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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 155, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/155\hfil Blowup time for a diffusion problem]
{Blowup of solutions to degenerate Kirchhoff-type
diffusion problems involving the fractional $p$-Laplacian}

\author[Y. Yang, X. Tian, M. Zhang, J. Shen \hfil EJDE-2018/155\hfilneg]
{Yanbing Yang, Xueteng Tian, Meina Zhang, Jihong Shen}

\address{Yanbing Yang (corresponding author) \newline
College of Science,
Harbin Engineering University,
Harbin 150001, China. \newline
Department of Mathematics,
University of Texas,
Arlington, TX 76019, USA}
\email{yangyanbheu@163.com}

\address{Xueteng Tian \newline
College of Science,
Harbin Engineering University,
Harbin 150001, China}
\email{1490758004@qq.com}

\address{Meina Zhang \newline
College of Science,
Harbin Engineering University,
Harbin 150001, China}
\email{meina\_zhang@163.com}

\address{Jihong Shen \newline
College of Science,
Harbin Engineering University,
Harbin 150001, China}
\email{shenjihong@hrbeu.edu.cn}

\dedicatory{Communicated by Vicentiu D. Radulescu}

\thanks{Submitted May 4, 2018. Published August 22, 2018.}
\subjclass[2010]{35R11, 35K55, 47G20}
\keywords{Kirchhoff-type problem; parabolic equation; fractional $p$-Laplacian;
\hfill\break\indent blow-up of solution; blow-up time}

\begin{abstract}
 We study an initial boundary value problem for
 Kirchhoff-type parabolic equation with the fractional $p$-Laplacian.
 We first discuss the blow up of solutions in finite time with three
 initial energy levels:  subcritical, critical and supercritical
 initial energy levels.  Then we estimate an upper bound of the blowup
 time for low and for high initial energies.
\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}\label{sec1}

In this article we  consider the  parabolic initial boundary value
problem involving the fractional $p$-Laplacian
\begin{equation}\label{py1.1a}
\begin{gathered}
\partial_{t}u+[u]_{s,p}^{(\lambda-1)p}\mathcal{L}_{K}^{p}u=|u|^{q-2}u,
\quad\text{in } \Omega\times \mathbb{R}^{+},\partial_{t}u=\partial u/\partial t,\\
u(x,0)=u_0(x),\quad \text{in } \Omega,\\
u(x,t)=0,\quad \text{in }   (\mathbb{R}^{N}\backslash \Omega)\times
 \mathbb{R}_0^{+},
\end{gathered}
\end{equation}
where $[u]_{s,p}=\big(\iint_{Q}|u(x,t)-u(y,t)|^{p}K(x-y)\,dx\,dy\big)^{1/p}$,
$p$ and $q$ satisfy $2<p\lambda<q<p_{s}^{*}$ with $\lambda\in[1,p_{s}^{*}/p)$
and $p_{s}^{*}:=Np/(N-sp)$, $s\in(0,1)$, $\Omega\subset \mathbb{R}^{N}$ is
a bounded domain with Lipschitz boundary $\partial\Omega$.
The initial function is $u_0\geq 0$ on $\Omega$, $\mathcal{L}_{K}^{p}$
is a nonlocal integro-differential operator, which is defined by
$$
\mathcal{L}_{K}^{p}\varphi(x)=2\lim_{\varepsilon\to 0^{+}}\int_{{\mathbb{R}^{N}
\backslash{B_{\varepsilon}(x)}}}|\varphi(x)-\varphi(y)|^{p-2}[\varphi(x)
-\varphi(y)]K(x-y)dy,
$$
for any $\varphi\in C_0^{\infty}(\mathbb{R}^{N})$, where $B_{\varepsilon}(x)$
denotes the ball in $\mathbb{R}^{N}$ with radius $\varepsilon>0$ centered at
$x\in \mathbb{R}^{N}$. The kernel $K:\mathbb{R}^{N}\setminus\{0\}\to \mathbb{R}^{+}$
satisfies the following assumptions
\begin{itemize}
\item[(A1)] $m(x)K\in L^{1}(\mathbb{R}^{N})$, where $ m(x)=\min\{|x|^{p},1\}$;
there exists $K_0>0$, such that $K(x)\geq K_0|x|^{-(N+ps)}$ for a.e.
$ x\in \mathbb{R}^{N}\setminus\{0\}$.
\end{itemize}
A typical example for $K$ is the singular kernel $K(x)=|x|^{-(N+ps)}$.
In this way, $\mathcal{L}_{K}^{p}\varphi(x)=(-\Delta)_{p}^{s}\varphi(x)$ for
all $\varphi(x)\in C_0^{\infty}(\mathbb{R}^{N})$.
 We refer the reader to \cite{r18,r19,r16,r20} for further details on
fractional Laplacian and the fractional Sobolev spaces. In this case,
$[u]_{s,p}$ becomes the celebrated Gagliardo semi-norm.
As well known, problem \eqref{py1.1a} has been used to model some physical
phenomena occurring in nonlocal reaction-diffusion problems,
non-Newtonian fluid, non-Newtonian filtration and turbulent flows of
a gas in a porous medium, and so on. In the non-Newtonian fluid theory,
the quantity $p$ is characteristic of the medium. Media with $p>2$ are
called dilatant fluid and those with $p<2$ are called pseudoplastics.
If $p=2$, they are Newtonian fluids.

To explain the motivation for problem \eqref{py1.1a}, let us  introduce a
prototype of nonlocal problem \eqref{py1.1a} in $\mathbb{R}^{N}\times \mathbb{R}^{+}_0$.
Nonlocal evolutions of the form
\begin{equation}\label{py1.6}
\partial_{t}u(x,t)=\int_{\mathbb{R}^{N}}[u(y,t)-u(x,t)]\mathcal{K}(x-y)dy,
\end{equation}
and its variants, have been recently  used to model diffusion processes.
 More precisely, as stated, if $u(x,t)$ is thought of as a density of population
at the point $x$ and time $t$ and $\mathcal{K}(x-y)$ is thought of as
the probability distribution of jumping from location $y$ to location $x$,
then $\int_{\mathbb{R}^{N}}u(y,t)\mathcal{K}(x-y)dy$ is the rate at which
individuals are arriving at position $x$ from all other places and
$\int_{\mathbb{R}^{N}}u(x,t)\mathcal{K}(x-y)dy$ is the rate at which
they are leaving location $x$ to travel to all other sites.
If we consider the effects of total population, then problem \eqref{py1.6} becomes
\begin{equation}\label{py1.7}
\begin{aligned}
\partial_{t}u(x,t)
&=M\Big(\iint_{\mathbb{R}^{2N}}|u(x,t)-u(y,t)|^2\mathcal{K}(x-y)
\,dx\,dy\Big) \\
&\quad \times\int_{\mathbb{R}^{N}}[u(y,t)-u(x,t)]\mathcal{K}(x-y)dy,
\end{aligned}
\end{equation}
where the coefficient $M:\mathbb{R}_0^{+}\to \mathbb{R}_0^{+}$ accounts for
the possible changes of total population in $\mathbb{R}^{N}$.
This signifies that the behavior of individuals is subject to total population,
such as the diffusion process of bacteria. As a matter of fact,
model \eqref{py1.7} is meaningful, since the way of measurements are usually
taken in average sense.
It is worthy pointing out that there are some papers dedicated to the study
of Kirchhoff-type parabolic problems. For example,
Gobbino in \cite{r29} investigated the properties of solutions for the
 degenerate parabolic equations of Kirchhoff type
\begin{equation}\label{py1.5}
u_{t}-M\Big(\int_{\mathbb{R}^{N}}|\nabla u|^2dx\Big)\Delta u=0,
\end{equation}
where the Kirchhoff function $M:\mathbb{R}_0^{+}\to \mathbb{R}_0^{+}$
is continuous, which have been studied by many authors, see \cite{r29} and
the references therein for more details; see also \cite{r24,r31} for wave
equations of Kirchhoff type.

In the classical case, let us sketch the recent advances concerning the
 equation
\begin{equation}\label{eq1.3a}
u_{t}-\Delta u=f(u).
\end{equation}
Liu and Zhao \cite{r1} considered the initial-boundary value problem with
initial data $J(u_0)<d$ for $I(u_0)<0$ and $I(u_0)\geq 0$, and initial data
 $J(u_0)=d$ for $I(u_0)\geq 0$. In \cite{r2} Xu studied the same
 problem with critical initial data $J(u_0)=d, I(u_0)<0$, and initial
data $J(u_0)>d, I(u_0)>0$. A powerful technique for treating the above problem
is the so-called potential well method, which was established by Payne
and Sattinger \cite{r3}. Gazzola and Weth \cite{r4} studied the initial-boundary
value problem of \eqref{eq1.3a}, where $f(u)=|u|^{p-1}u$. They proved finite
time blow-up of solutions with high initial energy $J(u_0)>d$ by the comparison
principle and variational methods. Xu and Su \cite{r15} studied the initial
boundary value problem of $u_{t}-\Delta u_{t}-\Delta u=u^{p}$.
More precisely, they used the family of the potential wells to prove the
nonexistence of solutions with initial energy $J(u_0)\leq d$, and obtained
finite time blowup with high initial energy $J(u_0)>d$ by comparison principle.
 Very recently, Xu et al. in \cite{r9} discussed the same problem and
established a new finite time blowup theorem for the solution of problem
for arbitrary high initial energy.

In the fractional case, Caffarelli and Silvestre \cite{r6} introduced the
 s-harmonic extension to define the fractional Laplacian operator.
Nezza et al.\ \cite{r16} established the corresponding Sobolev inequality
and Poincar\'e inequality on the cone Sobolev spaces.
Fu and Pucci in \cite{r12} proved the existence of global solutions with
exponential decay and showed the blow-up in finite time of solutions to
the space-fractional diffusion problem
\begin{equation}\label{py1.4a}
\begin{gathered}
u_{t}+(-\Delta)^{s}u=|u|^{p-1}u,   \quad x\in\Omega,t>0,\\
u(x,0)=u_0(x),\quad  x\in\Omega,\\
u(x,t)=0,\quad  x\in \mathbb{R}^{n}\setminus\Omega,t\geq0,
\end{gathered}
\end{equation}
 provided that $M\equiv1$ and $p$ satisfies $1<p\leq2_{s}^{\ast}-1=\frac{n+2s}{n-2s}$.
 More works on fractional equations can be found in \cite{ACGP, r10,r21,r11} and
the references therein.

In recent years, a lot of interest has grown about Kirchhoff-type problems,
see for example \cite{r24, r14, PXZ1, r13}.
In these papers,  to obtain the existence of weak solutions, the authors
always assume that the Kirchhoff function $M:\mathbb{R}_0^{+}\to \mathbb{R^{+}}$
is a continuous and nondecreasing function and satisfies the following conditions:
\begin{equation}\label{py1.2a}
\text{there exists $m_0>0$ such that $M(t)\geq m_0$ for all $t\in\mathbb{R}_0^{+}$}.
\end{equation}
A typical example  is  $M(t)=m_0+bt^{m}$ with $m_0>0$, $b\geq0$ for all
$t\in \mathbb{R}_0^{+}$.
Naturally, we distinguish the problem into non-degenerate and degenerate cases
in accordance with $M(0)>0$ and $M(0)=0$ respectively. It is worthwhile
pointing out that the degenerate case is rather interesting and is treated
in well-known papers in Kirchhoff theory, see for example \cite{r33}.
From a physical point of view, the fact that $M(0)=0$ means that the base
tension of the string is zero. For some recent results in the degenerate case,
see for instance \cite{r34,BM, r35,r37,r36,r38} and the references therein.
In these papers, the Kirchhoff function $M$ was assumed to fulfill more general
conditions which cover the degenerate case. In this paper, we assume that $M$
is the  simple power function
$M(t)=t^{\lambda-1}$ with $\lambda\in [1,p_{s}^{*}/p)$ for all
$t\in\mathbb{R}_0^{+}$,
which implies  problem \eqref{py1.1a} is degenerate, see
 \cite{r32,r25,r26} for more results about this type.
Pan et al.\ \cite{r7} first studied the global solutions for degenerate
 Kirchhoff-type wave problem in the setting of fractional Laplacian by
combing the Galerkin method with potential well theory.  Pan et al.\ \cite{r5}
investigated for the first time the existence of a global solution for degenerate
Kirchhoff-type diffusion problems involving fractional $p$-Laplacian by
combing the Galerkin method with potential well theory.
Recently, Xiang et al.\ \cite{r8} studied a diffusion model of Kirchhoff-type
driven by a nonlocal integro-differential operator, and obtained the
existence of nonnegative local solutions.
Also, they showed that  the nonnegative local  solutions blow up in finite time
with arbitrary negative initial energy. In particular, the authors gave
an estimate for the lower and upper bounds of the blow-up time under certain
hypotheses on $M$ which cover the degenerate case $M(0)=0$.

 Zhou and Yang \cite{r23} studied an evolution $m$-Laplace equation involving
variable source in which the upper bound of the blowup time for the blow-up
solutions with positive initial energy was estimated. Xu et al.\ \cite{r9}
discussed the initial boundary value problem of $u_{t}-\Delta u_{t}-\Delta u=u^{p}$,
and estimated the upper bound of the blowup time for arbitrary high initial energy.

Motivated by the above works, we complete the picture of weak solutions for problem
\eqref{py1.1a} in the setting of fractional $p$-Laplacian by potential well
theory and concave function method. More precisely,  we shall prove the finite
time blow-up of solutions for problem \eqref{py1.1a}  at three different 
energy levels:  $J(u_0)<d$, $J(u_0)=d$, $J(u_0)>d$.
Furthermore, we will estimate the upper bound of the blowup time at low initial
energy and arbitrary high initial energy.

 The outline of this paper is as follows. In Section 2, we recall some
necessary definitions and properties of the fractional Sobolev spaces and
introduce the family of potential wells. In Section 3, we prove the finite
time blow-up for problem \eqref{py1.1a} with low initial energy $J(u_0)<d$
and estimate the upper bound of the blowup time.
In Section 4, we show the finite time blow-up for problem \eqref{py1.1a}
with critical energy $J(u_0)=d$. In Section 5, we establish a new finite time
blowup theorem for the solution of problem \eqref{py1.1a} for arbitrary high
initial energy and estimate the upper bound of the blowup time.

\section{Preliminaries}\label{sec2}

\subsection{Functional spaces}\label{2.1}
In this section, we first recall some  definitions and properties of the
fractional Sobolev spaces, see  \cite{r22,r16,r13} for further details.

Let $0<s<1<p<\infty$ be real numbers and the fractional critical exponent
$p_{s}^{\ast}$ be defined as
\begin{equation}
 p_{s}^{*}= \begin{cases}
\frac{Np}{N-sp},& \text{if } sp<N,\\
\infty,  &\text{if } sp\geq N.
\end{cases}
\end{equation}
In the following, we denote $Q=\mathbb{R}^{2N}\setminus\mathcal{G}$, where
$$
\mathcal{G}=\mathcal{C}(\Omega)\times\mathcal{C}(\Omega)\subset \mathbb{R}^{2N},
$$
and $\mathcal{G}=\mathbb{R}^{N}\setminus\Omega$.
 $W$ is a linear space of Lebesgue measurable functions from $\mathbb{R}^{N}$
to $\mathbb{R}$ such that the restriction to $\Omega$ of any function $u$
in $W$ belongs to $L^{p}(\Omega)$ and
$$
\iint_{Q}|u(x)-u(y)|^{p}K(x-y)\,dx\,dy<\infty.
$$
The space $W$ is equipped with the norm
$$
\|u\|_{W}=\Big(\|u\|_{L^{p}(\Omega)}+\iint_{Q}|u(x)-u(y)|^{p}K(x-y)\,dx\,dy
 \Big)^{1/p}.
$$
It is easy to get that $\|\cdot\|_{W}$ is a norm on $W$, see \cite{r13}.
We shall work in the closed linear subspace
\begin{equation}
W_0=\{u\in W:u(x)=0  \text{ a.e. in} \ \mathbb{R}^{N}\setminus \Omega\}.
\end{equation}

For any $p\in[1,+\infty)$,  we define the fractional Sobolev space $W^{s,p}(\Omega)$
as follows
$$
W^{s,p}(\Omega)=\big\{u\in L^{p}(\Omega):\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}
\in L^{p}(\Omega\times\Omega)\big\},
$$
endowed with the norm
$$
\|u\|_{W^{s,p}(\Omega)}=\Big(\|u\|_{L^{p}(\Omega)}+\iint_{\Omega}
\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy\Big)^{1/p}.
$$

\begin{lemma}[{\cite[Lemma 2.3]{r13}}]\label{lemma2.2}
Let $K:\mathbb{R}^{N}\setminus\{0\}\to \mathbb{R}^{+}$ satisfy assumption(A1). Then
there exists a positive constant $C_0=C_0(N,p,s)$ such that for any
$v\in W_0$ and $q\in [1,p_{s}^{*}]$,
\begin{align*}
\|v\|_{L^q(\Omega)}^{p}
&\leq C_0\iint_{\Omega\times\Omega}\frac{|v(x)-v(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy\\
&\leq \frac{C_0}{K_0}\iint_{Q}|v(x)-v(y)|^{p}K(x-y)\,dx\,dy.
\end{align*}
\end{lemma}

\begin{definition}\label{definition2.1} \rm
Let $p\geq1$ and $W$ be a reflexive Banach space. A function $f$ defined and
measurable in $Q$ belongs to the space $L^{p}(0,T;W)$, if
$$
\|f\|_{L^{p}}(0,T;W)=\Big(\int_0^{T}\|f(x,t)\|_{W}^{p}dt\Big)^{1/p}<\infty,
$$
Using \cite{r12}, we can get an equivalent norm on $W_0$ defined as
$$
\|v\|_{W_0(\Omega)}=\Big(\iint_{Q}|v(x)-v(y)|^{p}K(x-y)\,dx\,dy\Big)^{1/p}.
$$
\end{definition}

\begin{definition}\label{definition2.2} \rm
A function $u\in L^{\infty}(0,\infty;W_0)$ is said to be a (weak) solution of
problem \eqref{py1.1a}, if $u_{t}\in L^2(0,\infty;L^2(\Omega))$ and for a.e.
$t>0$,
$$
\int_{\Omega}\partial_{t}u(x,t)\phi dx+\langle u,\phi\rangle_{W_0}
=\int_{\Omega}|u|^{q-2}u\phi dx,
$$
where
\begin{align*}
\langle u,\phi\rangle_{W_0}
&=M(\|u\|_{W_0}^{p})\iint_{Q}|u(x,t)-u(y,t)|^{p-2}[u(x,t)-u(y,t)] \\
&\quad\times [\phi(x)-\phi(y)]K(x-y)\,dx\,dy,
\end{align*}
for any $\phi\in W_0$.
\end{definition}

Then we introduce some functionals
\begin{gather}\label{py1.3a}
J(u)=\frac{1}{p\lambda}\|u\|_{W_0}^{p\lambda}-\frac{1}{q}\|u\|_{q}^q, \\
\label{py1.3b}
I(u)=\|u\|_{W_0}^{p\lambda}-\|u\|_{q}^q,
\end{gather}
and the potential well
\begin{gather*}
\mathcal{W}=\{u\in W_0\mid I(u)>0,J(u)<d\}\cup\{0\},\\
\mathcal{V}=\{u\in W_0\mid I(u)<0,J(u)<d\},\quad
d=\inf_{u\in\mathcal{N}}J(u).
\end{gather*}
The Nehari manifold
$$
\mathcal{N}=\{u\in W_0: I(u)=0,\|u\|_{W_0}\neq0\},
$$
separates the two unbounded sets
$$
\mathcal{N_{+}}=\{u\in W_0\mid I(u)>0\},\quad
\mathcal{N_{-}}=\{u\in W_0\mid I(u)<0\}.
$$

\subsection{Family of potential wells} \label{sec2.2}

In this section, we  introduce a family of potential wells
 $\mathcal{W}_{\delta}$ and its corresponding sets $\mathcal{V}_{\delta}$,
and give a series of their properties for problem \eqref{py1.1a}.
Firstly, let the definitions of functionals $J(u),I(u)$ and the potential
well $\mathcal{W}$ with its depth $d$ given above hold. Next, we give some
properties of above sets and functionals.

For $\delta>0$, we define
\begin{gather*}
I_{\delta}(u)=\delta\|u\|_{W_0}^{p\lambda}-\|u\|_{q}^q,\quad
d(\delta)=\inf_{u\in\mathcal{N}_{\delta}}J(u),\\
\mathcal{N}_{\delta}=\{u\in W_0\mid I_{\delta}(u)=0,\|u\|_{W_0}\neq0\},\quad
r(\delta)=\Big(\frac{\delta}{C_{\ast}^q}\Big)^{\frac{1}{q-p\lambda}},
\end{gather*}
where $C_{*}$ is the embedding constant from $W_0$ into $L^q(\Omega)$.

For $0<\delta<q/(p\lambda)$, we define
\begin{gather}
\mathcal{W}_{\delta}=\{u\in W_0\mid I_{\delta}(u)>0,J(u)<d(\delta)\}\cup\{0\},
\nonumber \\
\mathcal{V}_{\delta}=\{u\in W_0\mid I_{\delta}(u)<0,J(u)<d(\delta)\}, \nonumber\\
\label{py3.1a}
\int_0^{t}\|u_{\tau}\|_2^2d\tau+J(u)\leq J(u_0)\,.
\end{gather}

\begin{lemma}\label{Lemma2}
Let $u\in W_0$. Then we have
\begin{itemize}
\item[(i)] If $I_{\delta}(u)<0$, then $\| u\|_{W_0}>r(\delta)$.
 In particular, if $I(u)<0$, then $\|u\|_{W_0}>r(1)$.

\item[(ii)] If $I_{\delta}(u)=0$, then $\|u\|_{W_0}\geq r(\delta)$ or
 $\|u\|_{W_0}=0$. In particular, if $I(u)=0$, then $\| u\|_{W_0}\geq r(1)$ or
 $\|u\|_{W_0}=0$.

\item[(iii)] If $I_{\delta}u=0$ and $\|u\|_{W_0}\neq0$, then $J(u)>0$ for
 $0<\delta<q/(p\lambda)$, $J(u)=0$ for $\delta=q/(p\lambda)$,
$J(u)<0$ for $\delta>q/(p\lambda)$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) It is easy to see that $\|u\|_{W_0}\neq 0$ thanks to $I_{\delta}(u)<0$.
Thus from
$$
\delta\|u\|^{p\lambda}_{W_0}<\|u\|^q_{q}\leq C_{*}^q\|u\|^q_{W_0}
=C_{\ast}^q\|u\|^{p\lambda}_{W_0}\|u\|^{q-p\lambda}_{W_0},
$$
we obtain $\|u\|_{W_0}>r(\delta)$.

(ii) On the one hand, if $\|u\|_{W_0}=0$, then $I_{\delta}(u)=0$.
 On the other hand, if $\|u\|_{W_0}\neq 0$ and $I_{\delta}(u)=0$, then by
$$
\delta\|u\|_{W_0}^{p\lambda}=\|u\|^q_{q}
\leq C_{*}^q\|u\|^{p\lambda}_{W_0}\|u\|^{q-p\lambda}_{W_0},
$$
we obtain $\|u\|_{W_0}\geq r(\delta)$.

(iii) The conclusion follows from Lemma \ref{Lemma2}(ii) and by $I_{\delta}(u)=0$,
we have
\begin{align*}
J(u)&=\Big(\frac{1}{p\lambda}-\frac{\delta}{q}\Big)\|u\|_{W_0}^{p\lambda}
 +\frac{\delta}{q}\|u\|_{W_0}^{p\lambda}-\frac{1}{q}\|u\|_{q}^q\\
&=\Big(\frac{1}{p\lambda}-\frac{\delta}{q}\Big)\|u\|_{W_0}^{p\lambda}
 +\frac{1}{q}I_{\delta}u\,,
\end{align*}
which implies (iii).
\end{proof}

\begin{lemma}\label{Lemma3}
$d(\delta)$ satisfies the following properties:
\begin{itemize}
\item[(i)]
$d(\delta)\geq a(\delta)r^{p\lambda}(\delta)$ for
$a(\delta)=1/(p\lambda)-\delta/q, 0<\delta<q/(p\lambda)$.

\item[(ii)]
$\lim_{\delta\to 0}d(\delta)=0, d(q/(p\lambda))=0$ and $d(\delta)<0$
for $\delta>q/(p\lambda)$.

\item[(iii)]
$d(\delta)$ is increasing on $0<\delta\leq1$, decreasing on
$1\leq\delta\leq q/(p\lambda)$ and takes the maximum $d=d(1)$ at $\delta=1$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) If $u\in\mathcal{N}$, then by lemma \ref{Lemma2}(ii) we have
$\|u\|_{W_0}\geq r(\delta)$. Hence from
\[
J(u)=\Big(\frac{1}{p\lambda}-\frac{\delta}{q}\Big)\|u\|_{W_0}^{p\lambda}
 +\frac{1}{q}I_{\delta}(u)
=a(\delta)\|u\|_{W_0}^{p\lambda}\geq a(\delta)r^{p\lambda}(\delta),
\]
it follows that $d(\delta)\geq a(\delta)r^{p\lambda}(\delta)$.

(ii) For any $u\in W_0, \|u\|_{W_0}\neq0$, we define $\theta=\theta(\delta)$ by
\begin{equation}\label{2a}
\delta\|\theta u\|_{W_0}^{p\lambda}=\|\theta u\|_{q}^q,
\end{equation}
i.e.
$\delta\|u\|_{W_0}^{p\lambda}=\theta^{q-p\lambda}\|u\|_{q}^q$.
Hence, for any $\delta>0$, there exists a unique
$$
\theta(\delta)=\Big(\frac{\delta\|u\|_{W_0}^{p\lambda}}{\|u\|_{q}^q}
\Big)^{\frac{1}{q-p\lambda}},
$$
satisfying \eqref{2a}, which implies that $\theta u\in\mathcal{N}_{\delta}$, we have
$\lim_{\delta\to 0}\theta(\delta)=0$.
It is easy to see that
\[
\lim_{\delta\to 0}J(\theta u)=\lim_{\theta\to 0}J(\theta u)=0
\]
and
$\lim_{\delta\to 0}d(\delta)=0$.
From lemma \ref{Lemma2} (iii), we can complete this proof.

(iii) It is enough to prove that for any $0<\delta'<\delta''<1$ or
$1<\delta''<\delta'<q/(p\lambda)$ and for any $u\in\mathcal{N}_{\delta''}$,
there exist a $v\in\mathcal{N}_{\delta'}$ and a constant
$\varepsilon(\delta',\delta'')$ such that $J(v)<J(u)-\varepsilon(\delta',\delta'')$.
In fact, for above $u$ we can define $\theta(\delta)$, then
 $I_{\delta}(\theta(\delta)u)=0$ and $\theta(\delta'')=1$.
Let $g(\theta)=J(\theta u)$, we obtain
\[
\frac{d}{d\theta}g(\theta)
=\frac{1}{\theta}\Big((1-\delta)\|\theta u\|_{W_0}^{p\lambda}+I_{\delta}(\theta u)
\Big)
=\theta^{p\lambda-1}(1-\delta)\|u\|_{W_0}^{p\lambda}.
\]
Taking $v=\theta(\delta')u$, then $v\in\mathcal{N}_{\delta'}$.
For $0<\delta'<\delta''<1$, we have
\begin{align*}
J(v)-J(u)=&g(1)-g(\theta(\delta'))\\
=&\int_{\theta(\delta')}^{1}\frac{d}{d\theta}(g(\theta))d\theta\\
=&\int_{\theta(\delta')}^{1}(1-\delta)\theta^{p\lambda-1}
 \|u\|_{W_0}^{p\lambda}d\theta\\
>&(1-\delta'')r^{p\lambda}(\delta'')\theta^{p\lambda-1}(\delta')
 \left(1-\theta(\delta')\right)
\equiv\varepsilon(\delta',\delta'').
\end{align*}
For $1<\delta''<\delta'<q/(p\lambda)$, we have
\begin{align*}
J(u)-J(v)=&g(1)-g(\theta(\delta'))\\
>&(\delta''-1)r^{p\lambda}(\delta'')\theta^{p\lambda-1}(\delta'')
\left(\theta(\delta')-1\right)
\equiv\varepsilon(\delta',\delta'').
\end{align*}
Therefore, the conclusion of (iii) is proved.
\end{proof}

\begin{lemma}\label{Lemma5}
 Assume $0<J(u)<d$ for some $u\in W_0$, and
$\delta_1<\delta_2$ are the two roots of equation $d(\delta)=J(u)$. Then the sign of $I_{\delta}(u)$ doesn't change for $\delta_1<\delta<\delta_2$.
\end{lemma}

\begin{proof}
$J(u)>0$ implies $\|u\|_{W_0}\neq 0$. If the sign of $I_{\delta}(u)$ is changeable for $\delta_1<\delta<\delta_2$, then we choose $\overline{\delta}\in (\delta_1,\delta_2)$ and $I_{\overline{\delta}}(u)=0$. Therefore, we can get $J(u)\geq d(\overline{\delta})$, which contradicts $J(u)=d(\delta_1)=d(\delta_2)<d(\overline{\delta})$.
\end{proof}

\section{Blow up with low initial energy $J(u_0)<d$}
\label{sec3}


\begin{definition}\label{T2} \rm
Let $u(t)$ be a weak solution of problem \eqref{py1.1a}.
 We define the maximal time existence $T_{\rm max}$ of $u(t)$ as follows:
\begin{itemize}
\item[(i)]
If $u(t)$ exists for $0\leq t<\infty$, then $T_{\rm max}=\infty$.
\item[(ii)]
If there exists a $t_0\in (0,\infty)$ such that $u(t)$ exists for
$0\leq t<t_0$, but does not exists at $t=t_0$, then $T_{\rm max}=t_0$.
\end{itemize}
\end{definition}

\begin{lemma}[Invariant set for $J(u_0)<d$] \label{thm8}
Let $u_0\in W_0$, $0<e<d$, $\delta_1<\delta_2$ be the two roots of equation
$d(\delta)=e$. Then All weak solutions $u$ of problem \eqref{py1.1a}
with $J(u_0)=e$ belong to $\mathcal{V}_{\delta}$ for $\delta_1<\delta<\delta_2$,
$0\leq t<T_{\rm max}$, provided $I(u_0)<0$, where $T_{\rm max}$ is the
maximal existence time of $u(t)$.
\end{lemma}

\begin{proof}
Let $u(t)$ be any weak solution of problem \eqref{py1.1a} with $J(u_0)=e$, $I(u_0)<0$.
  From $J(u_0)=e$, $I(u_0)<0$ and Lemma
\ref{Lemma5}, it follows $I_{\delta}(u_0)<0$ and $J(u_0)<d(\delta)$.
Then $u_0(x)\in \mathcal{V}_{\delta}$ for $\delta_1<\delta<\delta_2$.

 We prove $u(t)\in V_{\delta}$ for $\delta_1<\delta<\delta_2$ and
$0<t<T_{\rm max}$. Arguing by contradiction, by time continuity of $I(u)$,
we suppose that there exists a $\delta_0\in (\delta_1, \delta_2)$ and
$t_0\in (0,T_{\rm max})$ such that $u(t_0)\in \partial \mathcal{V}_{\delta_0}$,
$I_{\delta_0}(u(t_0))=0$ or $J(u(t_0))=d(\delta_0)$. From
\begin{equation}\label{py2.5b}
\int^t_0\|u(\tau)\|_2^2d\tau+J(u)\leq J(u_0)<d(\delta),\quad
 \delta_1<\delta<\delta_2,\; 0\leq t<T_{\rm max},
\end{equation}
we can see that $J(u(t_0))\neq d(\delta_0)$. Assume $I_{\delta_0}(u(t_0))=0$
and $t_0$ is the first time such that $I_{\delta_0}(u(t_0))=0$, then
$I_{\delta_0}(u(t))<0$ for $0\leq t<t_0$. By Lemma \ref{Lemma2}(i) we have
$\|u(t_0)\|_{W_0}>r(\delta_0)$ for
$0\leq t<t_0$. Hence $\|u(t_0)\|_{W_0}> r(\delta_0)$, then $\|u(t_0)\|_{W_0}\neq0$.
From $u(t_0)\in\mathcal{N}_{\delta_0}$ and $J(u(t_0))\neq d(\delta_0)$,
we have $J(u(t_0))>d(\delta_0)$, which contradicts \eqref{py2.5b}.
\end{proof}

\begin{remark}\label{re3.3} \rm
If the assumption $J(u_0)=e$ is replaced by $0<J(u_0)\leq e$ in Lemma \ref{thm8},
then the conclusion of Lemma \ref{thm8} still  holds.
\end{remark}

\subsection{Finite time blow-up at low initial energy}\label{2.4}
In this section, we establish the finite time blow-up of solutions of
problem \eqref{py1.1a}. By Lemma \ref{lemma2.2} we know that $W_0$ is continuously
embedding in $L^2(\Omega)$, let $S$ be the best embedding constant.
Then the main result of this section is stated as follows.

\begin{theorem}[Blow-up for $J(u_0)<d$]\label{thm11}
 Suppose that $u_0\in W_0,J(u_0)<d$ and $I(u_0)<0$. then any nontrivial
solution of problem \eqref{py1.1a} must blowup in finite time. There exists a
$T>0$ such that
\begin{equation}\label{py2.9a}
\lim_{t\to  T}\int_0^{t}\|u\|_2^2d\tau=+\infty.
\end{equation}
\end{theorem}

\begin{proof}
Let $u(t)$ be any weak solution of problem \eqref{py1.1a} with $J(u_0)<d$
and $I(u_0)<0$.  We define
$$
M(t)=\int_0^{t}\|u\|_2^2d\tau,
$$
then
$M'(t)=\|u\|_2^2$, and
\begin{equation}\label{py2.9b}
M''(t)=2(u,u_{t})=2\int_{\Omega}u_{t}udx
=2\|u\|_{q}^q-2\|u\|_{W_0}^{p\lambda}
=-2I(u).
\end{equation}
Notice that
\[
J(u)=\frac{1}{p\lambda}\|u\|_{W_0}^{p\lambda}-\frac{1}{q}\|u\|_{q}^q
=\Big(\frac{1}{p\lambda}-\frac{1}{q}\Big)\|u\|_{W_0}^{p\lambda}+\frac{1}{q}I(u);
\]
thus
\[
I(u)=qJ(u)-\frac{q-p\lambda}{p\lambda}\|u\|_{W_0}^{p\lambda}.
\]
Applying the basic inequality $s\leq s^{\alpha}+1$ for any $s\geq0$ and
$\alpha\geq1$, we can get
\begin{align*}
M''(t)=&\frac{2(q-p\lambda)}{p\lambda}\|u\|_{W_0}^{p\lambda}-2qJ(u)\\
\geq&\frac{2(q-p\lambda)}{p\lambda}(\|u\|_{W_0}^2-1)
 +2q\int_0^{t}\|u_{\tau}\|_2^2d\tau-2qJ(u_0)\\
\geq&\frac{2C(q-p\lambda)}{p\lambda}\|u\|_2^2
 +2q\int_0^{t}\|u_{\tau}\|_2^2-\Big(2qJ(u_0)+\frac{2(q-p\lambda)}{p\lambda}\Big)\\
=&\frac{2C(q-p\lambda)}{p\lambda}M'(t)
 +2q\int_0^{t}\|u_{\tau}\|_2^2d\tau-\Big(2qJ(u_0)
 +\frac{2(q-p\lambda)}{p\lambda}\Big),
\end{align*}
where $C=S^2$.
Note that
\begin{align*}
\Big(\int_0^t(u_\tau,u)d\tau \Big)^2
=&\Big(\frac{1}{2}\int_0^t\frac{d}{d\tau}\|u\|_2^2\Big)^2 \\
=&\Big(\frac{1}{2}\|u\|_2^2-\frac{1}{2}\|u_0\|_2^2\Big)^2\\
=&\frac{1}{4}\big(\|u\|_2^4-2\|u\|_2^2\|u_0\|_2^2+\|u_0\|_2^4\big) \\
=&\frac{1}{4}\big((M'(t))^2-2M'(t)\|u_0\|_2^2+\|u_0\|_2^4\big)\,.
\end{align*}
It follows that
\begin{equation}\label{py2.9c}
\left(M'(t)\right)^2=4\Big(\int_0^{t}\int_{\Omega}u_{\tau}u\,dx\,d\tau\Big)^2
+2M'(t)\|u_0\|_2^2-\|u_0\|_2^4 \,.
\end{equation}
Using the Cauchy-Schwartz inequality, we have
\begin{align*}
&M''(t)M(t)-\frac{q}{2}\left(M'(t)\right)^2\\
&\geq2q\int_0^{t}\|u_{\tau}\|_2^2d\tau\int_0^{t}\|u\|_2^2d\tau
 -2q\Big(\int_0^{t}\int_{\Omega}u_{\tau}u\,dx\,d\tau\Big)^2+\frac{q}{2}\|u_0\|_2^4\\
&\quad -\Big(2qJ(u_0)+\frac{2(q-p\lambda)}{p\lambda}\Big)M(t)
 +\frac{2C(q-p\lambda)}{p\lambda}M'(t)M(t)-q\|u_0\|_2^2M'(t)\\
&\geq \frac{2C(q-p\lambda)}{p\lambda}M'(t)M(t)
 -\Big(2qJ(u_0)+\frac{2(q-p\lambda)}{p\lambda}\Big)M(t)-q\|u_0\|_2^2M'(t).
\end{align*}
We discuss the following two cases:

(i) If $J(u_0)\leq 0$, then
\begin{align*}
&M(t)M''(t)-\frac{q}{2}(M'(t))^2\\
&\geq \frac{2C(q-p\lambda)}{p\lambda}M(t)M'(t)-q\|u_0\|_2^2M'(t)
 -\frac{2(q-p\lambda)}{p\lambda}M(t).
\end{align*}
Now we prove $I(u)<0$ for $t>0$. If it is false, we must be allowed to
choose a $t_0>0$ such that $I(u(t_0))=0$ and $I(u)<0$ for $0\leq t<t_0$.
From Lemma \ref{Lemma2}(i), we have $ \|u\|_{W_0}>r(1)$ for $0\leq t<t_0$,
$\|u(t_0)\|_{W_0}\geq r(1)$ and $J(u(t_0))\geq d$, which contradicts
\eqref{py3.1a}. From \eqref{py2.9b}, we can get $M''(t)>0$ for $t\geq 0$.
 From $M'(0)=\|u_0\|_2^2\geq 0$,  we can see that there exists a $t_0\geq 0$
such that $M'(t_0)>0$. For $t\geq t_0$ we have
$$
M(t)\geq M'(t_0)(t-t_0)+M(t_0)>M'(0)(t-t_0).
$$
Therefore, for sufficiently large $t$, we obtain
\begin{gather*}
\frac{C(q-p\lambda)}{p\lambda}M(t)>q\|u_0\|_2^2, \\
\frac{C(q-p\lambda)}{p\lambda}M'(t)>\frac{2(q-p\lambda)}{p\lambda},
\end{gather*}
then
$$
M(t)M''(t)-\frac{q}{2}(M'(t))^2>0.
$$

(ii) If $0<J(u_0)<d$,  then by Lemma \ref{thm8} we have
$u(t)\in \mathcal{V}_{\delta}$ for $1<\delta<\delta_2$, $t\geq0$ and
$I_{\delta}(u)<0$,  $\| u\|_{W_0}>r(\delta)$ for $1<\delta<\delta_2$, $t\geq 0$,
where $\delta_2$ is the larger root of equation $d(\delta)=J(u_0)$.
Hence, $I_{\delta_2}(u)\leq 0$ and $\|u\|_{W_0}>r(\delta_2)$ for $t\geq 0$.
 By \eqref{py2.9b} we have
\begin{gather*}
\begin{aligned}
M''(t)&=-2I(u)=2(\delta_2-1)\|u\|_{W_0}^{p\lambda}-2I_{\delta_2}(u)\\
& \geq 2(\delta_2-1)\|u\|_{W_0}^{p\lambda}\geq  2(\delta_2-1)r^{p\lambda}(\delta_2),
\quad  t\geq 0,
\end{aligned}\\
M'(t)\geq 2(\delta_2-1)r^{p\lambda}(\delta_2)t+M'(0)
\geq2(\delta_2-1)r^{p\lambda}(\delta_2)t,\quad  t\geq 0,
\\
M(t)\geq 2(\delta_2-1)r^{p\lambda}(\delta_2)t^2,\ \ t\geq 0.
\end{gather*}
Therefore, for sufficiently large $t$, we have
\begin{gather*}
\frac{C(q-p\lambda)}{p\lambda}M(t)>q\|u_0\|_2^2, \\
\frac{C(q-p\lambda)}{p\lambda}M'(t)>2qJ(u_0)+\frac{2(q-p\lambda)}{p\lambda}.
\end{gather*}
Consequently,
\begin{align*}
&M(t)M''(t)-\frac{q}{2}(M'(t))^2\\
&\geq \frac{2C(q-p\lambda)}{p\lambda}M'(t)M(t)
 -\Big(2qJ(u_0)+\frac{2(q-p\lambda)}{p\lambda}\Big)M(t)-q\|u_0\|_2^2M'(t)\\
&=\Big(\frac{C(q-p\lambda)}{p\lambda}M(t)-q\|u_0\|_2^2\Big)M'(t)\\
&\quad +\Big(\frac{C(q-p\lambda)}{p\lambda}M'(t)-2qJ(u_0)
 -\frac{2(q-p\lambda)}{p\lambda}\Big)M(t)>0.
\end{align*}
The remainder of the proof is the same as that in \cite{r15}.
\end{proof}

\subsection{Blow up time with low initial energy}

We  give an upper bound for the blow up time.
By Lemma \ref{lemma2.2}, we know that the Sobolev space
$W_0\hookrightarrow L^q(\Omega)$ continuously. Let $C_{*}$ be the optimal
constant of the embedding then
\begin{gather}\label{2py2.1}
\|u\|_{q}\leq C_{*}\|u\|_{W_0}, \\
\label{2py2.3}
\alpha_1:=C_{*}^{-\frac{q}{q-p\lambda}}, \\
\label{2py2.4}
J_1=\frac{q-p\lambda}{p\lambda q}C_{*}^{-\frac{p\lambda q}{q-p\lambda}}
=\frac{q-p\lambda}{p\lambda q}\alpha_1^{p\lambda}.
\end{gather}
By \cite[Lemma 3.4]{r5}, we know that
\[
J_1=\frac{q-p\lambda}{p\lambda q}\frac{1}{C_{*}^{\frac{p\lambda q}{q-p\lambda}}}=d.
\]
Then the main result of this article reads as follows.

\begin{theorem}\label{thm2.1}
Suppose $q>p\lambda$, $q>2$. Then the solution of problem \eqref{py1.1a} will
blow up in finite time if the initial value $u_0$ is chosen to ensure that
$J(u_0)<d$ and $\|u_0\|_{W_0}>\alpha_1$.
Moreover, the blow-up time $T$ can be estimated from above by $T^{*}$, where
\begin{equation}\label{2py2.5}
T^{*}=\frac{q\big(-\hspace{-0.33cm}\int_{\Omega}u_0^2(x)\big)^{\frac{2-q}{2}}}
{(q-2)(q-p\lambda)\Big(1-\big(\big(\frac{1}{p\lambda}-J(u_0)\alpha_1^{-p\lambda}
\big)q\big)^{-\frac{q}{q-p\lambda}}\Big)}
\end{equation}
and
$$
-\hspace{-0.38cm}\int_{\Omega}f(x)dx=\frac{1}{|\Omega|}\int_{\Omega}f(x)dx
$$
where $|\Omega|$ is the Lebesgue measure of $\Omega$.
\end{theorem}

\begin{lemma}\label{lem2.3a}
The energy defined in \eqref{py1.3a} is nonincreasing with
\begin{equation}\label{2py2.6a}
J(u(t))=J(u_0)-\int_0^{t}\|u_{\tau}\|_2^2d\tau.
\end{equation}
\end{lemma}

\begin{proof}
From \eqref{py1.3a}, we have
\begin{align*}
J'(u(t))
=&\frac{d}{dt}\Big(\frac{1}{p\lambda}\|u\|_{W_0}^{p\lambda}
 -\frac{1}{q}\|u\|_{q}^q\Big)\\
=&-\int_{\Omega}|u|^{q-2}uu_{t}dx
 +\int_{\Omega}[u]_{s,p}^{(\lambda-1)p}(-\Delta)_{p}^{s}uu_{t}dx\\
=&-\int_{\Omega}\left(|u|^{q-2}u-[u]_{s,p}^{(\lambda-1)p}(-\Delta)_{p}^{s}u\right)
 u_{t}dx\\
=&-\int_{\Omega}u_{t}^2dx,
\end{align*}
which yields \eqref{2py2.6a}.
\end{proof}

We deduce from \eqref{py1.3a} and \eqref{2py2.1} that
\begin{equation}\label{2py2.6}
J(u(t))=\frac{1}{p\lambda}\|u\|_{W_0}^{p\lambda}-\frac{1}{q}\|u\|_{q}^q
\geq\frac{1}{p\lambda}\alpha^{p\lambda}-\frac{1}{q}(C_{*}\alpha)^q,
\end{equation}
where $\alpha(t)=\|u(\cdot,t)\|_{W_0}$.

\begin{lemma}\label{lem2.3b}
Let $g:[0,\infty)\mapsto \mathbb{R}$ be defined by
$$
g(\alpha)=\frac{1}{p\lambda}\alpha^{p\lambda}-\frac{1}{q}C_{*}^q\alpha^q.
$$
Then the following properties hold under the assumptions of Theorem \ref{thm2.1}:
\begin{itemize}
\item[(i)]
$g$ is increasing for $0<\alpha<\alpha_1$ and decreasing for $\alpha\geq\alpha_1$;
\item[(ii)]
$ \lim_{\alpha\to \infty}g(\alpha)=-\infty$ and $g(\alpha_1)=J_1$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) The first derivative of $g(\alpha)$ is
\begin{align*}
g'(\alpha)=\alpha^{p\lambda-1}-C_{*}^q\alpha^{q-1}.
\end{align*}
Note that $g'(\alpha)=0$ implied that $\alpha_1=C_{*}^{-\frac{q}{q-p\lambda}}$,
hence (i) follows.

(ii) Since $p\lambda<q$, we have that $\lim_{\alpha\to \infty}g(\alpha)=-\infty$.
$\alpha_1$ is the  extreme point and a routine computation gives rise to
 $g(\alpha_1)=J_1$. Then (ii) holds.
\end{proof}

\begin{lemma}\label{lem2.4c}
Under the assumptions of Theorem \ref{thm2.1}, there exists a positive
constant $\alpha_2>\alpha_1$ such that
\begin{gather}\label{2py2.7}
\|u(\cdot,t)\|_{W_0}\geq \alpha_2,\quad  t\geq 0, \\
\label{2py2.8}
\int_{\Omega}|u|^qdx\geq (C_{*}\alpha_2)^q, \\
\label{2py3.12}
\frac{\alpha_2}{\alpha_1}
\geq\Big(\Big(\frac{1}{p\lambda}-J(0)\alpha_1^{-p\lambda}\Big)q
\Big)^{\frac{1}{q-p\lambda}}>1.
\end{gather}
\end{lemma}

\begin{proof}
Since $J(u_0)<J_1$, it follows from Lemma \ref{lem2.3b} that there exists
a positive constant $\alpha_2>\alpha_1$ such that $J(u_0)=g(\alpha_2)$.
Let $\alpha_0=\|u_0\|_{W_0}$, by \eqref{2py2.6}, we have
$g(\alpha_0)\leq J(u_0)=g(\alpha_2)$. Since $\alpha_0,\alpha_2\geq\alpha_1$,
it follows from Lemma \ref{lem2.3b}(i) that $\alpha_0\geq\alpha_2$ so \eqref{2py2.7}
holds for $t=0$.

Now we prove \eqref{2py2.7} by contradiction. Suppose that
$\|u(\cdot,t_0)\|_{W_0}<\alpha_2$ for some $t_0>0$. By the continuity of
$\|u(\cdot,t)\|_{W_0}$ and $\alpha_1<\alpha_2$, we may choose $t_0$ such that
 $\|u(\cdot,t_0)\|_{W_0}>\alpha_1$. Then it follows from \eqref{2py2.6} that
$$
J(u_0)=g(\alpha_2)<g(\|u(\cdot,t_0)\|_{W_0})\leq J(u(t_0)),
$$
which contradicts Lemma \ref{lem2.3a}, and \eqref{2py2.7} follows.

By \eqref{py1.3a} and Lemma \ref{lem2.3a}, we obtain
\begin{align*}
\int_{\Omega}\frac{1}{q}|u|^qdx\geq \frac{1}{p\lambda}\|u\|_{W_0}^{p\lambda}-J(u_0)
\geq\frac{1}{p\lambda}\alpha_2^{p\lambda}-J(u_0)
=\frac{1}{q}(C_{*}\alpha_2)^q,
\end{align*}
and \eqref{2py2.8} follows.

Since $J(u_0)<J_1$, by a straightforward computation, we can check
$$
\Big(\frac{1}{p\lambda}-J(u_0)\alpha_1^{-p\lambda}\Big)q>1.
$$

Denote $\beta=\alpha_2/\alpha_1$, then $\beta>1$ by the fact that
$\alpha_2>\alpha_1$. So it follows from $J(u_0)=g(\alpha_2)$ and \eqref{2py2.3}
that
\begin{equation}\label{2py7}
\begin{aligned}
J(u_0)=&g(\beta\alpha_1) \\
=&\frac{1}{p\lambda}(\beta\alpha_1)^{p\lambda}-\frac{1}{q}C_{*}^q
 (\beta\alpha_1)^q \\
\geq&\alpha_1^{p\lambda}\Big(\frac{1}{p\lambda}
 -\frac{\beta^{q-p\lambda}}{q}C_{*}^q\alpha_1^{q-p\lambda}\Big)\\
=&\alpha_1^{p\lambda}\Big(\frac{1}{p\lambda}-\frac{\beta^{q-p\lambda}}{q}\Big),
\end{aligned}
\end{equation}
which implies that the inequality in \eqref{2py3.12}.
\end{proof}

\begin{lemma}\label{lem2.5d}
Under the assumptions of Theorem \ref{thm2.1},
we have the estimate
\begin{equation}\label{2py2.11}
0<H(0)\leq H(t)\leq \frac{1}{q}\int_{\Omega}|u|^qdx,
\end{equation}
 where $H(t)=J_1-J(u(t))$ for $t\geq 0$.
\end{lemma}

\begin{proof}
From Lemma \ref{lem2.3a}, we know that $H(t)$ is nondecreasing in $t$. Thus
\begin{equation}\label{2py2.12}
H(t)\geq H(0)=J_1-J(u_0)>0,\ t\geq 0.
\end{equation}
Combining  \eqref{py1.3a}, \eqref{2py2.4} and \eqref{2py2.7}, $J(u(t))>0$ and
$\alpha_2>\alpha_1$, we have
\[
H(t)=J_1-J(u(t))
\leq J_1-\frac{1}{p\lambda}\alpha_1^{p\lambda}+\frac{1}{q}\int_{\Omega}|u|^qdx
\leq \frac{1}{q}\int_{\Omega}|u|^qdx.
\]
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
Let
$$
M(t)=\frac{1}{2}\int_{\Omega}u^2(x,t)dx.
$$
Then by the definition of $J(u(t))$ and $H(t)$, the derivative of $M(t)$ satisfies
\begin{equation}\label{2py2.13}
\begin{aligned}
M'(t)=&\int_{\Omega}u u_{t}dx    \\
=&-\|u\|_{W_0}^{p\lambda}+\|u\|_{q}^q    \\
=& \|u\|_{q}^q-p\lambda J(u(t))-\frac{p\lambda}{q}\|u\|_{q}^q  \\
=&\frac{q-p\lambda}{q}\|u\|_{q}^q-p\lambda J_1+p\lambda H(t).
\end{aligned}
\end{equation}
From \eqref{2py2.3}, \eqref{2py2.4} and \eqref{2py2.8}, we obtain
\begin{equation}\label{2py2.14}
\begin{aligned}
p\lambda J_1&=\frac{q-p\lambda}{q}C_{*}^{-\frac{p\lambda q}{q-p\lambda}}
=\frac{q-p\lambda}{q}(C_{*}\alpha_1)^q  \\
&=\frac{q-p\lambda}{q}\Big(\frac{\alpha_1}{\alpha_2}\Big)^q(C_{*}\alpha_2)^q  \\
&\leq\frac{q-p\lambda}{q}\Big(\frac{\alpha_1}{\alpha_2}\Big)^q\int_{\Omega}|u|^qdx.
\end{aligned}
\end{equation}
So, we have
\begin{equation}\label{2py2.15}
M'(t)\geq \tilde{C}\|u\|_{q}^q,
\end{equation}
where
$$
\tilde{C}=\Big(1-\big(\frac{\alpha_1}{\alpha_2}\big)^q\Big)\frac{q-p\lambda}{q}.
$$
By H\"older's inequality, we have
\begin{equation}\label{2py2.16}
M^{q/2}(t)\leq \bar{C}\int_{\Omega}|u|^qdx,
\end{equation}
where
$$
\bar{C}=2^{-q/2}|\Omega|^{\frac{q-2}{2}},
$$
and $|\Omega|$ is the Lebesgue measure of $\Omega$.
Then it follows from \eqref{2py2.15} and \eqref{2py2.16} that
$$
M'(t)\geq \frac{\tilde{C}}{\bar{C}}M^{q/2}(t),
$$
which means that
\begin{equation}\label{2py2.17}
M(t)=\Big(\Big(\frac{1}{2}\int_{\Omega}|u_0|^2dx\Big)^{\frac{2-q}{2}}
-\frac{(q-2)\tilde{C}}{2\bar{C}}t\Big)^{-\frac{2}{q-2}}.
\end{equation}
Let
\begin{equation}\label{2py2.18}
\tilde{T}:=\frac{2^{q/2}\bar{C}}{(q-2)\tilde{C}}
\Big(\int_{\Omega}|u_0|^2dx\Big)^{\frac{2-q}{2}}\in(0,\infty)\,.
\end{equation}
Then $M(t)$ blows up at time $\tilde{T}$. Therefore, $u(x,t)$ ceases to
exist at some finite time $T\leq\tilde{T}$, that is to say,
$u(x,t)$ blows up at a finite time $T$.

Next, we estimate $T$. By \eqref{2py3.12} and the values of $\tilde{C}, \bar{C}$,
we have
\begin{align*}
\frac{2^{q/2}\bar{C}}{(q-2)\tilde{C}}
\leq\frac{|\Omega|^{\frac{q-2}{2}}}{(q-2)
\Big(1-\big((\frac{1}{p\lambda}-J(u_0)\alpha_1^{-p\lambda})q
\big)^{\frac{q-p\lambda}{q}}\Big)\frac{q-p\lambda}{q}}.
\end{align*}
The above inequalities combined with \eqref{2py2.18} give $T\leq\tilde{T}\leq T^{*}$,
where $T^{*}$ is defined in \eqref{2py2.5}.
The remainder of the proof is the same as that in \cite{r23}.
\end{proof}

\section{Blow up with critical initial energy $J(u_0)=d$}
\label{sec4}

In this section, we prove the finite time blow-up of solution for
 problem \eqref{py1.1a} with the critical initial condition $J(u_0)=d$.

\begin{theorem}\label{thm14}
Suppose that $u_0\in W_0$, $J(u_0)=d$ and $I(u_0)<0$.
Then any nontrivial solution of problem \eqref{py1.1a} must blow up in finite time.
\end{theorem}

\begin{proof}
Let $u(t)$ be any weak solution of problem \eqref{py1.1a} with $J(u_0)=d$
and $I(u_0)<0$,  $T$ being the existence time of $u(t)$.
We prove that $T<\infty$. Arguing by contradiction, we assume that $T=\infty$.
 Now we define
$$
M(t)=\int_0^{t}\|u\|_2^2d\tau.
$$
By Theorem \ref{thm11} and $J(u_0)=d$ we have
\begin{align*}
M''(t)=&\frac{2(q-p\lambda)}{p\lambda}\|u\|_{W_0}^{p\lambda}-2qJ(u)\\
=&\frac{2(q-p\lambda)}{p\lambda}\|u\|_{W_0}^{p\lambda}
 +2q\int_0^{t}\|u_{\tau}\|d\tau-2qJ(u_0)\\
\geq&\frac{2(q-p\lambda)}{p\lambda}(\|u\|_{W_0}^2-1)
 +2q\int_0^{t}\|u_\tau\|_2^2d\tau-2qJ(u_0)\\
\geq&\frac{2C(q-p\lambda)}{p\lambda}\|u\|_2^2
 +2q\int_0^{t}\|u_{\tau}\|_2^2d\tau
 -\Big(2qJ(u_0)+\frac{2(q-p\lambda)}{p\lambda}\Big)\\
=&\frac{2C(q-p\lambda)}{p\lambda}M'(t)+2q\int_0^{t}\|u_{\tau}\|_2^2d\tau
 -\Big(2qJ(u_0)+\frac{2(q-p\lambda)}{p\lambda}\Big).
\end{align*}
According to the estimate of the $(M'(t))^2$ in Theorem \ref{thm11} which
is \eqref{py2.9c}, we obtain
\begin{align*}
&M''(t)M(t)-\frac{q}{2}\left(M'(t)\right)^2\\
&\geq 2q\int_0^{t}\|u_{\tau}\|_2^2d\tau\int_0^{t}\|u\|_2^2d\tau
 -2q\Big(\int_0^{t}\int_{\Omega}u_{\tau}u\,dx\,d\tau\Big)^2\\
&\quad -\Big(2qJ(u_0)+\frac{2(q-p\lambda)}{p\lambda}\Big)M(t)
 +\frac{2C(q-p\lambda)}{p\lambda}M'(t)M(t) \\
&\quad -q\|u_0\|_2^2M'(t)+\frac{q}{2}\|u_0\|_2^4\\
&\geq\frac{2C(q-p\lambda)}{p\lambda}M'(t)M(t)
 -\Big(2qJ(u_0)+\frac{2(q-p\lambda)}{p\lambda}\Big)M(t)-q\|u_0\|_2^2M'(t).
\end{align*}
By using the Cauchy-Schwartz inequality, we obtain
\begin{equation}\label{py2.10b}
\begin{aligned}
&M(t)M''(t)-\frac{q}{2}(M'(t))^2 \\
&\geq\frac{2C(q-p\lambda)}{p\lambda}M'(t)M(t)
 -\Big(2qJ(u_0)+\frac{2(q-p\lambda)}{p\lambda}\Big)M(t)-q\|u_0\|_2^2M'(t) \\
&=\Big(\frac{C(q-p\lambda)}{p\lambda}M(t)-q\|u_0\|_2^2\Big)M'(t) \\
&\quad +\Big(\frac{C(q-p\lambda)}{p\lambda}M'(t)-2qJ(u_0)
 -\frac{2(q-p\lambda)}{p\lambda}\Big)M(t).
\end{aligned}
\end{equation}
On the other hand, from $J(u_0)=d>0$, $I(u_0)<0$ and the continuity of $J(u)$
and $I(u)$ with respect to $t$, it follows that there exists a sufficiently
small $t_1>0$ such that $J(u(t_1))>0$ and $I(u)<0$ for $0\leq t\leq t_1$.
 Hence $(u_{t},u)=-I(u)>0$, $u_{t}\neq0, \|u_{t}\|>0$ for $0\leq t\leq t_1$.
From this and the continuity of $\int_0^{t} \|u_{\tau}\|_2^2d\tau$,
we can choose a $t_1$ such that
$$
0<J(u(t_1))=d_1=d-\int_0^{t_1}\|u_{\tau}\|_2^2d\tau<d.
$$
Thus we take $t=t_1$ as the initial time, then we know that
$u(t)\in \mathcal{V}_{\delta}$ for $\delta\in (\delta_1,\delta_2)$,
$t_1\leq t<\infty$,  where $(\delta_1,\delta_2)$ is the maximal interval
including $\delta=1$ such that $d(\delta)>d_1$ for
$\delta\in (\delta_1, \delta_2)$. Hence we have $I_{\delta}(u)<0$ and
$\|u\|_{W_0}>r(\delta)$ for $\delta\in (1,\delta_2)$, $t_1\leq t<\infty$,
and $I_{\delta_2}(u)\leq 0, \|u\|_{W_0}\geq r(\delta_2)$ for $t_1\leq t<\infty$.
 Thus from \eqref{py2.9b} we obtain
\begin{gather}\label{py2.10c}
\begin{aligned}
M''(t)&=-2I(u)=2(\delta_2-1)\|u\|_{W_0}^{p\lambda}-2I_{\delta_2}(u)\\
&\geq 2(\delta_2-1)\|u\|_{W_0}^{p\lambda}\\
&\geq  2(\delta_2-1)r^{p\lambda}(\delta_2)\equiv C(\delta_2),\quad t_1\leq t<\infty,
\end{aligned} \\
\label{py2.10d}
M'(t)\geq C(\delta_2)(t-t_1)+M'(t_1)\geq C(\delta_2)(t-t_1),\quad t_1\leq t<\infty,\\
\label{py2.10e}
M(t)\geq \frac{1}{2} C(\delta_2)(t-t_1)^2+M(t_1)>\frac{1}{2} C(\delta_2)(t-t_1)^2,
\quad t_1\leq t<\infty.
\end{gather}
From \eqref{py2.10d} and \eqref{py2.10e} it follows that for sufficiently large
$t$ we have
$$
\frac{C(q-p\lambda)}{p\lambda}M(t)>q\|u_0\|_2^2,
$$
and
$$
\frac{C(q-p\lambda)}{p\lambda}M'(t)>2qd+\frac{2(q-p\lambda)}{p\lambda},\quad
 t_1\leq t<\infty.
$$
Thus \eqref{py2.10b} yields
$$
M(t)M'(t)-\frac{q}{2}(M'(t))^2>0,
$$
which gives
$$
(M^{-\alpha}(t))''
=\frac{-\alpha}{M^{\alpha+2}}(t)\left(M(t)M'(t)-(\alpha+1)(M'(t))^2\right)\leq 0, \quad
 \alpha=\frac{q-2}{2}.
$$
From this it follows that there exists a $T_1>0$ such that
$$
\lim_{t\to  T_1}M^{-\alpha}(t)=0,\quad\text{and}\quad
\lim_{t\to  T_1}M(t)=+\infty,
$$
which contradicts  that $T=+\infty$.
\end{proof}


\section{Blow up time with high initial energy $J(u_0)>0$}
\label{sec5}

In this section, we establish a finite time blowup theorem for the solution
of problem \eqref{py1.1a} with arbitrary high initial energy.
 At the same time, we estimate the upper bound of the blowup time.

\begin{theorem} \label{thm5.1}
Let $u(x,t)$ be a weak solution to problem \eqref{py1.1a}, $u_0\in W_0$.
Suppose that $J(u_0)>0$ and
\begin{equation}\label{3py2.3}
\frac{p\lambda q}{q-p\lambda}J(u_0)<B\|u_0\|^{p\lambda}_2
\end{equation}
hold. Then the solution $u(x,t)$ blows up in finite time,
where $B$ is best constant of inequality
$\|u\|_{W_0}^{p\lambda}\geq B\|u\|_2^{p\lambda}$ with $B=S^{p\lambda}$.
In addition there exists a $t_1$ as
$$
0<t_1\leq\frac{2\varphi(0)}{(\alpha-1)\varphi'(0)},
$$
such that
\begin{equation}\label{5.2}
\lim_{t\to  t_1}\int_0^{t}\|u\|_2^2d\tau=+\infty,
\end{equation}
where
\begin{gather}\label{5.3}
\varphi(t)=\Big(\int_0^{t}\|u\|_2^2d\tau\Big)
+\varepsilon^{-1}\|u_0\|_2^2\int_0^{t}\|u\|_2^2d\tau+c, \\
\label{3py2.11}
1<\alpha<\frac{B(q-p\lambda)\|u_0\|^{p\lambda}_2}{p\lambda qJ(u_0)}, \\
\label{5.5}
0<\varepsilon<\frac{1}{p\lambda\alpha\|u_0\|_2^2}
\Big(\frac{2B(q-p\lambda)}{q}\|u_0\|_2^2-2p\lambda\alpha J(u_0)
 -\frac{2(q-p\lambda)}{q}\Big), \\
\label{3py2.18}
c>\frac{1}{4}\varepsilon^{-2}\|u\|^4_2.
\end{gather}
\end{theorem}

\begin{lemma}[\cite{r28}] \label{lemma 5.1}
Suppose that a positive, twice-differentiable function $\psi(t)$ satisfy
the inequality
$$
\psi''(t)\psi(t)-(1+\theta)(\psi'(t))^2\geq 0,\quad t>0,
$$
where $\theta>0$ is a constant. If $\psi(0)>0$ and $\psi'(0)>0$, then
there exists $0<t_1\leq \frac{\psi(0)}{\theta\psi'(0)}$ such that
$\psi(t)$ tends to $\infty$ as $t\to  t_1$.
\end{lemma}

To prove the high energy blowup, we first establish the following lemma.

\begin{lemma}\label{lemma 5.2}
Assume that $u_0\in W_0$ satisfies \eqref{3py2.3}.
Then $u\in \mathcal{N}_-=\{u\in W_0|I(u)<0\}$.
\end{lemma}

\begin{proof}
Let $u(t)$ be any weak solution of problem \eqref{py1.1a}. Multiplying
\eqref{py1.1a} by $u_t(t)$ and integrating on $\Omega$, then we have
$$
\|u_t(t)\|_2^2=-\frac{1}{p\lambda}\frac{d}{dt}\|u\|_{W_0}^{p\lambda}
+\frac{1}{q}\frac{d}{dt}\|u\|_{q}^q;
$$
that is,
$$
-\|u_t(t)\|^2_2=\frac{d}{dt}\Big(\frac{1}{p\lambda}\|u\|_{W_0}^{p\lambda}
-\frac{1}{q}\|u\|_{q}^q\Big).
$$
Then, we could obtain
\begin{equation}\label{3py2.4}
\frac{d}{dt}J(u)=-\|u_t(t)\|_2^2\leq 0.
\end{equation}
Multiplying \eqref{py1.1a} by $u$ and integrate on $\Omega\times(0,t)$, we have
$$
\frac{1}{2}\|u\|_2^2-\frac{1}{2}\|u_0\|_2^2
+\int_0^t(\|u\|_{W_0}^{p\lambda}-\|u\|_{q}^q)d\tau=0\,;
$$
that is,
\begin{equation}\label{3py2.5}
\frac{1}{2}\frac{d}{dt}\|u\|^2_2=-I(u).
\end{equation}
Note that
\begin{align*}
J(u_0)=&\frac{q-p\lambda}{p\lambda q}\|u_0\|_{W_0}^{p\lambda}+\frac{1}{q}I(u_0)\\
\geq&\frac{B(q-p\lambda)}{p\lambda q}\|u_0\|_2^{p\lambda}+\frac{1}{q}I(u_0)\,.
\end{align*}
Then \eqref{3py2.3} indicates that $I(u_0)<0$.

Next, we prove $u(t)\in \mathcal{N}_-$ for all $t\in [0,T)$.
Arguing by contradiction, by the continuity of $I(t)$ in $t$,
we assume that there exists an $s\in(0,T)$ such that $u(t)\in \mathcal{N}_-$
for $0\leq t<s$ and $u(s)\in \mathcal{N}$, then by \eqref{3py2.5} we have
\begin{equation}\label{3py2.6}
\frac{d}{dt}\|u(t)\|_2^2=-2I(u)>0,\quad \text{for all } t\in [0,s),
\end{equation}
which implies that
$\|u_0\|_2^2<\|u(s)\|_2^2$.
Then, we have
\begin{equation}\label{3py2.7}
\|u_0\|_2^{p\lambda}<\|u(s)\|_2^{p\lambda}.
\end{equation}
From \eqref{3py2.4} it follows that
\begin{equation}\label{3py2.8}
J(u(s))\leq J(u_0) \quad \text{for all } t\in [0,s).
\end{equation}
By the definition of $J(u)$ and $u(s)\in \mathcal{N}$, we arrive to
\[
J(u(s))=\frac{q-p\lambda}{p\lambda q}\|u\|_{W_0}^{p\lambda}+\frac{1}{q}I(u(s))
\geq\frac{B(q-p\lambda)}{p\lambda q}\|u\|_2^{p\lambda}.
\]
Combining \eqref{3py2.3} and \eqref{3py2.8}, we obtain
$$
\frac{B(q-p\lambda)}{p\lambda q}\|u\|_2^{p\lambda}\leq J(u_0)
<\frac{B(q-p\lambda)}{p\lambda q}\|u_0\|^{p\lambda}_2;
$$
that is
$$
\|u(s)\|_2^{p\lambda}<\|u_0\|_2^{p\lambda}.
$$
This contradicts \eqref{3py2.7}.
\end{proof}

Now we show high energy blowup and estimate the upper bound of the blowup
time of solutions for problem\eqref{py1.1a}.

\begin{proof}
Arguing by contradiction, we assume the existence time of solutions $T=+\infty$.
Integrating of \eqref{3py2.4} with from $0$ to $t$,
\begin{equation}\label{3py2.9}
J(u)+\int_0^t\|u_{\tau}\|^2_2d\tau=J(u_0).
\end{equation}
From \eqref{3py2.5} we have
\begin{equation}\label{3py2.10}
\begin{aligned}
\frac{d}{dt}\|u\|^2_2
&=-2I(u) \\
&=-2(\|u\|_{W_0}^{p\lambda}-\|u\|_{q}^q) \\
&=-2p\lambda\Big(\frac{1}{p\lambda}\|u\|_{W_0}^{p\lambda}
 -\frac{1}{q}\|u\|_{q}^q\Big)+\Big(2-\frac{2p\lambda}{q}\Big)\|u\|_{q}^q \\
&=-2p\lambda J(u)+\frac{2q-2p\lambda}{q}\|u\|_{q}^q.
\end{aligned}
\end{equation}
In the rest of the proof, we consider the following two cases.

(i) $J(u)\geq 0$, for all $t>0$.
From \eqref{3py2.3}, we choose $\alpha$ satisfying \eqref{3py2.11}.
Substituting \eqref{3py2.9} into \eqref{3py2.10}, as $J(u)\geq0$ in this
case we obtain
\begin{equation}\label{3py2.12}
\begin{aligned}
\frac{d}{dt}\|u\|^2_2
&=2p\lambda(\alpha-1)J(u)-2p\lambda\alpha J(u)+\frac{2(q-p\lambda)}{q}\|u\|_{q}^q \\
&\geq-2p\lambda\alpha J(u_0)+2p\lambda\alpha\int_0^t\|u_\tau\|^2_2d\tau
+\frac{2(q-p\lambda)}{q}\|u\|_{q}^q.
\end{aligned}
\end{equation}
From Lemma \ref{lemma 5.2}, we know that
$\|u\|_{W_0}^{p\lambda}<\|u\|_{q}^q$.
Therefore, applying the basic inequality $s\leq s^{\alpha}+1$ for any $s\geq0$
and $\alpha\geq1$, we  obtain
\begin{equation}\label{3py2.13}
\begin{aligned}
&\frac{d}{dt}\|u\|^2_2\\
&\geq -2p\lambda\alpha J(u_0)+2p\lambda\alpha \int_0^t\|u_\tau\|^2_2d\tau
 +\frac{2(q-p\lambda)}{q}\|u\|_{q}^q \\
&> -2p\lambda\alpha J(u_0)+2p\lambda\alpha \int_0^t\|u_\tau\|^2_2d\tau
 +\frac{2(q-p\lambda)}{q}\|u\|_{W_0}^{p\lambda}\\
&> -2p\lambda\alpha J(u_0)+2p\lambda\alpha \int_0^t\|u_\tau\|^2_2d\tau
 +\frac{2(q-p\lambda)}{q}(\|u\|_{W_0}^2-1) \\
&> -2p\lambda\alpha J(u_0)+2p\lambda\alpha \int_0^t\|u_\tau\|^2_2d\tau
 +\frac{2B(q-p\lambda)}{q}\|u\|_2^2-\frac{2(q-p\lambda)}{q} .
\end{aligned}
\end{equation}
Then
\begin{equation}\label{3py2.14}
\frac{d}{dt}\|u\|^2_2-\frac{2B(q-p\lambda)}{q}\|u\|^2_2
>-2p\lambda\alpha J(u_0)-\frac{2(q-p\lambda)}{q},
\end{equation}
which yields
\begin{equation}\label{3py2.15}
\begin{aligned}
\|u\|^2_2&>\|u_0\|^2_2e^{\frac{2B(q-p\lambda)}{q}t} \\
&\quad +\frac{q}{B(q-p\lambda)}\Big(p\lambda\alpha J(u_0)
 +\frac{q-p\lambda}{q}\Big)
\Big(1-e^{\frac{2B(q-p\lambda)}{q}t}\Big).
\end{aligned}
\end{equation}
Next, we define $y(t)=\int_0^t\|u(\tau)\|^2_2d\tau$. Since the solution $u(x,t)$
is global, thus the function $y(t)$ is bounded for all $t\geq 0$. Then we have
$$
y'(t)=\|u(t)\|^2_2,\quad  y''(t)=\frac{d}{dt}\|u\|^2_2.
$$
Substituting \eqref{3py2.15} into \eqref{3py2.13}, we obtain
\begin{equation}\label{3py2.16}
\begin{aligned}
y''(t)&>\Big(\frac{2B(q-p\lambda)}{q}\|u_0\|^2_2
 -2p\lambda\alpha J(u_0)-\frac{2(q-p\lambda)}{q}\Big)e^{\frac{2B(q-p\lambda)}{q}t} \\
&\quad + 2p\lambda\alpha\int_0^t\|u_\tau\|_2^2d\tau \\
&>p\lambda\alpha\varepsilon\|u_0\|_2^2
 +2p\lambda\alpha\int_0^{t}\|u_{\tau}\|_2^2d\tau \\
&=A(t).
\end{aligned}
\end{equation}
By \eqref{3py2.11}, we can take $\varepsilon>0$ small enough such that
\begin{equation}\label{3py2.17}
\varepsilon<\frac{1}{p\lambda\alpha \|u_0\|^2_2}
\Big(\frac{2B(q-p\lambda)}{q}\|u_0\|^2_2-2p\lambda\alpha J(u_0)
 -\frac{2(q-p\lambda)}{q}\Big),
\end{equation}
then we pick $c>0$ large enough such that
\begin{equation}\label{3py2.18b}
c>\frac{1}{4}\varepsilon^{-2}\|u\|^4_2.
\end{equation}
We now define the auxiliary function
$\varphi(t)=y^2(t)+\varepsilon^{-1}\|u_0\|^2_2y(t)+c$. Hence
\begin{gather}\label{3py2.19}
\varphi'(t)=\left(2y(t)+\varepsilon^{-1}\|u_0\|^2_2\right)y'(t), \\
\label{3py2.20}
\varphi''(t)=\left(2y(t)+\varepsilon^{-1}\|u_0\|^2_2\right)y''(t)+2(y'(t))^2.
\end{gather}
Set $\delta=4c-\varepsilon^{-2}\|u_0\|^4_2$, because of \eqref{3py2.18}, $\delta>0$.
Now, from \eqref{3py2.19} we can write
\begin{equation}\label{3py2.21}
\begin{aligned}
(\varphi'(t))^2
&=\left(2y(t)+\varepsilon^{-1}\|u_0\|^2_2\right)^2(y'(t))^2 \\
&=\left(4y^2(t)+4\varepsilon^{-1}\|u_0\|^2_2y(t)
 +\varepsilon^{-2}\|u_0\|^4_2\right)(y'(t))^2 \\
&=\left(4y^2(t)+4\varepsilon^{-1}\|u_0\|^2_2y(t)+4c-\delta\right)(y'(t))^2\\
&=(4\varphi(t)-\delta)(y'(t))^2.
\end{aligned}
\end{equation}
The above equality yields
\begin{equation}\label{3py2.22}
4\varphi(t)(y'(t))^2=(\varphi'(t))^2+\delta(y'(t))^2.
\end{equation}
By integrating
\begin{equation}\label{3py2.23}
\frac{1}{2}\frac{d}{dt}\|u(t)\|^2_2=(u,u_t)
\end{equation}
from $0$ to $t$,
we obtain
$$
\frac{1}{2}\left(\|u(t)\|^2_2-\|u_0\|^2_2\right)=\int_0^t(u,u_\tau)d\tau.
$$
Hence
$$
\|u(t)\|^2_2=\|u_0\|^2_2+2\int_0^t(u,u_\tau)d\tau.
$$
This equality along with the H\"{o}lder and Young's inequality give
\begin{equation}\label{3py2.24}
\begin{aligned}
&(y'(t))^2 \\
&=\|u(t)\|^4_2 \\
&=\Big(\|u_0\|^2_2+2\int_0^t(u,u_\tau)d\tau\Big)^2\\
&\leq\Big(\|u_0\|^2_2+2\Big(\int_0^t\|u\|^2_2d\tau\Big)^{1/2}
 \Big(\int_0^t\|u_\tau\|^2_2d\tau\Big)^{1/2}\Big)^2 \\
&\leq\|u_0\|^4_2+4y(t)\int_0^t\|u_\tau\|^2_2d\tau+2\varepsilon\|u_0\|^2_2y(t)
 +2\varepsilon^{-1}\|u_0\|^2_2\int_0^t\|u_\tau\|d\tau \\
&=B(t).
\end{aligned}
\end{equation}
From \eqref{3py2.20} and \eqref{3py2.22}, we obtain
\begin{equation}\label{3py2.25}
\begin{aligned}
2\varphi(t)\varphi''(t)
=&2\left(\left(2y(t)+\varepsilon^{-1}\|u_0\|^2_2\right)y''(t)+2(y'(t))^2\right)
 \varphi(t) \\
=&2\left(2y(t)+\varepsilon^{-1}\|u_0\|^2_2\right)y''(t)\varphi(t)
 +4(y'(t))^2\varphi(t)\\
=&2\left(2y(t)+\varepsilon^{-1}\|u_0\|^2_2\right)y''(t)\varphi(t)
 +(\varphi'(t))^2+\delta(y'(t))^2.
\end{aligned}
\end{equation}
By \eqref{3py2.17} and the fact that $e^{\frac{2C(q-p\lambda)}{q}}>1$ and
$\varphi>0$, we obtain
\begin{align*}
&2\varphi(t)\varphi''(t)-(1+\alpha)(\varphi'(t))^2\\
&>2\varphi(t)\left(2y(t)+\varepsilon^{-1}\|u_0\|^2_2\right)
 \Big(2p\lambda\alpha\int_0^t\|u_\tau\|^2_2d\tau+p\lambda\alpha\varepsilon
 \|u_0\|^2_2\Big)-4\alpha\varphi(t)B(t)\\
&>2p\lambda\alpha\varphi(t)\big(2y(t)+\varepsilon^{-1}\|u_0\|^2_2\big)
 \Big(2\int_0^t\|u_\tau\|^2_2d\tau+\varepsilon\|u_0\|^2_2\Big)
 -4\alpha\varphi(t)B(t)\\
&=2p\lambda\alpha B(t)\varphi(t)-4\alpha B(t)\varphi(t)
>0;
\end{align*}
that is,
$$
\varphi(t)\varphi''(t)-\frac{1+\alpha}{2}(\varphi'(t))^2>0,\quad t\in[0,T]
,$$
which implies that
$$
(\varphi^{-\beta}(t))''=-\frac{\beta}{\varphi^{\beta+2}}(\varphi''(t)\varphi(t)
-(\beta+1)(\varphi'(t))^2)<0, \quad \beta=\frac{\alpha-1}{2}>0.
$$
Since $\varphi(0)>0$ and $\varphi'(0)>0$, by Lemma \ref{lemma 5.1},
 there exists  $t_*$ such that
$$
0<t_*\leq \frac{2\varphi(0)}{(\alpha-1)\varphi'(0)},
$$
such that
$$
\lim_{t\to  t_*}\varphi^{-\beta}(t)=0,\quad\text{and}\quad
\lim_{t\to  t_*}\varphi(t)=+\infty,
$$
which contradicts $T=+\infty$. Now, by considering the continuity of $\varphi$
 with respect to $y$, we can conclude that $y(t)$ tends to $\infty$ at some
finite time which is a contradiction.

(ii) There exist some $\tilde{t}$ such that $J(u(\tilde{t}))<0$.
Since $J(u_0)>0$, by the continuity of $J(u)$ in $t$, we can assume that
there exists a first time $t_0>0$ such that $J(u(t_0))=0$ and $J(u(\hat{t}))<0$
for some $\hat{t}>t_0$. We take $u(\hat{t})$ as a new initial datum,
then from Lemma \ref{lemma 5.2}, we have $u(t)\in \mathcal{N}_{-}$ for
 $t>\hat{t}$. Then similar to the proof of Theorem \ref{thm11}, we can
prove the finite time blowup of the solution.

Combining the above two cases, we conclude that $u(x,t)$ blows up in finite time.
\end{proof}


\subsection*{Acknowledgements}

This work was supported by the Heilongjiang Postdoctoral Foundation (LBH-Z15036),
by the Fundamental Research Funds for the Central Universities,
by the China Scholarship Council (201706685064).
Y. Yang specially appreciates Prof. Yue Liu's invitation of visit to UTA.


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