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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 10, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/10\hfil Critical exponent for asymptotic behavior]
{Critical exponent for the asymptotic behavior of rescaled solutions to the
porous medium equation}

\author[L. W. Wang, J. X. Yin \hfil EJDE-2017/10\hfilneg]
{Liangwei Wang, Jingxue Yin}

\address{Liangwei Wang (corresponding author) \newline
Key Laboratory for Nonlinear Science and System Structure,
College of Mathematics and Statistics, Chongqing Three Gorges University,
Chongqing 404000, China}
\email{wanglw08@163.com}

\address{Jingxue Yin \newline
School of Mathematical Sciences,
South China Normal University,
Guangzhou 510631, China}
\email{yjx@scnu.edu.cn}

\thanks{Submitted November 17, 2016. Published January 10, 2017.}
\subjclass[2010]{35B40, 35K65}
\keywords{Complexity; asymptotic behavior; porous medium equation}

\begin{abstract}
 In this article, we find that $\mu_c\equiv 2N/(N(m-1)+2)$ is the critical
 exponent for the asymptotic behavior  of rescaled solutions
 $t^{\mu/2}u(t^\beta x,t)$ for the porous medium
 equation.
\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}\label{1}

In this article, we consider the asymptotic behavior of  solutions to the
Cauchy problem of the porous medium equation
\begin{gather}
\frac{\partial u}{\partial t}-\Delta u^{m}=0 \quad\text{in }\mathbb{R}^{N}\times
(0,\infty),\label{1-1}\\
u(x,t)=u_{0}(x)  \quad\text{in } \mathbb{R}^{N}. \label{1-2}
\end{gather}
Here the initial value satisfies
\[
u_0\in C_0^{+}(\mathbb{R}^N)\equiv\{\varphi\in C(\mathbb{R}^N);
 \lim_{|x|\to\infty}\varphi(x)=0 \text{ and }
 \varphi(x)\geq0\}
\]
 and $m>1$ is a physical constant.

Asymptotic behavior of solutions for the porous medium equation has
attracted much attention of mathematicians for a long time and many
interesting results have been obtained, see \cite{10,3,4,5,6,8,9,0,2',7,1,2}.

 Friedman and Kamin \cite{4} first revealed the fact that if the
nonnegative initial value $u_0\in L^{1}(\mathbb{R}^N)$, then the solution
$u(x,t)$ of  problem \eqref{1-1}--\eqref{1-2} satisfies
\begin{align*}
\lim_{t\to\infty}t^{\frac{N}{N(m-1)+2}}
\|u(\cdot,t)-U_M(\cdot,t)\|_{L^{\infty}(\mathbb{R}^N)}
=0,
\end{align*}
where $U_M(x,t)$ is the source-type solution with the same mass $M$
as that of $u_0$; see also \cite{5,8}.

This result means that
if $0\leq u_0\in L^1(\mathbb{R}^N)$, then the $\omega$-limit set of rescaled
solutions $t^{\mu/2}u(t^{\beta}x, t)$ with $\mu=\frac{2N}{N(m-1)+2}$ and
$\beta=\frac{1}{N(m-1)+2}$ contains one point; that is, the rescaled
solutions $t^{\frac{N}{N(m-1)+2}}u(t^{\frac{1}{N(m-1)+2}}x, t)$
possess the simple asymptotic behavior (KV point in Figure \ref{f-1}).
However, for $u_0\in L^\infty(\mathbb{R}^N)$, in 2002, V\'azquez and Zuazua \cite{9}
found that the $\omega$-limit set of the rescaled solutions
$t^{\mu/2}u(t^{\beta}x, t)$ of  problem \eqref{1-1}--\eqref{1-2}
 with $\mu=0$ and $\beta=1/2$ may contain infinite points, i.e.,
$u(t^{1/2}\cdot,t)$ (VZ point in Figure \ref{f-1}) possess complicated
 asymptotic behavior.

Such phenomena that the different exponents of rescaled solutions
$t^{\mu/2}u(t^{\beta}x, t)$ show different asymptotic behaviors for the
porous medium equation have been studied in \cite{10,9,0,1,2}, for other
evolution equations, one can see \cite{11,12,13,14,15,15'}.

For the $\omega$-limit set of the rescaled solutions
$t^{\mu/2}u(t^{\beta}\cdot,t)$ of  problem \eqref{1-1}--\eqref{1-2}
in $C_0(\mathbb{R}^N)$, we showed in our  previous paper \cite{1} that
 if $(\mu,\beta)\in \rm I$ ( $0<\mu<\frac{2N}{N(m-1)+2}$ and
$\beta>\beta(\mu)=\frac{2-\mu(m-1)}{4}$,
then there exists $u_0\in C_0^{+}(\mathbb{R}^N)$ such that this $\omega$-limit
set  contains infinite points; see Figure \ref{f-1}).
In  another paper \cite{2}, we revealed that if $\mu$ and $\beta$ in the
line segment $\beta(\mu)=\frac{2-\mu(m-1)}{4}$
($0<\mu<\frac{2N}{N(m-1)+2}$, see Figure \ref{f-1}), then there also exists
 $u_0\in C_0^{+}(\mathbb{R}^N)$
such that this $\omega$-limit set  contains infinite points.
While in this paper, we will reveal the different fact that if
$(\mu,\beta)\in II$ ($\mu\geq\frac{2N}{N(m-1)+2}$, $\beta>0$,
then for any $u_0\in C_0^{+}(\mathbb{R}^N)$, this $\omega$-limit set contains
at most one point, see Figure \ref{f-1}), i.e., the complicated asymptotic behavior of the
 rescaled solutions cannot happen.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1} 
\end{center}
\caption{The $\mu$-$\beta$ Parameters Plane}\label{f-1}
\end{figure}

\begin{remark}  \label{rmk1.1} \rm
From the above results, we can find that $\mu_{c}=2N/(N(m-1)+2)$
is the critical exponent of $\mu$ on the asymptotic behavior of the rescaled
solutions $t^{\mu/2}u(t^{\beta}x,t)$. It is not clear whether the rescaled
solutions $t^{\mu/2}u(t^{\beta}x,t)$ with $(\mu,\beta)\in$ {\rm III}
($0<\mu<2N/(N(m-1)+2)$ and $0<\beta<(2-\mu(m-1))/4$,
see Figure \ref{f-1}) possess complicated asymptotic behavior, so the
problem of the critical exponent for $\beta$ still has not been solved.
\end{remark}

The rest of this article is organized as following. In the next section,
we introduce some definitions and concepts to give a series of lemmas.
In the last of this paper, we give and prove our results.

\section{Preliminaries}

Before introducing the main results of this paper, we give some concepts as in
\cite{16,17,18}. For $f\in L^{1}_{\rm{loc}}(\mathbb{R}^N)$ and $r> 0$, let
$$
\||f\||_r =\sup_{R\geq r}
R^{-\frac{N(m-1)+2}{m-1}}\int_{|x|\leq R}|f(x)|dx.
$$
Then we define the space $X=X(\mathbb{R}^N)$ by
$$
X\equiv\{f\in L^{1}_{\rm loc}(\mathbb{R}^N) ; \||f\||_1<\infty\},
$$
and equip this space with the norm $\||\cdot\||_1$. Hence it is a
Banach space, and any norm $\||\cdot\||_r$, $r>0$, is an equivalent
norm. For $f\in X$, we define
$$
\ell(f)=\lim_{r\to\infty}|\|f\||_r.
$$
The space $X_0 = X_0(\mathbb{R}^N)$ is  defined by
$$
X_0\equiv\{f\in X ; \ell(f) =0 \}.
$$
Notice that $L^1(\mathbb{R}^N)\subset X_0 \subset X \subset L^{1}_{
\rm loc}(\mathbb{R}^N)$ with continuous inclusions. Similarly,
$L^{\infty}(\mathbb{R}^N)\subset X_0$ with continuous inclusion.
We now give the definition of solutions for  problem \eqref{1-1}--\eqref{1-2}
with the initial value $u_0\in X_0$.

\begin{definition} \label{def2.1} \rm
A nonnegative measurable function $u=u(x,t)$ defined in
$S_T=[0,T)\times\mathbb{R}^N$, $T>0$, is a solution of \eqref{1-1}--\eqref{1-2} if
\begin{itemize}
\item[(I)] $ u\in C([0,T); L^{1}_{\rm loc}(\mathbb{R}^N))\bigcap L^{\infty}(0,T; X)$;

\item[(II)] $u^m\in L^{1}((0,T)\times B_r(0))$ for any
 $B_r(0)\equiv\{x\in \mathbb{R}^N;|x|<r,\ r>0\}$;

\item[(III)] for every test function $ \phi\in C_c^{2,1}(S_T)$, it holds
$$
\iint_{S_T}(u\phi_t+u^m
\Delta\phi)\,dx\,dt+\int_{\mathbb{R}^N}u_0(x)\phi(x,0)dx=0.
$$
\end{itemize}
\end{definition}

For any $u_0\in X_0$, the existence and uniqueness of the
solution is well established in \cite{16,17,18}. Moreover,  problem
\eqref{1-1}--\eqref{1-2} generates a bounded continuous semigroup in the
space $X_0$ given by
\begin{align}\label{2-1}
S(t) : u_0 \to u(x, t);
\end{align}
that is, $S(t)u_0\in C([0,\infty); X_0)$, see \cite{17,18}. We now
introduce the definitions of scalings and present the commutative
relations between the semigroup operators and the dilation operators
as in \cite{1,2}.  For $\lambda$, $\mu$, $\beta>0$ and
$u_0\in X_0$, the space-time dilation $\Gamma^{\mu,\beta}_{\lambda}$
is defined as following:
$$
\Gamma^{\mu,\beta}_{\lambda}[u_0](x)\equiv
D^{\mu,\beta}_{\lambda}[S(\lambda^{2}t)u_0(x)]=
\lambda^{\mu}u(\lambda^{2\beta}x,\lambda^{2}t),
$$
where the dilation $D^{\mu,\beta}_{\lambda}$ is defined as
$$
D^{\mu,\beta}_{\lambda}w(x)\equiv\lambda^{\mu}w(\lambda^{2\beta}x)
$$
and $S(t)$ is the PME semigroup given by \eqref{2-1}.  From the
definitions of $D^{\mu,\beta}_{\lambda}$ and $S(t)$,  we can get the
following commutative relations between the semigroup operators
$S(t)$ and the dilation operators $D^{\mu,\beta}_{\lambda}$,
\[
\Gamma_{\lambda}^{\mu,\beta}u_0(x)=D^{\mu,\beta}_{\lambda}
[S(\lambda^{2}t)u_0(x)]
=S(\lambda^{2-4\beta-\mu(m-1)}t)[D^{\mu,\beta}_{\lambda}u_0](x).
\]
In particular,
\begin{equation} \label{2-2}
\Gamma_{\sqrt{t}}^{\mu,\beta}u_0(x)=
S(t^{\frac{2-4\beta-\mu(m-1)}{2}})[D^{\mu,\beta}_{\sqrt{t}}u_0](x),
\end{equation}
see details in \cite{1,2}.
The  set of functions
\[
\omega^{\mu,\beta}(u_0)\equiv\{f\in C_{0}^{+}(\mathbb{R}^N);\exists
t_n\to\infty\text{ s.t. }
D^{\mu,\beta}_{\sqrt{t_n}}[S(t_n)u_0](\cdot)\xrightarrow{t_n\to\infty}
f  \text{ in } L^{\infty}(\mathbb{R}^N)\}
\]
is called $\Omega$-limit set.  We also
introduce the following symbol to denote the positive set of
$u(x,t)$ at time $t$,
$$
\Omega(t)\equiv\{x\in\mathbb{R}^N;\, u(x,t)>0\}.
$$
The $\rho$-neighborhood of the set $\Omega(t)$ is defined as
$$
\Omega_{\rho}(t)\equiv\{x\in\mathbb{R}^N;\ d(x,\Omega(t))\leq \rho\},
$$
where $d(x,\Omega(t))$ is the distance from $x$ to $\Omega(t)$.
We now list some important properties of the solutions.

\begin{lemma}[\cite{17}]\label{L-4}
If $0\leq u_0\in L^1(\mathbb{R}^N)$, then the solution $u(x, t)$ satisfies the
$ L^1$-$L^\infty$ smoothing effect: for every $t>0$,
$$
\|u(\cdot, t)\|_{L^{\infty}(\mathbb{R}^N)}
\leq C_1\|u_0\|_{L^1(\mathbb{R}^N)}^{\frac{2}{N(m-1)+2}}t^{-\frac{N}{N(m-1)+2}},
$$
where $C_1$ is a constant dependent on $m$ and $N$.
\end{lemma}

The following lemma was proved in \cite{1}, we give here a different proof
for the sake of completeness.

\begin{lemma}[\cite{1}] \label{L-1}
Let $u(x,t)$ be a nonnegative solution of
\eqref{1-1}--\eqref{1-2} with the initial value $u_0$ such that
$0\leq u_0\in L^{1}(\mathbb{R}^N)$.
Then for any $0\leq t_1<t_2<\infty$,
 $$
 \Omega(t_2)\subset\Omega_{\rho(t_2-t_1)}(t_1),
 $$
 where
 $$
 \rho(t_2-t_1)=C_{2}(t_2-t_1)^{\frac{1}{N(m-1)+2}}\|u_0\|_{L^{1}
 (\mathbb{R}^N)}^{\frac{m-1}{N(m-1)+2}}
 $$
 and
 $C_2$ is a constant dependent on $m$ and $N$.
\end{lemma}

\begin{proof}
To prove this lemma, we need the fact that if $u(x,t)$ is a nonnegative
solution of \eqref{1-1}--\eqref{1-2} with the initial data $u_0$
satisfying
$$
0\leq u_0\in L^{\infty}(\mathbb{R}^N),
$$
then
\begin{align} \label{a-0}
 \Omega(t_2)\subset\Omega_{\rho(t_2-t_1)}(t_1) \quad \text{for }
 0\leq t_1<t_2<\infty,
 \end{align}
 where
\[
 \rho(t_2-t_1)=C(t_2-t_1)^{1/2}\|u_0\|_{L^{\infty}
 (\mathbb{R}^N)}^{\frac{m-1}{2}}.
\]
In fact, for any given $x_0\in \mathbb{R}^N$
with $d(x_0)>0$, if $R\geq d(x_0)$, then
\begin{align*}
R^{-\frac{N(m-1)+2}{m-1}}\int_{B_R(x_0)}u_0(y)dy
&\leq C\|u_0\|_{L^{\infty}(\mathbb{R}^N)}R^{-\frac{N(m-1)+2}{m-1}}R^N\\
&=C\|u_0\|_{L^{\infty}(\mathbb{R}^N)}R^{-\frac{2}{m-1}}\\
&\leq C\|u_0\|_{L^{\infty}(\mathbb{R}^N)}d(x_0)^{-\frac{2}{m-1}};
\end{align*}
or if $R <d(x_0)$, then
$$
\int_{B_R(x_0)}u_0(y)dy=0,
$$
where $B_R(x_0)=\{y;|x_0-y|<R\}$.
So
\begin{equation} \label{0-1}
B(x_0)\equiv\sup_{R\geq d(x_0)}
R^{-\frac{N(m-1)+2}{m-1}}\int_{B_R(x_0)}u_0(y)dy
 \leq C\|u_0\|_{L^{\infty}(\mathbb{R}^N)}d(x_0)^{-\frac{2}{m-1}}.
\end{equation}
The condition $0\leq u_0\in L^{\infty}(\mathbb{R}^N)\subset X_0$ implies
that if $|x|\leq R$ and $r\leq R$, then
\[
u(x,t)\leq Ct^{-\frac{N}{N(m-1)+2}}R^{\frac{2}{m-1}}\||u_0\||_{r}
^{\frac{2}{N(m-1)+2}}\quad  \text{for } 0<t<\infty,
\]
see \cite{16,17}. This result and \eqref{0-1} imply that
$$
u(x_0,t)=0\ \text{for all}\ 0\leq t\leq
C\|u_0\|_{L^{\infty}(\mathbb{R}^N)}^{-(m-1)}d(x_0)^{2}.
$$
This implies $\Omega(t)\subset\Omega_{\rho(t)}(0)$,
where 
\[
\rho(t)=C\|u_0\|_{L^{\infty}(\mathbb{R}^N)}^{\frac{m-1}{2}}t^{1/2}.
\]
From this, we can get the desired result.

We now discuss the case that $0\leq u_0\in L^1(\mathbb{R}^N)$ to complete
the proof.
Without loss of generality, we can restrict our
consideration to the case of $t_1=0$. For any $0< t<\infty$, we
select a sequence of times
$$
t_k=2^{-k}t\to 0\quad\text {as } k\to\infty.
$$
We then consider the evolution in the time intervals
$I_k=[t_k,t_{k-1}]$; that is, we will estimate the increase of the
support in these time intervals. From the $L^{1}$-$L^{\infty}$
smoothing effect, at each initial time $t=t_k$, we have
\begin{align} \label{a-2}
\|u(t_k)\|_{L^\infty(\mathbb{R}^N)}\leq
C(p,N)\|u_0\|_{L^1(\mathbb{R}^N)}^{\frac{2}
{N(m-1)+2}}t_{k}^{-\frac{N}{N(m-1)+2}}.
\end{align}
Therefore, we can deduce from \eqref{a-0} that
$$
\Omega(t_{k-1})\subset\Omega_{\rho(t_{k-1}-t_k)}(t_k),
$$
where $\rho(t_{k-1}-t_k)=C\|u(t_k)\|_{L^{\infty}(\mathbb{R}^N)}^
{\frac{m-1}{2}}(t_{k-1}-t_k)^{1/2}$. Iterating, we have
$$
\Omega(t)\subset\Omega_{\rho(t)}(0),
$$
where
\begin{align*}
\rho(t)&=C\sum_{k=1}^{\infty}\|u(t_k)\|_{L^{\infty}
(\mathbb{R}^N)}^{\frac{m-1}{2}}(t_{k-1}-t_k)^{1/2}
\leq C\sum_{k=1}^{\infty}\|u_0\|_{L^{1}
(\mathbb{R}^N)}^{\frac{m-1}{N(m-1)+2}}t_k^{\frac{1}{N(m-1)+2}}\\
&=C\|u_0\|_{L^{1}(\mathbb{R}^N)}^{\frac{m-1}
{N(m-1)+2}}t^{\frac{1}{N(m-1)+2}}
\sum_{k=1}^{\infty}2^{-\frac{k}{N(m-1)+2}}
\leq C\|u_0\|_{L^{1}(\mathbb{R}^N)}^
{\frac{m-1}{N(m-1)+2}}t^{\frac{1}{N(m-1)+2}}.
\end{align*}
Here we have used the estimates \eqref{a-2}.
The proof is complete.
\end{proof}

The next lemma is called Aleksandrov's reflection (see \cite{17}). We
introduce some notation to give this principle. Any $H$, hyperplane
of $\mathbb{R}^N$, divides $\mathbb{R}^N$ into two half spaces
$\Omega_1(H)$ and $\Omega_2(H)$. We denote by $\pi=\pi_H$ the
specular symmetry that maps a point $x\in \Omega_1(H)$ into its
symmetric image with respect to $H$, $\pi_H(x)\in\Omega_2(H)$.

\begin{lemma}[Aleksandrov's Reflection Principle \cite{17}] \label{L-2}
Let $u\geq 0$ be a  solution of  problem \eqref{1-1}--\eqref{1-2} with initial
value $u_0\in X_0$. Suppose that for a given hyperplane $H$ and all
$x\in\Omega_1(H)$,
\begin{align*}
 u_0(\pi_H(x))\leq u_0(x).
 \end{align*}
Then, for all times $0\leq t<\infty$,
 \begin{align*}
 u(\pi_H(x), t)\leq u(x, t),\quad x\in \Omega_1(H).
 \end{align*}
 \end{lemma}

The following lemma depends on Lemma \ref{L-1} and \ref{L-2}.

\begin{lemma}\label{L-3}
Suppose $u(x,t)$ is a non-negative solution of
\eqref{1-1}--\eqref{1-2} with initial-value $u_0\in
C_{0}^{+}(\mathbb{R}^N)$ and $u_0\not\equiv0$. Let
\[
M(t)=\int_{|x|\leq t^{\frac{1}{2N(m-1)+4}}}u_0(x)dx.
\]
Then there exists a $0<t_0<\infty$ such that for  $t\geq t_0$,
\[
 u(0, t)\geq C t^{-\frac{N}{N(m-1)+2}}M(t)^{\frac{2}{N(m-1)+2}}.
\]
\end{lemma}

\begin{proof}
Since the nonnegative initial value $u_0\not\equiv 0$ and $u_0\in C(\mathbb{R}^N)$,
then there exist constants $t_1, C_3>0$ such that
$$
\int_{B_{t_1}}u_0(x){\rm d}x\geq C_3.
$$
Now let
\begin{gather*}
t_2=C_2^{-\frac{2}{N(m-1)+2}}C_3^{-2m+2}, \\
t_3=(2^{N+1} C_1|B_1|)^{\frac{2N(m-1)+4}{N}}C_3^{-2m+2}
\end{gather*}
where $C_1$, $C_2$ are the constants given in Lemma \ref{L-4} and
Lemma \ref{L-1} respectively.
Let $t_0=\max{(t_1,t_2,t_3)}$. Then for any  $t\geq t_0$,
using comparison principle,  we can suppose that $u_0$
is supported in the ball $B_t=\{x;|x|\leq
t^{\frac{1}{2N(m-1)+4}}\}$.
In fact, for general $u_0$, suppose $\eta_{t}(x)$ is a cut-off function
compactly supported in $B_t$ and less than one with
$$
\int_{B_t}\eta_{t}(x)u_0(x)dx\geq\frac{1}{2}M(t),
$$
then $u_0\eta_t$ is lesser than $u_0$. Therefore, if $v$ is the
solution with initial data $u_0\eta_t$, then
\begin{align*}
v(x,s)\leq u(x,s)\quad\text{for all}\quad s>0.
\end{align*}
Hence, if this lemma holds for $v(x,t)$, then
\begin{align*}
u(0, t)\geq v(0, t)\geq C
(\frac{1}{2}M(t))^{\frac{2}{N(m-1)+2}}t^{-\frac{N}{N(m-1)+2}}.
\end{align*}
Therefore, in the next part of this proof, we assume that
 $\operatorname{supp}u_0\subset B_t$.
So,
$$
M(t)=\int_{\mathbb{R}^N}u_0(x){\rm d}x\geq C_3.
$$
The $L^1$-$L^\infty$ smoothing effect implies that for any $s>0$,
\begin{align*}
0 \leq u(x, s)\leq C_1
M(t)^{\frac{2}{N(m-1)+2}}s^{-\frac{N}{N(m-1)+2}}.
\end{align*}
 The conservation of mass means that for all $s\geq0$,
\begin{align*}
\int_{\mathbb{R}^N}u_0(x) dx
&=\int_{\mathbb{R}^N}u(x, s)dx \\
&= \int_{|x|\geq 2t^{\frac{1}{2N(m-1)+4}}}u(x, s)dx
 + \int_{|x|\leq 2t^{\frac{1}{2N(m-1)+4}}} u(x, s) dx,
\end{align*}
the last term can be estimated as
\begin{equation} \label{b-1}
\int_{|x|\leq 2t^{\frac{1}{2N(m-1)+4}}}u(x,s)dx
\le2^N C_1 |B_{1}|M(t)^{\frac{2}{N(m-1)+2}}
s^{-\frac{N}{N(m-1)+2}}t^{\frac{N}{2N(m-1)+4}},
\end{equation}
where $|B_1|$ is the measure of the unit ball $B_1$ in $\mathbb{R}^N$.
Since $\operatorname{supp} u_0\subset B_{t}$, then Lemma \ref{L-1} indicates
that for all $s>0$,
\[
\operatorname{supp} u(x,s)\subset B_{R_1(s)},
\]
where $R_1(s) = t^{\frac{1}{2N(m-1)+4}}+C_2
M(t)^{\frac{m-1}{N(m-1)+2}}s^{\frac{1}{N(m-1)+2}}$.
Let $s=t$ and
$$
R(t)=4C_2
M(t)^{\frac{m-1}{N(m-1)+2}}t^{\frac{1}{N(m-1)+2}}.
$$
Notice that $t\geq t_0\geq t_2=
C_{2}^{-\frac2{N(m-1)+2}}C_3^{-2m+2}$ and $M(t)\geq C_3$.
So
\begin{equation} \label{b-2}
R(t)> 2R_{1}(t)\geq 4t^{\frac{1}{2N(m-1)+4}}.
\end{equation}
The hypothesis  $\operatorname{supp} u_0\subset B_{t}$ implies, via the
Aleksandrov reflection principle (Lemma \ref{L-2}),  that for all
$|x|\geq 2t^{\frac{1}{2N(m-1)+4}}$ and  $s\geq 0$,
\[
u(0, s)\geq u(x,s).
\]
So, from \eqref{b-2}, we have
\begin{align*}
u(0,t)R(t)^N
&\geq u(0,t)(R(t)^N-2^Nt^{\frac{N}{2N(m-1)+4}})\\
&=\frac{1}{|B_1|}\int_{2t^{\frac{1}{2N(m-1)+4}} \leq |x|\leq R(t)} u(0, t)dx\\
&\geq \frac{1}{|B_1|}\int_{2t^{\frac{1}{2N(m-1)+4}}\leq |x|\leq R(t)}
u(x, t)dx \\
&=\frac{1}{|B_1|}\int_{|x|\geq 2t^{\frac{1}{2N(m-1)+4}}}u(x, t)dx\\
&=\frac{1}{|B_1|}\int_{\mathbb{R}^N}u(x, t)dx-
\frac{1}{|B_1|}\int_{|x|<2t^{\frac{1}{2N(m-1)+4}}}u(x, t)dx.
\end{align*}
Now using estimate \eqref{b-1} and $t\geq t_0\geq t_3$, we obtain
\[
u(0,t)R(t)^N\geq
\frac{1}{|B_1|}[M(t) -2^N C_1|B_1| M(t)^{\frac{2}{N(m-1)+2}}
t^{-{\frac{N}{2N(m-1)+4}}}]
\geq \frac{1}{2|B_1|}M(t).
\]
It follows from the definition of $R(t)$ that
\[
 u(0, t)\geq Ct^{-\frac{N}{N(m-1)+2}}M(t)^{\frac{2}{N(m-1)+2}}.
\]
The proof is complete.
\end{proof}

\section{Results and their proofs}\label{2}

\begin{theorem}\label{T-6}
Let $u_0\in C_0^{+}(\mathbb{R}^N)$, $u_0 \not\equiv0$. If
there exist $0\not\equiv v\in C_0(\mathbb{R}^N)$,
$\mu_0\geq\frac{2N}{N(m-1)+2}$, $\beta_0>0$ and a sequence
$\{t_n\}_{n=1}^{\infty}$ with $\lim_{n\to\infty}t_n=+\infty$ such that
\begin{equation} \label{6-2}
\Gamma^{\mu_0,\beta_0}_{\sqrt{t_n}}u_0=t_{n}^{\frac{\mu_0}2}
[S(t_n)u_0](t_{n}^{\beta_0}\cdot)\xrightarrow{t_n\to\infty} v\quad\text{in}\quad
C_0(\mathbb{R}^N),
\end{equation}
then
\begin{gather*}
u_0\in L^1(\mathbb{R}^N),\quad \mu_0=\frac{2N}{N(m-1)+2}, \\
\beta_0=\frac{2-\mu_0[m-1]}{4}=\frac{1}{N(m-1)+2}.
\end{gather*}
 In other words,  if
$\mu>\frac{2N}{N(m-1)+2}$, or if $\mu= \frac{2N}{N(m-1)+2}$ and
$\beta\neq\frac{1}{N(m-1)+2}$, then
$$
\omega(u_0)=\emptyset,\quad\text{or}\quad \omega(u_0)=\{0\}.$$
\end{theorem}

\begin{proof}
It follows from \eqref{6-2} and Lemma \ref{L-3} that if $n$ sufficiently large,
then
\begin{equation} \label{6-3}
v(0)+1\geq [\Gamma^{\mu_0,\beta_0}_{\sqrt{t_n}}u_0](0)=t_n^{\frac{\mu_0}2}
[S(t_n)u_0](0)\geq
 C t_{n}^{\frac{\mu_0-\frac{2N}{N(m-1)+2}}{2}}M(t_{n})^{\frac{2}{N(m-1)+2}}.
\end{equation}
 Here $M(t)$ is given by Lemma \ref{L-3}. Letting
$n\to\infty$, we conclude that
$$
\mu_0=\frac{2N}{N(m-1)+2}
$$
and $ u_0\in L^1(\mathbb{R}^N)$.
 Notice also that $u_0\geq 0$. This gives
\begin{equation} \label{6-4}
D^{\frac{2N}{N(m-1)+2},\frac{1}{N(m-1)+2}}_{\sqrt{t}}S(t)u_0(x)=
t^{\frac{N}{N(m-1)+2}}u(t^{\frac{1}{N(m-1)+2}}x,t)\to
U_{M}(x,1)
\end{equation}
uniformly on $\mathbb{R}^N$ as $t\to\infty$.
Here $U_M(x,t)$ is the source-type solution with the same mass
as that of $u_0$, where
$M=\int_{\mathbb{R}^N}u_0(x){\rm d}x$,
see \cite{5,8}. Therefore,
\begin{equation} \label{6-5}
D^{\frac{2N}{N(m-1)+2},\beta_0}_{\sqrt{t_n}}S(t_n)u_0(x)-U_M(xt_{n}^
{\beta_0-\frac{1}{N(m-1)+2}},1)\xrightarrow{n\to\infty} 0
\end{equation}
uniformly on $\mathbb{R}^N$. The expression of the source-type solution
clearly means
$$
\operatorname{supp}(U_{M}(x,1))\subset \{x;|x|\leq
CM^{\frac{m-1}{N(m-1)+2}}\},
$$
so that if $\beta_0>\frac{1}{N(m-1)+2}$, then
$$
U_M(xt_{n}^
{\beta_0-\frac{1}{N(m-1)+2}},1)\to 0\quad\text{for all } x\neq0
$$
as $t_n\to\infty$. Notice also that
$v\not\equiv0$, so \eqref{6-5} is compatible with \eqref{6-2} only if
$$
\beta_0\leq\frac{1}{N(m-1)+2}.
$$
On the other hand, from \eqref{6-2} and \eqref{6-4} we deduce that
\begin{align}\label{6-6}
D^{\frac{2N}{N(m-1)+2},\frac{1}{N(m-1)+2}}_{\sqrt{t_n}}S(t_n)u_0(x)-
v(t_{n}^{\frac{1}{N(m-1)+2}-\beta_0}x)\to0
\end{align}
uniformly on $\mathbb{R}^N$ as $t_n\to\infty$. The hypothesis
that $v\in C_0(\mathbb{R}^N)$ clearly implies that if
$\beta_0<\frac{1}{N(m-1)+2}$, then
\begin{align*}
 v(t_{n}^{\frac{1}{N(m-1)+2}-\beta_0}x)\to 0
 \quad\text{for all } x\not=0
\end{align*}
as $t_n\to\infty$.
Recall that $u_0\not\equiv0$, so
$U_M\not\equiv0$.
Therefore,
\eqref{6-6} is compatible with \eqref{6-4} only if
$$
\beta_0\geq\frac{1}{N(m-1)+2}.
$$
Hence
$$
\beta_0=\frac{1}{N(m-1)+2}.
$$
 So that $\omega^{\mu,\beta}(u_0)=\emptyset$ if
$\mu>\frac{2N}{N(m-1)+2}$,  or if $\mu= \frac{2N}{N(m-1)+2}$ and
$\beta\neq\frac{1}{N(m-1)+2}$. This completes the proof.
\end{proof}

\begin{theorem}\label{T-7}
Let
$$
\mu=\frac{2N}{N(m-1)+2}, \quad\text{and}\quad
\beta=\frac{1}{N(m-1)+2}.
$$
If $u_0\in C^{+}_0(\mathbb{R}^N)$, then
$$
\omega^{\mu,\beta}(u_0)=\emptyset,
\quad\text{or}\quad \omega^{\mu,\beta}(u_0)=\{U_M(x,1)\},
$$
where $U_M(x,t)$ is source-type solution with the same mass $M$
as that of $u_0$.
\end{theorem}

\begin{proof}
If $u_0\in C^{+}_0(\mathbb{R}^N)$, then $u_0\in L^{1}(\mathbb{R}^N)$,
or else $u_0\in L^{1}_{\rm loc}(\mathbb{R}^N)$
with $\|u_0\|_{L^{1}(\mathbb{R}^N)}=\infty$.
 If $u_0\in
L^{1}(\mathbb{R}^N)$,
then
\begin{align}\label{6-7}
\lim_{t\to\infty} t^{\frac{N}{N(m-1)+2}}
u(t^{\frac{1}{N(m-1)+2}}x,t)=U_M(x,1)\quad\text{in }
L^{\infty}(\mathbb{R}^N).
\end{align}
So
$$
\omega^{\mu,\beta}(u_0)=\{U_M(x,1)\}.
$$
If $u_0\in L^{1}_{\rm loc}(\mathbb{R}^N)$ and
$\|u_0\|_{L^{1}(\mathbb{R}^N)}=\infty$, approximating $u_0$ by an
increasing sequence of integrable data $u_{0n}$, applying
\eqref{6-7} and passing to the limit, we have
\[
\lim_{t\to\infty} t^{\frac{N}{N(m-1)+2}}
u(t^{\frac{1}{N(m-1)+2}}x,t)=\infty\quad\text{in}\quad
L^{\infty}(\mathbb{R}^N).
\]
Hence
$\omega^{\mu,\beta}(u_0)=\emptyset$.
The proof is complete.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
As we had showed in \cite{1,2} that for $0<\mu<2N/(N(m-1)+2)$,
if $\beta=(2-\mu(m-1))/4$, then there exists an initial value
$u_0\in C_0^{+}(\mathbb{R}^N)$ such that the $\Omega$-limit set $\omega^{\mu,\beta}(u_0)$ contains the set
$$
S(1)C_{0}^{+}(\mathbb{R}^N)
\equiv\{S(1)\varphi; \varphi\in C_{0}^{+}(\mathbb{R}^N)\},
$$
or if $\beta> \frac{2-\mu(m-1)}4$, then there also exists an initial value
$u_0\in C_0^{+}(\mathbb{R}^N)$ such that the
$\Omega$-limit set $\omega^{\mu,\beta}(u_0)$ contains the set
$$
C_{0}^{+,0}(\mathbb{R}^N)\equiv\{\varphi\in C_{0}^{+}(\mathbb{R}^N); \varphi(0)=0\}.
$$
Therefore,
$$
\mu_{c}=\frac{2N}{N(m-1)+2}
$$
is the critical exponent of $\mu$ on the asymptotic behavior of the rescaled
solutions $t^{\mu/2}u(t^{\beta}\cdot,t)$.
\end{remark}

\subsection*{Acknowledgements}
This research was  supported by the NSFC (11071099 and
11371153), Natural Science Foundation Project of CQ (cstc2016jcyjA0596),
Scientific and Technological Research Program of Chongqing Municipal Education
 Commission (KJ1401003, KJ1601006), and Innovation Team Building at Institutions
of Higher Education in Chongqing (CXTDX201601035).

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\end{document}