac{c^{2}}{2aM} \exp \Big\{-\frac12\|b\|_{L^\infty([0,+\infty))}T_*\Big\}.
\]
\end{theorem}

\begin{proof}
For each $0<s<T_*$, multiplying \eqref{a-1.1} by $\frac{\partial u}{\partial t}$ 
and then integrating over $(0,a)\times(0,s)$ by parts
with \eqref{a-1.2}, one gets
\begin{align*}
&\int_0^s\int_0^a\Big(\frac{\partial u}{\partial t}\Big)^2\,dx\,dt
 +\frac12\int_0^s\int_0^a\frac{\partial }{\partial t}\Big(\frac{\partial u}{\partial x}\Big)^2\,dx\,dt
 + \int_0^s\int_0^a b(x)\frac{\partial u}{\partial x}\frac{\partial u}{\partial t} \,dx\,dt\\
&=\int_0^s\int_0^a\frac{\partial }{\partial t}F(u)\,dx\,dt
\end{align*}
with
\[
F(\omega) = \int_0^\omega f(y)\,dy \quad (\omega\geq 0 ),
\]
which, together with \eqref{a-1.3}, the Young inequality 
and the Schwarz inequality, lead to
\begin{align*}
\int_0^a\Big(\frac{\partial u}{\partial x}(x,s)\Big)^2dx
&\le 2\int_0^a F(u(x,s)) dx
+\frac12\|b\|_{L^\infty([0,+\infty))}\int_0^s\int_0^a\Big(\frac{\partial u}{\partial x}(x,t)
 \Big)^2\,dx\,dt
\\
&\le 2aM+\frac12\|b\|_{L^\infty([0,+\infty))}\int_0^s\int_0^a
 \Big(\frac{\partial u}{\partial x}(x,t)\Big)^2\,dx\,dt.
\end{align*}
Then, the Gronwall inequality shows
\begin{equation} \label{www-5}
\int_0^a\Big(\frac{\partial u}{\partial x}(x,s)\Big)^2dx
\le 2aM \exp \Big\{\frac12\|b\|_{L^\infty([0,+\infty))}T_*\Big\},\quad
t\in(0,T_*).
\end{equation}
By \eqref{www-5}, \eqref{a-1.2} and the Schwarz inequality, one gets
\begin{align*}
u(x,t)&=\int^x_0 \frac{\partial u}{\partial x}(y,t) dy \\
&\leq x^{1/2} \Big(\int_0^a\Big(\frac{\partial u}{\partial x}(y,t)\Big)^2dy\Big)^{1/2}\\
&\leq (2aMx)^{1/2} \exp \Big\{\frac14\|b\|_{L^\infty([0,+\infty))}T_*\Big\},
\quad (x,t)\in[0,a/2]\times(0,T_*)
\end{align*}
and
\begin{align*}
u(x,t)&=-\int_x^a \frac{\partial u}{\partial x}(y,t) dy\leq(a-x)^{1/2}
\Big(\int_0^a\Big(\frac{\partial u}{\partial x}(y,t)\Big)^2dy\Big)^{1/2}
\\
&\leq (2aM(a-x))^{1/2}	\exp \Big\{\frac14\|b\|_{L^\infty([0,+\infty))}T_*\Big\},
 \quad (x,t)\in[a/2,a]\times(0,T_*),
\end{align*}
which show that there is no quenching point in $[0,\delta)\cup(a-\delta,a]$.
\end{proof}

\begin{theorem} \label{th3.4}
Assume that $f\in C^2([0,c))$ satisfies \eqref{aa0},
$\int_0^{c}f(s)ds<+\infty$ and $f''\ge0$ in $(0,c)$.
Let $u$ be the solution to the problem \eqref{a-1.1}--\eqref{a-1.3}
quenching at a finite time $T_*$.
Then the solution $u$ to  \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{th3.3},  there exist $0<x_1<x_2<x_3<x_4<a$ such that
\begin{equation} \label{www-b1}
\lim_{t\to T_*^-}\sup_{(x_2,x_3)}u(\cdot,t)=c,\quad
\sup_{(0,x_2)\times(0,T_*)} u<c,\quad \sup_{(x_3,a)\times(0,T_*)}u<c.
\end{equation}
Set
\[
v(x,t)=\frac{\partial u}{\partial t}(x,t),\quad(x,t)\in[0,a]\times[0,T_*),
\]
which solves
\begin{equation} \label{www-b3}
\frac{\partial v}{\partial t}-\frac{\partial^2 v}{\partial x^2}+b(x)\frac{\partial v}{\partial x}=f'(u)v,\quad (x,t) \in (0,a)\times(0,T).
\end{equation}
Proposition \ref{existencere} gives
\begin{equation} \label{www-b4}
v(x,t)>0,\quad(x,t) \in (0,a)\times(0,T_*).
\end{equation}
Let $z$ be the solution to the linear problem
\begin{gather} \label{www-b5}
\frac{\partial z}{\partial t}-\frac{\partial^2 z}{\partial x^2}+b(x)
\frac{\partial z}{\partial x}=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}
with $\delta=\min_{(x_1,x_4)}v(\cdot,T_*/2)$.
Owing to \eqref{www-b3} and \eqref{www-b4}, $v$ is a supersolution to
\eqref{www-b5}--\eqref{www-b7}.
The comparison principle and the maximum principle give
\begin{equation}\label{www-b9}
v(x,t)\ge z(x,t)\ge\gamma,\quad(x,t)\in(x_1,x_4)\times(T_*/2,T_*)
\end{equation}
with some $\gamma>0$.
Set
$$
w(x,t)=v(x,t)-\kappa f(u(x,t)),\quad(x,t)\in[x_2,x_3]\times[T_*/2,T_*).
$$
By \eqref{www-b1} and \eqref{www-b9},
there exists $\kappa>0$ such that
\begin{equation} \label{www-b10}
w(x,t)\ge 0,\quad(x,t)\in \{x_2,x_3\}\times[T_*/2,T_*)
\cup [x_1,x_2]\times\{T_*/2\}.
\end{equation}
Thanks to \eqref{a-1.1} and \eqref{www-b3}, $v$ solves
\begin{gather*}
\frac{\partial w}{\partial t}-\frac{\partial^2 w}{\partial x^2}+b(x)\frac{\partial w}{\partial x}-f'(u)w
=\kappa f''(u)\Big(\frac{\partial w}{\partial x}\Big)^2\ge0,\\
(x,t)\in(x_2,x_3)\times(T_*/2,T_*).
\end{gather*}
Then, it follows from the maximal principle with \eqref{www-b10} that
$w\ge0$ in $(x_2,x_3)\times[T_*/2,T_*)$,
which, together with \eqref{www-b1}, yields
$\lim_{t\to T_*^-}\sup_{(x_2,x_3)}v(\cdot,t)=+\infty$.
\end{proof}

\begin{remark} \label{remark3} \rm
As in Remark \ref{remark2}, we note that
Theorems \ref{th3.3} and \ref{th3.4}   remain valid  if $b\in L^\infty([0,+\infty))$.
\end{remark}


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\end{document}

First author:
First name is Qian, Last name is Zhou,
her e-mail address is zhouqian@jlu.edu.cn;

Second author:
First name is Yuanyuan, Last name is Nie,
his e-mail address is nieyy@jlu.edu.cn;

Third author(the corresponding author):
First name is Xu, Last name is Zhou,
his e-mail address is zhouxu0001@163.com;

Last author:
First name is Wei, Last name is Guo,
her e-mail address is guoweijilin@163.com.