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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 116, pp. 1--19.\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-2018/116\hfil Initial boundary value problem]
{Initial boundary value problem for a mixed pseudo-parabolic
$p$-Laplacian type equation with logarithmic nonlinearity}

\author[Y. Cao, C. Liu \hfil EJDE-2018/116\hfilneg]
{Yang Cao, Conghui Liu}

\address{Yang Cao (corresponding author) \newline
School of Mathematical Sciences, Dalian University of
Technology, Dalian 116024, China}
\email{mathcy@dlut.edu.cn}

\address{Conghui Liu \newline
School of Mathematical Sciences,
Dalian University of Technology,
 Dalian 116024, China}
\email{conghuil@sina.com}

\thanks{Submitted December 26, 2017. Published May 14, 2018.}
\subjclass[2010]{35K58, 35K35, 35B40}
\keywords{ Pseudo-parabolic; $p$-Laplacian; logarithmic nonlinearity; 
\hfill\break\indent long time behavior}

\begin{abstract}
 We consider the initial boundary value problem for a mixed pseudo-parabolic
 $p$-Laplacian type equation with logarithmic nonlinearity.
 Constructing a family of potential wells and using the logarithmic Sobolev
 inequality, we establish the existence of global weak solutions.
 we consider two cases: global boundedness and blowing-up at $\infty$.
 Moreover, we discuss the asymptotic behavior of solutions and give some
 decay estimates and growth estimates.
\end{abstract}

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


\section{Introduction}

In this article we study the following initial-boundary value problem
for a nonlinear evolution equation with logarithmic source
\begin{equation}\label{qq1}
 \begin{gathered}
 u_t-\operatorname{div}(|\nabla u|^{p-2}\nabla u)-k \triangle u_t=|u|^{p-2}u\log|u|, \quad
 \Omega \times(0,T),\\
 u(x,t)=0, \quad\partial \Omega \times (0,T),\\
 u(x,0)=u_0(x), \quad \Omega\,,\\
 \end{gathered}
\end{equation}
where $1<p<2$, $u_0 \in H_0^{1}(\Omega)$, $T\in(0,+\infty]$, $k\geq0$,
$\Omega \subset \mathbb{R}^{n}(n\geq 1)$
is a bounded domain with smooth boundary $\partial \Omega$.

Problem \eqref{qq1} is a mixed pseudo-parabolic $p$-Laplacian type equation,
whose abstract form was first considered by Showalter \cite{Showalter1},
and sometimes referred to as Showalter equation \cite{Al'shin}.
When $k=0$, \eqref{qq1} is the classical fast diffusive $p$-Laplacian,
which appears to be relevant in the theory of non-Newtonian fluids.
When $k>0$, \eqref{qq1} belongs to the pseudo-parabolic equations,
which are characterized by the occurrence of first-order partial 
derivative in time of the highest order term \cite{ShowalterTing}.
These equations arise from a variety of important physical processes,
such as the flows of fluids through fissured rock \cite{Barenblatt1},
nonlinear dispersive long waves \cite{BBM},
the heat conduction involving two temperatures \cite{Chen},
the aggregation of populations \cite{Padron}, etc.
Particularly, \eqref{qq1} is from shearing flows of incompressible simple 
fluids \cite{Barenblatt2}.
The quantity $|\nabla u|^{p-2}\nabla u+k\nabla u_t$ can be viewed as 
approximation to the stress functional
in such a flow, and $k\nabla u_t$ can be interpreted as viscous relaxation effects.
On the other hand, when considering the influence of many factors, such as 
the molecular and ion effects,
the nonlinear term $\nabla(|\nabla u|^{p-2}\nabla u)$ appears to replace 
$\Delta u$ in pseudo-parabolic models.

Let us introduce the research on the asymptotic behavior of solutions that 
related to our work.
We mainly review the following three aspects.

(i) For the fast diffusive $p$-Laplacian equations, Jin et al \cite{Yin1} considered
the initial boundary value problem of the  equation
\begin{equation*}
 u_t-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=u^q,
\end{equation*}
with $0<p<2$ and $q>0$. They determined both the critical extinction exponent
$q_0=p-1$ and the critical blow-up exponent $q_c=1$.
Lately, Qu et al \cite{QBZ} and Li et al \cite{gao} extended the critical 
exponent results to the sign-changing solutions
for $p$-Laplacian equations with nonlocal source 
$|u|^{q}-\frac{1}{|\Omega|}\int_{\Omega}|u|^{q}dx$.

(ii) For the pseudo-parabolic equation
\begin{equation}
\label{pseudo}
u_t-\Delta u_t-\Delta u=u^q,
\end{equation}
Cao et al \cite{Cao-Cauchy} studied the Cauchy problem of \eqref{pseudo}
and obtained the complete Fujita type result with showing $q_c=1+\frac{2}{n}$.
For the initial boundary value problem of \eqref{pseudo},
via the potential well method, Xu et al \cite{XS} also confirmed the
Fujita exponent $q_c=\infty$ ($n=1,2$)
and $q_c=\frac{n+2}{n-2}$ ($n\geq3$) with bounded initial energy.
Lately, Chen et al. \cite{CHT} carried out the research on pseudo-parabolic equations with logarithmic source
\begin{equation}\label{q1}
 u_t-\Delta u_t-\Delta u=u\log|u|,
\end{equation}
and found the blowing-up at $\infty$ of the solutions, which with \cite{XS} reveal
that the polynomial nonlinearity is an important condition for the solutions 
to be blow-up in finite time.

(iii) Recently, Le et al \cite{Le} investigated \eqref{qq1} with $p>2$.
Owing to the slow diffusion, there exist both global existence and
blowing-up in finite time of the weak solutions, under the same 
conditions in \cite{CHT}.
Moreover, Le et al gave the large time decay of the global weak solutions.

In this article, we would like to reveal the effect from fast diffusive, 
pseudo-parabolic viscosity and logarithmic nonlinearity on the asymptotic 
behavior of solutions.
First, different from the case $p>2$, we prove that the weak solutions of 
\eqref{qq1} are global and can not blow up in finite time. This means that 
the fast diffusion is dominant, and the logarithmic source is not strong 
enough to cause blowing-up in finite time.
Next, similar to \cite{CHT}, we find the sufficient conditions to divide
the global boundedness and blowing-up at $\infty$ of the weak solutions 
(Theorems \ref{th1} and \ref{th2}).
Moreover, we derive some decay estimates of the global bounded solutions, 
namely Theorem \ref{th1'},
as while as some growth estimates of the unbounded solutions, namely
 Theorem \ref{th2'}.
From Theorem \ref{th1'}, the global bounded solutions
of the 1-D case decay exponentially,
which is the same as the case $p=2$, while different from the algebraical 
decay of the case $p>2$.
Theorem \ref{th1'} also tells us that {the upper bound of the decay rate
are proportional to $k$}, which seems that the pseudo-parabolic viscosity 
slows down the decay.
From Theorem \ref{th2'} and Theorem \ref{th3}, the weak solutions that 
blow up at $\infty$ grow algebraically.
Theorem \ref{th2'} also indicates that the lower bound of growth estimates 
is smaller than that of the case $p=2$,
which is caused by the fast diffusion.

Here we exploit the potential well method which was proposed by Sattinger 
et al \cite{DHS}. Liu et al \cite{YCL,YCL1} generalized and improved the method
by introducing a family of potential wells which include the known potential 
well as a special case.
Nowadays, it is one of the most useful method for proving global existence 
and nonexistence of solutions,
and vacuum isolating of solutions for parabolic equations \cite{CLL,RZX}.

This article is organized as follows.
In Section 2, we prove the global existence and uniqueness of the weak solution.
Section 3 gives some preliminary lemmas of the potential wells.
In Section 4, we treat the global bounded case and the decay estimates.
Section 5 is devoted to the blow-up at $\infty$ and the growth estimates.

\section{Global existence and uniqueness}

We start this section with the definition of the weak solutions. Set
\begin{equation*}
E=\left\{u \in C(0,T; H_0^1(\Omega));u_t \in L^2(0,T; H^{1}_0(\Omega))\right\}.
\end{equation*}

\begin{definition} \label{def1} \rm
A function $u(x,t)$ is said to be a weak solution of \eqref{qq1},
if $u \in E$, $u(x,0)=u_0(x) \in H^{1}_0(\Omega)$,
it holds
\begin{equation} \label{weaksol}
(u_t,\varphi)_2+(|\nabla u|^{p-2}\nabla u,\nabla \varphi)_2
+k(\nabla u_t,\nabla \varphi)_2=(|u|^{p-2}u\log|u|,\varphi)_2,
\end{equation}
for any $\varphi\in H^{1}_0(\Omega)$, and for a.e. $t \in(0,T)$,
where $(\cdot,\cdot)_2$ means the inner product of ${L^2(\Omega)}$.
\end{definition}

\begin{lemma}[Imbedding inequality] \label{Glemma}
For any function $u\in W_0^{1,q}(\Omega)$, we have the inequality
$$
\|u\|_p\leq C(p,q,n,\Omega)\|\nabla u\|_q,
$$
for all $1\leq p\leq q^*$, where $q^*=\frac{nq}{n-q}$ if $n>q$
and $q^*=\infty$ if $n{=}q$.
\end{lemma}

\begin{theorem}[Global existence and uniqueness] \label{th3}
Assume that $u_0(x)\in H_0^1(\Omega)$. Then for any $T>0$,
the problem \eqref{qq1} admits a unique weak solution.
\end{theorem}

\begin{proof}
 Here we use the Galerkin approximation method to prove the existence
of the global weak solutions for \eqref{qq1}.
\smallskip

\noindent\textbf{Step 1: Approximation problem.}
Let $\{w_j(x)\}$ be the orthogonal basis in $H^{1}_0(\Omega)$,
 which is also orthogonal in $L^2(\Omega)$.
We look for the approximate solutions of the following form
 \[
 u^{m}(x,t)=\sum_{j=1}^{m}g_j^m(t)w_j(x), \quad m=1,2,...,
 \]
where the coefficients $g_j^m(t)=(u^{m},w_j)_2$, satisfy the system of ODEs
\begin{equation} \label{G1}
\begin{gathered}
\begin{aligned}
&(u^m_t,w_j)_2+ (|\nabla u^{m}|^{p-2}\nabla u^{m},\nabla w_j)_2
+k(\nabla u^m_t,\nabla w_j)_2 \\
&=(|u^{m}|^{p-2}u^m\log|u^{m}|,w_j)_2,
\end{aligned}
\\
u_0^{m}(x)=\sum_{j=1}^{m}g_j^m(0)w_j(x) \to u_0,\quad \text{in }
 H_0^{1}(\Omega),
\end{gathered}
\end{equation}
for $j=1,2,\dots, m$.
The standard theory of ODEs, e.g. Peano's theorem, yields that 
$g_j^m(t)\in C^1[0,\infty)$.
Thus $u^m\in C^1([0,\infty);H_0^1(\Omega))$.
\smallskip

\noindent\textbf{Step 2: A priori estimates.}
We need some a priori estimates of the approximate solutions $u^m$.
Multiplying the first equality of \eqref{G1} by $g^{m}_j(t)$ and summing
for $j$, we have
\begin{equation}\label{G2}
\frac{1}{2}\frac{d}{dt}\|u^m\|_2^2
+\frac{k}{2}\frac{d}{dt}\|\nabla u^m\|_2^2
+\|\nabla u^m\|_p^p=\int_{\Omega}|u^m|^p \log|u^m|dx.
\end{equation}
Via a direct calculation and Lemma \ref{Glemma}, it holds
\begin{equation}\label{G3}
\begin{split}
\int_{\Omega}|u^m|^p \log|u^m|dx
\leq\frac{1}{e\alpha_0}\int_{\Omega}|u^m|^{p+\alpha_0}dx
\leq \frac{1}{e\alpha_0}\Big(\int_{\Omega}|\nabla u^m|^2dx
 \Big)^{\frac{p+\alpha_0}{2}},
\end{split}
\end{equation}
where $\alpha_0$ satisfies $1\leq p+\alpha_0<2$, e.g. we can choose 
$\alpha_0=\frac{2-p}{2}$.
Substituting \eqref{G3} into \eqref{G2}, we can deduce that
\begin{equation*}
\frac{d}{dt}\|u^m\|_2^2+k\frac{d}{dt}\|\nabla u^m\|_2^2dx
\leq\frac{2}{e\alpha_0 k^{(p+\alpha_0)/2}}\left(\|u^m\|_2^2
+k\|\nabla u^m\|_2^2\right)^{\frac{p+\alpha_0}{2}},
\end{equation*}
which implies 
\begin{equation}\label{G4}
\begin{aligned}
&\|u^m\|_2^2+k\|\nabla u^m\|_2^2 \\
&\leq\Big(\frac{2(1-\frac{p+\alpha_0}{2})t}{e\alpha_0 k^{(p+\alpha_0)/2}}
+\big(\|u_0^m\|_2^2+k\|\nabla u_0^m\|_2^2\big)^{1-\frac{p+\alpha_0}{2}}
\Big)^{\frac{1}{1-\frac{p+\alpha_0}{2}}}.
\end{aligned}
\end{equation}

Multiplying the first equality of \eqref{G1} by $\frac{d}{dt}g^{m}_j(t)$, summing for $j$,
and integrating with respect to time from $0$ to $t$, we obtain
\begin{equation}\label{G5}
\begin{split}
&\int_0^{t}\|u^m_{\tau}\|^2_2 d\tau
 + k\int_0^{t}\|\nabla u^m_{\tau}\|^2_2 d\tau
+\frac{1}{p}\|\nabla u^m\|_p^p+\frac{1}{p^2}\|u^m\|^p_p\\
&=\frac{1}{p}\|\nabla u^m_0\|_p^p
 -\frac{1}{p}\int_{\Omega}|u^m_0|^p \log|u^m_0| dx
 +\frac{1}{p^2}\|u^m_0\|^p_p
 +\frac{1}{p}\int_{\Omega}|u^m|^p \log|u^m| dx,
\end{split}
\end{equation}
On the one hand, the convergence of $u_0^m(x)$ gives 
\[
\frac{1}{p}\|\nabla u^m_0\|_p^p-\frac{1}{p}\int_{\Omega}|u^m_0|^p \log|u^m_0| dx
+\frac{1}{p^2}\|u^m_0\|^p_p
\leq C(u_0),
\]
for sufficiently large $m$,
with 
\[
C(u_0)=\frac{1}{p}\|\nabla u_0\|_p^p-\frac{1}{p}\int_{\Omega}|u_0|^p \log|u_0| dx
+\frac{1}{p^2}\|u_0\|^p_p+1.
\]
On the other hand, \eqref{G3} and \eqref{G4} tell us that
\[
\frac{1}{p}\int_{\Omega}|u^m|^p \log|u^m| dx\leq C(u_0,t)
\]
with
\[
C(u_0,t)=\frac{1}{pe\alpha_0 k^{(p+\alpha_0)/2}}
\Big(\frac{2(1-\frac{p+\alpha_0}{2})t}{e\alpha_0 k^{(p+\alpha_0)/2}}
+\left(\|u_0^m\|_2^2+k\|\nabla u_0^m\|_2^2\right)^{1-\frac{p+\alpha_0}{2}}
\Big)^{\frac{1}{\frac{2}{p+\alpha_0}-1}}.
\]
Substituting the above two inequalities into \eqref{G5}, we obtain
\begin{equation}\label{G6}
\int_0^{t}\|u^m_{\tau}\|^2_2 d\tau
+ k\int_0^{t}\|\nabla u^m_{\tau}\|^2_2 d\tau
+\frac{1}{p}\|\nabla u^m\|_p^p+\frac{1}{p^2}\|u^m\|^p_p
\leq C(u_0)+C(u_0,t).
\end{equation}
\smallskip

\noindent\textbf{Step 3: Passing to the limit.}
Therefore, from \eqref{G4} and \eqref{G6}, for any $T>0$,
there exist $u\in L^{\infty}(0,T;H_0^1(\Omega))$ and a subsequence of $u^m$,
which is still denoted by itself, such that when sending $m\to\infty$,
\begin{gather*}
 u^{m}\to u
 \quad\text{weak$\star$ in } L^{\infty}(0,T;H_0^{1}(\Omega))\text{ and a.e. in }
\Omega\times[0,T), \\
 u^m_t\to u_t
 \quad\text{weakly in } L^2(0,T;H^{1}_0(\Omega)), \\
 |\nabla u^{m}|^{p-2} \nabla u^{m}\to \chi
 \quad\text{weak$\star$ in } L^{\infty}(0,T;L^{\frac{p}{p-1}}(\Omega)).
\end{gather*}
Since the convergence of $u^m$ and $u^m_t$, it follows from Aubin-Lions 
compactness theorem that
\[
 u^{m}\to u  \quad\text{strongly in } C(0,T;L^2(\Omega)),
\]
which implies 
\[
 |u^{m}|^{p-2}u^m\log|u^{m}|\to |u|^{p-2}u\log|u|
 \quad\text{a.e. in } \Omega\times[0,T).
\]
For $j$ fixed, we can pass to the limit in \eqref{G1} to get
 \begin{equation*}
 (u_t,w_j)_2+(\chi,\nabla w_j)_2
 +k(\nabla u_t,\nabla w_j)_2=(|u|^{p-2}u\log|u|,w_j)_2.
 \end{equation*}
Then for any $\varphi\in H_0^1(\Omega)$, it holds
 \begin{equation}  \label{cy6}
 (u_t,\varphi)_2+(\chi,\nabla \varphi)_2
 +k(\nabla u_t,\nabla \varphi)_2=(|u|^{p-2}u\log|u|,\varphi)_2.
 \end{equation}

We only need to prove that $\chi=|\nabla u|^{p-2}\nabla u$ in the weak sense,
namely
\begin{equation}
\label{cy5}
(\chi,\nabla\varphi)_2=(|\nabla u|^{p-2}\nabla u,\nabla\varphi)_2,
\quad \forall\varphi\in H_0^1(\Omega).
\end{equation}
In fact, for any $v\in L^{\infty}(0,T;W_0^{1,p}(\Omega))$, $\psi\in H_0^1(\Omega)$,
$0\le\psi\le1$, we have
$$
\int_{\Omega}\psi\left(|\nabla u^m|^{p-2}\nabla u^m-|\nabla v|^{p-2}\nabla v\right)\nabla (u^m-v) dx\ge0,
$$
namely
\begin{align*}
&\int_{\Omega}\psi |\nabla u^m|^{p-2}|\nabla u^m|^2 dx
-\int_{\Omega}\psi|\nabla u^m|^{p-2}\nabla u^m \nabla v dx \\
&-\int_{\Omega}\psi|\nabla v|^{p-2}\nabla v\nabla (u^m-v) dx
\ge0.
\end{align*}
Letting $m\to \infty$ in the above equation and noticing that
\begin{align*}
&\int_{\Omega}\psi|\nabla u^m|^{p-2}|\nabla u^m|^2 dx
\\
&=-\int_{\Omega}\operatorname{div}(|\nabla u^m|^{p-2}\nabla u^m)u^m\psi dx
 -\int_{\Omega}|\nabla u^m|^{p-2}\nabla u^m u^m \nabla \psi dx\\
&=-\int_{\Omega}u^m_tu^m\psi dx
 -k\int_{\Omega}\nabla u^m_t\nabla u^m\psi dx
 -k\int_{\Omega}\nabla u^m_t u^m \nabla\psi dx \\
 &\quad +\int_{\Omega}|u^m|^p\log|u^m|\psi dx
 -\int_{\Omega}|\nabla u^m|^{p-2}\nabla u^mu^m\nabla\psi dx,
\end{align*}
we have
\begin{equation}\label{u-k}
\begin{split}
&-\int_{\Omega}u_tu\psi dx
-k\int_{\Omega}\nabla u_t\nabla u\psi dx
-k\int_{\Omega}\nabla u_t u \nabla\psi dx
+\int_{\Omega}|u|^p\log|u|\psi dx\\
&-\int_{\Omega}\chi u \nabla\psi dx
-\int_{\Omega}\psi\chi \nabla v dx
-\int_{\Omega}\psi|\nabla v|^{p-2}\nabla v\nabla(u-v) dx\ge0.
\end{split}
\end{equation}
Choosing $\varphi=u\psi$ in \eqref{cy6}, we obtain
\begin{equation} \label{chi-k-1}
\begin{aligned}
&\int_{\Omega}u_t u\psi dx+\int_{\Omega}\chi\nabla u\psi dx
+\int_{\Omega}\chi\nabla\psi u dx \\
&+k\int_{\Omega}\nabla u_t\nabla u\psi dx
+k\int_{\Omega}\nabla u_t u \nabla\psi dx\\
&=\int_{\Omega}|u|^p\log|u|\psi dx.
\end{aligned}
\end{equation}
Combining \eqref{chi-k-1} with \eqref{u-k}, we obtain
$$
\int_{\Omega}\psi\left(\chi-|\nabla v|^{p-2}\nabla v\right)\nabla(u-v) dx\ge0.
$$
Choosing $v=u-\lambda\varphi$, $\lambda\ge0$, $\varphi\in H_0^1(\Omega)$ in
the above inequality, we arrive at
$$
\int_{\Omega}\psi\left(\chi-|\nabla(u-\lambda\varphi)|^{p-2}
\nabla(u-\lambda\varphi)\right)\nabla\varphi {dx}\ge0.
$$
Taking $\lambda\to 0$, we have
$$
{\int_{\Omega}}\psi\left(\chi-|\nabla u|^{p-2}\nabla u\right)\nabla\varphi dx\ge0,
\quad\forall\varphi\in H_0^1(\Omega).
$$
Obviously, if we choose $\lambda\le 0$, we can deduce the similar inequality
replacing
``$\ge$" by ``$\le$". Hence, \eqref{cy5} holds.
On the other hand, from \eqref{G1} we obtain $u(x,0)=u_0(x)$ in $H_0^{1}(\Omega)$.
Thus $u$ is a global weak solution of \eqref{qq1}.
\smallskip

\noindent\textbf{Step 4: Uniqueness.}
Suppose \eqref{qq1} admits two weak solutions $u_1$ and $u_2$.
Set $w=u_1-u_2$, then $w$ satisfies
\begin{equation}\label{02.1} 
 \begin{gathered}
 w_t-\operatorname{div}((p-1)|\nabla \overline{w}|^{p-2}\nabla w)-k \Delta w_t
 =((p-1)\log|\tilde{w}|+1)|\tilde{w}|^{p-2}w, \quad\Omega \times(0,T),\\
 w(x,t)=0, \quad \partial \Omega \times (0,T),\\
 w(x,0)=0, \quad \Omega\,,\\
 \end{gathered}
\end{equation}
where $\overline{w}=\theta_1u_1+(1-\theta_1)u_2$,
$\tilde{w}=\theta_2u_1+(1-\theta_2)u_2$ with $\theta_1, \theta_2 \in [0,1]$.

Multiplying \eqref{02.1} by $w$ and integrating on $\Omega$, we have
\begin{align*}
&\frac{1}{2}\frac{d}{dt}\int_{\Omega}w^2dx
+\int_{\Omega}(p-1)|\nabla\overline{w}|^{p-2}|\nabla w|^2dx
+\frac{k}{2}\frac{d}{dt}\int_{\Omega}|\nabla w|^2dx\\
&=\int_{\Omega}((p-1)\log|\tilde{w}|+1)|\tilde{w}|^{p-2}w^2dx.
\end{align*}
For any $t\in(0,T)$, integrating both side of the above equation on $(0,t)$ 
and noticing that $w(x,0)=0$, we can get
\[
\frac{1}{2}\int_{\Omega}w^2dx+\frac{k}{2}\int_{\Omega}|\nabla w|^2dx
\leq \int_0^{t}\int_{\Omega}((p-1)\log|\tilde{w}|+1)
|\tilde{w}|^{p-2}w^2dx{d\tau}.
\]
In fact, since when $1<p<2$, it holds
$$
\lim_{f\to+\infty}((p-1)\log f+1)f^{p-2}=0,\quad
\lim_{f\to0^+}((p-1)\log f+1)f^{p-2}<0;
$$
thus
$((p-1)\log f+1)f^{p-2}\leq C$ with 
$f=e^{\frac{2p-3}{(2-p)(p-1)}}$ as the maximum point,
and
$((p-1)\log f+1)f^{p-2}<0$ with  $0<f<e^{-\frac{1}{p-1}}$.
Thus we can find a positive constant $C$ independent of $u_1$ and $u_2$,
such that
\[
\frac{1}{2}\int_{\Omega}w^2dx+\frac{k}{2}\int_{\Omega}|\nabla w|^2dx
\leq C \int_0^{t}\int_{\Omega}w^2dx{d\tau}.
\]
It follows from Gronwall's inequality that
\[
\int_{\Omega}w^2dx=0, \quad\text{a.e. }(0,t).
\]
Thus $w=0$ a.e in $\Omega\times(0,T)$.
\end{proof}

\section{Potential wells}

We define the following two functionals on $H_0^{1}(\Omega)$:
\begin{equation} \label{1.1}
 \begin{gathered}
 J(u)=\frac{1}{p}\|\nabla u\|_p^p
 -\frac{1}{p}\int_{\Omega}|u|^p \log|u| dx
 +\frac{1}{p^2}\|u\|^p_p, \\
 I(u)=\|\nabla u\|_p^p
 -\int_{\Omega}|u|^p \log|u| dx.
 \end{gathered}
\end{equation}
It is obvious that
\begin{equation} \label{1.3}
 J(u)=\frac{1}{p}I(u) + \frac{1}{p^2}\|u\|^p_p.
\end{equation}

\begin{remark} \rm
Since $u\in E$ and $1<p<2$, we can use the H\"{o}lder inequality and 
Lemma \ref{Glemma} to derive that
\begin{gather*}
\|u\|_p+\|\nabla u\|_p \leq C(p,\Omega)(\|u\|_2+\|\nabla u\|_2),\\
\int_{\Omega}|u|^p \log|u|dx
\leq \frac{1}{e\alpha}\|\nabla u\|_2^{p+\alpha}, 
\end{gather*}
where $\alpha$ satisfies $1\leq p+\alpha<2^*$,
which imply that $J(u)$ and $I(u)$ are well-defined in $H_0^{1}(\Omega)$ and
$W_0^{1,p}(\Omega)$.
Further, similar to the Step 4 of Theorem \ref{th3}, one can prove that
$$
u\mapsto \int_{\Omega}|u|^p \log|u|dx
$$
is continuous from $H_0^1(\Omega)$ to $\mathbb{R}$.
It follows that $J(u)$ and $I(u)$ are continuous.
\end{remark}

Let
\begin{equation} \label{1.4}
 d=\inf\{\sup_{\lambda\geq0}J(\lambda u)|u \in H_0^{1}(\Omega),
\|\nabla u\|_p^p\neq 0\},
\end{equation}
and
$$
N=\{u \in H_0^1(\Omega) | I(u)=0,\|\nabla u\|_p^p\neq 0\}.
$$
Then Lemma \ref{lem2.1} and Lemma \ref{lem2.3} below tell us that
$$
d=\inf_{u\in N}J(u)\geq M
=\frac{1}{p^2}(\frac{p^2 e}{n\mathcal{L}_p})^{n/p},
$$
where $\mathcal{L}_p$ can be found in \eqref{cy0}.
Thus we can define
\begin{gather*}
 W=\{u\in H_0^1(\Omega)|I(u)>0, J(u)<d\} \cup \{0\}, \\
 V=\{u\in H_0^1(\Omega)|I(u)<0,\,J(u)<d\}.
\end{gather*}
For $\delta >0$,\, we introduce
\begin{gather}\label{2.1}
 I_{\delta}(u)=\delta\|\nabla u\|_p^p
 -\int_{\Omega}|u|^p \log|u| dx, \\
\label{2.4}
 N_{\delta}=\{u \in H_0^1(\Omega)|I_{\delta}(u)=0,\|\nabla u\|_p^p\neq 0\},\\
\label{delta}
 d(\delta)=\inf_{u \in N_{\delta}}J(u), \\
\label{2.9}
 W_{\delta}=\{u\in H_0^1(\Omega)|I_{\delta}(u)>0, J(u)<d(\delta)\} \cup \{0\}, \\
 V_{\delta}=\{u\in H_0^1(\Omega)|I_{\delta}(u)<0, J(u)<d(\delta)\}.
\end{gather}

To handle the logarithmic nonlinearity $|u|^{p-2}u\log|u|$, we need
the following $L^p$ logarithmic Sobolev inequality

\begin{lemma}[\cite{MDG,MD}]
For any $u \in W^{1,p}(\mathbb{R}^{n})$ with $p\in(1,+\infty)$, $u\neq 0$, 
and any $\mu >0$, 
$$
p\int_{\mathbb{R}^{n}}|u|^p\log(\frac{|u|}{\|u\|_p})dx+
\frac{n}{p}\log(\frac{p\mu e}{n \mathcal{L}_p})\int_{\mathbb{R}^{n}}|u|^pdx
\leq \mu \int_{\mathbb{R}^{n}}|\nabla u|^pdx,
$$
where
\begin{equation} \label{cy0}
\mathcal{L}_p=\frac{p}{n}(\frac{p-1}{e})^{p-1}
\pi^{-\frac{p}{2}}\Big[\frac{\Gamma
(\frac{n}{2}+1)}{\Gamma(n\frac{p-1}{p}+1)}\Big]^{p/n}.
\end{equation}
\end{lemma}

For $u\in W^{1,p}(\Omega)$, we can define $u=0$ for
$x\in\mathbb{R}^n\setminus\Omega$, such that 
$u\in W^{1,p}(\mathbb{R}^n)$. Thus it holds the $L^p$ logarithmic
Sobolev inequality for bounded domain $\Omega$
\begin{equation}\label{ineq1}
p\int_{\Omega}|u|^p\log(\frac{|u|}{\|u\|_p})dx+
\frac{n}{p}\log(\frac{p\mu e}{n
\mathcal{L}_p})\int_{\Omega}|u|^pdx \leq\mu \int_{\Omega}|\nabla
u|^pdx.
\end{equation}

Lemmas \ref{lem2.1}, \ref{lem2.2}, \ref{lem2.3} and \ref{lem2.4}
are similar to \cite[Lemmas 2.1, 2.2, 2.3 and 2.4]{CHT},
so we omit most of their proofs.

\begin{lemma}\label{lem2.1}
Assume $\lambda>0$, $u\in H_0^1(\Omega)$ and
$\|u\|_p\neq 0$, then we have
\begin{itemize}
\item[(1)] $J(\lambda u)$ strictly increases on $0<\lambda\leq\lambda^{*}$,
strictly decreases on $\lambda^{*}\leq\lambda<\infty$ and takes the maximum
at $\lambda=\lambda^{*}$. Further $\lim_{\lambda\to
0}J(\lambda u)=0$, and $\lim_{\lambda\to
+\infty}J(\lambda u)=-\infty$;

\item[(2)] $I(\lambda u)>0$ on $0<\lambda<\lambda^{*}$, $I(\lambda^* u)=0$ and
$I(\lambda u)<0$ on $\lambda^{*}<\lambda<\infty$,
where
$$
\lambda^{*}=\exp\{\frac{\|\nabla u\|_p^p-\int_{\Omega}|u|^p
\log|u|dx}{\|u\|_p^p}\}.
$$
\end{itemize}
\end{lemma}

\begin{lemma}\label{lem2.2}
Let $u \in W_0^{1,p}(\Omega)$ and $\|u\|_p\neq 0$. Then we have
\begin{itemize}
\item[(1)] if $ 0< \|\nabla u\|_p\leq r(\delta)$, then $I_{\delta}(u)\geq 0$;

\item[(2)] if $I_{\delta}(u)<0$, then $\|\nabla u\|_p>r(\delta)$;

\item[(3)] if $I_{\delta}(u)=0$, then $\|\nabla u\|_p \geq r(\delta)$,
\end{itemize}
where $r(\delta)={\lambda_1^{1/p}}(\frac{p^2\delta
e}{n\mathcal{L}_p})^{\frac{n}{p^2}}$, and $\lambda_1$ is
the first eigenvalue of the problem
\begin{gather*}
 -\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\lambda |u|^{p-2}u, \quad x\in\Omega, \\
 u=0, \quad x\in\partial \Omega.
\end{gather*}
\end{lemma}

\begin{proof} (1) Using the $L^p$ Sobolev logarithmic inequality
\eqref{ineq1}, for any $\mu>0$, we have
\begin{equation}\label{2.2}
I_{\delta}(u)\geq (\delta-\frac{\mu}{p})\|\nabla u\|^p_p
+(\frac{n}{p^2} \log(\frac{p\mu e}{n\mathcal{L}_p})-\log\|u\|_p)\|u\|^p_p.
\end{equation}
Taking $\mu=p\delta$ in \eqref{2.2}, we obtain that
\begin{equation}\label{2.3}
I_{\delta}(u)\geq (\frac{n}{p^2}\log(\frac{p^2\delta e}{n\mathcal{L}_p})
-\log\|u\|_p)\| u\|^p_p.
\end{equation}
By the Poincar\'e inequality, if $0<\|\nabla u\|_p\leq r(\delta)$,
then $0<\|u\|_p\leq \lambda_1^{-\frac1p}\|\nabla u\|_p
\leq(\frac{p^2\delta e}{n\mathcal{L}_p})^{\frac{n}{p^2}}$.
Thus $I_{\delta}(u)\geq 0$.

The proof for (2) and (3) is similar to that of \cite[Lemma 2.2 ]{CHT}, 
so we omit it here.
\end{proof}

\begin{lemma}\label{lem2.3}
For $d(\delta)$ in \eqref{delta}, we have
\begin{itemize}
\item[(1)] $d(\delta)\geq \frac{1}{p}(1-\delta)
r^p(\delta)+\frac{1}{p^2}(\frac{p^2\delta e}{n\mathcal{L}_p})^{n/p}$.
In particular,
$d(1)\geq\frac{1}{p^2}(\frac{p^2e}{n\mathcal{L}_p})^{n/p}=:
M$;

\item[(2)] there exists a unique $b$, $b \in
(1,1+\frac{1}{p{\lambda_1}}]$ such that $d(b)=0$, and
$d(\delta)>0$ for $1\leq \delta < b$;

\item[(3)] $d(\delta)$ is strictly increasing on $0< \delta\leq 1$,
decreasing on $1\leq \delta\leq b$, and takes the maximum $d=d(1)$
at $\delta=1$.
\end{itemize}
\end{lemma}

Now, we can define
 \begin{equation}
 d_0=\lim_{\delta\to 0^{+}} d(\delta),
 \end{equation}
where $d_0 \geq 0$ from Lemma \ref{lem2.3}.

\begin{lemma}\label{lem2.4}
Let $d_0<J(u)<d $ for some $u \in H^{1}_0(\Omega)$,
and $\delta_1<1<\delta_2$ are the two roots of the equation $d(\delta)=J(u)$.
Then the sign of $I_{\delta}(u)$ is unchangeable for 
$\delta_1<\delta<\delta_2$.
\end{lemma}

In what follows, we prove that when $0<J(u_0)<d$, $W_\delta$ and $V_\delta$ 
are the invariant sets of \eqref{qq1}.
The discussion is divided into two parts: $J(u_0)$ being in the monotonous 
interval of $d(\delta)$,
and $J(u_0)$ being in the non-monotonous interval of $d(\delta)$.

\begin{proposition}\label{prop2.1}
Assume $u_0\in H_0^{1}(\Omega)$, $u$ is a weak solution of
{\rm\eqref{qq1}} with $J(u_0)=\sigma$. Then we have the following results.
\begin{itemize}
\item[(1)] If $0<\sigma\leq d_0$, then there exists a unique $\bar{\delta}\in(1,b)$ 
such that
$d(\bar{\delta})=\sigma$, where $b$ is the constant in Lemma {\rm\ref{lem2.3}} (2).
Furthermore, if $I(u_0)>0$, then $u\in W_{\delta}$ for any
$1\leq\delta<\bar{\delta}$;
else if $I(u_0)<0$, then $u\in V_{\delta}$ for any $1\leq\delta<\bar{\delta}$.

\item[(2)] If $d_0<\sigma<d$, then there exists $\delta_1$ and $\delta_2$ such that
$\delta_1<1<\delta_2$ and $d(\delta_1)=d(\delta_2)=\sigma$. Furthermore,
if $I(u_0)>0$, then $u\in W_{\delta}$ for any $\delta_1<\delta<\delta_2$;
else if $I(u_0)<0$, then $u\in V_{\delta}$ for any $\delta_1<\delta<\delta_2$.
\end{itemize}
\end{proposition}

\begin{proof}
 Case 1. $0<J(u_0)=\sigma\leq d_0$, namely $J(u_0)$ is in the monotonous interval 
of $d(\delta)$.
From Lemma \ref{lem2.3}, there exists a unique $\bar{\delta}\in(1,b)$ such that 
$d(\bar{\delta})=\sigma$.
For any $\delta\in[1,\bar{\delta})$, we have
\begin{equation}
\label{cy1}
I_{\delta}(u_0)=(\delta-1)\|{\nabla} u_0\|_p^p+I(u_0)\geq I(u_0),
\quad
J(u_0)=\sigma=d(\bar{\delta})<d(\delta).
\end{equation}
Multiplying both sides of \eqref{qq1} by $u_t$ and integrating on 
$\Omega\times[0,t]$, it holds
\begin{equation} \label{cy2}
\int_0^t(\|u_\tau\|_2^2+k\|\nabla u_\tau\|_2^2)d\tau+J(u)
=J(u_0)=d(\bar{\delta})<d(\delta),
\end{equation}
for all $t\in(0,T)$ and all $\delta\in[1,\bar{\delta})$,
where $T$ is the maximal existence time.

If $I(u_0)>0$, then \eqref{cy1} means that $u_0\in W_{\delta}$ for 
$\delta\in[1,\bar{\delta})$.
We assert that $u\in W_{\delta}$ for $t\in(0,T)$ and $\delta\in[1,\bar{\delta})$.
If it is false, then there exists $\delta^*\in[1,\bar{\delta})$ and $t_0\in(0,T)$,
such that $u\in W_{\delta^*}$ for $t\in(0,t_0)$, but
 $u(x,t_0)\in\partial W_{\delta^*}$, namely
$$
I_{\delta^{*}}(u(t_0))=0,\quad \|\nabla u(t_0)\|_p^p\neq0,
\quad\text{or}\quad J(u(t_0))=d(\delta^*).
$$
In fact, \eqref{cy2} shows that $J(u(t_0))\leq J(u_0)<d(\delta^*)$,
which implies $I_{\delta^{*}}(u(t_0))=0$ and $\|\nabla u(t_0)\|_p^p\neq0$, 
namely $u(x,t_0)\in N_{\delta^*}$.
Thus from the definition of $d(\delta^*)$, we have $J(u(t_0))\geq d(\delta^*)$,
 which is a contradiction.

Next, we prove that if $I(u_0)<0$, then $u_0\in V_{\delta}$ for
 $\delta\in[1,\bar{\delta})$,
and $u\in V_{\delta}$ for $t\in(0,T)$ and $\delta\in[1,\bar{\delta})$.
If the assertion of $u_0$ is false, then \eqref{cy1} shows that there exists 
$\delta_*\in[1,\bar{\delta})$
being the first number such that $u_0\in V_{\delta}$ for 
$\delta\in[1,\delta_*)$ and $u_0\in\partial V_{\delta_*}$,
namely
$$
I_{\delta_*}(u_0)=0,
\quad\text{or}\quad J(u_0)=d(\delta_*).
$$
Since $J(u_0)$ is in the strictly decreasing interval of $d(\delta)$, then
$J(u_0)=d(\bar{\delta})<d(\delta_*)$, which indicates that $I_{\delta_*}(u_0)=0$.
Since $I_{\delta}(u_0)<0$ for $\delta\in[1,\delta_*)$,
then Lemma \ref{lem2.2} (2) gives $\|\nabla u_0\|_p>r(\delta)>0$,
which indicates that $u_0\in N_{\delta_*}$.
By the definition of $d(\delta_*)$, we have 
$J(u_0)=d(\bar{\delta})\geq d(\delta_{*})$,
which is contradict with the monotonicity of $d(\delta)$.
If the assertion of $u$ is false, then there exists 
$\delta^*_*\in[1,\bar{\delta})$ and $t_0\in(0,T)$,
such that $u\in V_{\delta_*^*}$ for $t\in(0,t_0)$, but 
$u(x,t_0)\in\partial V_{\delta_*^*}$, namely
$$
I_{\delta{_*^*}}(u(t_0))=0,\quad\text{or}\quad J(u(t_0))=d(\delta_*^*).
$$
In fact, \eqref{cy2} shows that $J(u(t_0))\leq J(u_0)<d(\delta_*^*)$,
which implies $I_{\delta{_*^*}}(u(t_0))=0$.
If $I_{\delta{_*^*}}(u(t_0))=0$, then from Lemma \ref{lem2.2} (3),
$\|\nabla u(t_0)\|_p\geq r(\delta)$, namely $u(x,t_0)\in N_{\delta_*^*}$.
Thus from the definition of $d(\delta_*^*)$, we have $J(u(t_0))\geq d(\delta_*^*)$, 
which is a contradiction.

Case 2. $d_0<J(u_0)=\sigma<d$, namely $J(u_0)$ is in the non-monotonous interval 
of $d(\delta)$. From Lemma \ref{lem2.3}, there exist $\delta_1<1<\delta_2$ being 
two roots of $d(\delta)=\sigma$,
and $d_0<J(u_0)=d(\delta_1)=d(\delta_2)<d(\delta)$ for 
$\delta\in(\delta_1, \delta_2)$.

If $I(u_0)>0$, then from Lemma \ref{lem2.4}, the sign of $I_{\delta}(u)$ 
is unchangeable for $\delta_1<\delta<\delta_2$.
Thus we have $I_{\delta}(u_0)>0$ for $\delta\in(\delta_1, \delta_2)$.
Therefore, $u_0\in W_{\delta}$ for $\delta\in(\delta_1,\delta_2)$.
The proof of $u\in W_{\delta}$ is similar to that in Case 1.

If $I(u_0)<0$, also from Lemma \ref{lem2.4}, we have $I_{\delta}(u_0)<0$ 
for $\delta\in(\delta_1, \delta_2)$,
which with $J(u_0)<d(\delta)$ for $\delta\in(\delta_1, \delta_2)$,
imply that $u_0\in V_{\delta}$ for $\delta\in(\delta_1,\delta_2)$.
The proof of $u\in V_{\delta}$ is similar to that in Case 1.
\end{proof}

\begin{proposition}\label{prop2.2}
Assume $u_0 \in H^{1}_0(\Omega)$ with $u_0\not\equiv0$, $J(u_0)=d$, $u$
is a weak solution of \eqref{qq1}.
If $I(u_0)>0$, then $I(u(t))\geq0$ for all $0<t<T$;
if $I(u_0)<0$, then $I(u(t))<0$ for all $0\leq t< T$,
where $T$ is the maximal existence time of $u$.
\end{proposition}

\begin{proof} We prove the proposition by contradiction.
When $I(u_0)>0$, if there exists $t_1\in(0,T)$ such that $I(u(t_1))<0$,
then we can find $t_0\in(0,t_1)$ being the first point satisfying $I(u)=0$,
namely
$$
I(u(t_0))=0,\quad \text{and}\quad I(u(t))>0\quad\text{for all }  0<t<t_0.
$$
Thus $\int_0^t(\|u_{\tau}\|_2^2+k\|\nabla u_{\tau}\|_2^2)d\tau>0$ for $0<t<t_0$.
Otherwise $u_t=0$ and $\nabla u_t=0$ a.e. $(x,t)\in\Omega\times(0,t_0)$,
which are contradict with the fact 
$I(u)=-\int_{\Omega}u_t udx-k\int_{\Omega}\nabla u_t\cdot\nabla udx>0$
for $0<t<t_0$. Thus
\begin{equation}
\label{cy3}
J(u(t))=J(u_0)-\int_0^t(\|u_{\tau}\|_2^2+k\|\nabla u_{\tau}\|_2^2)d\tau<d,
\quad\text{for all } 0<t\leq t_0.
\end{equation}
Also $I(u(t_0))=0$ imply that $u(x,t_0)=0$ or $\|\nabla u(t_0)\|_p^p\geq r(1)\neq0$.
If $u(x,t_0)=0$, then from the uniqueness of solutions, $u(x,t)=0$ for $t>t_0$,
which is a contradiction, since $I(u(t_1))<0$.
If $\|\nabla u(t_0)\|_p^p\neq0$, then by the definition of $d(\delta)$,
we have $J(u(t_0))\geq d$, which is contradict with \eqref{cy3}.

When $I(u_0)<0$, if there exists $t_1\in(0,T)$ such that $I(u(t_1))=0$,
and $I(u(t))<0$ for all $0<t<t_1$. Similar to the proof of \eqref{cy3}, we have
\begin{equation} \label{cy4}
J(u(t))=J(u_0)-\int_0^t(\|u_{\tau}\|_2^2+k\|\nabla u_{\tau}\|_2^2)d\tau<d,
\quad\text{for all } 0<t\leq t_1.
\end{equation}
Also from Lemma \ref{lem2.2} and $I(u(t))<0$ for all $0\leq t<t_1$,
then $\|\nabla u(t_0)\|_p^p\geq r(1)\neq0$. By the definition of $d(\delta)$,
we have $J(u(t_0))\geq d$, which is contradict with \eqref{cy4}.
\end{proof}


\section{Global boundedness and decay estimation}

In this section, we treat the globally bounded case, especially including the 
decay estimates.
First we need to point out that if $u$ is a solution of \eqref{qq1} with 
$J(u_0)\leq d$, $I(u_0)\geq0$, and there exists $t_2>0$ such that
$\|\nabla u(t_2)\|_p=0$, then from the uniqueness of the solution, $u=0$ 
for all $t\geq t_2$.
So in what follows, we do not consider this type of solutions.

\begin{theorem} \label{th1}
When $J(u_0)\leq d$ and $I(u_0)\geq 0$,
the weak solution of \eqref{qq1} is globally bounded.
\end{theorem}
\smallskip

\noindent\textbf{Step 1: $J(u_0)<d$.}
Actually, we only need to focus on the case $0<J(u_0)<d$ \& $I(u_0)>0$,
irrespectively of other cases.
The reasons are that the case $J(u_0)<0$ \& $I(u_0)\geq0$ is contradict
with \eqref{1.3};
the case $0<J(u_0)<d$ \& $I(u_0)=0$ is contradict with the definition of $d$;
if $J(u_0)=0$ and $I(u_0)\geq0$, then $u_0\equiv0$, which is a trivial case.

Multiplying the first equation of \eqref{qq1} by $u_t$ and integrating with 
respect to time from $0$ to $t$, we obtain
\begin{equation} \label{3.3}
\int_0^{t} \|u_{\tau}\|^2_2 d\tau + k\int_0^{t}\|\nabla u_{\tau}\|^2_2 d\tau
+ J(u(t))=J(u(0))<d,\quad\text{for }t>0.
\end{equation}
We assert that $u(x,t)\in W$ for any $t>0$.
If it is false, then there exists $t_0>0$ such that $u(x,t_0) \in \partial W$,
then
 \[
 I(u(t_0))=0, \|\nabla u(t_0)\|_p\neq 0,\quad\text{or}\quad  J(u(t_0))=d.
 \]
On the one hand, \eqref{3.3} indicates that $J(u(t_0))=d$ is not true.
On the other hand, if $I(u(t_0))=0, \|\nabla u(t_0)\|_p\neq 0$,
then by the definition of $d$, we have $J(u(t_0))\geq d$,
which is also contradict with \eqref{3.3}.
Thus we have $u(x,t)\in W$, which with \eqref{1.3} deduce that
 \begin{equation} \label{3.4}
 \|u\|^p_p < p^2 d.
 \end{equation}
Taking $\mu=\frac{p}{2}$ in \eqref{ineq1}, we have
\begin{equation}\label{3.5}
\begin{split}
\|\nabla u\|^p_p
&=I(u)+ \int_{\Omega}|u|^p \log|u| dx \\
&= 2I(u)+ 2\int_{\Omega}|u|^p \log|u| dx -\|\nabla u\|^p_p \\
&\leq 2I(u) + 2\|u\|^p_p \log\|u\|_p
-\frac{2n}{p^2} \log(\frac{p^2 e}{2n\mathcal{L}_p})\|u\|^p_p \\
&=2pJ(u)+(2\log\|u\|_p-\frac{2}{p}-\frac{2n}{p^2}
 \log(\frac{p^2 e}{2n\mathcal{L}_p}))\|{u}\|^p_p \\
&\leq Cd.
\end{split}
\end{equation}
Also, \eqref{3.3} implies
\begin{equation} \label{3.6}
 \int_0^{t} \|u_{\tau}\|^2_2 d\tau
+ k\int_0^{t} \|\nabla u_{\tau}\|^2_2 d\tau  <d.
\end{equation}
From \eqref{3.4}, \eqref{3.5} and \eqref{3.6}, we have 
\begin{equation}\label{3.7}
\int_0^{t}\|u_{\tau}\|^2_2 d\tau + k\int_0^{t}\|\nabla u_{\tau}\|^2_2 d\tau
+\frac{1}{p}\|\nabla u\|_p^p+\frac{1}{p^2}\|u\|^p_p
\leq \big(2+\frac{C}{p}\big)d.
\end{equation}

Multiplying the first equation of \eqref{qq1} by $u$, we have
\begin{equation}
\label{cy7}
\frac{1}{2}\frac{d}{dt}\int_{\Omega}|u|^2dx
+\frac{k}{2}\frac{d}{dt}\int_{\Omega}|\nabla u|^2dx+I(u)
=0
\end{equation}
Combining \eqref{cy7} and the fact that $u(x,t)\in W$ for any $t>0$,
we find that
$$
\frac{1}{2}\frac{d}{dt}\int_{\Omega}|u|^2dx
+\frac{k}{2}\frac{d}{dt}\int_{\Omega}|\nabla u|^2dx<0,
$$
which means that
\begin{equation}
\label{cy8}
\|u\|_2^2+\|\nabla u\|_2^2\leq C(\|u_0\|_2^2+\|\nabla u_0\|_2^2).
\end{equation}
Thus \eqref{3.7} and \eqref{cy8} show that $u$ is globally bounded in $E$.
\smallskip

\noindent\textbf{Step 2: $J(u_0)=d$.}
Let $\mu_{m}=1-\frac{1}{m}$ and $u_{m0}=\mu_{m}u_0$ for $m\geq2$.
We consider the following problem:
\begin{equation}\label{qq2} 
 \begin{gathered}
 u_t-\operatorname{div}(|\nabla u|^{p-2}\nabla u)-k \triangle u_t=|u|^{p-2}u \log|u|, 
\quad \Omega \times(0,T), \\
 u(x,t)=0, \quad \partial \Omega \times (0,T),  \\
 u(x,0)=u_{m0}(x), \quad \Omega.
 \end{gathered}
\end{equation}

We assert $J(u_{m0})<d$ and $I(u_{m0})>0$.
If $\|u_0\|_p=0$, then from \eqref{1.3} and $J(u_0)=d$,
we have $I(u_0)=pJ(u_0)=pd$.
Thus $I(u_{m0})=\mu{_m^p}I(u_0)=\mu{_m^p}pd>0$,
$J(u_{m0})=\mu{_m^p}J(u_0)=\mu{_m^p}d<d$.
If $\|u_0\|_p\neq0$, then from $I(u_0)\geq 0$ and Lemma \ref{lem2.1},
we have $\lambda^{*} \geq 1$. We can also deduce that
$I(u_{m0})=I(\mu_{m} u_0)>0$, and
$J(u_{m0})=J(\mu_{m}u_0)<J(u_0)=d$.

Using the similar arguments as in Theorem \ref{th3} and Step 1,
\eqref{qq2} admits a unique global bounded weak solution $u_m\in E$.
Since the initial data $u_{m0}(x)\to u_0$ strongly in $H_0^1(\Omega)$,
then via a standard procedure, $u_m\to u$ strongly in $E$. Thus $u$ is 
globally bounded in $E$.


\begin{theorem} \label{th1'}
Let $u=u(x,t)$ be the global bounded weak solution in Theorem \ref{th1}.

(1) If $J(u_0)<M$ and $I(u_0)\geq 0$, then we have
\begin{equation}\label{R3}
\lim_{t\to\infty}(\|u\|^p_p + k\|\nabla u\|^p_p)=0.
\end{equation}
Furthermore, when $n=1$, there exists time $t_\beta>0$ such that
$$
\|u(t)\|^2_2+k\|\nabla u(t)\|^2_2
\leq (\|u(t_\beta)\|_2^2+k\|\nabla u(t_\beta)\|^2_2)e^{\frac{1}{2}-C\alpha_1t},
\quad \text{for all} \quad t\geq t_\beta,
$$
where
$$
\alpha_1=\min\{\frac{1}{k}(1-\frac{\mu}{p}),
\frac{n}{p^2}\log(\frac{p\mu e}{n \mathcal{L}_p})
-\frac{1}{p}\log(p^2J(u_0))\}>0,
$$
for any $\mu\in([p^2J(u_0)]^{p/n} \frac{n \mathcal{L}_p}{pe}, p)$
and $\mathcal{L}_p$ is \eqref{cy0}.

(2) If $J(u_0)=M$ and $I(u_0)> 0$, then
$$
\lim_{t\to\infty}(\|u\|^p_p + k\|\nabla u\|^p_p)=0.
$$
Furthermore, when $n=1$, there exists time $t_{\gamma}>0$, such that
\[
\|u(t)\|^2_2+k\|\nabla u(t)\|^2_2
\leq (\|u(t_\gamma)\|_2^2+k\|\nabla u(t_\gamma)\|^2_2)e^{\frac{1}{2}-C\alpha_2t},
\quad \text{for all }  t\geq t_\gamma,
\]
where
$$
\alpha_2=\min\{\frac{1}{k}(1-\frac{\mu}{p}),
\frac{n}{p^2}\log(\frac{p\mu e}{n \mathcal{L}_p})
-\frac{1}{p}\log(p^2(M-\gamma))\}>0,
$$
for any $\mu\in([p^2(M-\gamma)]^{p/n} \frac{n \mathcal{L}_p}{pe}, p)$
and $\mathcal{L}_p$ is \eqref{cy0}.
\end{theorem}

\begin{remark} \rm
When $p>2$, under  similar conditions as in Theorem \ref{th1'},
the global {bounded} solutions decay algebraically \cite{Le}.
However, if $p<2$, Theorem \ref{th1'} shows that the {global} bounded solutions decay
exponentially, which is the same as the results in \cite{CHT} for $p=2$.
Further, Theorem \ref{th1'} tells us that {the upper bound of the decay rate
 $e^{-\alpha_1t}$ and $e^{-\alpha_2t}$
are proportional to $k$}, which seems that the pseudo-parabolic viscosity
 slows down the decay.
\end{remark}

To prove the theorem, we need to introduce the following two lemmas.

\begin{lemma}[{\cite[Lemma 3.1]{CHT}}] \label{LemmaR2}
Let $y(t):\mathbb{R}^+\to \mathbb{R}^+$ be a nonincreasing function.
Assume that there is a constant $A>0$ such that
$$
\int_t^{+\infty}y(s)ds\leq Ay(t), \quad 0\leq t<+\infty.
$$
Then $y(t)\leq y(0)e^{1-\frac{t}{A}}$, for all $t>0$.
\end{lemma}

\begin{lemma}[{\cite[Prop. 6.2.3]{Wang}}] \label{LemmaR1}
Assume that $a$ is a positive constant, $g(t), h(t)\in C^1([a,\infty))$,
$h(t)\geq0$ and $g(t)$ is bounded blow. If there exists a positive $b$ and 
$C$, such that
$$
g'(t)\leq-bh(t),\quad h'(t)\leq C,\quad t\in[a,\infty),
$$
then $\lim_{t\to\infty}h(t)=0$.
\end{lemma}

\begin{proof} \textbf{Case 1. Decay estimates for $J(u_0)<M$.}
Let $u=u(x,t)$ be the global bounded solution of \eqref{qq1}
with $J(u_0)<M\leq d$ and $I(u_0)\geq0$. As in the proof for Theorem \ref{th1},
we only need to discuss the case $0<J(u_0)<M$ and $I(u_0)>0$.
Proposition \ref{prop2.1} reveals that $u\in W_{\delta}$ for $1\leq \delta <\bar{\delta}$
or $\delta_1<\delta<\delta_2$ with $\delta_1<1<\delta_2$ and particularly $I(u)>0$.
Then from \eqref{1.3} and \eqref{cy2}, we have
 \begin{equation} \label{3.8}
 \|u\|^p_p<p^2J(u) \leq p^2J(u_0)< p^2M.
 \end{equation}
Because $J(u_0)<M=\frac{1}{p^2}(\frac{p^2 e}{n\mathcal{L}_p})^{n/p}$,
 for $\mu\in([p^2J(u_0)]^{p/n} \frac{n \mathcal{L}_p}{pe}, p)$,
we obtain the following inequality from \eqref{ineq1} and \eqref{3.8},
\begin{equation} \label{3.9}
 \begin{aligned}
 I(u) 
&\geq \|\nabla u\|^p_p-\|u\|^p_p\log\|u\|_p
 +\frac{n}{p^2}\log(\frac{p\mu e}{n \mathcal{L}_p})
 \|u\|^p_p-\frac{\mu}{p}\|\nabla u\|^p_p\\
&\geq(1-\frac{\mu}{p})\|\nabla u\|^p_p
 +(\frac{n}{p^2}\log(\frac{p\mu e}{n \mathcal{L}_p})
 -\frac{1}{p}\log(p^2J(u_0)))\|u\|_p^p\\
&\geq \alpha_1 (\|u\|^p_p+k\|\nabla u\|^p_p),
 \end{aligned}
\end{equation}
where
$$
\alpha_1=\min\{\frac{1}{k}(1-\frac{\mu}{p}),
 \frac{n}{p^2}\log(\frac{p\mu e}{n \mathcal{L}_p})
 -\frac{1}{p}\log(p^2J(u_0))\}>0.
$$
Combining \eqref{3.9} with
$$
I(u)=-\frac{1}{2}\frac{d}{dt}\|u\|^2_2-\frac{k}{2}\frac{d}{dt}\|\nabla u\|^2_2,
$$
it holds
\begin{equation}
\label{R4}
 \frac{1}{2}\frac{d}{dt}\|u\|^2_2+\frac{k}{2}\frac{d}{dt}\|\nabla u\|^2_2
 \leq -\alpha_1 (\| u\|^p_p+k\|\nabla u\|^p_p).
\end{equation}


Next we first prove that $\|u\|^p_p+k\|\nabla u\|^p_p$ decays to $0$ as 
$t\to\infty$. For this purpose, Lemma \ref{LemmaR1} is useful.
Set
$$
g(t)=\|u\|^2_2+k\|\nabla u\|^2_2,\quad h(t)=\|u\|^p_p+k\|\nabla u\|^p_p.
$$
Then it is sufficient to prove $h'(t)\leq C$.
Multiplying the first equation of \eqref{qq1} by $u_t$ and using the Young 
inequality, we can obtain
\begin{equation}\label{R1}
\begin{aligned}
&\int_{\Omega}|u_t|^2dx+k\int_{\Omega}|\nabla u_t|^2dx
 +\frac{d}{dt}\int_{\Omega}\frac{|\nabla u|^p}{p}dx\\
&\leq \frac{1}{2}\int_{\Omega}|u_t|^2dx+\frac{1}{2}
 \int_{\Omega}|u|^{2p-2}(\log|u|)^2dx.
\end{aligned}
\end{equation}
Since
$$
\lim_{f\to+\infty}f^{-\alpha}\log f=0,\quad
\lim_{f\to0^+}f^{\alpha}\log f=0, \quad\text{for}\quad 0<\alpha<1,
$$
then we can deduce that
$$
\int_{\Omega}|u|^{2p-2}(\log|u|)^2dx\leq C\int_{\Omega}|u|^2dx+C,
$$
which with \eqref{R1} and \eqref{cy8} indicate that
$$
\int_{\Omega}|u_t|^2dx+k\int_{\Omega}|\nabla u_t|^2dx
+\frac{d}{dt}\int_{\Omega}|\nabla u|^pdx \leq C.
$$
Thus we  find that
\begin{align*}
h'(t)
&=\int_{\Omega}p|u|^{p-2}uu_tdx+\frac{d}{dt}\int_{\Omega}k|\nabla u|^pdx\\
&\leq \frac12\int_{\Omega}p^2|u|^{2p-2}dx+\frac12\int_{\Omega}|u_t|^2dx
 +\frac{d}{dt}\int_{\Omega}k|\nabla u|^pdx
\leq C.
\end{align*}
Then from Lemma \ref{LemmaR1} and \eqref{R4}, we can prove \eqref{R3}.

Next, we deal with the decay estimates of the solutions for the 
$1$-Dimensional case.
On the one hand, \eqref{R3} and the Sobolev imbedding inequality imply that
\begin{equation}\label{R2}
|u|_{0;\Omega}=\sup_{\Omega}|u|\to 0, \quad\text{as } t\to\infty.
\end{equation}
On the other hand, multiplying the first equation of \eqref{qq1} by $\Delta u$ 
and integrating on $\Omega$, we have
\begin{align*}
&\frac{d}{dt}\int_{\Omega}(\frac{1}{2}|\nabla u|^2+\frac{k}{2}|\Delta u|^2)dx
+(p-1)\int_{\Omega}|\nabla u|^{p-2}|\Delta u|^2dx \\
&=\int_{\Omega}|u|^{p-2}((p-1)\log|u|+1)|\nabla u|^2dx,
\end{align*}
which with \eqref{R2} indicate that there exists a $t_{\beta}>0$,
such that
$$
|u|_{0;\Omega}<e^{-\frac{1}{p-1}}\quad \text{and}\quad
\int_{\Omega}|\Delta u|^2dx\leq C,\quad \text{for } t\geq t_{\beta}.
$$
Using the Sobolev imbedding inequality again, we have that
\begin{equation}
\label{R5}
|\nabla u|_{0;\Omega}=\sup_{\Omega}|\nabla u|<C.
\end{equation}
Substituting \eqref{R2} and \eqref{R5} into \eqref{R4} gives
\begin{align*}
\frac{d}{dt}\|u\|^2_2+\frac{d}{dt}k\|\nabla u\|^2_2
&\leq -2\alpha_1 (\|u\|^p_p+k\|\nabla u\|^p_p)  \\
&=-2\alpha_1 (\int_{\Omega}|u|^2|u|^{p-2}dx
 +k\int_{\Omega}|\nabla u|^2|\nabla u|^{p-2}dx) \\
&\leq -2C\alpha_1(\|u\|^2_2+k\|\nabla u\|^2_2).
\end{align*}
Integrating the above inequality from $t$ to $T$ with $t\geq t_\beta$,
we have
\begin{align*}
\int_t^T(\|u\|^2_2+k\|\nabla u\|^2_2)ds
&\leq \frac{1}{2C\alpha_1}(\|u(t)\|^2_2+k\|\nabla u(t)\|^2_2
 -(\|u(T)\|^2_2+k\|\nabla u(T)\|^2_2)) \\
&\leq \frac{1}{2C\alpha_1}(\|u(t)\|^2_2+k\|\nabla u(t)\|^2_2).
\end{align*}
Let $T\to\infty$ and from Lemma \ref{LemmaR2}, we can find
\[
\|u(t)\|^2_2+k\|\nabla u(t)\|^2_2
\leq (\|u(t_\beta)\|_2^2+k\|\nabla u(t_\beta)\|^2_2)e^{\frac{1}{2}-C\alpha_1t},
\]
for all $ t\geq t_\beta$.
\smallskip


\textbf{Case 2. Decay estimates for $J(u_0)=M$.}
Let $u=u(x,t)$ be the global bounded solution of the problem \eqref{qq1} 
with $J(u_0)=M\leq d$ and $I(u_0)>0$.
From Propositions \ref{prop2.1} and  \ref{prop2.2},
we know that
\begin{equation}
\label{cy9}
I(u)=-(u_t,u)-k(\nabla u_t,\nabla u)\geq0, \quad\text{for all } t>0,
\end{equation}
and there exists a $t_0>0$, such that
$$
I(u(t_0))=0,\quad\text{and}\quad I(u(t))>0,\quad\text{for }0<t<t_0,
$$
which implies 
$$
\int_0^{t} (\|u_{\tau}\|^2_2 + k \|\nabla u_{\tau}\|^2_2) d\tau>0,
\quad 0<t<t_0.
$$
Thus we can choose some time $0<t_{\gamma}<t_0$, such that
$$
\int_0^{t_\gamma} (\|u_{\tau}\|^2_2+k \|\nabla u_{\tau}\|^2_2) d\tau=\gamma,
$$
where $\gamma$ is a sufficiently small positive number.
If we take $t_\gamma$ as the initial time, then we have
\begin{gather*}
I(u(t_{\gamma}))>0,\\
J(u(t_{\gamma}))=J(u_0)-\int_0^{t_\gamma}(\|u_{\tau}\|^2_2+k\|\nabla u_{\tau}\|^2_2) d\tau
=M-\gamma <M,
\end{gather*}
which is the same as  Case 1. 
Similar to the proof for Case 1,
we can choose $t_\gamma$ large enough such that
\[
\|u(t)\|^2_2+k\|\nabla u(t)\|^2_2
\leq (\|u(t_\gamma)\|_2^2+k\|\nabla u(t_\gamma)\|^2_2)e^{\frac{1}{2}-C\alpha_2t},
\quad \text{for all }  t\geq t_\gamma,
\]
where
$$
\alpha_2=\min\Big\{\frac{1}{k}(1-\frac{\mu}{p}),\,
\frac{n}{p^2}\log(\frac{p\mu e}{n \emph{L}}_p)
-\frac{1}{p}\log(p^2(M-\gamma))\Big\}>0,
$$
for all $\mu\in([p^2(M-\gamma)]^{p/n} \frac{n \mathcal{L}_p}{pe}, p)$.
\end{proof}

\section{Blow-up at $+\infty$ and growth estimation}

Actually, the estimation \eqref{G4} in Theorem \ref{th3} tells us that
the solution of \eqref{qq1} would not blow up at any finite time $T>0$.
However, in this section, we prove that the solution may blow up at $+\infty$
and further give some growth estimates of the solution.

\begin{theorem} \label{th2}
When $J(u_0)\leq d$ and $I(u_0)<0$,
then the weak solution of \eqref{qq1} blows up at $+\infty$, namely
\[
\lim_{t\to+\infty}(\|u\|^2_2+k\|\nabla u\|^2_2)=+\infty.
\]
\end{theorem}

\begin{remark} \rm
Under the similar conditions, when $p>2$, the weak solutions blow up in finite
 time \cite{Le}.
However, when $p\leq 2$, the weak solutions blow up at $\infty$.
\end{remark}

\begin{proof} \textbf{Step 1: $J(u_0)<d$.}
From Proposition \ref{prop2.1}, we obtain for all $t\geq0$,
$u\in V_{\delta}$ for any $1\leq\delta<\bar{\delta}$ or $\delta_1<\delta<\delta_2$
 with $\delta_1<1<\delta_2$.
Then by $I_{\delta}(u)<0$ and Lemma \ref{lem2.2}, we obtain
 $\|\nabla u\|_p^p>r^p(\delta)=\lambda_1
(\frac{p^2\delta e}{n\mathcal{L}_p})^{n/p}$ for all $t\geq0$.
Set
\begin{equation*}
 G(t)= \int^{t}_0(\|u\|^2_2 + k\|\nabla u\|^2_2)d\tau.
\end{equation*}
A simple calculation indicates that
\begin{align*}
G''(t)&= -2I(u)=2(\delta-1)\|\nabla u\|_p^p-2I_{\delta}(u)\\
&>2(\delta-1)\|\nabla u\|_p^p \\
&>2(\delta-1)r^p(\delta),\quad\text{for all}\,\,t\geq0.
\end{align*}
Thus setting $\delta>1$, we can have
\begin{equation}
\label{3.16}
G'(t)=G'(0)+\int^{t}_0G''(\tau)d\tau>2(\delta-1)
\lambda_1(\frac{p^2\delta e}{n\mathcal{L}_p})^{n/p}t,
\quad\text{for all } t\geq0,
\end{equation}
namely
\[
\|u(t)\|^2_2+k\|\nabla u(t)\|^2_2>2(\delta-1)
\lambda_1(\frac{p^2\delta e}{n\mathcal{L}_p})^{n/p}t,
\quad\text{for all }t>0,
\]
where $\delta>1$ in Proposition \ref{prop2.1},
$\lambda_1$ can be found in Lemma \ref{lem2.2}
and $\mathcal{L}_p$ is \eqref{cy0}.
This means that the weak solution $u$ will blow up at $+\infty$.
\smallskip

\noindent\textbf{Step 2: $J(u_0)=d$.}
From Proposition \ref{prop2.2},
we know $I(u)=-(u_t,u)-k(\nabla u_t,\nabla u)<0$ for $t\geq 0$,
and then $\int_0^{t} (\|u_{\tau}\|^2_2 + k\|\nabla u_{\tau}\|^2_2) d\tau$ is
strictly positive for $t>0$.
For any sufficiently small positive number $t_1$, we have
\begin{equation*}
 J(u(t_1))=J(u_0)- \int_0^{{t_1}} (\|u\|^2_2+k\|\nabla u_{\tau}\|^2_2)
 d\tau <d.
\end{equation*}
If we take $t=t_1$ as the initial time, then similar to Step 1,
we can obtain that the weak solution $u$ blows up at $+\infty$.
\end{proof}

\begin{theorem} \label{th2'}
Let $u=u(x,t)$ be the weak solution in Theorem \ref{th2}.
If $J(u_0)\leq M$ and $I(u_0)<0$, then for any $\alpha_{3}\in(0,1)$,
there exist $t_{\alpha_{3}}>0$ such that
\begin{equation}\label{1.5}
\|u\|^2_2+k\|\nabla u\|^2_2\geq C_{\alpha_{3}}(t-t_{\alpha_{3}}
)^{\frac{1}{1-\frac{p\alpha_{3}}{2}}-1},
\quad\text{for all }t\geq t_{\alpha_3},
\end{equation}
where
$$
C_{\alpha_{3}}=((1-\frac{p\alpha_{3}}{2})
G^{-\frac{p\alpha_{3}}{2}}(t_{\alpha_3})
G'(t_{\alpha_3}))^{\frac{1}{1-\frac{p\alpha_{3}}{2}}}
$$
with $G(t)=\int_0^t(\|u\|_2^2+k\|\nabla u\|_2^2)d\tau$.
\end{theorem}

\begin{remark} \rm
From \eqref{1.5} and \eqref{G4}, the weak solutions that blow up at $\infty$ 
grow algebraically.
 \eqref{1.5} also indicates that the lower bound of growth estimates is
 smaller than that of the case $p=2$,
which is caused by fast diffusion.
\end{remark}

\begin{proof}
 Let $u=u(x,t)$ be the weak solution of \eqref{qq1} with $J(u_0)\leq M$ and 
$I(u_0)<0$.
Then Propositions \ref{prop2.1} and  \ref{prop2.2} tell us that
$u\in V$ and $I(u)<0$ for all $t\geq0$.
Taking $\mu=p$ in \eqref{ineq1} and noticing $I(u)<0$,
we can obtain
\begin{equation} \label{3.14}
 \|u\|_p^p\geq (\frac{p^2 e}{n\mathcal{L}_p})^{n/p}=p^2M,
 \quad\text{for all } t\geq0,
\end{equation}
which also implies $\|u\|^2_2>0$ for all $0 \leq t<T$. Thus
\[
G'(t)= \|u\|^2_2 + k\|\nabla u\|^2_2 >0
\quad\text{and}\quad G''(t)= -2I(u) >0,\quad\text{for all}\,\,t\geq0.
\]
Furthermore, from \eqref{3.14}, we obtain
\begin{equation} \label{3.15}
\begin{split}
G''(t)&=-2I(u)=-2pJ(u)+\frac{2}{p}\|u\|^p_p\\
&=-2pJ(u_0)+2p\int^{t}_0(\|u_\tau\|^2_2+k\|\nabla u_\tau\|^2_2)d\tau
+\frac{2}{p}\|u\|^p_p \\
&\geq 2p(M-J(u_0))+2p\int^{t}_0(\|u_\tau\|^2_2+k\|\nabla u_\tau\|^2_2)d\tau,
\quad\text{for all}\,\,t\geq0.
\end{split}
\end{equation}
Since
\begin{equation} \label{3.17}
\begin{split}
\Big(\int^{t}_0((u_{\tau},u)_2+k(\nabla u_{\tau},\nabla u)_2)d\tau\Big)^2
&=\frac{1}{4}(\int^{t}_0\frac{d}{d\tau}(\|u\|^2_2+k\|\nabla u\|^2_2)d\tau)^2 \\
&=\frac{1}{4}(G'(t)-G'(0))^2 \\
&=\frac{1}{4}(G'^2(t)-2G'(t)G'(0)+G'^2(0)),
\end{split}
\end{equation}
then combining \eqref{3.15} and \eqref{3.17}, and using the H\"older inequality,
 we can calculate
\begin{equation} \label{3.18}
\begin{split}
&G(t)G''(t)-\frac{p}{2}G'^2(t)\\
&\geq 2p(M-J(u_0))G(t)+2p\int^{t}_0(\|u_\tau\|^2_2+k\|\nabla u_\tau\|^2_2)d\tau
\int^{t}_0(\|u\|^2_2 + k\|\nabla u\|^2_2)d\tau
\\
&\quad-2p(\int^{t}_0((u_{\tau},u)_2+k(\nabla u_{\tau},\nabla u)_2)d\tau)^2
-pG'(t)(\|u_0\|^2_2+k\|\nabla u_0\|^2_2)
\\
&\quad+\frac{p}{2}(\|u_0\|^2_2+k\|\nabla u_0\|^2_2)^2
\\
& \geq2p(M-J(u_0))G(t)-pG'(t)(\|u_0\|^2_2+k\|\nabla u_0\|^2_2)
\\
& \geq-p(\|u_0\|^2_2+k\|\nabla u_0\|^2_2)G'(t).
\end{split}
\end{equation}
Then for $0<\alpha_{3}<1$, we obtain
\[
G(t)G''(t)-\frac{p\alpha_{3}}{2}G'^2(t)
\geq(1-\alpha_{3})\frac{p}{2}G'^2(t)-p(\|u_0\|^2_2+k\|\nabla u_0\|^2_2) G'(t).
\]
From \eqref{3.16}, there exists $t_1>0$ such that $G'(t)$ is large enough and
\begin{equation} \label{3.19}
 G(t)G''(t)-\frac{p\alpha_{3}}{2}G'^2(t)>0,\quad\text{for all } t\geq t_1.
\end{equation}
Then we have
\begin{gather*}
(G^{1-\frac{p\alpha_{3}}{2}}(t))'
=(1-\frac{p\alpha_{3}}{2})G^{-\frac{p\alpha_{3}}{2}}(t)G'(t), \\
(G^{1-\frac{p\alpha_{3}}{2}}(t))''=(1-\frac{p\alpha_{3}}{2})
G^{-\frac{p\alpha_{3}}{2}-1}(t)
(G(t)G''(t)-\frac{p\alpha_{3}}{2}G'^2(t))>0.
\end{gather*}
Now, we take $t_{\alpha_{3}}\geq t_1$ satisfying $G(t_{\alpha_{3}})>0$.
Then for $t\geq t_{\alpha_3}$,
\begin{equation}\label{3.20}
\begin{aligned}
G(t)&=(G^{1-\frac{p\alpha_{3}}{2}}(t))^{\frac{1}{1-\frac{p\alpha_{3}}{2}}}
\\
&=(G^{1-\frac{p\alpha_{3}}{2}}(t_{\alpha_{3}})
+\int^t_{t_{\alpha_3}}(1-\frac{p\alpha_{3}}{2})G^{-\frac{p\alpha_{3}}{2}}
(\tau)G'(\tau)d\tau)^{\frac{1}{1-\frac{p\alpha_{3}}{2}}}
\\
&\geq C_{\alpha_{3}}(t-t_{\alpha_{3}})^{\frac{1}{1-\frac{p\alpha_{3}}{2}}}
\end{aligned}
\end{equation}
with
$$
C_{\alpha_{3}}=((1-\frac{p\alpha_{3}}{2})G^{-\frac{p\alpha_{3}}{2}}
(t_{\alpha_3})G'(t_{\alpha_3}))^{\frac{1}{1-\frac{p\alpha_{3}}{2}}}.
$$
Since $G''(t)>0$ for all $t\geq0$, then we have $\int_0^t G'(\tau)d\tau\leq tG'(t)$, 
namely
$$
t(\|u\|_2^2+k\|\nabla u\|_2^2)\geq G(t),\quad\text{for all}\,\,t\geq0,
$$
which combining with \eqref{3.20} we deduce that for $0<\alpha_{3}<1$ and
 $t\geq t_{\alpha_3}$,
\[
\|u\|^2_2+k\|\nabla u\|^2_2\geq C_{\alpha_{3}}
(t-t_{\alpha_{3}})^{\frac{1}{1-\frac{p\alpha_{3}}{2}}-1}.
\]
\end{proof}

\subsection*{Acknowledgments}
This research was supported by the National Natural Science Foundation
of China (11571062, 11671155),
and by the Fundamental Research Funds for the Central Universities
(DUT16LK01)

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