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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 298, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/298\hfil Asymptotic behavior of solutions]
{Asymptotic behavior of solutions to a degenerate quasilinear
parabolic equation with a gradient term}

\author[H. Li, X. Wang, Y. Nie, H. He \hfil EJDE-2015/298\hfilneg]
{Huilai Li, Xinyue Wang, Yuanyuan Nie, Hong He}

\address{Huilai Li \newline
School of Mathematics, Jilin University,
Changchun 130012, China}
\email{lihuilai@jlu.edu.cn}

\address{Xinyue Wang \newline
Experimental School of the Affiliated
Middle School to the Jilin University,
Changchun 130021, China}
\email{xinyuewang0000@163.com}

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

\address{Hong He \newline
School of Mathematics, Jilin University,
Changchun 130012, China}
\email{honghemath@163.com}

\thanks{Submitted October 27, 2015. Published December 3, 2015.}
\subjclass[2010]{35K65, 35K59, 35B33}
\keywords{Critical Fujita exponent; degenerate; quasilinear; gradient term}

\begin{abstract}
 This article concerns the asymptotic behavior of solutions
 to the Cauchy problem of a degenerate quasilinear parabolic equations
 with a gradient term. A blow-up theorem of Fujita type is established
 and the critical Fujita exponent is formulated by the spacial dimension
 and the behavior of the coefficient of the gradient term at $\infty$.
\end{abstract}

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

\section{Introduction}

In this article, we study the asymptotic behavior of solutions
to the Cauchy problem
\begin{gather}\label{eq}
\frac{\partial u}{\partial t}=\Delta u^m
+b(|x|)x\cdot\nabla u^m+u^p, \quad x\in\mathbb{R}^n,\;t>0,\\
\label{initial}
u(x,0)=u_0(x),\quad x\in\mathbb{R}^n,
\end{gather}
where $p>m>1$, $b\in C^{0,1}([0,+\infty))$ and 
$0\le u_0\in L^\infty(\mathbb{R}^n)$.
The equation \eqref{eq} is a typical quasilinear parabolic equaton
which is called the Newtonian filtration equation.
It is noted that \eqref{eq} is degenerate at the points where $u=0$.
In the semilinear case $m=1$, \eqref{eq} is the heat equation.


The studies on asymptotic behavior of solutions
to diffusion equations with nonlinear reaction began in 1966 by Fujita \cite{fujita}.
There it was proved that for  \eqref{eq}-\eqref{initial}
with $m=1$ and $b\equiv0$,
there does not exist a nontrivial nonnegative global
solution if $1<p<p_c=1+2/n$, whereas if $p>p_c$, there exist both
nontrivial nonnegative global and blow-up solutions.
This result shows that the exponent $p$ of the nonlinear reaction
affects the properties of solutions directly.
We call $p_c$ the critical Fujita exponent and such a
result a blow-up theorem of Fujita type.

The elegant work of Fujita revealed a new phenomenon of nonlinear
evolution equations. There have been a number of extensions of
Fujita's results in several directions since then, including
similar results for numerous of quasilinear parabolic equations
and systems in various of geometries (whole spaces, cones and
exterior domains) with nonlinear reactions or nonhomogeneous boundary
conditions, and even degenerate equations in domains with
non-compact boundary \cite{andr2,andr}. We refer to the survey
papers \cite{deng,levine} and the references therein, and more recent works
\cite{fira,zhang1,qi,semi2,wang,winkler,zhang2,zheng2,zheng,Nie}.
Among those extensions, it is Galaktionov \cite{gala1,gala2} who first
investigated the blow-up theorem of Fujita type for \eqref{eq}-\eqref{initial}
with $b\equiv0$ and obtained that $p_c=m+2/n$.
As to nonlinear evolution equations with gradient terms,
there are some studies for the semilinear case.
In 1990, Meier \cite{meier} studied the critical Fujita exponent
for the Cauchy problem of
\begin{equation} \label{newin-1}
\frac{\partial u}{\partial t}=\Delta u+\vec b(x)\cdot\nabla u+u^p,\quad
x\in\mathbb{R}^n,\;t>0,
\end{equation}
where $\vec b\in L^\infty(\mathbb{R}^n;\mathbb{R}^n)$.
It was proved that
$$
p_c=1+\frac1{\lambda^*},
$$
where $\lambda^*$ is the maximal decay rate for solutions to
\begin{equation}
\label{newin-2}
\frac{\partial w}{\partial t}=\Delta w+\vec b(x)\cdot\nabla w,\quad
x\in\mathbb{R}^n,\;t>0,
\end{equation}
i.e.
\begin{align*}
\lambda^*=\sup\big\{&\lambda\in\mathbb{R}: \text{ there exists a nontrivial solution
$w$ of \eqref{newin-2}}\\
&\text{such that }
\limsup_{t\to+\infty} t^\lambda\|w(\cdot,t)\|_{L^\infty(\mathbb{R})}<+\infty\big\}.
\end{align*}
If $\vec b$ is constant, it is clear that $\lambda^*={n}/2$ and $p_c=1+2/n$.
However, for nonconstant $\vec b\in L^\infty(\mathbb{R}^n;\mathbb{R}^n)$,
$\lambda^*$ and $p_c$ are unknown generally.
In 1993, Aguirre and Escobedo \cite{ag} considered
the Cauchy problem of
\[
\frac{\partial u}{\partial t}=\Delta u+\vec b_0\cdot\nabla u^q+u^p, \quad
x\in\mathbb{R}^n,\;t>0,
\quad(0\neq \vec b_0\in \mathbb{R}^n,\,q>1)
\]
and proved that
$$
p_c=\min\big\{1+\frac{2}{n},1+\frac{2q}{n+1}\big\}.
$$
In \cite{Nie}, the semilinear problem \eqref{eq}-\eqref{initial} with $m=1$ was studied
and it was shown that if $b$ satisfies
\begin{gather}
\label{b1}
\lim_{s\to+\infty}s^2b(s)=\kappa,\quad(-\infty\le\kappa\le+\infty),\\
\label{b2}
\inf\{s^2b(s):s>0\}>-n\text{ in the case } -n<\kappa\le+\infty,
\end{gather}
then the critical Fujita exponent is
\[
p_c=\begin{cases}
1, &\kappa=+\infty, \\
1+2/(n+\kappa),&-n<\kappa<+\infty,\\
+\infty,&-\infty\le\kappa\le-n.
\end{cases}
\]
As to the quasilinear parabolic equations with gradient terms,
Suzuki \cite{suzuki} in 1998 considered the Cauchy problem of
\[
\frac{\partial u}{\partial t}=\Delta u^m+\vec b_0\cdot\nabla u^q+u^p,
\quad x\in\mathbb{R}^n,\;t>0, \quad(m\ge1,\,0\neq \vec b_0\in \mathbb{R}^n,\,p,q>1)
\]
and proved that
if $q>m-1$ and $\max\{m,q\}\le p<\min\{m+2/n,m+2(q-m+1)/(n+1)\}$,
then there does not exist any nontrivial nonnegative global solutions.
In \cite{zheng}, the case
$$
b(s)=\frac{\kappa}{s^2},\quad s>0,\quad(-\infty<\kappa<+\infty)
$$
was studied.
Since such a function is singular at $0$ when $\kappa\neq 0$, the authors
considered the Neumann exterior problem
\begin{gather*}
\frac{\partial u}{\partial t}=\Delta u^m+\frac{\kappa}{|x|^2}x\cdot\nabla u^m+u^p,
\quad x\in\mathbb{R}^n\setminus B_1,\;t>0,
\\
\frac{\partial u^m}{\partial\nu}(x,t)=0,\quad x\in\partial B_1,\;t>0,
\\
u(x,0)=u_0(x),\quad x\in\mathbb{R}^n\setminus\overline B_1
\end{gather*}
and showed that its critical Fujita exponent is
\[
p_c=\begin{cases}
m+2/(n+\kappa), &-n<\kappa<+\infty,\\
+\infty, &-\infty<\kappa\le-n,
\end{cases}
\]
where $B_1$ is the unit ball in $\mathbb{R}^n$
and $\nu$ is the unit inner normal vector to $\partial B_1$.
Also they considered a special case for the Dirichlet exterior problem.

In this article, we study the asymptotic behavior of solutions
to the Cauchy problem \eqref{eq}-\eqref{initial},
where $b$ satisfies \eqref{b1} and \eqref{b2}.
It is proved that the critical Fujita exponent to \eqref{eq}-\eqref{initial}
can be formulated as
\begin{equation}
p_c=\begin{cases}
m,&\kappa=+\infty, \\
m+2/(n+\kappa),&-n<\kappa<+\infty,\\
+\infty,&-\infty\le\kappa\le-n.
\end{cases} \label{pc}
\end{equation}
That is to say, if $m<p<p_c$, there does not exist any nontrivial nonnegative global
solution to   \eqref{eq}-\eqref{initial},
whereas if $p>p_c$, there exist both
nontrivial nonnegative global and blow-up solutions to
 \eqref{eq}-\eqref{initial}.
It is shown from \eqref{pc} that
the behavior of the coefficient of the gradient term at $\infty$,
together with the spacial dimension, determines precisely
the critical Fujita exponent to  \eqref{eq}-\eqref{initial}.
The technique used in this paper is mainly inspired by \cite{semi1,semi2,zheng,Nie}.
To prove the blow-up of solutions, we determine the interactions among the
diffusion, the gradient and the reaction by a precise energy integral estimate
instead of pointwise comparisons. The key is to choose a suitable weight for
the energy integral.
For the  existence of global nontrivial solutions,
we construct a global nontrivial supersolution.
Noting that \eqref{eq} does not possess a self-similar construct,
we have to seek a complicated supersolution and do some precise calculations.
By the way, \eqref{b2} is used only for constructing a global nontrivial supersolution
and it seems necessary when one constructs such a supersolution.


This article is organized as follows.
We give some preliminaries in \S 2,
such as the well-posedness of  \eqref{eq}-\eqref{initial} and
some auxiliary lemmas. The blow-up theorems of Fujita type for
 \eqref{eq}-\eqref{initial}
are obtained in \S 3.


\section{Preliminaries}

Equation \eqref{eq} is degenerate at the points where $u=0$.
So, weak solutions are considered at those points in this paper.

\begin{definition} \label{ddef1} \rm
Let $0<T\le+\infty$.
A nonnegative function $u$ is called a super (sub) solution to the problem
\eqref{eq}, \eqref{initial} in $(0,T)$, if
$$
u\in C([0,T),L^m_{\rm loc}(\mathbb{R}^n))\cap
L^\infty_{\rm loc}(0,T;L^\infty(\mathbb{R}^n))
$$
and the integral inequality
\begin{align*}
&\int_0^T\int_{\mathbb{R}^n}u(x,t)\frac{\partial\varphi}{\partial t}(x,t) \,dx\,dt
+\int_0^T\int_{\mathbb{R}^n}u^m(x,t)\Delta\varphi(x,t)\,dx\,dt \\
&-\int_0^T\int_{\mathbb{R}^n} u^m(x,t)
\operatorname{div}(b(|x|)\varphi(x,t) x)\,dx\,dt
\\
& +\int_0^T\int_{\mathbb{R}^n}u^p(x,t)\varphi(x,t) \,dx\,dt
+\int_{\mathbb{R}^n}u_0(x)\varphi(x,0) dx\le(\ge)0
\end{align*}
is satisfied for each $0\le\varphi\in C^{1,1}(\mathbb{R}^n\times[0,T))$
vanishing when $t$ near $T$ or $|x|$ being sufficiently large.
A nonnegative function $u$ is called a solution to the problem
\eqref{eq}, \eqref{initial} in $(0,T)$, if it is both a supersolution
and a subsolution.
\end{definition}

\begin{definition} \label{def2.2} \rm
A solution $u$ to the problem
\eqref{eq}, \eqref{initial} is said to blow up in a finite time
$0<T<+\infty$, if
$$
\|u(\cdot,t)\|_{L^\infty(\mathbb{R}^n)}\to+\infty\quad\text{as } t\to T^-.
$$
Otherwise, $u$ is said to be global.
\end{definition}

For $0\le u_0\in L^\infty(\mathbb{R}^n)$,
by using the theory on the Newtonian filtration equations (see, e.g.,
\cite{kala,lady,wu}), one can establish the
existence, uniqueness and the comparison principle to the solutions of
\eqref{eq}-\eqref{initial} locally in time.
Moreover, it can be proved that

\begin{lemma} \label{lemmaw1}
Assume that $u$ is a solution to \eqref{eq}-\eqref{initial} in $(0,T)$
with $0<T\le+\infty$.
Then, for each $\psi\in C^{1,1}_0(\mathbb{R}^n)$,
\begin{align*}
\frac{d}{dt}\int_{\mathbb{R}^n}u(x,t)\psi(x) dx
&=\int_{\mathbb{R}^n}u^m(x,t)\Delta\psi(x)\,dx\,dt
-\int_{\mathbb{R}^n} u^m(x,t)\operatorname{div}(b(|x|)\psi(x) x)dx\\
&\quad +\int_{\mathbb{R}^n}u^p(x,t)\psi(x) dx
\end{align*}
in the distribution sense.
\end{lemma}

To investigate the blow-up property of solutions to  \eqref{eq}-\eqref{initial},
we need the following auxiliary lemma.

\begin{lemma}\label{lemma1}
Assume that $b\in C^{0,1}([0,+\infty))$ satisfies \eqref{b1} with
 $-\infty\le\kappa<+\infty$. Let $u$ be a solution to the problem \eqref{eq},
\eqref{initial}. Then there exist three numbers $R_0>0$, $\delta>1$ and $M_0>0$
depending only on $n$ and $b$, such that for each
$R>R_0$,
\begin{equation} \label{Nle1}
\begin{aligned}
\frac{d}{dt}\int_{\mathbb{R}^n}u(x,t)\psi_R(|x|) dx
&\ge-M_0R^{-2}\int_{B_{\delta R}\setminus B_R} u^m(x,t)\psi_R(|x|) dx\\
&\quad +\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|)dx,
\end{aligned}
\end{equation}
for $t>0$, where
$$
\psi_R(r)=\begin{cases}
h(r),&0\le r\le R,\\
\frac12h(r)\Big(1+\cos\frac{(r-R)\pi}{(\delta-1)R}\Big),\quad&R<r<\delta R,\\
0,&r\ge\delta R
\end{cases}
$$
with
$$
h(r)=\exp \Big\{\int_0^r sb(s)ds\Big\},\quad r\ge 0,
$$
while $B_r$ denotes the open ball in $\mathbb{R}^n$
with radius $r$ and centered at the origin.
\end{lemma}

\begin{proof}
It is clear that $\psi_R\in C^{1,1}([0,+\infty))$
with $\psi'_R(0)=0$.
Choosing $\psi(x)=\psi_R(|x|)$ in Lemma \ref{lemmaw1},
one gets that
\begin{equation} \label{Nle1-1}
\begin{aligned}
&\frac{d}{dt}\int_{\mathbb{R}^n}u(x,t)\psi_R(|x|) dx\\
&=\int_{\mathbb{R}^n}u^m(x,t)\Delta\psi_R(|x|)dx
-\int_{\mathbb{R}^n}u^m(x,t)\operatorname{div}(b(|x|)\psi_R(|x|)x)dx\\
&\quad +\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|) dx\\
&=\int_{B_{\delta R}}u^m(x,t)\big(\Delta\psi_R(|x|)
 -\operatorname{div}(b(|x|)\psi_R(|x|)x)\big)dx\\
&\quad  +\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|) dx.
\end{aligned}
\end{equation}
As shown in \cite{Nie}, there exist three numbers $R_0>0$, $\delta>1$ and $M_0>0$
depending only on $n$ and $b$, such that for each $R>R_0$,
\begin{equation} \label{new-4}
\Delta\psi_R(|x|)-\operatorname{div}(b(|x|)\psi_R(|x|)x)
\ge-M_0R^{-2}\psi_R(|x|),\quad |x|>0.
\end{equation}
Substituting \eqref{new-4} into \eqref{Nle1-1} leads to \eqref{Nle1}.
\end{proof}

\begin{remark} \label{remark1} \rm
Lemma \ref{lemma1} still holds if \eqref{b1} is replaced by
$$
\limsup_{s\to+\infty} s^2b(s)=\kappa.
$$
\end{remark}

\begin{remark} \label{remark1-1} \rm
The proof of Lemma \ref{lemma1} is invalid if $\kappa=+\infty$.
In this case, \eqref{Nle1} holds for each fixed $R>0$, but
$\delta>1$ and $M_0$ depend also on $R$.
\end{remark}

Next, we study self-similar supersolutions of \eqref{eq} of the form
\begin{equation} \label{self}
u(x,t)={(t+T)^{-\alpha}}v((t+T)^{-\beta}|x|),\quad
x\in\mathbb{R}^n,\, t\ge0,
\end{equation}
where
$$
\alpha=\frac{1}{p-1},\quad\beta=\frac{p-m}{2(p-1)},
$$
$T\ge1$ will be determined.
If $v\in C^{0,1}([0,+\infty))$ with $v^m\in C^{1,1}([0,+\infty))$ solves
\begin{equation} \label{ode}
\begin{aligned}
&(v^m)''(r)+\frac{n-1}{r}(v^m)'(r)+
(t+T)^{2\beta}b((t+T)^{\beta}r)r(v^m)'(r)\\
&+\beta rv'(r)+\alpha v(r)+v^p(r)\le0,\quad r>0
\end{aligned}
\end{equation}
for each $t>0$, then $u$ given by \eqref{self} is a supersolution to \eqref{eq}.

\begin{lemma} \label{lemma3}
Assume that $b\in C^{0,1}([0,+\infty))$ satisfies \eqref{b1} and \eqref{b2}
with $-n<\kappa\le+\infty$, and $p>p_c$. Choose $-n<\kappa_1<\kappa_2<\kappa$
such that
\begin{equation} \label{ww0}
\inf\{s^2b(s):s>0\}>\kappa_1,\quad \kappa_2>\frac1{p-m}-n.
\end{equation}
Let $T=\eta^{-(m+2)/\beta}$ and
\begin{equation} \label{vm1}
v(r)=(\eta-A(r))_+^{1/(m-1)},\quad r\ge 0,
\end{equation}
where $s_+=\max\{0,s\}$,
$0<\eta<1$ will be determined, while $A\in C^{1,1}([0,+\infty))$ satisfies
$A(0)=0$ and
\[
A'(r)=\begin{cases}
A_1r, &0\le r\le \eta^{m+1},\\
A_2r+(A_1-A_2)\frac{\eta^{(m+1)(n+\kappa_2)}}{r^{n+\kappa_2-1}},
& \eta^{m+1}<r<\eta^{m},\\
A_2r+(A_1-A_2)\eta^{n+\kappa_2}r,& r\ge \eta^{m}
\end{cases}
\]
with
\begin{gather*}
A_1=\max\Big\{\frac{2(m-1)}{m(n+\kappa_1)(p+p_c-2)},A_2\Big\},
\\
A_2=\frac{m-1}{2m(n+\kappa_2)(p-1)}+\frac{(m-1)(p-m)}{4m(p-1)}.
\end{gather*}
Then, there exists sufficiently small $0<\eta<1$ such that
$u$ given by \eqref{self} and \eqref{vm1} is a supersolution to \eqref{eq}.
\end{lemma}

\begin{proof}
Choose $\eta_0\in(0,1)$ such that for each $0<\eta<\eta_0$,
\begin{equation} \label{ww1}
A(r)<r,\quad 0<r<\eta_0.
\end{equation}
By the first formula in \eqref{ww0}, one has
\begin{equation} \label{ww12}
(t+T)^{2\beta}b((t+T)^{\beta}r)r\ge\frac{\kappa_1}{r},\quad r>0,\;t>0.
\end{equation}
Therefore, for each $0<\eta<\eta_0$,
it follows from \eqref{ww1} and \eqref{ww12} that
\begin{align*}
&(v^m)''(r)+\frac{n-1}{r}(v^m)'(r)+
(t+T)^{2\beta}b((t+T)^{\beta}r)r(v^m)'(r)\\
&+\beta rv'(r)+\alpha v(r)+v^p(r)
\\
&\le\frac{m}{(m-1)^2}(A'(r))^2v^{2-m}(r)
+\Big(\alpha-\frac{m}{m-1}A''(r)-\frac{m(n+\kappa_1-1)}{m-1}\frac{A'(r)}{r}\Big)v(r)\\
&\quad +v^p(r)
\\
&=\frac{A_1^2m}{(m-1)^2}r^2v^{2-m}(r)
+\Big(\alpha-\frac{A_1m(n+\kappa_1)}{m-1}\Big)v(r)+v^p(r)
\\
&\leq \Big(\frac{A_1^2m}{(m-1)^2} 
\eta^{2m-1}(1-\eta^{m-1})^{-1}
+\Big(\alpha-\frac{A_1m(n+\kappa_1)}{m-1}\Big)
+\eta^{(p-1)/(m-1)}\Big)v(r),
\end{align*}
for $0<r<\eta^{m+1}$ and $t>0$.
The choice of $A_1$ implies
$$
\alpha<\frac{A_1m(n+\kappa_1)}{m-1}.
$$
Thus, there exists $0<\eta_1<\eta_0$ such that
for each $0<\eta<\eta_1$,
\begin{equation} \label{ww2}
\begin{gathered}
\begin{aligned}
&(v^m)''(r)+\frac{n-1}{r}(v^m)'(r)+
(t+T)^{2\beta}b((t+T)^{\beta}r)r(v^m)'(r)\\
&+\beta rv'(r)+\alpha v(r)+v^p(r) \le0,
\end{aligned}\\
0<r<\eta^{m+1},\quad t>0.
\end{gathered}
\end{equation}
By \eqref{b1} and $\kappa_2<\kappa$,
there exists $0<\eta_2<\eta_0$ such that
\begin{align*}
s^2b(s)\ge\kappa_2,\quad s>\frac{1}{\eta_2},
\end{align*}
which, together with the choice of $T$, implies
that for each $0<\eta<\eta_2$,
\begin{equation} \label{ww3}
(t+T)^{2\beta}b((t+T)^{\beta}r)r\ge\frac{\kappa_2}{r},\quad r>\eta^{m+1},\;t>0.
\end{equation}
For each $0<\eta<\eta_2$,
it follows from \eqref{ww3} and \eqref{ww1} that
\begin{align*}
&(v^m)''(r)+\frac{n-1}{r}(v^m)'(r)+
(t+T)^{2\beta}b((t+T)^{\beta}r)r(v^m)'(r)\\
&+\beta rv'(r)+\alpha v(r)+v^p(r)
\\
&\leq \frac{m}{(m-1)^2}(A'(r))^2v^{2-m}(r)
+\Big(\alpha-\frac{m}{m-1}A''(r)-\frac{m(n+\kappa_2-1)}{m-1}\frac{A'(r)}{r}\Big)v(r)\\
&\quad +v^p(r)
\\
&= \frac{m}{(m-1)^2}\Big(A_2+(A_1-A_2)\frac{\eta^{(m+1)
 (n+\kappa_2)}}{r^{n+\kappa_2}}\Big)^2r^2v^{2-m}(r) \\
&\quad +\Big(\alpha-\frac{A_2m(n+\kappa_2)}{m-1}\Big)v(r)+v^p(r)
\\
&\leq \Big(\frac{mA_1^2}{(m-1)^2}
\eta^{2m-1}(1-\eta^{m-1})^{-1}
+\Big(\alpha-\frac{A_2m(n+\kappa_2)}{m-1}\Big)
+\eta^{(p-1)/(m-1)}\Big)v(r),
\end{align*}
for $\eta^{m+1}<r<\eta^{m}$ and $t>0$.
The choice of $A_2$ implies
$$
\alpha<\frac{A_2m(n+\kappa_2)}{m-1}.
$$
Thus, there exists $0<\eta_3<\eta_2$ such that
for each $0<\eta<\eta_3$,
\begin{equation} \label{ww4}
\begin{aligned}
&(v^m)''(r)+\frac{n-1}{r}(v^m)'(r)+
(t+T)^{2\beta}b((t+T)^{\beta}r)r(v^m)'(r)\\
&+\beta rv'(r)+\alpha v(r)+v^p(r) \leq 0,
\quad \eta^{m+1}<r<\eta^{m},\;t>0.
\end{aligned}
\end{equation}
For each $0<\eta<\eta_0$, \eqref{ww1} yields
$A^{-1}(\eta)>\eta$.
Thus, for each $0<\eta<\eta_2$,
it follows from \eqref{ww3} that
\begin{align*}
&(v^m)''(r)+\frac{n-1}{r}(v^m)'(r)+
(t+T)^{2\beta}b((t+T)^{\beta}r)r(v^m)'(r)\\
&+\beta rv'(r)+\alpha v(r)+v^p(r)
\\
&\leq \frac{1}{m-1}rA'(r)\Big(\frac{m}{m-1}\frac{A'(r)}{r}-\beta\Big)v^{2-m}(r)\\
&\quad +\Big(\alpha-\frac{m}{m-1}A''(r)-\frac{m(n+\kappa_2-1)}{m-1}
\frac{A'(r)}{r}\Big)v(r)+v^p(r)
\\
&= \frac{A_2}{m-1}r^2\Big(\frac{mA_2}{m-1}-\beta\Big)v^{2-m}(r)
+\Big(\alpha-\frac{A_2m(n+\kappa_2)}{m-1}\Big)v(r)+v^p(r),
\end{align*}
for $\eta^{m}<r<A^{-1}(\eta)$ and $t>0$.
The choice of $A_2$ implies
$$
\frac{mA_2}{m-1}<\beta,\quad
\alpha<\frac{A_2m(n+\kappa_2)}{m-1}.
$$
Thus, there exists $0<\eta_4<\eta_2$ such that
for each $0<\eta<\eta_4$,
\begin{equation} \label{ww5}
\begin{aligned}
&(v^m)''(r)+\frac{n-1}{r}(v^m)'(r)+
(t+T)^{2\beta}b((t+T)^{\beta}r)r(v^m)'(r)\\
&+\beta rv'(r)+\alpha v(r)+v^p(r) \leq 0,
\quad \eta^{m}<r<A^{-1}(\eta),\;t>0.
\end{aligned}
\end{equation}
Summing up,  from \eqref{ww2}, \eqref{ww4} and \eqref{ww5},
for $0<\eta<\min\{\eta_1,\eta_3,\eta_4\}$, the function
$u$ given by \eqref{self} and \eqref{vm1} is a supersolution of \eqref{eq}.
\end{proof}


\begin{remark} \label{remark1-2} \rm
Lemma \ref{lemma3} still holds if \eqref{b1} is replaced by
$$
\liminf_{s\to+\infty} s^2b(s)=\kappa.
$$
\end{remark}

\begin{remark} \rm
In Lemma \ref{lemma3}, \eqref{b2} is necessary to get a supersolution to \eqref{eq}
of the form \eqref{self} and \eqref{vm1}.
But, it is not clear at this moment whether \eqref{eq}
admits a global supersolution of other form if \eqref{b2} is invalid.
\end{remark}


\section{Blow-up theorem of Fujita type}

In this section, we establish the blow-up theorem of Fujita
type for the problem \eqref{eq}, \eqref{initial} by using Lemmas
\ref{lemma1} and \ref{lemma3}.
First consider the case $m<p<p_c$ with $-\infty\le\kappa<+\infty$.

\begin{theorem}\label{theorem1}
Assume that $b\in C^{0,1}([0,+\infty))$ satisfies \eqref{b1}
with $-\infty\le\kappa<+\infty$.
Let $m<p<p_c$. Then for each nontrivial $0\le u_0\in L^\infty(\mathbb{R}^n)$,
the solution to the problem \eqref{eq}, \eqref{initial} must blow up in a finite time.
\end{theorem}

\begin{proof}
Let $\psi_R$, $h$, $R_0$, $\delta$ and $M_0$ be given by Lemma \ref{lemma1}.
Owing to $-\infty\le\kappa<+\infty$ and $1<p<p_c$,
\begin{align*}
\kappa<\frac{2}{p-m}-n.
\end{align*}
Fix $\tilde\kappa$ to satisfy
\begin{align}
\label{nn0}
\kappa<\tilde\kappa<\frac{2}{p-m}-n.
\end{align}
By \eqref{b1} and \eqref{nn0}, there exists $R_1>0$ such that
\begin{align*}
s^2b(s)<\tilde\kappa,\quad s>R_1.
\end{align*}
For each $R>R_1$, one obtain
\begin{equation} \label{nn1}
\begin{aligned}
\int_{\mathbb{R}^n}\psi_R(|x|) dx
&\leq n\omega_n\int_0^{\delta R}r^{n-1}h(r)dr
\\
&\leq \omega_n(\delta R)^n\exp \Big\{\int_0^{\delta R} sb(s)ds\Big\}
\\
&\leq {\omega_n}(\delta R)^n\exp \Big\{\int_0^{R_1} sb(s)ds\Big\}
\exp \Big\{\tilde\kappa\int_{R_1}^{\delta R}\frac1s ds\Big\}
\\
&= M_1R^{n+\tilde\kappa},
\end{aligned}
\end{equation}
where $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n$,
while $M_1>0$ depends only on $n$, $b$, $R_1$, $\delta$ and $\tilde\kappa$.
Let $u$ be the solution to the problem \eqref{eq}, \eqref{initial}.
Denote
$$
w_R(t)=\int_{\mathbb{R}^n}u(x,t)\psi_R(|x|) dx,\quad t\ge0.
$$
For each $R>\max\{R_0,R_1\}$, it follows from Lemma \ref{lemma1} that
\begin{equation} \label{th1-2-1}
\frac{d}{dt}w_R(t)
\ge -M_0R^{-2}\int_{\mathbb{R}^n} u^m(x,t)\psi_R(|x|) dx
+\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|)dx,
\end{equation}
for $t>0$.
The H\"older inequality and \eqref{nn1} yield
\begin{equation} \label{th1-2-2}
\begin{aligned}
&\int_{\mathbb{R}^n} u^m(x,t)\psi_R(|x|) dx
\\
&\leq \Big(\int_{\mathbb{R}^n}\psi_R(|x|)dx\Big)^{(p-m)/p}
\Big(\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|)dx\Big)^{m/p}
\\
&\leq M_1^{(p-m)/p}R^{(p-m)(n+\tilde\kappa)/p}
\Big(\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|)dx\Big)^{m/p},\quad t>0.
\end{aligned}
\end{equation}
Substitute \eqref{th1-2-2} into \eqref{th1-2-1} to obtain
\begin{equation} \label{th1-2-3}
\begin{aligned}
\frac{d}{dt}w_R(t)
&\geq \Big(\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|)dx\Big)^{m/p}
\Big(-M_0M_1^{(p-m)/p}R^{-2+(p-m)(n+\tilde\kappa)/p}
\\
&\quad+\Big(\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|)dx\Big)^{(p-m)/p}\Big),\quad t>0.
\end{aligned}
\end{equation}
It follows from the H\"older inequality and \eqref{nn1} that
\begin{align*}
\int_{\mathbb{R}^n} u(x,t)\psi_R(|x|) dx
&\leq \Big(\int_{\mathbb{R}^n}\psi_R(|x|)dx\Big)^{(p-1)/p}
\Big(\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|)dx\Big)^{1/p}
\\
&\leq M_1^{(p-1)/p}R^{(p-1)(n+\tilde\kappa)/p}
\Big(\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|)dx\Big)^{1/p},
\end{align*}
for $t>0$, which implies
\begin{equation} \label{th1-2-4}
\int_{\mathbb{R}^n}u^p(x,t)\psi_R(|x|)dx
\ge M_1^{-(p-1)}R^{-(p-1)(n+\tilde\kappa)}w_R^p(t),\quad t>0.
\end{equation}
Substituting \eqref{th1-2-4} into \eqref{th1-2-3}, one gets that
for each $R>\max\{R_0,R_1\}$,
\begin{equation} \label{th1-2}
\begin{aligned}
\frac{d}{dt}w_R(t)
&\geq M_1^{-m(p-1)/p}R^{-m(p-1)(n+\tilde\kappa)/p}w_R^m(t) \\
&\quad\times \Big(-M_0M_1^{(p-m)/p}R^{-2+(p-m)(n+\tilde\kappa)/p}
\\
&\quad+M_1^{-(p-1)(p-m)/p}R^{-(p-1)(p-m)(n+\tilde\kappa)/p}w_R^{(p-m)}(t)\Big),
\quad t>0.
\end{aligned}
\end{equation}
Note that \eqref{nn0} implies $2>(p-m)(n+\tilde\kappa)$,
while $w_R(0)$ is nondecreasing with respect to
$R\in(0,+\infty)$ and
$$
\sup\{w_R(0):R>0\}>0.
$$
Therefore, there exists $R_2>0$ such that for each $R>R_2$,
\begin{equation} \label{th1-3}
\begin{aligned}
&M_0M_1^{(p-m)/p}R^{-2+(p-m)(n+\tilde\kappa)/p}\\
&\le\frac12M_1^{-(p-1)(p-m)/p}R^{-(p-1)(p-m)(n+\tilde\kappa)/p}w_R^{(p-m)}(0).
\end{aligned}
\end{equation}
Fix $R>\max\{R_0,R_1,R_2\}$. Then, \eqref{th1-2} and \eqref{th1-3} yield
\[
\frac{d}{dt}w_R(t)
\geq \frac12M_1^{-p(p-1)/p}R^{-p(p-1)(n+\tilde\kappa)/p}w_R^{p}(t),\quad t>0.
\]
Since $p>m>1$, there exists $T>0$ such that
$$
w_R(t)=\int_{\mathbb{R}^n}
u(x,t)\psi_R(|x|) dx \to+\infty\quad\text{as } t\to T^-.
$$
Noting that $\operatorname{supp}\psi_R(|x|)$ is bounded, one gets
$$
\|u(\cdot,t)\|_{L^\infty(\mathbb{R}^n)}\to+\infty\quad\text{as } t\to T^-.
$$
That is to say, $u$ blows up in a finite time.
\end{proof}

Turn to the case $p>p_c$ with $-n<\kappa\le+\infty$.

\begin{theorem} \label{theorem2}
Assume that $b\in C^{0,1}([0,+\infty))$ satisfies \eqref{b1} and \eqref{b2}
with $-n<\kappa\le+\infty$.
Let $p>p_c$. Then there exist
both nontrivial nonnegative global and blow-up solutions of
 problem \eqref{eq}-\eqref{initial}.
\end{theorem}

\begin{proof}
The comparison principle and Lemma \ref{lemma3} yield
that  problem \eqref{eq}-\eqref{initial} admits a global nontrivial solution.
Let us show there also exists a blow-up solutions to \eqref{eq}-\eqref{initial}.
Fix $R>R_0$. Assume that $u$ is a solution to  \eqref{eq}-\eqref{initial}.
It follows from Lemma \ref{lemma1} (the case $-n<\kappa<+\infty$),
Remark \ref{remark1-1} (the case $\kappa=+\infty$) that
there exist a nontrivial function
$0\le \psi\in C^{1,1}_0(\mathbb{R}^n)$, 
with $\|\psi\|_{L^1(\mathbb{R}^n)}\leq 1$,
and a constant $M>0$,
both depending only on $n$, $R$ and $b$, such that
\begin{equation} \label{N1}
\frac{d}{dt}w(t)
\ge-M\int_{\mathbb{R}^n} u^m(x,t)\psi(x) dx
+\int_{\mathbb{R}^n}u^p(x,t)\psi(x)dx,\quad t>0,
\end{equation}
where
$$
w(t)=\int_{\mathbb{R}^n}u(x,t)\psi(x) dx,\quad t\ge0.
$$
The H\"older inequality yields
\begin{equation} \label{N2}
\int_{\mathbb{R}^n} u^m(x,t)\psi(x) dx
\le\Big(\int_{\mathbb{R}^n}\psi(x)dx\Big)^{(p-m)/p}
\Big(\int_{\mathbb{R}^n}u^p(x,t)\psi(x)dx\Big)^{m/p},
\end{equation}
for $t>0$.
Substitute \eqref{N2} into \eqref{N1} to get
\begin{equation} \label{N3}
\frac{d}{dt}w(t)
\ge\Big(\int_{\mathbb{R}^n}u^p(x,t)\psi(x)dx\Big)^{m/p}
\Big(-M+\Big(\int_{\mathbb{R}^n}u^p(x,t)\psi(x)dx\Big)^{(p-m)/p}\Big),
\end{equation}
for $t>0$. It follows from the H\"older inequality that
\begin{align*}
\int_{\mathbb{R}^n} u(x,t)\psi(x) dx
&\leq \Big(\int_{\mathbb{R}^n}\psi(x)dx\Big)^{(p-1)/p}
\Big(\int_{\mathbb{R}^n}u^p(x,t)\psi(x)dx\Big)^{1/p},\quad t>0,
\end{align*}
which implies
\begin{equation} \label{N4}
\int_{\mathbb{R}^n}u^p(x,t)\psi(x)dx
\ge\Big(\int_{\mathbb{R}^n}\psi(x)dx\Big)^{-(p-1)}w^p(t),\quad t>0.
\end{equation}
Substituting \eqref{N4} into \eqref{N3}, one gets that
\begin{equation} \label{N5}
\begin{aligned}
\frac{d}{dt}w(t)
&\ge\Big(\int_{\mathbb{R}^n}\psi(x)dx\Big)^{-m(p-1)/p}w^m(t)
\Big(-M\\
&\quad +\Big(\int_{\mathbb{R}^n}\psi(x)dx\Big)^{-(p-m)(p-1)/p}w^{p-m}(t)\Big),\quad t>0.
\end{aligned}
\end{equation}
If $u_0$ is so large that
\[
w(0)=\int_{\mathbb{R}^n}u_0(x)\psi(x)dx
\ge (2M)^{1/{(p-m)}}\Big(\int_{\mathbb{R}^n}\psi(x)dx\Big)^{(p-1)/p}.
\]
Then, \eqref{N5} leads to
\begin{align*}
\frac{d}{dt}w(t)
\ge\frac12\Big(\int_{\mathbb{R}^n}\psi(x)dx\Big)^{-p(p-1)/p}w^p(t),\quad t>0.
\end{align*}
By the same argument as in the end of the proof of Theorem \ref{theorem1},
$u$ must blow up in a finite time.
\end{proof}

\begin{remark} \label{remark2-2} \rm
In Theorem \ref{theorem1}, $b$ need not to satisfy \eqref{b2}
even if $-n<\kappa<+\infty$.
However, \eqref{b2} is needed in the proof of Lemma \ref{lemma3} and
thus in the proof of Theorem \ref{theorem2}.
\end{remark}

According to Remarks \ref{remark1} and \ref{remark1-2}, one gets the following
statement.

\begin{remark} \label{remark2-1-1}
Theorem \ref{theorem1} still holds if \eqref{b1} is replaced by
$$
\limsup_{s\to+\infty} s^2b(s)=\kappa,
$$
while Theorem \ref{theorem2} still holds if \eqref{b1} is replaced by
$$
\liminf_{s\to+\infty} s^2b(s)=\kappa.
$$
\end{remark}

From Theorems \ref{theorem1} and \ref{theorem2} we have the following statement.

\begin{remark} \label{remark2-3} \rm
For  problem \eqref{eq}-\eqref{initial},
$p_c=m$ if $\lim_{s\to+\infty}s^2b(s)=+\infty$,
while $p_c=+\infty$ if $\limsup_{s\to+\infty} s^2b(s)\le-n$.
In particular, $p_c=m$ for the Cauchy problems of
\[
\frac{\partial u}{\partial t}=\Delta u^m+x\cdot\nabla u^m+u^p,\quad
x\in\mathbb{R}^n,\;t>0
\]
and
\[
\frac{\partial u}{\partial t}=\Delta u^m+\frac{x}{|x|+1}\cdot\nabla u^m+u^p,\quad
x\in\mathbb{R}^n,\;t>0,
\]
while $p_c=+\infty$ for the Cauchy problems of
\[
\frac{\partial u}{\partial t}=\Delta u^m-x\cdot\nabla u^m+u^p,\quad
x\in\mathbb{R}^n,\; t>0
\]
and
\[
\frac{\partial u}{\partial t}=\Delta u^m-\frac{x}{|x|+1}\cdot\nabla u^m+u^p,\quad
x\in\mathbb{R}^n,\;t>0.
\]
\end{remark}

\subsection*{Acknowledgments}
 This research was supported by the National Natural Science Foundation 
of China (No. 11271154).

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\end{document}