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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 17, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/17\hfil Life span of blow-up solutions]
{Life span of blow-up solutions for higher-order semilinear
parabolic equations}

\author[F. Sun \hfil EJDE-2010/17\hfilneg]
{Fuqin Sun}

\address{Fuqin Sun \newline
School of Science,
Tianjin University of Technology and Education,
Tianjin 300222, China}
\email{sfqwell@163.com}

\thanks{Submitted August 17, 2009. Published January 27, 2010.}
\thanks{Supported by grant 10701024 from
the National Natural Science Foundation of China}
\subjclass[2000]{35K30, 35K65}
\keywords{Higher-order parabolic equation;
 critical exponent; life span; \hfill\break\indent test function method}

\begin{abstract}
 In this article, we study the higher-order semilinear parabolic
 equation
 \begin{gather*}
 u_t+(-\Delta)^m u=|u|^p, \quad (t,x)\in
 \mathbb{R}^1_+\times \mathbb{R}^N,\\
 u(0,x)= u_0(x),\quad x\in \mathbb{R}^N.
 \end{gather*}
 Using the test function method, we derive the blow-up
 critical exponent. And then based on integral inequalities,
 we  estimate the life span of blow-up solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

This article  concerns the cauchy problem for the
higher-order semilinear parabolic equation
\begin{equation}
\begin{gathered}
 u_t+(-\Delta)^m u=|u|^p, \quad (t,x)\in \mathbb{R}^1_+\times
 \mathbb{R}^N,\\
 u(0,x)=u_0(x),\quad x\in \mathbb{R}^N,
 \end{gathered} \label{1.1}
\end{equation}
where $m, p>1$. Higher-order semilinear and quasilinear heat
equations appear in numerous applications such as thin film theory,
flame propagation, bi-stable phase transition and higher-order
diffusion. For examples of these mathematical models, we refer the
reader to the monograph \cite{PT}. For studies of higher-order heat
equations we refer also to \cite{GC, SC, EVVP1, EVVP2, GP, WSP} and
the  references therein.

 In  \cite{GP}, under the assumption that $u_0\in
L^1(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$,
$u_0\not\equiv0$ and
\begin{equation}
 \int_{\mathbb{R}^N}u_0(x){\rm d}x\geq0, \label{1.2}
 \end{equation}
Galaktionov and Pohozaev  studied the Fujita critical exponent of
problem \eqref{1.1} and showed that $p_F=1+2m/N$.  The critical
exponents $p_F$ is calculated from both sides:
\begin{itemize}
\item[(i)] blow-up of any solutions with \eqref{1.2} for $1<p\leq p_F$
and

\item[(ii)] global existence of small solutions for $p>p_F$.
\end{itemize}

Egorov et al \cite{EVVP2} studied the asymptotic behavior
of global solutions  with suitable initial data in the supercritical
Fujita range $p>p_F$ by constructing self-similar solutions of
higher-order parabolic operators and through a stability analysis of
the autonomous dynamical system. For other studies of the problem,
we refer to \cite{EVVP1} where global non-existence was proved for
$p\in(1,p_F]$ by using the test function approach, and \cite{GC}
where a general situation was discussed with nonlinear function
$h(u)$ in place of $|u|^p$.

 In a recent paper \cite{WSP}, we discussed the system
  \begin{equation}
\begin{gathered}
  u_t+(-\Delta)^m u=|v|^p, \quad (t,x)\in \mathbb{R}^1_+\times
  \mathbb{R}^N,\\
 v_t+(-\Delta)^m v=|u|^q, \quad (t,x)\in \mathbb{R}^1_+\times
  \mathbb{R}^N,\\
  u(0,x)=u_0(x),\ \ v(0, x)=v_0(x),  \quad x\in \mathbb{R}^N.
  \end{gathered} \label{1.3}
  \end{equation}
It is proved  that if $N/(2m>\max \big\{\frac{1+p}{pq-1},
\frac{1+q}{pq-1}\big\}$ then solutions of \eqref{1.3} with small
initial data exist globally in time. Moreover the decay estimates
$\|u(t)\|_\infty\leq C(1+t)^{-\sigma_1}$ and
$\|v(t)\|_\infty\leq C(1+t)^{-\sigma_2}$ with
$\sigma_1>0$ and $ \sigma_2>0$ are also
satisfied. On the other hand, under the assumption that
 \[
 \int_{\mathbb{R}^N}u_0(x){\rm d}x>0, \quad
 \int_{\mathbb{R}^N}v_0(x){\rm d}x>0,
 \]
if $N/(2m)\leq\max\big\{\frac{1+p}{pq-1},\frac{1+q}{pq-1}\big\}$
then every solution of \eqref{1.3} blows up in finite time.

 In our present  work,  exploiting the test function
method, we shall give the life span of blow-up solution for some
special initial data. The main idea comes from \cite{HK} for
discussing cauchy problem of the second order equation
 \begin{equation}
\begin{gathered}
\rho(x)u_t-\Delta u^m=h(x,t)u^{1+p}, \quad
 (t,x)\in \mathbb{R}^1_+\times \mathbb{R}^N,\\
 u(0,x)=u_0(x),\quad x\in \mathbb{R}^N.
 \end{gathered} \label{1.5}
\end{equation}
Using the test function  method, the author gave  the blow-up type
critical exponent and the estimates for life span $[0,T)$ like that
in \cite{LN}. For the construction of a test function, the author
mainly based on the eigenfunction $\Phi$ corresponding to the
principle eigenvalue $\lambda_1$ of the Dirichlet problem on unit
ball $B_1$,
\begin{gather*}
  -\Delta w(x)=\lambda_1 w(x), \quad x\in B_1,\\
  w(x)=0,\quad x\in \partial B_1.
\end{gather*}
However, for the operator $(-\Delta)^m$,  the eigenfunction $\Phi$
corresponding to the principal eigenvalue $\lambda_1$ of the
Dirichlet problem may change sign (see \cite{EV}). We will use a
non-negative smooth function $\Phi$ constructed in \cite{GC} and
\cite{GP}. The organization of this paper is as follows. In section
2, by the test function method, we derive some integral inequalities
and reacquire the Fujita critical exponent $p_F$ obtained in the
paper \cite{GP}. Section 3 is for the estimate of life span of
blow-up solution.

\section{Fujita  critical exponent}

 In this section, we shall use the test
function method to derive the Fujita critical exponent and some
useful inequalities. From the reference  \cite{GP}, we know that if
$u_0\in  L^1(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)$, then the
solution $u(t, \cdot)\in  C^1([0,T]; L^1(\mathbb{R}^N)\cap
L^{\infty}(\mathbb{R}^N))$ for some $T>0$. Therefore, without loss
of generality, we may consider $u_0(x)$ concentrated around the
origin and bounded below by a positive constant in some neighborhood
of  origin.  Further, $u_0(x)\to 0$ as
$|x|\to\infty$. With these choices,
the solution $u$ and its spatial derivatives vanish as
$|x|\to\infty$ for $t>0$.

 First we construct a test function. For this aim, we shall
use a non-negative smooth function $\Phi$ which was constructed in
the papers \cite{GC} and \cite{GP}.

 Let
 \[
 \Phi(x)=\Phi(|x|)>0,\quad \Phi(0)=1;\quad
 0<\Phi(r)\leq1\quad \mbox{for }  r>0,
 \]
where $\Phi(r)$ is decreasing and $\Phi(r)\to 0$ as
$r\to \infty$ sufficiently fast. Moreover, there exists a
constant $\lambda_1>0$ such that
  \begin{equation}
  |\Delta^m \Phi|\leq \lambda_1\Phi,\quad x \in  \mathbb{R}^N,\label{2.1}
  \end{equation}
and such that
  $$
  \|\Phi\|_1=\int_{\mathbb{R}^N}\Phi(x){\rm d}x=1.
  $$
This can be done by letting $\Phi(r)=e^{-r^\nu}$ for $r\gg1$ with
$\nu\in (0,1]$, and then extending $\Phi$ to $[0, \infty)$ by a
smooth approximation. Take $\theta>p/(p-1)$, and define
 \[
 \phi(t)=\begin{cases}
 0,& t>T,\\
(1-(t-S)/(T-S))^\theta, & 0\leq t\leq T,\\
 1, & t<S,
 \end{cases}
 \]
where $0\leq S<T$. Now set
 $$
 \xi(t,x)=\phi(t/R^{2m})\Phi(x/R),\quad R>0.
 $$

 Suppose that $u$ exists in $[0, t_*)\times \mathbb{R}^N$.
For $TR^{2m}<t_*$, multiply both sides of equation \eqref{1.1} by
$\xi$ and integrate over $[0, TR^{2m})\times \mathbb{R}^N$ by parts
to obtain
 \begin{equation}
 \int_0^{TR^{2m}}\!\!\!\int_{\mathbb{R}^N}|u|^p  \xi{\rm d}x{\rm d}t
+\int_{\mathbb{R}^N}u_0(x)\xi(0,x) {\rm d}x
\leq\int_0^{TR^{2m}}\!\!\!\int_{\mathbb{R}^N}|u|
 \{|\xi_t| +|\Delta^m\xi|\}{\rm d}x{\rm d}t.\label{2.2}
\end{equation}
Denote
 \[
I(S, T)=\int_{SR^{2m}}^{TR^{2m}}\!\!\!\int_{\mathbb{R}^N}|u|^p
 \phi(t/R^{2m})\Phi(x/R){\rm  d}x{\rm d}t,\quad
 J=\int_{\mathbb{R}^N}u_0(x)\Phi(x/R)  {\rm d}x.
 \]
We now estimate $I(0,T)+J$.
Using the H\"older inequality, since $\phi'(t)=0$ except
 on $(S, T)$,  we obtain
\begin{equation}
\begin{aligned}
  \int_{0}^{TR^{2m}}\!\!\!\int_{\mathbb{R}^N}|u||\xi_t|
  {\rm d}x{\rm d}t
&= \int_{SR^{2m}}^{TR^{2m}}\!\!\!\int_{\mathbb{R}^N}|u|\phi(t/R^{2m})^{1/p}
 |\phi'(t/R^{2m})|\\
&\quad\times \phi(t/R^{2m})^{-1/p}\ \Phi(x/R)R^{-2m}  {\rm d}x{\rm d}t\\
&\leq  I(S, T)^{1/p}R^{-2m}\Big(\int_{SR^{2m}}^{TR^{2m}}
  \!\!\!\int_{\mathbb{R}^N}|\phi'(t/R^{2m})|^{p/(p-1)}\\
&\quad  \times \phi(t/R^{2m})^{-1/(p-1)}\Phi(x/R)
  {\rm d}x{\rm d}t\Big)^{(p-1)/p}.\label{2.4}
\end{aligned}
\end{equation}
Since $\Delta_x^m \Phi(x/R)=R^{-2m}\Delta_y^m \Phi(y)$
for $y=x/R$, using the H\"older inequality and
\eqref{2.1} we have
  \begin{equation}
\begin{aligned}
&\int_0^{TR^{2m}}\!\!\!\int_{\mathbb{R}^N}|u||\Delta^m\xi|
  {\rm d}x{\rm d}t\\
&= \int_0^{TR^{2m}}\!\!\!\int_{\mathbb{R}^N}|u|\phi(t/R^{2m})|
  \Delta^m\Phi(x/R)|
  {\rm d}x{\rm d}t\\
&=  R^{-2m}\int_0^{TR^{2m}}\!\!\!\int_{\mathbb{R}^N}|u|\phi(t/R^{2m})
  |\Delta_{x/R}^m\Phi(x/R)|{\rm d}x{\rm d}t\\
&\leq  \lambda_1R^{-2m}\int_0^{TR^{2m}}\!\!\!\int_{\mathbb{R}^N}|u|
  \phi(t/R^{2m})\Phi(x/R){\rm d}x{\rm d}t\\
&\leq  I(0,T)^{1/p}\lambda_1R^{-2m}\Big(\int_0^{TR^{2m}}
  \int_{\mathbb{R}^N}\phi(t/R^{2m})\Phi(x/R)
  {\rm d}x{\rm d}t\Big)^{(p-1)/p}.
\end{aligned}\label{2.5}
  \end{equation}
Making the change of variables $\tau=t/R^{2m}$ and $\eta=x/R$,
 from \eqref{2.2}, \eqref{2.4} and \eqref{2.5} we deduce that
  \begin{equation}
\begin{aligned}
&I(0,T)+J\\
&\leq  I(S,T)^{1/p}R^{s}\Big(\int_{SR^{2m}}^{TR^{2m}}
  \int_{\mathbb{R}^N}|\phi'(\tau)|^{p/(p-1)}\phi(\tau)^{-1/(p-1)}\Phi(\eta)
  {\rm d}\eta{\rm d}\tau\Big)^{(p-1)/p}\\
&\quad +I(0,T)^{1/p}\lambda_1R^{s}\Big(\int_0^T
  \int_{\mathbb{R}^N}\phi(\tau)\Phi(\eta)
  {\rm d}\eta{\rm d}\tau\Big)^{(p-1)/p},
\end{aligned}\label{2.6}
  \end{equation}
where $s=-2m+(2m+N)(p-1)/p$.
Set
\begin{gather*}
 A(S,T)=\Big(\int_S^T
 \int_{\mathbb{R}^N}|\phi'(\tau)|^{p/(p-1)}\phi(\tau)^{-1/(p-1)}\Phi(\eta)
 {\rm d}\eta{\rm d}\tau\Big)^{(p-1)/p},\\
 B(T)=\Big(\int_0^T
 \int_{\mathbb{R}^N}\phi(\tau)\Phi(\eta)
 {\rm d}\eta{\rm d}\tau\Big)^{(p-1)/p}.
\end{gather*}
Thus \eqref{2.6} can be simply written as
 \begin{equation}
 I(0,T)+J\leq
 R^{s}[I(S,T)^{1/p}A(S,T)+\lambda_1I(0,T)^{1/p}B(T)].\label{2.7}
 \end{equation}
We have the following result:

\begin{theorem}[Fujita critical exponent] \label{thm1}
 If
$$
     \int_{\mathbb{R}^N} u_0(x) {\rm d}x\geq 0,\,\, u_0(x)\not\equiv0
$$
and $s\leq0$, that is to say   $ p\leq p_c= 1+2m/N$,
  then  \eqref{1.1} has no global solution.
\end{theorem}

\begin{proof}
 By slightly shifting  the origin in time, we may
 assume
  \begin{equation}
  \int_{\mathbb{R}^N}u_0(x){\rm d}x>0.\label{dd}
  \end{equation}
Let $u$ be a global solution with  $u_0$ satisfying \eqref{dd}, then
  \[
  \int_0^{\infty}\int_{\mathbb{R}^N}|u|^p{\rm d}x{\rm d}t>0.
  \]
Suppose $s<0$. Letting  $R$ tend to infinity  in \eqref{2.7} to
obtain
  $$
  \int_0^{\infty}\int_{\mathbb{R}^N}|u|^p{\rm d}x{\rm d}t
  +\int_{\mathbb{R}^N}u_0(x){\rm d}x=0.
  $$
Hence $u\equiv0$, a contradiction.

Suppose $s=0$. We first show  $J\geq 0$ for all $R>0$. In fact, from
the assumptions on initial datum, there exists $\varepsilon_0>0$
such that $u_0(x)\geq\delta>0$ for $|x|\leq \varepsilon_0$. Set
  \begin{align*}
J&= \int_{|x|\leq \varepsilon_0}u_0(x)\Phi(x/R){\rm d}x
  +\int_{|x|> \varepsilon_0}u_0(x)\Phi(x/R){\rm d}x \\
&>  \delta\int_{|x|\leq \varepsilon_0}\Phi(x/R){\rm d}x
  +\int_{|x|>\varepsilon_0}u_0(x)\Phi(x/R){\rm d}x\\
&=  \delta R^N\int_{|\eta|\leq \varepsilon_0/R}\Phi(\eta){\rm
  d}\eta +\int_{|x|>\varepsilon_0}u_0(x)\Phi(x/R){\rm d}x\\
&\geq \int_{|x|>\varepsilon_0}u_0(x)\Phi(x/R){\rm d}x.
\end{align*}
By the choice of $\Phi$, we have
    \[
   \lim_{R\to0}\int_{|x|>\varepsilon_0}u_0(x)\Phi(x/R){\rm
    d}x=0.
    \]
And so  there exists  $R_0>0$ such that
 $ J\geq 0$ for all $0<R< R_0$.
On the other hand,  there exists $M>0$ such that
 $$
 \int_{|x|\leq R_0M}u_0(x){\rm d}x >\int_{|x|>R_0M}|u_0(x)|{\rm d}x.
 $$
In addition, by a slight modification of $\Phi$, we may set
$\Phi(x)\equiv 1$ in $\{x:\ |x|\leq M \}$. Note that since
 $0\leq\Phi\leq1$
we have, for $R\geq R_0$,
 \begin{align*}
J&= \int_{|x|\leq R_0M}u_0(x)\Phi(x/R){\rm d}x
 +\int_{|x|>R_0M}u_0(x)\Phi(x/R){\rm d}x\\
&\geq  \int_{|x|\leq R_0M}u_0(x){\rm d}x
 -\int_{|x|>R_0 M}|u_0(x)|\Phi(x/R){\rm d}x\\
&\geq  \int_{|x|\leq R_0M}u_0(x){\rm d}x
 -\int_{|x|>R_0 M}|u_0(x)|{\rm  d}x>0.
\end{align*}
Now we are in the position to complete the proof of case $s=0$.
Since
 $$
 A(S, T)=\frac{\theta(T-S)^{-1/p}}{[\theta-1/(p-1)]^{(p-1)/p}},\quad
 B(T)=\Big[S+\frac{T-S}{\theta+1}\Big]^{(p-1)/p},
 $$
we may choose $S$ small and $\theta$ large, $T-S$ bounded, such
that
\begin{equation}  B(T)\leq \int_{\mathbb{R}^N}u_0(x){\rm
  d}x/\Big[2\lambda_1\Big(\int_0^{\infty}\int_{\mathbb{R}^N}|u|^p{\rm
  d}x{\rm d}t\Big)^{1/p} \Big].\label{2.8}
\end{equation}
Moreover, note that $J\geq0$, from \eqref{2.7} we get that
$I(0,T)$ is uniformly bounded for all $R>0$. Then, keeping $T-S$
bounded,
 \begin{equation}
 \lim_{R\to \infty}I(S,T)^{1/p}A(S, T)=0.\label{2.9}
 \end{equation}
Letting $R\to\infty$,  \eqref{2.7}--\eqref{2.9} give
  $$
\int_0^{\infty}\int_{\mathbb{R}^N}|u|^p{\rm
  d}x{\rm d}t+\frac12\int_{\mathbb{R}^N}u_0(x){\rm  d}x=0,
  $$
which  also implies  $u\equiv0$.
\end{proof}

  Let $\sigma$ be an arbitrary positive number.
 For $x\in [0, \infty)$ and $0<\omega<1$, define
  $$
  \Psi(\omega; \sigma):=\max_x(\sigma x^{\omega}-x).
  $$
It is easy to check that $\Psi(\omega;
\sigma)=(1-\omega)\omega^{\frac{\omega}{1-\omega}}\sigma^{\frac1{1-\omega}}$.
Set
   $$
   A(T)=A(0,T),\quad  S(T)=A(T)+\lambda_1B(T).
   $$
We have the following result.

\begin{theorem}\label{thm2}
If $u$ is a solution of  \eqref{1.1} defined on
 $[0, t_*)\times \mathbb{R}^N$. Then, for $R>0$ and $0\leq\tau \leq
 t_*R^{-2m}$, we have
  \begin{equation}
  \int_{\mathbb{R}^N}u_0(x)\Phi(x/R){\rm d}x\leq
  \Psi\Big(\frac1p;\ S(T)R^s\Big).\label{2.10}
  \end{equation}
 Moreover, if $u$
is a global solution  of  {\rm\eqref{1.1}}, then
 \begin{equation}
 \lim_{R\to \infty}\sup R^{-\hat{s}}\int_{\mathbb{R}^N}u_0(x)\Phi(x/R){\rm
 d}x\leq  \lambda_1^{1/(p-1)},\label{2.11}
 \end{equation}
where $\hat{s}=sp/(p-1)$.
\end{theorem}


\begin{proof}
 Denote $I(T)=I(0, T)$. Firstly, by the definition of
$\Psi$, from \eqref{2.7} we know that
   $$
   J\leq I(T)^{1/p}S(T)R^s-I(T) \leq\Psi\Big(\frac1p;\ S(T)R^s\Big).
   $$
This is exactly \eqref{2.10}.
By means of \eqref{2.10}, we deduce that
  \begin{equation}
\begin{aligned}
  \int_{\mathbb{R}^N}u_0(x)\Phi(x/R){\rm d}x
&\leq  \Psi\Big(\frac1p;\ S(T)R^s\Big)\\
&=  (1-1/p)(1/p)^{\frac{1/p}{1-1/p}}[S(T)R^s]^{\frac1{1-1/p}}\\
&=  (p-1)p^{p/(1-p)}R^{sp/(p-1)}S(T)^{\frac{p}{p-1}},
\end{aligned} \label{2.12}
  \end{equation}
which leads to
  \begin{equation}
  \lim_{R\to \infty}\sup
  R^{-\hat{s}}\int_{\mathbb{R}^N}u_0(x)\Phi(x/R){\rm
  d}x\leq (p-1)p^{p/(1-p)}[\inf_T S(T)]^{\frac{p}{p-1}}.\label{2.13}
  \end{equation}
To estimate $S(T)$, we need estimate $A(T)$ and $B(T)$
respectively.
Denote
  \[
  a_p=\frac{\theta}{[\theta-1/(p-1)]^{(p-1)/p}},\quad
  b_p=\frac{\lambda_1}{(\theta+1)^{(p-1)/p}}.
  \]
We obtain
   $$
   S(T)=a_pT^{-1/p}+b_pT^{(p-1)/p}.
   $$
Since
\begin{align*}
\min_T S(T)&= p[a_p/(p-1)]^{(p-1)/p}b_p^{1/p}\\
&= \frac{p(p-1)^{-(p-1)/p}\lambda_1^{1/p}\theta^{(p-1)/p}}
 {[\theta-1/(p-1)]^{(p-1)^2/p^2}(1+\theta)^{(p-1)/p^2}},
\end{align*}
we have
 \begin{equation}
 \lim_{\theta\to\infty}\min_T
 S_p(T)=p(p-1)^{-(p-1)/p}\lambda_1^{1/p}.\label{2.14}
 \end{equation}
Combining \eqref{2.13} and \eqref{2.14}, we obtain  \eqref{2.11}.
The proof is complete.
\end{proof}

\section{Life span of blow-up solutions}

 In this section, we shall estimate the life span of the blow-up
solution with some special initial datum.
 To this aim, we assume that $u_0$ satisfies
\begin{itemize}
\item[(H)] There exist positive constants $C_0, L$ such
that
  \[
  u_0(x) \geq \begin{cases}
\delta, & |x|\leq \varepsilon_0,\\
C_0|x|^{-\kappa}, & |x|>\varepsilon_0,
\end{cases}
  \]
where $\delta$ and $\varepsilon_0$ are  as in the proof
of Theorem \ref{thm1}, and  $N< \kappa< 2m/(p-1)$ if $p<1+2m/N$;
$0<\kappa<N$ if  $p=1+2m/N$.

\end{itemize}
Now we state the main result.

 \begin{theorem}\label{thm3}
Let {\rm (H)} be fulfilled and $u_\varepsilon$ be the solution
of \eqref{1.1} with initial data
 $u_\varepsilon(0,x)=\varepsilon u_0(x)$, where $\varepsilon>0$.
Denote $[0,T_\varepsilon)$ be the life span of
 $u_\varepsilon$.
 Then there exists a positive constant
 $C$  such that
    $ T_\varepsilon\leq  C\varepsilon^{1/\hat{\beta}}$,
 where
  \[
  \hat{\beta}=\frac{\kappa}{2m}-\frac1{p-1}<0.
  \]
  \end{theorem}

\begin{remark}\label{rmk3} \rm
  When  $p=1+2m/N$, note that $\hat{\beta}=(\kappa-N)/(2m)$.
 \end{remark}

\begin{proof}
Choose $R$ such that $R\geq R^0>0$.  By the definition of $J$ and
the assumptions of initial data, we have
\begin{equation}
\begin{aligned}
  J&= \varepsilon\int_{\mathbb{R}^N}u_0(x)\Phi(x/R){\rm d}x\\
&\geq  \varepsilon \int_{|x|>\varepsilon_0} u_0(x)\Phi(x/R){\rm
  d}x\\
&= \varepsilon R^{N}\int_{|\eta|>\varepsilon_0/R }u_0(R\eta)\Phi(\eta){\rm
  d}\eta\\
&\geq \varepsilon C_0R^{N-\kappa}\int_{|\eta|>\varepsilon_0/R}|\eta|^{-\kappa}\Phi(\eta){\rm
  d}\eta\\
&\geq \varepsilon C_0R^{N-\kappa}\int_{|\eta|>\varepsilon_0/R^0}|\eta|^{-\kappa}\Phi(\eta){\rm
  d}\eta\\
&= \widetilde{C}R^{N-\kappa}.
\end{aligned}\label{4.1}
  \end{equation}
Using \eqref{2.12}, we know from \eqref{4.1} that,
for $0<\tau<T_{\varepsilon}$,
  \begin{equation}
\begin{aligned}
  \varepsilon &\leq  R^{\kappa-N}\widetilde{C}^{-1}(p-1)p^{p/(1-p)}[R^sS(T)]^{p/(p-1)}
  \\
  &= \widetilde{C}^{-1}(p-1)p^{p/(1-p)}H(\tau, R),
\end{aligned}\label{4.2}
  \end{equation}
where
$  H(\tau, R)=R^{\kappa-N}[S(\tau  R^{-2m})R^{s}]^{p/(p-1)}$.
 We  write
  \[
  H(\tau,  R)=[a_p\tau^{-1/p}R^{\alpha_1}+b_p\tau^{(p-1)/p}R^{-\alpha_2}]^{p/(p-1)},
  \]
where
  $  \alpha_1=(p-1)\kappa/p$, $\alpha_2= 2m-(p-1)\kappa/p$.
The choice of $\kappa$ implies $\alpha_1,\alpha_2>0$.

  Now we derive some estimates on  $H(\tau, R)$.
 If we can find a function $G(\tau)$  such that
      $$
      H(\tau, R)\geq G(\tau),\quad \forall \ \tau>0,
      $$
and for each value of $R\geq R^0$ there exists a value of $\tau_R$
such that $H(\tau_R, R)=G(\tau_R)$, then \eqref{4.2} holds for
all $R\geq R^0$ if and only if
   \begin{equation}
   \varepsilon\leq \widetilde{C}^{-1}(p-1)p^{p/(1-p)}G(\tau).
   \label{4.3}
   \end{equation}
Set
     $$
     y=R^{\alpha_1+\alpha_2}=R^{2m},\quad
     \beta_1=\alpha_2/(\alpha_1+\alpha_2)=\alpha_2/(2m).
     $$
Then
      $$
      H(\tau,R)=\tau^{-1/(p-1)}h(\tau, y)^{p/(p-1)}
      $$
with $h(\tau, y)=a_py^{1-\beta_1}+b_p y^{-\beta_1}\tau$.
Denote
  $$
  \sigma=a_pb_p^{-1}(1-\beta_1)\beta_1^{-1}y,\quad
  G(\tau)=\tau^{-1/(p-1)}g(\tau)^{p/(p-1)},
  $$
where
  $$
  g(\tau)=[a_py^{1-\beta_1}\sigma^{\beta_1-1}
+b_py^{-\beta_1}\sigma^{\beta_1}]  \tau^{1-\beta_1}.
  $$
It is easy to check that $0<\beta_1<1$. Then, $\zeta=g(\tau)$ ia a
concave curve. Furthermore, $\zeta=h(\tau, y)$ is a tangent line
of $\zeta=g(\tau)$ at the point of $( \sigma, g(\sigma))$.
Therefore, we get that
  $h(\tau,y)\geq g(\tau)$, for all  $\tau>0$.
Hence $H(\tau, R)\geq G(\tau)$,  for all $\tau>0$. Moreover,
$H(\tau,R_{\tau})=G(\tau)$ with
 $$
 \tau_R=a_pb_p^{-1}(1-\beta_1)\beta_1^{-1}R^{2m}.
 $$
By computations,
  \begin{equation}
   G(\tau)=\tau^{-1/(p-1)}g(\tau)^{p/(p-1)}= C_1\tau^{\hat{\beta}}.
  \label{4.4}
  \end{equation}
for some positive constant $C$, where
    \[
    \hat{\beta}=\frac \kappa{2m}-\frac{1}{p-1}.
    \]
The choice of $\kappa$ implies that $\hat{\beta}<0$.
 Combining \eqref{4.3} and \eqref{4.4}, we find that
     \begin{equation}
     \varepsilon\leq K\tau^{\hat{\beta}}
     \label{4.5}
     \end{equation}
for some $K>0$.
 From \eqref{4.5}, it follows that
      \[
      \tau\leq C\varepsilon^{1/\hat{\beta}}
      \]
for some $C>0$.
The proof  is complete.
\end{proof}

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