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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 18, pp. 1--27.\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-2018/18\hfil Asymptotic stability and blow-up of solutions]
{Asymptotic stability and blow-up of solutions for an edge-degenerate wave
equation with singular potentials and several nonlinear source terms
of different sign}

\author[F. Jiang, Y. Luan, G. Li \hfil EJDE-2018/18\hfilneg]
{Feida Jiang, Yue Luan, Gang Li}

\address{Feida Jiang \newline
College of Mathematics and Statistics,
Nanjing University of Information Science and Technology,
Nanjing 210044, China}
\email{jfd2001@163.com}

\address{Yue Luan \newline
College of Mathematics and Statistics,
Nanjing University of Information Science and Technology,
Nanjing 210044, China}
\email{yluan\_sunflower@163.com}

\address{Gang Li \newline
College of Mathematics and Statistics,
Nanjing University of Information Science and Technology,
Nanjing 210044, China}
\email{ligang@nuist.edu.cn}

\dedicatory{Communicated by Zhaosheng Feng}

\thanks{Submitted May 12, 2017. Published January 13, 2018.}
\subjclass[2010]{35L20, 35L80, 33B44} 
\keywords{Edge-degenerate; potential function; blow-up; decay; global solution}

\begin{abstract}
 We study the initial boundary value problem of an edge-degenerate wave equation.
 The operator $\Delta_{\mathbb{E}}$ with edge degeneracy on the boundary
 $\partial E$ was investigated in the literature. We give the invariant sets
 and the vacuum isolating behavior of solutions by introducing a family of
 potential wells. We prove that the solution is global in time and exponentially
 decays when the initial energy satisfies $E(0)\leq d$ and $I(u_0)>0$.
 Moreover, we obtain the result of blow-up with initial energy $E(0)\leq d$
 and $I(u_0)<0$, and give a lower bound for the blow-up time $T^*$.
\end{abstract}

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


\section{Introduction}

We consider the following initial-boundary value problem of an edge-degenerate wave 
equation with singular potentials and several nonlinear source terms 
of different sign:
\begin{equation}\label{1.1}
\begin{gathered}
\begin{aligned}
&\partial_{tt}u-\Delta _{\mathbb{E}}u+{\partial_tu}+Vu\\
&=\sum_{k=1}^la_k|u|^{p_k-1}u-\sum_{j=1}^sb_j|u|^{q_j-1}u \quad\text{in }
 \operatorname{int}(\mathbb{E})\times(0,T), 
\end{aligned}\\
 u(0)=u_0,  \quad {\partial_tu(0)}=u_1\quad\text{in }
 \operatorname{int}(\mathbb{E}), \\
 u=0 \quad\text{on }\partial\mathbb{E}\times(0,T). 
\end{gathered}
\end{equation}
where $ u_0\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$,
$u_1\in \mathcal{L}_2({\mathbb{E}})\cap \mathcal{L}_2(Q_T)$,
$Q_T={\mathbb{E}}\times[0,T]$, $T\in(0,\infty)$ and
$ N=1+n+h\geq3$, $a_k>0$ for $1\leq k\leq l$, $b_j>0$ for $1\leq j\leq s$, 
$p_k$ and $q_j$ satisfy 
\begin{equation*}
1<q_s<q_{s-1}<\dots<q_1=q<p<p_l<p_{l-1}<\dots<p_1<\frac{N+2}{N-2}\quad \text{if }
 N\geq 3,
\end{equation*}
and $V$ is a positive potential function which can be unbounded on the edge 
manifold $\mathbb{E}$. Here, $X$ is a closed compact $C^{\infty}$-smooth 
submanifold of dimension $n$ embedded in the unit sphere of $\mathbb{R}^{n+1}$ 
and $Y$ is a bounded subset in $\mathbb{\mathbb{R}}^h$ containing the origin. 
Write $\mathbb{E}=[0,1)\times X\times Y$, which can be regarded as the local 
model near the boundary of stretched edge-manifolds. Denote $\mathbb{E}_0$ 
for the interior of $\mathbb{E}$, and the boundary of $\mathbb{E}$ by 
$\partial\mathbb{E}={0}\times X\times Y$. The edge-Laplacian operator is defined as
\begin{equation*}
\Delta_{\mathbb{E}}
=\nabla_{\mathbb{E}}^2=(\omega \partial_{\omega} )^2
+\partial_{x_1}^2+\ldots+\partial_{x_n}^2+(\omega \partial_{y_1} )^2
+\ldots+(\omega \partial_{y_h} )^2,
\end{equation*}
where $\nabla_{\mathbb{E}}=(\omega \partial_{\omega},\partial_{x_1},\ldots,
\partial_{x_n},\omega \partial_{y_1},\ldots,\omega \partial_{y_h} )$ 
denotes the gradient operator with edge degeneracy on the boundary 
$\partial {\mathbb{E}}$. Moreover, the degenerate elliptic operator 
$\Delta_{\mathbb{E}}$ is regarded as a spacial case of type degenerate 
differential operators on a stretched edge manifold.

In the case of edge degenerate, Chen and Liu \cite{CL2016} considered the 
following initial-boundary value problem for semi-linear parabolic equations
 with singular potential term:
\begin{gather*}
 \partial_tu-\Delta _{\mathbb{E}}u-Vu=|u|^{p-1}u \quad\text{in }
  \operatorname{int}({\mathbb{E}})\times(0,T), \\
 u(0)=u_0 \quad\text{in } \operatorname{int}({\mathbb{E}}), \\
 u=0 \quad\text{on }\partial\mathbb{E}\times(0,T),
\end{gather*}
and derived a threshold of the existence of global solutions with exponential 
decay and the blow-up in finite time, with low initial energy case and 
critical initial energy case.  Chen et al.\  \cite{CL2012} first established 
the corresponding Sobolev inequality and Poincar{\'e} inequality on the 
cone Sobolev spaces, and then proved the existence of non-trivial weak 
solution for the following Dirichlet boundary value problem for a class 
of non-linear elliptic equation on manifolds with conical singularities:
\begin{gather*}
 -\Delta _{\mathbb{B}}u=u|u|^{p-1}, \quad  \text{for } 
  1<p<\frac{n+2}{n-2}\text{ in }  \operatorname{int}({\mathbb{B}}), \\
 u=0 \quad\text{on }\partial\mathbb{B}\times(0,T).
\end{gather*}
Here $\mathbb{B} = [0,1)\times X$ and $X$ is an $(n-1)$-dimensional closed 
compact manifold. The local model $\mathbb{B}$ is regarded as the local 
model near the conical points, and $\partial\mathbb{B} = \{ 0 \} \times X$. 
Moreover, the operator $\Delta_{\mathbb{B}}$ in the above equation is defined 
by $(x_{1}\partial_{x_1})^{2} + \partial_{x_2}^{2} + \dots + \partial_{x_n}^2$,
which is an elliptic operator with conical degeneration on the boundary 
$x_1 = 0$, and corresponding gradient operator is denoted by 
$\nabla_{\mathbb{B}} = (x_{1}\partial_{x_1}, \partial_{x_2}, \ldots,
\partial_{x_n})$.
 Alimohammady et al.\ I\cite{AK2017} applied the family of potential wells 
to the following initial boundary value problem of semilinear hyperbolic
 equations on the cone Sobolev spaces:
\begin{gather*}
 u_{tt}-\Delta _{\mathbb{B}}u+V(x)u+\gamma u_t=f(x,u) \quad\text{in }
 \operatorname{int}({\mathbb{B}})\times(0,T), \\
 u(x,0)=u_0(x),\ u_t(x,0)=u_1(x) \quad\text{in } \operatorname{int}({\mathbb{B}}), \\
 u(t,x)=0 \quad\text{on }\partial\mathbb{B}\times(0,T),
\end{gather*}
They not only gave some results of  existence and nonexistence of global solutions,
 but also obtained the vacuum isolating of solutions and showed blow-up in 
finite time of solutions on a manifold with conical singularities. 
For different problems which take into account degenerate spaces,
 we refer to \cite{CL22012,CL32012,CL2011,GY2017}.

For a more general problem, Jiang \cite{JY2011} considered equation
\begin{equation}\label{1.3}
\phi_{tt}-\Delta\phi+V(x)\phi+\phi_t=\phi|\phi|^{p-1} \quad \text{in } 
  \mathbb{R}^N\times(0,T)
\end{equation}
with initial data
\begin{equation*}
\phi(0,x)=\phi_0(x), \quad \phi_t(0,x)=\phi_1(x) \quad  \text{in }  \mathbb{R}^N,
\end{equation*}
where $1<p\leq\frac{N}{N-2}$ for $N>2$ $(p>1,\ N=1,2)$. He established new 
stable and unstable sets for the initial data, and proved that the solution 
blows up in finite time when the evolution enters into the unstable set and 
the solution is global existence when the evolution enters into the stable set. 
For the case of linear damping and potential terms, Levine \cite{HA1974} showed 
that the solution of \eqref{1.3} with negative initial energy blows up for the
 abstract version. Zhou \cite{YZ2005} proved that the solution of \eqref{1.3} 
with the intial data $(\phi_0,\phi_1)$ satisfies 
$\int_{\mathbb{R}^N}\phi_0\phi_1\geq 0$ and $V(x)=0$  blows up in finite 
time even for vanishing initial energy.
Jiang and Zhang \cite{JZ2013} considered the Cauchy problem of a nonlinear 
wave equation with damping and source terms. When the term works as the 
damping in the case $m=1$, the equation is equivalent to the problem \eqref{1.3}
 with the case $V(x)=0$. Galakhov in \cite{EI2009} discussed the semi-linear wave 
equation with a potential, which is different from \eqref{1.3} because of the 
damping term. For more papers related to the semi-linear hyperbolic problem,
you can refer to \cite{GS2006,KS2001,XW2013} and the references therein.
 Liu and Xu \cite{LX2007} studied the following initial boundary value problem 
of wave equations and reaction-diffusion equations with several nonlinear 
source terms of different sign:
\begin{equation*}
u_{tt}-\Delta u=f(u)\equiv\sum_{k=1}^la_k|u|^{p_k-1}u
-\sum_{j=1}^sb_j|u|^{q_j-1}u\quad   \text{in }  \Omega\times[0,T],
\end{equation*}
with initial boundary conditions
\begin{gather*}
u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad  x\in\Omega,\\
u(x,t)=0,\quad x\in\partial\Omega,\quad t\geq0.
\end{gather*}
The authors introduced a family of potential wells and corresponding outside 
sets and who also proved the invariance of some sets under the flow of the 
equation and vacuum isolating of solutions. Then, they got the threshold 
result of existence and nonexistence of global solution.

 Motivated by the above results, in this paper, we intend to study the initial 
boundary value problem for an edge-degenerate wave equation. 
We consider the existence of global solution by Faedo-Galerkin approximation 
method \cite{XS2013} and discuss the blow-up of solutions by means of a 
a convexity argument \cite{WJ2010}. 
The main difficulty in carrying out this paper is considering the problem 
with singular potentials in an edge type Sobolev space. We introduce stable 
and unstable sets and construct some functionals. By giving a family of 
potential wells, we obtain vacuum isolating of solutions. Then we derive a 
threshold of the existence of global solutions with exponential decay if 
the initial data are in the stable set, otherwise, we obtain the blow-up result.

The remaining part of our paper is organized as follows. In Section \ref{se2}, 
we give some notation and basic facts which are needed for our work. 
In Section \ref{se5}, we introduce a family of potential wells, and give 
the invariance of some sets under the flow of problem \eqref{1.1} and the 
vacuum isolating behavior of the problem. The existence of global solutions 
and exponential decay result are given in Section \ref{se3}. 
In Section \ref{se4}, we discuss the blow-up in finite time of solution 
and seek a lower bound for the blow-up time $T^*$.

\section{Edge type Sobolev spaces}\label{se2}

In this section, we introduce some definitions and propositions which will 
be used in this paper. Now, we recall the Edge type weighted $p$-Sobolev 
space \cite{CL2016}.

\begin{definition}\label{d.2.1} \rm % \label{d1} 
For $(\omega,x,y)\in {\mathbb{R}}_{+}^{N}$ with $N=n+h+1$, assume 
$(\omega,x,y)\in \mathcal{D'}({\mathbb{R}}_{+}^{N})$, we say that 
$u(\omega,x,y)\in \mathcal{L}_p({\mathbb{R}}_{+}^{N},\,d\sigma)$ if
\begin{equation*}
\|u\|_{\mathcal{L}_p}
=\Big(\int_{{\mathbb{R}}_{+}^{N}}\omega^N|u(\omega,x,y)|^p\,d\sigma\Big)^{1/p}<+\infty,
\end{equation*}
where $ \,d\sigma=\frac{d\omega}{\omega}dx_1\ldots
dx_n\frac{dy_1}{\omega}\ldots\frac{dy_h}{\omega}$.
 Moreover, the weighted $\mathcal{L}_p$ spaces with weight $\gamma\in R$ 
is denoted by $\mathcal{L}_p^{\gamma}(\mathbb{R}_{+}^{N},\,d\sigma)$,
which consists of functions $u(\omega,x,y)$ such that
\begin{equation*}
\|u\|_{\mathcal{L}_p^{\gamma}}
=\Big(\int_{\mathbb{R}_{+}^{N}}\omega^N|\omega^{-\gamma}u(\omega,x,y)|^p
\,d\sigma\Big)^{1/p}<+\infty.
\end{equation*}
\end{definition}

\begin{definition}\label{d.2.2} \rm % \label{d2} 
For $m\in N$, $\gamma\in \mathbb{R}$ and $N=n+h+1$, the spaces 
$\mathcal{H}_p^{m,\gamma}(\mathbb{R}_{+}^{N})$ is defined by
\begin{equation*}
\mathcal{H}_p^{m,\gamma}(\mathbb{R}_{+}^{N})
:=\Big\{u\in \mathcal{D'}( \mathbb{R}_{+}^{N})|\omega^{\frac{N}{p}
-\gamma}(\omega \partial_{\omega})^k \partial_x^{\alpha}
(\omega \partial_y)^{\beta}: u\in \mathcal{L}_p(\mathbb{R}_{+}^{N},\,d\sigma)\Big\},
\end{equation*}
for $k\in N$, multi-index $\alpha\in N^n$, $\beta\in N^q$ with 
$k+|\alpha|+|\beta|\leq m$. Therefore, 
$\mathcal{H}_p^{m,\gamma}(\mathbb{R}_{+}^{N})$ is a Banach space with the 
following norm:
\begin{equation*}
\|u\|_{\mathcal{H}_p^{m,\gamma}(\mathbb{R}_{+}^{N})}
=\sum_{k+|\alpha|+|\beta|\leq m}\Big(\int_{\mathbb{R}_{+}^{N}}\omega^N
|\omega^{-\gamma}(\omega \partial_{\omega})^k \partial_x^{\alpha}
(\omega \partial_y)^{\beta}u(\omega,x,y)|^p\,d\sigma\Big)^{1/p}.
\end{equation*}
Moreover, the subspace $\mathcal{H}_{p,0}^{m,\gamma}(\mathbb{R}_{+}^{N})$ of 
$\mathcal{H}_p^{m,\gamma}(\mathbb{R}_{+}^{N})$ denotes the closure of 
$C_0^{\infty}(\mathbb{R}_{+}^{N})$ in $\mathcal{H}_p^{m,\gamma}(\mathbb{R}_{+}^{N})$.
\end{definition}
If $u\in \mathcal{L}_p^{\frac{n+1}{p}}({\mathbb{E}})$ and 
$v\in \mathcal{L}_{p'}^{\frac{n+1}{p'}}({\mathbb{E}})$, where 
$p,p'\in (1,\infty)$ and $ \frac{1}{p}+\frac{1}{p'}=1$, then we have the 
following edge type H$\ddot{\mathrm{o}}$lder inequality:
\begin{equation*}
\int_{\mathbb{E}}\omega^h|uv|\,d\sigma
\leq\Big(\int_{\mathbb{E}}\omega^h|u|^p\,d\sigma\Big)^{1/p}
\Big(\int_{\mathbb{E}}\omega^h|v|^{p'}\,d\sigma\Big)^{1/p'}.
\end{equation*}
When $p=2$, we get the corresponding edge type Schwarz inequality
\begin{equation*}
\int_{\mathbb{E}}\omega^h|uv|\,d\sigma
\leq\Big(\int_{\mathbb{E}}\omega^h|u|^2\,d\sigma\Big)^{1/2}
\Big(\int_{\mathbb{E}}\omega^h|v|^{2}\,d\sigma\Big)^{1/2}.
\end{equation*}
For convenience we denote
\begin{equation*}
(u,v)_2=\int_{\mathbb{E}}\omega^huv\,d\sigma,\quad
\|u\|_{\mathcal{L}_p^{\frac{n+1}{p}}({\mathbb{E}})}
=Big(\int_{\mathbb{E}}\omega^h|u|^p\,d\sigma\Big)^{1/p}.
\end{equation*}

For  $t\in(0,T)$, we introduce the  energy functional
\begin{align*}
E(t)
=&\frac{1}{2}\int_{\mathbb{E}}\omega^h|\partial_tu|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^hV|u|^2\,d\sigma \\
&-\sum_{k=1}^l\frac{a_k}{p_k+1}\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 +\sum_{j=1}^s\frac{b_j}{q_j+1}\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma.
\end{align*}
Then
\begin{align*}
E(0)
=&\frac{1}{2}\int_{\mathbb{E}}\omega^h|u_1|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u_0|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^hV|u_0|^2\,d\sigma \\
&-\sum_{k=1}^l\frac{a_k}{p_k+1}\int_{\mathbb{E}}\omega^h|u_0|^{p_k+1}
 \,d\sigma+\sum_{j=1}^s\frac{b_j}{q_j+1}\int_{\mathbb{E}}\omega^h|u_0|^{q_j+1}\,d\sigma.
\end{align*}
An elementary calculation shows
\begin{equation*}
 E(t)=E(0)-\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2\,d\tau.
\end{equation*}
Next we define the following functionals on the edge Sobolev space 
$\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$:
\begin{gather*}
\begin{aligned}
J(u)&=\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma
 -\sum_{k=1}^l\frac{a_k}{p_k+1}\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma \\
&\quad +\sum_{j=1}^s\frac{b_j}{q_j+1}\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma,
\end{aligned} \\
\begin{aligned}
I(u)
&=\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma
 -\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma \\
&\quad +\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma,
\end{aligned} \\
\begin{aligned}
I_{\delta}(u)
&=\delta\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\delta\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma
 -\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma\\
&\quad +\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma.
\end{aligned}
\end{gather*}
Here, $J(u)$, $I(u)$ and $I_{\delta}(u)$ are well-defined and belong to 
$C^1(\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}),\mathbb{R})$. 
Then, we introduce the  potential well
\begin{equation*}
W=\big\{u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}):
I(u)>0,J(u)<d\big\}\cup\{0\},
\end{equation*}
and the outside set of the corresponding potential well is defined as
\begin{equation*}
V=\big\{u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}):
I(u)<0,J(u)<d\big\}.
\end{equation*}
Now, we define
\begin{gather*}
\mathcal{N}=\Big\{u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}))\setminus 
\{0\} : I(u)=0,\; \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
\neq 0 \Big\}, \\
d=\inf\{J(u),u\in \mathcal{N}\}.
\end{gather*}
And for $\delta\geq0$, we define
\begin{gather*}
\mathcal{N}_{\delta}=\Big\{u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}
({\mathbb{E}}))\setminus \{0\}: I_{\delta}(u)=0,\; 
\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma \neq 0 \Big\}, \\
d_{\delta}=\inf\{J(u),u\in \mathcal{N}_{\delta}\}.
\end{gather*}

To facilitate our calculations, we state the following propositions.

\begin{proposition}[\cite{CL2016}]\label{prop3} 
For $1<l<2^*=\frac{2N}{N-2}$, the embedding
 $\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})\hookrightarrow 
\mathcal{H}_{l,0}^{1,\frac{n+1}{l}}({\mathbb{E}})
=\mathcal{L}_l^\frac{n+1}{l}({\mathbb{E}})$ is compact.
\end{proposition}

\begin{proposition}[Edge type Poincar\'e inequality \cite{CL2016}]\label{prop2} 
Let ${\mathbb{E}}\subset[0,1)\times X\times Y$ be a bounded subset in 
$\mathbb{R}_{+}^N$ with $N=1+n+h$. 
If $u(\omega,x,y)\in\mathcal{H}_{p,0}^{m,\gamma}(\mathbb{R}_{+}^{N})$ 
for $1<p<+\infty$, $\gamma\in \mathbb{R}_{+}$, then
\begin{equation*}
\|u(\omega,x,y)\|_{\mathcal{L}_p^{\gamma}({\mathbb{E}})}
\leq d_{\mathbb{E}}\|\nabla _{\mathbb{E}}u(\omega,x,y)
\|_{\mathcal{L}_p^{\gamma}({\mathbb{E}})},
\end{equation*}
where $d_{\mathbb{E}}$ is the diameter of ${\mathbb{E}}$.
\end{proposition}

\begin{proposition}[Edge type Hardy's inequality \cite{CL2016}]\label{prop1} 
For all $u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$,
\begin{equation*}
\int_{\mathbb{E}} \omega^hV|u|^2\,d\sigma
\leq C\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2.
\end{equation*}
\end{proposition}

For $u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$, we define
\begin{equation*}
C^*=\inf\Big\{\frac{\sqrt V\|u\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}}{\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}}: u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})\Big\}.
\end{equation*}
Next, we give some preparatory work, so that there are some lemmas which will 
be used in this paper.

\begin{lemma}[Sobolev-Poincar\'e \cite{CL2016}] \label{le3.1}  
Assume that $u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$ and that \\
$\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}\neq 0$. 
If $ 1<p<2^*-1=\frac{N+2}{N-2}$  and $N=n+h+1\geq3$, then there exist a constant 
$C_*$ such that
\begin{equation*}
\|u\|_{\mathcal{L}_{p+1}^{\frac{n+1}{p+1}}({\mathbb{E}})}\leq C_*\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}.
\end{equation*}
\end{lemma}

By Proposition \ref{prop3} we can obtain the following constants:
\begin{gather*}
C_k=\sup\Big\{\frac{\|u\|_{\mathcal{L}_{p_k+1}^{\frac{n+1}{p_k+1}}
({\mathbb{E}})}}{\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}};u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})\Big\},\\
C_j=\sup\Big\{\frac{\|u\|_{\mathcal{L}_{q_j+1}^{\frac{n+1}{q_j+1}}
({\mathbb{E}})}}{\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}};u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})\Big\}.
\end{gather*}

\section{Invariant sets and vacuum isolating}\label{se5}

In this section, we shall introduce a family of Nehari functionals in edge 
type Sobolev spaces and the family of potential wells sets. 
Now, we give the corresponding lemmas, which will help us to demonstrate 
the invariant sets and the vacuum isolating behavior of solutions 
for the problem.

\subsection{Properties of potential wells}
In this subsection, we introduce a family of potential wells and give a 
series of their properties which are used to prove of our main results.

\begin{lemma}\label{le3.2}
Assume that $u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$ and that
$\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}\neq 0$, 
we have
\begin{enumerate}
\item $\lim_{\lambda\to 0}J(\lambda u)=0$, and
 $\lim_{\lambda\to+\infty}J(\lambda u)=-\infty$;

\item In the interval $0<\lambda<\infty$, there is a unique
  $\lambda^*=\lambda^*(u)>0$, such that $\frac{d}{d\lambda}J(\lambda^* u)=0$;

\item $J(\lambda u)$ is increasing on $0\leq\lambda<\lambda^*$, decreasing on 
 $\lambda>\lambda^*$ and takes the maximum at $\lambda=\lambda^*$;

\item $I(\lambda u)>0$ for $0<\lambda<\lambda^*$, $I(\lambda u)<0$ for 
$\lambda^*<\lambda<\infty$ and $I(\lambda^* u)=0$.
\end{enumerate}
\end{lemma}

\begin{proof}
(1) From the definition of $J(u)$, we know that
\begin{align*}
J(\lambda u)
=&\frac{\lambda^2}{2}\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\frac{\lambda^2}{2}\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma
 -\sum_{k=1}^l\frac{a_k\lambda^{p_k+1}}{p_k+1}\int_{\mathbb{E}}
 \omega^h|u|^{p_k+1}\,d\sigma\\
&+\sum_{j=1}^s\frac{b_j\lambda^{q_j+1}}{q_j+1}\int_{\mathbb{E}}
 \omega^h|u|^{q_j+1}\,d\sigma,
\end{align*}
which gives $\lim_{\lambda\to 0}J(\lambda u)=0$, and 
$\lim_{\lambda\to+\infty}J(\lambda u)=-\infty$.

(2) An easy calculation shows that
\begin{equation}\label{3.2}
\begin{aligned}
 \frac{d}{d\lambda}J(\lambda u)
=  &  \lambda\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2
 \,d\sigma+\lambda\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma \\
& -\lambda^{p_k}\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma 
  +\lambda^{q_j}\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma=0,
\end{aligned}
\end{equation}
which is equivalent to
\begin{align*}
&\sum_{k=1}^la_k\lambda^{p_k-1}\int_{\mathbb{E}}\omega^h |u|^{p_k+1}\,d\sigma
-\sum_{j=1}^sb_j\lambda^{q_j-1}\int_{\mathbb{E}}\omega^h |u|^{q_j+1}\,d\sigma \\
& =\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma.
\end{align*}
Then we let
\begin{align*}
g(\lambda)
=&\sum_{k=1}^la_k\lambda^{p_k-1}\int_{\mathbb{E}}\omega^h |u|^{p_k+1}\,d\sigma
 -\sum_{j=1}^sb_j\lambda^{q_j-1}\int_{\mathbb{E}}\omega^h |u|^{q_j+1}\,d\sigma\\
=&\lambda^{p-1}\Big(\sum_{k=1}^ja_k\lambda^{p_k-p}\int_{\mathbb{E}}\omega^h|u|^{p_k+1}
 \,d\sigma-\sum_{j=1}^sb_j\lambda^{q_j-p}\int_{\mathbb{E}}\omega^h |u|^{q_j+1}
 \,d\sigma\Big)\\
=&: \lambda^{p-1}g^*(\lambda).
\end{align*}
Note that $g^*(\lambda)$ is increasing on $0<\lambda<\infty$, and that
$\lim_{\lambda\to0^+}g^*(\lambda)=-\infty$, 
$\lim_{\lambda\to+\infty}g^*(\lambda)=+\infty$. Hence there exists a unique 
$\lambda_0>0$ such that $g^*(\lambda_0)=0$, therefore $g(\lambda_0)=0$,
 $g(\lambda)<0$ for $0<\lambda<\lambda_0$, $g(\lambda)>0$ 
for $\lambda_0<\lambda<\infty$ and $g(\lambda)$ is increasing on 
$\lambda_0<\lambda<\infty$. Hence for any 
$\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}\geq 0$, 
there exists a unique $\lambda^*>\lambda_0$ such that
\begin{align*}
&\sum_{k=1}^la_k\lambda^{p_k-1}\int_{\mathbb{E}}\omega^h |u|^{p_k+1}\,d\sigma
-\sum_{j=1}^sb_j\lambda^{q_j-1}\int_{\mathbb{E}}\omega^h |u|^{q_j+1}\,d\sigma\\
&=\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\,.
\end{align*}

(3) Note that 
\[
 \frac{d}{d\lambda}J(\lambda u)
=\lambda(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma-g(\lambda)).
\]
 From the proof of (2), it follows that if $0<\lambda\leq\lambda_0$ then
 $g(\lambda)\leq0$, if $\lambda_0<\lambda<\lambda^*$ then 
\[
 0<g(\lambda)<\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma,
\]
 if $\lambda^*<\lambda<\infty$ then 
$ g(\lambda)>\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma$. Hence, we have
 $\frac{d}{d\lambda}J(u)>0$ for $0<\lambda<\lambda^*$, 
$\frac{d}{d\lambda}J(u)<0$ for $\lambda^*<\lambda<\infty$. 
From this, the conclusion of (3) follows.

(4) The conclusion follows from the proof of (3) and
\begin{align*}
I(\lambda u)
=&\lambda^2\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\lambda^2\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma
 -\lambda^{p_k+1}\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma \\
 &+\lambda^{q_j+1}\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma \\
=&\lambda \frac{d}{d\lambda}J(\lambda u).
\end{align*}
\end{proof}

\begin{lemma}\label{le4.1}
Let $\delta>0$, if $ 0<\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\leq \gamma^2(\delta)$, then
 $I_{\delta}(u)>0$. In particular, if 
$ 0<\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\leq \gamma^2(1)$,
then $I(u)>0$, where $\gamma(\delta)$ is the unique real root of equation 
$\varphi(\gamma)=\delta$, and
\begin{equation*}
\varphi(\gamma)=\sum_{k=1}^l a_kC_*^{p_k+1}\gamma^{p_k-1}.
\end{equation*}
\end{lemma}

\begin{proof}
From $ 0<\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma+\int_{\mathbb{E}}
\omega^h V|u|^2\,d\sigma\leq \gamma^2(\delta)$, we have
 $ \int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma>0, \ 1\leq j\leq s$ and by
\begin{align*}
&\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 -\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma \\
&<\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma\\
&<\sum_{k=1}^la_kC_*^{p_k+1}\int_{\mathbb{E}}\omega^h|\nabla_{\mathbb{E}}u|^{p_k+1}
 \,d\sigma\\
&=\varphi(\int_{\mathbb{E}}\omega^h|\nabla_{\mathbb{E}}u|\,d\sigma)
 \int_{\mathbb{E}}\omega^h|\nabla_{\mathbb{E}}u|^2\,d\sigma
 \leq\delta\int_{\mathbb{E}}\omega^h|\nabla_{\mathbb{E}}u|^2\,d\sigma,
\end{align*}
we get $I_{\delta}(u)>0$.
\end{proof}

\begin{lemma}\label{le4.2}
Let $\delta>0$, if $I_{\delta}(u)<0$, then 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma> \gamma^2(\delta)$.
In particular, if  $I_{\delta}(u)>0$, then 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma> \gamma^2(1)$.
\end{lemma}

\begin{proof}
From $I_{\delta}(u)<0$, we have
\begin{align*}
&\delta\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\delta\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma \\
&<\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 -\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\\
&<\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
\leq \varphi(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|\,d\sigma)
\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})},
\end{align*}
which implies $ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma>\gamma^2(\delta)$.
\end{proof}

\begin{lemma}\label{le4.3}
let $\delta>0$, if $I_{\delta}(u)=0$, then 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma> \gamma^2(\delta)$ or
\[
 \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma=0.
\]
 In particular, if $I(u)=0$, then 
\[
 \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma> \gamma^2(1)
\]
 or 
\[
 \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma=0.
\]
\end{lemma}

\begin{proof}
If $ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma=0$, then $I_{\delta}(u)=0$.
If $I_{\delta}(u)=0$ and 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma=0$, then $I_{\delta}(u)\neq0$, then by
\begin{align*}
&\delta(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma)\\
&=\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma-\sum_{j=1}^sb_j
\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\\
&<\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
\leq\varphi(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|\,d\sigma)
\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})},
\end{align*}
we get $ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma> \gamma^2(\delta)$.
\end{proof}

\begin{theorem}\label{th0}
The function of $\delta$, $d(\delta)$ possesses the following properties:
\begin{enumerate}
\item $d(\delta)>a(\delta)\gamma^2(\delta)$ for 
 $ 0<\delta<\frac{(p+1)(1+C^{*2})}{2}$ where 
 $ a(\delta)=\frac{1}{2}(1+C^{*2})-\frac{\delta}{p+1}$.

\item  $ \lim_{\delta\to0}d(\delta)>0$ and there exists a unique
 $ \delta_0>\frac{(p+1)(1+C^{*2})}{2}$ such that $d(\delta_0)=0$ and 
$d(\delta)>0$ for $0\leq\delta<\delta_0$.

\item  $d(\delta)$ is increasing on $0\leq\delta\leq1$, decreasing on 
$1\leq\delta\leq\delta_0$ and takes the maximum $d=d(1)$ at $\delta=1$.
\end{enumerate}
\end{theorem}

\begin{proof}
(1) If  $I_{\delta}(u)=0$ and 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\neq0$, then by Lemma \ref{le4.3}, we have
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma>\gamma^2(\delta)$ and
\begin{align*}
J(u)=&\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\frac{1}{2}\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\\
&-\sum_{k=1}^l\frac{a_k}{p_k+1}\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 +\sum_{j=1}^s\frac{b_j}{q_j+1}\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\\
\geq&\frac{1}{2}(1+C^{*2})\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma\\
&-\frac{1}{p+1}\Big(\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 -\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\Big)\\
=&\Big(\frac{1}{2}(1+C^{*2})-\frac{\delta}{p+1}\Big)
 \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\frac{1}{p+1}I_{\delta}(u)\\
=&a(\delta)\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
>a(\delta)\gamma^2(\delta).
\end{align*}
If $I_{\delta}(u)=0$ and $ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2
\,d\sigma+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma=0$,
by the definition of $d(\delta)$, which is a contradiction.

(2) First $I_0(u)=0$ implies 
\[
 \sum_{k=1}^l\frac{a_k}{p_k+1}\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
=\sum_{j=1}^s\frac{b_j}{q_j+1}\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma
\]
 and
\begin{align*}
J(u)=&\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\\
&-\sum_{k=1}^l\frac{a_k}{p_k+1}\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 +\sum_{j=1}^s\frac{b_j}{q_j+1}\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\\
\geq&\frac{1}{2}(1+C^{*2})\int_{\mathbb{E}}\omega^h|\nabla_{\mathbb{E}}u|^2\,d\sigma
 -\frac{1}{p+1}\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma\\
& +\frac{1}{q+1}\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma)\\
=&\frac{1}{2}(1+C^{*2})\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\frac{p-q}{(p+1)(q+1)}\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma\\
=&\frac{1}{2}(1+C^{*2})\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\frac{p-q}{(p+1)(q+1)}\sum_{k=1}^lb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma.
\end{align*}
Next, we prove that $d(0)>0$. Let
\begin{equation*}
J_1(u)=\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\frac{1}{2}\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma
+\frac{p-q}{(p+1)(q+1)}\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma,
\end{equation*}
and $ d_1(0)=\inf_{u\in\mathcal{N}_0}J_1(u)$. The fact that $J(u)\geq J_1(u)$ 
for $u\in\mathcal{N}_0$ implies $d(0)\geq d_1(0)$. In what follows, we prove
 $d_1(0)>0$. Let
\begin{equation*}
L_1(\nabla_{\mathbb{E}}u,u)=J(u)=\frac{1}{2}|\nabla _{\mathbb{E}}u|^2
+\frac{1}{2}V|u|^2+\frac{p-q}{(p+1)(q+1)}\sum_{k=1}^la_k|u|^{p_k+1}.
\end{equation*}
Then $ J_1(u)=\int_{\mathbb{E}}\omega^hL_1(\nabla_{\mathbb{E}}u,u)\,d\sigma$.
Since $L_1(\nabla_{\mathbb{E}}u,u)$ is convex in $|\nabla_{\mathbb{E}}u|$ and 
satisfies
\begin{equation*}
L_1(\nabla_{\mathbb{E}}u,u)>\alpha|\nabla_{\mathbb{E}}u|-\beta
\end{equation*}
for $ \alpha=\frac{1}{2}(1+C^{*2})$ and any $\beta>0$, from the theory of 
functional minimization, it follows that there exists a $\mu\in\mathcal{N}_0$ 
such that $ d_1(0)=\inf_{u\in\mathrm{N}_0}J_1(u)=J_1(\mu)>0$, hence we have
 $d(0)>0$. Next, for any $u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}(\mathbb{E})$, 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma+\int_{\mathbb{E}}
\omega^h V|u|^2\,d\sigma\neq 0$ and $\delta>0$, we define
$\lambda=\lambda(\delta)$ such that
\begin{equation}\label{50}
\begin{aligned}
&\delta\Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}\lambda u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|\lambda u|^2\,d\sigma\Big) \\
&=\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|\lambda u|^{p_k+1}\,d\sigma
-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|\lambda u|^{q_j+1}\,d\sigma.
\end{aligned}
\end{equation}
Then $I_{\delta}(\lambda u)=0$ and
\begin{equation}\label{51}
\begin{aligned}
  &   \delta\Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\Big) \\
&=    \sum_{k=1}^la_k\lambda^{p_k-1}\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 -\sum_{j=1}^sb_j\lambda^{q_j-1}\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma.
\end{aligned}
\end{equation}
From the proof of (2) in Lemma \ref{le3.2}, it follows that for any $\delta>0$, 
there exists a unique $\lambda=\lambda(\delta)$ such that \eqref{50} and 
\eqref{51} hold. Again by the proof of Lemma \ref{le3.2}, we have
 $\lim_{\delta\to\infty}\lambda(\delta)=+\infty$. 
Hence, by (1) of Lemma \ref{le3.2}, we get 
$\lim_{\delta\to\infty}J(\delta u)=\lim_{\lambda\to\infty}J(\lambda u)=-\infty$. 
From this and (1) of this theorem, it follows that there exists a unique 
$ \delta_0>\frac{(p+1)(1+C^{*2})}{2}$ such that $d(\delta_0)=0$ and $d(\delta)>0$ 
for $0\leq\delta<\delta_0$.

(3) We prove that $d(\delta')<d(\delta'')$ for any $0\leq\delta'<\delta''<1$ or 
$1<\delta''<\delta'<b$ and any 
$u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$, $I_{\delta''}(u)=0$ 
and $\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}\neq0$ 
there exists a 
$v\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$, $I_{\delta'}(v)=0$ and 
$\|\nabla_{\mathbb{E}}v\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}\neq0$, 
a constant $\varepsilon(\delta',\delta'')>0$ such that 
$J(v)<J(u)-\varepsilon(\delta',\delta'')$. In fact, for above $u$, we also define 
$\lambda(\delta)$ by \eqref{50}, then $I_{\delta}(\lambda(\delta)u)=0$, 
$\lambda(\delta'')=1$ and \eqref{51} holds. 
Assume that $\varphi(\lambda)=J(\lambda u)$, we obtain
\begin{align*}
\frac{d}{d\lambda}\varphi(\lambda)
=&\frac{1}{\lambda}\Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma
 -\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma\\
 &+\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\Big)\\
=&\frac{1}{\lambda}\Big((1-\delta)\Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2d
 \sigma+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\Big)+I_{\delta}(\lambda u)\Big)\\
=&(1-\delta)\lambda\Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\Big).
\end{align*}
Taking $v=\lambda(\delta')u$, we have $I_{\delta'}(v)=0$ and 
$\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}\neq 0$.
If $0<\delta'<\delta''<1$, then
\begin{equation*}
J(u)-J(v)=\varphi(1)-p(\lambda(\delta'))>(1-\delta'')\gamma^2(\delta'')
\lambda(\delta')(1-\lambda(\delta'))\equiv \varepsilon(\delta',\delta'').
\end{equation*}
If $1<\delta''<\delta'<\delta_0$, then
\begin{equation*}
J(u)-J(v)=\varphi(1)-p(\lambda(\delta'))>(\delta''-1)
\gamma^2(\delta'')\lambda(\delta')(\lambda(\delta')-1)
\equiv \varepsilon(\delta',\delta'').
\end{equation*}
\end{proof}

\begin{lemma}\label{le4.4}
Let $ 0<\delta<\frac{(p+1)(1+C^{*2})}{2}$, assume that 
$J(u)\leq d(\delta)$ and $I_{\delta}(u)>0$. Then
\begin{equation*}
0<\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
<\frac{d(\delta)}{a(\delta)}.
\end{equation*}
In particular, if $J(u)\leq d$ and $I_{\delta}(u)>0$, one has
\begin{equation*}
0<\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
<\frac{2(p+1)}{(p-1)(1+C^{*2})}d.
\end{equation*}
\end{lemma}

The above lemma follows from
\begin{equation*}
a(\delta)\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2+\frac{1}{p+1}I_{\delta}(u)<J(u)\leq d(\delta).
\end{equation*}
Now we define a family of potential wells as follows
\begin{gather*}
W_{\delta}=\big\{u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}):
I_{\delta}(u)>0,J(u)<d(\delta)\big\}\cup\{0\}, \quad 0<\delta<\delta_0;\\
\bar{W}_{\delta}=\big\{u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}):
I_{\delta}(u)>0,J(u)<d(\delta)\big\}, \quad 0<\delta<\delta_0,
\end{gather*}
and
\begin{gather*}
V_{\delta}=\big\{u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}):
I_{\delta}(u)<0,J(u)<d(\delta)\big\},\quad 0<\delta<\delta_0;\\
B_{\delta}=\big\{u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}):
\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}\lambda u|^2\,d\sigma
 +\int_{\mathbb{E}}\omega^h V|\lambda u|^2\,d\sigma<\gamma^2(\delta)\big\},\quad
 0<\delta<\delta_0;\\
\bar{B}_{\delta}=B_{\delta}\cup\partial B_{\delta}
=\big\{u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}):
\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}\lambda u|^2\,d\sigma
 +\int_{\mathbb{E}}\omega^h V|\lambda u|^2\,d\sigma\leq \gamma^2(\delta)\big\},\\
 0<\delta<\delta_0;\\
B_{\delta}^c=\big\{u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}):
\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}\lambda u|^2\,d\sigma
 +\int_{\mathbb{E}}\omega^h V|\lambda u|^2\,d\sigma>\gamma^2(\delta)\big\},\quad
 0<\delta<\delta_0.
\end{gather*}

\subsection{Characteristics of solutions}

In this subsection, we state the invariance of some sets under the flow
 of \eqref{1.1} and the vacuum isolating behavior of problem \eqref{1.1}.

\begin{definition}[Maximal existence time]\label{de4} \rm
Let $u(t)$ be a weak solution of problem \eqref{1.1}. We define the maximal 
existence time $T_{\rm max}$ of $u(t)$ as follows:
\begin{enumerate}
\item If $u(t)$ exists for $0\leq t<\infty$, then $T_{\rm max}=+\infty$.
\item If there exists a $t_0\in(0,\infty)$ such that $u(t)$ exists
for $0\leq t<t_0$, but doesn't exist at $t = t_0$, then $T_{\rm max} = t_0$.
\end{enumerate}
\end{definition}

Now, we discuss the invariance of some sets corresponding to problem \eqref{1.1}.

\begin{theorem}\label{th1}
Let $u_0\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}(\mathbb{E})$,
$u_{1}\in \mathcal{L}_2^{\frac{n+1}{2}}(\mathbb{E})$, and $0<e<d$,
 $\delta_{1}<\delta_2$ are two roots of equation $d(\delta) = e$. Then:
\begin{enumerate}
\item All weak solutions of problem \eqref{1.1} with $0<J(u_0)\leq e$
belong to $W_{\delta}$ for $\delta_{1}<\delta<\delta_2$, $0\leq t < T_{\rm max}$,
provided $I(u_0)>0$ or
$\| \nabla_{\mathbb{E}}u_0\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})}  = 0$.
\item All weak solutions of problem \eqref{1.1} with $0<J(u_0)\leq e$ belong to
$V_{\delta}$ for $\delta_{1}<\delta<\delta_2$, $0\leq t < T_{\rm max}$,
 provided $I(u_0) < 0$,
\end{enumerate}
where $T_{\rm max}$ is the maximal existence time of $u(t)$.
\end{theorem}

\begin{proof}
(1) Let $u(t)$ be any weak solution of problem \eqref{1.1} with
 $J(u_0)\leq e$, $I(u_0)>0$ or
$\| \nabla_{\mathbb{E}}u\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})} = 0$.
$T_{\rm max}$ is the existence time of $u(t)$. 
If $\| \nabla_{\mathbb{E}}u_0\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})}  = 0$,
then $u_0(x)\in W_{\delta}$. If $I(u_0) > 0$, by the definition of $d(\delta)$, 
it follows $I_{\delta}(u_0)>0$ and $J(u_0)<d(\delta)$. 
Then $u_0(x) \in W_{\delta}$ for $\delta_1 < \delta < \delta_2$. Next, we 
should prove $u(t)\in W_{\delta}$ for $\delta_1 < \delta < \delta_2$ and 
$0<t<T_{\rm max}$. Arguing by contradiction, by the continuity of 
$I(u)$ we suppose that there must exist a $\delta_0\in (\delta_{1},\delta_2)$ 
and $t_0\in(0,T_{\rm max})$ such that $u(t_0)\in \partial W_{\delta_0}$, and 
$I_{\delta_0}(u(t_0)) = 0$, 
$\| \nabla_{\mathbb{E}}u\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})} \neq 0$ or 
$J(u(t_0)) = d(\delta_0)$. From
\begin{equation} \label{48}
  \int_0^{t}\| u_{\tau}\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})}^{2}\,d\tau
+ \int_0^{t}\| \nabla_{\mathbb{E}}u\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})}^{2}\,d\tau
+ J(u(t)) \leq J(u_0)<d(\delta),
\end{equation}
 $\delta_1 < \delta < \delta_2$,  $0\leq t<T_{\rm max}$.
we can see that $J(u(t_0))\neq d(\delta_0)$.
If $I_{\delta_0}(u(t_0)) = 0$,
$\| \nabla_{\mathbb{E}}u\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})}^{2} \neq 0$,
then by the definition of $d(\delta)$, we have $J(u(t_0))\geq d(\delta_0)$,
 which contradicts \eqref{48}.

(2) Let $u(t)$ be a weak solution of problem \eqref{1.1} with $0<J(u_0)\leq e < d$,
 $I(u_0)<0$. From $J(u_0) \leq e$, $I(u_0)<0$ and
\begin{equation*}
\frac{1}{2}\| u_1\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})}^{2}+J(u_0)
=E(0)=d(\delta_1)=d(\delta_2)<d(\delta),\quad \delta_1<\delta<\delta_2,
\end{equation*}
it follows $I_{\delta}(u_0)<0$ and $J(u_0)<d(\delta)$. Then 
$u_0(x)\in V_{\delta}$ 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 must exist a $\delta_0\in(\delta_{1},\delta_2)$ and 
$t_0\in(0,T_{\rm max})$ such that $u(t_0)\in \partial V_{\delta_0}$, and 
$I_{\delta_0}(u(t_0)) = 0$ or $J(u(t_0)) = d(\delta_0)$. By \eqref{48} 
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 $I_{\delta_0}(u(t)) = 0$, then
 $I_{\delta_0}(u(t)) < 0$ for $0\leq t < t_0$. By Lemma \ref{le4.2} we have 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma> \gamma^2(\delta_0)$ for $0\leq t< t_0$.
 Hence $ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma \geq \gamma^2(\delta_0)$, then
$\| u(t_0) \|_{\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}(\mathbb{E})}\neq 0$. 
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 to \eqref{48}.
\end{proof}

To discuss the invariance of the solutions with negative level energy, 
we introduce the following results.

\begin{proposition}\label{bP1}
All nontrivial solutions of problem \eqref{1.1} with $J(u_0) = 0$ belong to
\begin{align*}
B_{\gamma_0}^{c} = \big\{  u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}(\mathbb{E}) | 
 \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\geq \gamma_0^2 \big\},
\end{align*}
where $\gamma_0$ is the unique real root of equation
\begin{align}
\sum_{k=1}^la_kC_k^{p_k+1}\gamma^{p_k-1} = \frac{1}{2}.
\end{align}
\end{proposition}

\begin{proof}
Let $u(t)$ be any solution of problem \eqref{1.1} with $J(u_0) = 0$, 
$T_{\rm max}$ is the maximal existence time of $u(t)$. From the energy equality
\begin{equation*}
\frac{1}{2}\|\partial_tu\|^2+J(u)\equiv E(0)=0,
\end{equation*}
we get $J(u)\leq 0$ for $0\leq t  < T_{\rm max}$. Hence from
\begin{align*}
&\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 -\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\\
\leq&\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 \leq \sum_{k=1}^la_kC_k^{p_k+1}\|\nabla_{\mathbb{E}}
 u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^{p+1}, \quad 0\leq t < T_{\rm max},
\end{align*}
it follows that either 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma = 0$ or
 $ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\geq \gamma_0^2$.
 If $ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma = 0$,
$\| \nabla_{\mathbb{E}}u\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})}\equiv 0$ for 
$0\leq t < T_{\rm max}$. 
Otherwise, there exists a $t_0 \in (0,T_{\rm max})$ such that 
$ 0 <\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma <\gamma_0^2$.
By a similar argument we can prove that if 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\geq \gamma_0^2$, then
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\geq \gamma_0^2$ for $0<t<T_{\rm max}$.
\end{proof}

\begin{theorem}\label{bth2}
Let $u_0\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}(\mathbb{E})$. Assume that 
$J(u_0)<0$ or $J(u_0) = 0$ and 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\neq 0$.
Then all solutions of problem \eqref{1.1} belong to $V_{\delta}$ for 
$ 0<\delta<\frac{(p+1)(1+C^{*2})}{2}$.
\end{theorem}

\begin{proof}
Let $u(t)$ be any solution of problem \eqref{1.1} with $J(u_0)<0$ or $J(u_0) = 0$ 
and $ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\neq 0$, $T_{\rm max}$
is the maximal existence time of $u(t)$. The energy inequality gives
\begin{equation}\label{49}
\begin{gathered}
a(\delta)\| \nabla_{\mathbb{E}}u\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})}^{2} 
+ \frac{1}{p+1}I_{\delta}(u) \leq J(u) \leq J(u_0),\\
0<\delta<\frac{(p+1)(1+C^{*2})}{2}.
\end{gathered}
\end{equation}
From \eqref{49} it follows that if $J(u_0) < 0$, then $I_{\delta}(u) < 0$ and 
$J(u) < 0 < d(\delta)$ for $ 0<\delta<\frac{(p+1)(1+C^{*2})}{2}$; 
if $J(u_0) = 0$ and 
$\| \nabla_{\mathbb{E}}u_0\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})}\neq 0$, 
then by Proposition \ref{bP1} we have
 $\| \nabla_{\mathbb{E}}u\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})}\geq \gamma_0$ 
for $0\leq t < T_{\rm max}$. Again by \eqref{49} we get 
$I_{\delta}(u) < 0$ and $J(u) < 0 < d(\delta)$ for 
$ 0<\delta<\frac{(p+1)(1+C^{*2})}{2}$. Hence for above two cases we always 
have $u(t)\in V_{\delta}$ for $ 0<\delta<\frac{(p+1)(1+C^{*2})}{2}$, 
$0\leq t < T_{\rm max}$.
\end{proof}

\begin{corollary}\label{bc1}
Let $u_0\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}(\mathbb{E})$. Assume that 
$J(u_0)<0$ or $J(u_0) = 0$ and 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\neq 0$.
Then all weak solutions of problem \eqref{1.1} belong to 
$ \bar{B}^{c}_{\frac{(p+1)(1+C^{*2})}{2}}$.
\end{corollary}

\begin{proof}
Let $u(t)$ be any weak solution of problem \eqref{1.1} with $J(u_0)<0$ or 
$J(u_0) = 0$ and
 $ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\neq 0$, $T_{\rm max}$
is the maximal existence time of $u(t)$. Then Theorem \ref{bth2} gives
\begin{align*}
u(t)\in V_{\delta} \quad\text{for } 
0< \delta<\frac{(p+1)(1+C^{*2})}{2}, \; 0\leq t<T_{\rm max}.
\end{align*}
From this and Lemma \ref{le4.2} we get 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\leq \gamma^2(\delta)$
for $ 0 < \delta <\frac{(p+1)(1+C^{*2})}{2}$, $0\leq t < T_{\rm max}$. 
Then, letting $ \delta \to \frac{(p+1)(1+C^{*2})}{2}$, we obtain 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma
\geq \gamma^2(\frac{(p+1)(1+C^{*2})}{2})$ for $0\leq t < T_{\rm max}$.
\end{proof}

Next, we discuss the vacuum isolating to problem \eqref{1.1} with $J(u_0)<d$.

\begin{theorem}\label{bth3}
Let $e\in (0,d)$. Suppose $\delta_{1},\delta_2$ are the two solutions of 
$d(\delta) = e$. Then for all weak solutions of problem \eqref{1.1} with 
$J(u_0)\leq e$, there is a vacuum region
\[
U_{e} = \big\{ u\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}(\mathbb{E})| I_{\delta}(u) 
= 0, \| \nabla_{\mathbb{E}}u\|_{L_2^{\frac{n+1}{2}}(\mathbb{E})} \neq 0,
 \delta_{1}<\delta<\delta_2  \big\},
\]
such that there is no any weak solution of problem \eqref{1.1} in $U_{e}$.
\end{theorem}

\begin{proof}
 Assume that $u(t)$ is any weak solution of problem \eqref{1.1} with 
$J(u_0) \leq e$, $T_{\rm max}$ is the maximal existence time of $u(t)$. 
We only need to prove that if 
$ \int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma \neq 0$ and $J(u_0) \leq e$,
then for all $\delta\in(\delta_{1},\delta_2)$, $u(t)\notin N_{\delta}$, 
i.e. $I_{\delta}(u(t))\neq 0$, for all $t\in[0,T_{\rm max})$.

At first, it is clear that $I_{\delta}(u_0)\neq 0$. Since if 
$I_{\delta}(u_0) = 0$, then $J(u_0)\geq d(\delta) > d(\delta_{1}) = d(\delta_2)$, 
which contradicts with $J(u_0) \leq e$.

Suppose there exists $t_{1} > 0$ such that $u(t_{1})\in U_{e}$. 
Namely, there must exist a $\delta_0 \in (\delta_{1},\delta_2)$ such that 
$u(t_{1})\in N_{\delta_0}$. From the definition of $d(\delta)$, 
we get $J(u_0)\geq J(u(t_{1})) \geq d(\delta) > J(u_0)$, 
which leads to a contradiction.
\end{proof}

\section{Global solution and exponential decay}\label{se3}

In this section, we prove the existence of global solutions by using 
the Galerkin approximation technique and the potential well theory. 
Meanwhile, we give an exponential decay result of the solution.

\begin{theorem}\label{th1.1}
Let $u_0\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$. 
Assume that $E(0)\leq d$ and $I(u)>0$. Then problem \eqref{1.1} 
admits a global weak solution 
$u\in L^{\infty}(0,T;\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}))$ with 
$\partial_tu\in L^{\infty}(0,T;\mathcal{L}_2^{\frac{n+1}{2}}({\mathbb{E}}))
\cap \mathcal{L}_2^{\frac{n+1}{2}}(Q_T)$, where $Q_T={\mathbb{E}}\times[0,T]$.
\end{theorem}

\begin{proof}
 (1) Low initial energy case ($E(0)<d$ and $I(u)>0$): 
Let $\left\{{\varphi}_j\right\}$ be a sequence of orthogonal basis of 
$\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$, we introduce the 
following approximate solutions of problem \eqref{1.1}:
\begin{equation*}
u_m(t,\omega,x,y)=\sum_{j=1}^m f_{jm}(t){\varphi}_j(\omega,x,y) \quad m=1,2,\ldots,
\end{equation*}
 which satisfies
\begin{equation} \label{2.4}
\begin{aligned}
&\int_{\mathbb{E}}\omega^h\partial_{tt}u_m\cdot {\varphi}_k\,d\sigma
 +\int_{\mathbb{E}}\omega^h\nabla_{\mathbb{E}}u_m\cdot
 \nabla_{\mathbb{E}}{\varphi}_k\,d\sigma+\int_{\mathbb{E}}\omega^h\partial_tu_m
 \cdot {\varphi}_k\,d\sigma \\
& +\int_{\mathbb{E}}\omega^h Vu_m\cdot {\varphi}_k\,d\sigma \\
&=\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u_m|^{p_k}u_m\cdot {\varphi}_k\,d\sigma
 -\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u_m|^{q_j}u_m\cdot {\varphi}_k\,d\sigma
\end{aligned}
\end{equation}
for $k=1,2,\ldots,m$.
As $m\to+\infty$,
\begin{gather}\label{2.5}
u_m(0,\omega,x,y)=\sum_{j=1}^mc_{jn}(0){\varphi}_j(\omega,x,y)
\to u_0(\omega,x,y)\quad \text{in }  \mathcal{H}_{2,0}^{1 \frac{N}{2}}, \\
\label{2.6}
u_{mt}(0,\omega,x,y)=\sum_{j=1}^md_{jn}(0){\varphi}_j(\omega,x,y)\to u_1(\omega,x,y)
\quad \text{in } \mathcal{L}_2^{\frac{N}{2}}.
\end{gather}
Multiplying \eqref{2.4} by $f_{im}'(t)$ and summing for $i$, we have
\begin{align*}
&\sum_{i=1}^m\int_{\mathbb{E}}\omega^h\partial_{tt}u_m\cdot f_{im}'(t)
 {\varphi}_k\,d\sigma
 +\sum_{i=1}^m\int_{\mathbb{E}}\omega^h\nabla_{\mathbb{E}}u_m\cdot f_{im}'(t)
 \nabla_{\mathbb{E}}{\varphi}_k\,d\sigma\\
&+\sum_{i=1}^m\int_{\mathbb{E}}\omega^h\partial_tu_m\cdot
 f_{im}'(t){\varphi}_k\,d\sigma+\sum_{i=1}^m\int_{\mathbb{E}}\omega^hVu_m\cdot
 f_{im}'(t){\varphi}_k\,d\sigma\\
&=\sum_{i=1}^m\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u_m|^{p_k}u_m\cdot
  f_{im}'(t){\varphi}_k\,d\sigma \\
&\quad -\sum_{i=1}^m\sum_{j=1}^sb_j
 \int_{\mathbb{E}}\omega^h|u_m|^{q_j}u_m\cdot f_{im}'(t){\varphi}_k\,d\sigma.
\end{align*}
Hence
\begin{equation}\label{2.7}
\begin{aligned}
& \frac{1}{2}\frac{d}{dt}\int_{\mathbb{E}}\omega^h\partial_tu_m\cdot
 \partial_tu_m \,d\sigma+\frac{1}{2}\frac{d}{dt}
 \int_{\mathbb{E}}\omega^h\nabla_{\mathbb{E}}u_m\cdot \nabla_{\mathbb{E}}u_m 
 \,d\sigma\\
& +\frac{1}{2}\frac{d}{dt}\int_{\mathbb{E}}\omega^h V|u_m|^2\,d\sigma
  -\sum_{k=1}^l\frac{a_k}{p_k+1}\frac{d}{dt}\int_{\mathbb{E}}
 \omega^h|u_m|^{p_k+1}\,d\sigma \\
& +\sum_{j=1}^s\frac{b_j}{q_j+1}
 \frac{d}{dt}\int_{\mathbb{E}}\omega^h|u_m|^{q_j+1}\,d\sigma 
+\int_{\mathbb{E}}\omega^h\partial_tu_m\cdot \partial_tu_m \,d\sigma
=   0.
\end{aligned}
\end{equation}
Integrating \eqref{2.7} with respect $t$ from $0$ to $t$, we get
\begin{align*}
&\frac{1}{2}\int_{\mathbb{E}}\omega^h|\partial_tu_m|^2 \,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla_{\mathbb{E}}u_m|^2\,d\sigma
+\frac{1}{2}\int_{\mathbb{E}}\omega^h V|u_m|^2\,d\sigma \\
&-\sum_{k=1}^l\frac{a_k}{p_k+1}\int_{\mathbb{E}}\omega^h|u_m|^{p_k+1}\,d\sigma
 +\sum_{j=1}^s\frac{b_j}{q_j+1}\int_{\mathbb{E}}\omega^h|u_m|^{q_j+1}\,d\sigma\\
&+\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^{\frac{n+1}{2}}
 ({\mathbb{E}})}^2\,d\tau\\
&=\frac{1}{2}\int_{\mathbb{E}}\omega^h|u_1|^2 \,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla_{\mathbb{E}}u_0|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^h V|u_0|^2\,d\sigma\\
&\quad -\sum_{k=1}^l\frac{a_k}{p_k+1}\int_{\mathbb{E}}\omega^h|u_0|^{p_k+1}\,d\sigma
 +\sum_{j=1}^s\frac{b_j}{q_j+1}\int_{\mathbb{E}}\omega^h|u_0|^{q_j+1}\,d\sigma
=E(0).
\end{align*}
The next result is given to show the invariant sets of the solutions for
problem \eqref{1.1}.
By \eqref{2.5} and \eqref{2.6}, we have $E(u_m(0))\to E(u_0)$.
Then for sufficiently large $m$, we have
\begin{equation*}
\int_0^t\|\partial_tu_m(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2\,d\tau
+E(u_m)=E(u_0)<d.
\end{equation*}
From $E(0)<d$, \eqref{2.5} and \eqref{2.6} we find $E_m(0)<d$ for sufficiently
large $m$. Hence we have
\begin{equation}\label{2.8}
\int_0^t\|\partial_tu_m(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2\,d\tau
+J(u_m)+\frac{1}{2}\int_{\mathbb{E}}\omega^h|\partial_tu_m|^2 \,d\sigma<d \quad
 0\leq t<\infty.
\end{equation}

For $0\leq t<\infty$, we have $u_m(t)\in W$ with $m$ sufficiently large. 
In fact, if it is false, then there exists $t_0>0$ such that 
$u_m(t_0)\in \partial W$ which implies that $I(u_m(t_0))=0$ or $E(u_m(t_0))=d$. 
Since $E(u_m(t))<E(u_0)<d$, we know that $I(u_m(t_0))=0$. 
From the definition of $d$, we have that $J(u_m(t_0))>d$, which is contradiction. 
Thus for sufficiently large $m$, $u_m(t)\in W$ and $I(u_m)>0$. Then we obtain
\begin{align*}
J(u_m)
&=\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u_m|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^h V|u_m|^2\,d\sigma
 -\sum_{k=1}^l\frac{a_k}{p_k+1}\int_{\mathbb{E}}\omega^h|u_m|^{p_k+1}\,d\sigma\\
&\quad +\sum_{j=1}^s\frac{b_j}{q_j+1}\int_{\mathbb{E}}\omega^h|u_m|^{q_j+1}\,d\sigma\\
&\geq\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u_m|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^h V|u_m|^2\,d\sigma\\
&\quad -\frac{1}{p+1}\Big(\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 -\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\Big)\\
&=\big(\frac{1}{2}-\frac{1}{p+1}\big)
\Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u_m|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u_m|^2\,d\sigma\Big)-\frac{1}{p+1}I(u_m)\\
&\geq\frac{p-1}{2(p+1)}(1+{C^*}^2)
\|\nabla_{\mathbb{E}}u_m\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2.
\end{align*}
From \eqref{2.8} we have that for $0\leq t<\infty$ and sufficiently large $m$,
\begin{align*}
&\frac{1}{2}\int_{\mathbb{E}}\omega^h|u_{mt}|^2 \,d\sigma
 +\frac{p-1}{2(p+1)}(1+{C^*}^2)\|\nabla_{\mathbb{E}}u_m\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2+\int_0^t\|\partial_tu(\tau)
 \|_{\mathcal{L}_2^{\frac{n+1}{2}}({\mathbb{E}})}^2\,d\tau\\
&\leq \frac{1}{2}\int_{\mathbb{E}}\omega^h|u_{mt}|^2\,d\sigma+J(u_m)
 +\int_0^t\|\partial_tu(\tau)
\|_{\mathcal{L}_2^{\frac{n+1}{2}}({\mathbb{E}})}^2\,d\tau<d.
\end{align*}
That means
\begin{equation*}
\|\nabla_{\mathbb{E}}u_m\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
<\frac{2(p+1)}{p-1}\big(1+{C^*}^2\big)^{-1}d.
\end{equation*}
From Lemma \ref{le3.1} we have
\begin{gather*}
\|u_m\|_{\mathcal{L}_2^\frac{p_k+1}{2}}\leq C_k^2\|\nabla_{\mathbb{E}}u_m\|^2
<C_k^2\frac{2(p+1)}{p-1}(1+{C^*}^2)^{-1}d, \\
\|u_m\|_{\mathcal{L}_2^\frac{q_j+1}{2}}\leq C_j^2\|\nabla_{\mathbb{E}}u_m\|^2
<C_j^2\frac{2(p+1)}{p-1}(1+{C^*}^2)^{-1}d, \\
\int_{\mathbb{E}}\omega^hV|u_m|^2\,d\sigma
\leq  C^2\|\nabla_{\mathbb{E}}u_m\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
<C^2\cdot\Big[ \frac{2(p+1)}{p-1}(1+{C^*}^2)^{-1}d\Big]^2, \\
\int_0^t\|\partial_tu_m(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}\,d\tau<d, \\
\int_{\mathbb{E}}\omega^h|\partial_tu_m|^2 \,d\sigma<2d.
\end{gather*}
Hence, there exist $u$ and a subsequence $\left\{u_m\right\}$ such that as 
$m\to\infty$,
$u_m\to u$ in $L^{\infty}(0,\infty;\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}))$
 wead star and a.e. in $[0,\infty)\times \operatorname{int}({\mathbb{E}})$,
$|u_m|^{p_k-1}u_m\to |u|^{p_k-1}u$ in 
$L^{\infty}(0,\infty;\mathcal{L}_{\frac{p_1+1}{q}}^{\frac{(n+1)q}{p_1+1}}
 ({\mathbb{E}}))$ weak star and a.e. in 
$[0,\infty)\times \operatorname{int}({\mathbb{E}})$,
$|u_m|^{q_j-1}u_m\to |u|^{q_j-1}u$ in 
$L^{\infty}(0,\infty;\mathcal{L}_{\frac{p_1+1}{q}}^{\frac{(n+1)q}{p_1+1}}
({\mathbb{E}}))$ wead star and a.e. in 
$[0,\infty)\times \operatorname{int}({\mathbb{E}})$,
$V|u_m|^2\to V|u|^2$ in 
$L^{\infty}(0,\infty;\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}))$ 
wead star and a.e. in $[0,\infty)\times \operatorname{int}({\mathbb{E}})$,
$\partial_tu_m\to \partial_tu$ in 
$L^2(0,\infty;\mathcal{L}_2^{\frac{n+1}{2}}({\mathbb{E}}))$ wead star.
In \eqref{2.4} we fixed $k$, then letting $m\to \infty$, we have
\begin{equation}\label{2.9}
\begin{aligned}
  &   \int_{\mathbb{E}}\omega^h\partial_{tt}u\cdot {\varphi}_k\,d\sigma
+\int_{\mathbb{E}}\omega^h\nabla_{\mathbb{E}}u\cdot \nabla_{\mathbb{E}}{\varphi}_kd
\sigma+\int_{\mathbb{E}}\omega^h\partial_tu\cdot {\varphi}_k\,d\sigma
+\int_{\mathbb{E}}\omega^h Vu\cdot {\varphi}_k\,d\sigma \\
&=     \sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u_m|^{p_k}u\cdot {\varphi}_k\,d\sigma
-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u_m|^{q_j}u\cdot {\varphi}_k\,d\sigma.
\end{aligned}
\end{equation}
Integrating \eqref{2.9} with respect $t$ from $0$ to $t$, we get
\begin{align*}
&(\partial_tu,{\varphi}_k)_2+\int_0^t(\nabla_{\mathbb{E}}u,
 \nabla_{\mathbb{E}}{\varphi}_k)_2\,d\tau+\int_0^t(Vu,{\varphi}_k)_2\,d\tau
 +\int_0^t(\partial_tu,{\varphi}_k)_2\,d\tau\\
&=\sum_{k=1}^la_k\int_0^t(|u|^{p_k}u,{\varphi}_k)_2\,d\tau
 -\sum_{j=1}^sb_j\int_0^t(|u|^{q_j}u,{\varphi}_k)_2\,d\tau+(u_1,{\varphi}_k)_2,
\end{align*}
Moreover, \eqref{2.5} and \eqref{2.6} give $u(0)=u_0$ in 
$\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$. By density, one has 
that $u$ is a global weak solution of problem \eqref{1.1} and $u(t)\in W$
 where $u\in L^{\infty}(0,\infty;\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}))$ 
with $\partial_tu\in L^{\infty}(0,T;\mathcal{L}_2^{\frac{n+1}{2}}({\mathbb{E}}))
\cap \mathcal{L}_2^{\frac{n+1}{2}}(Q_T)$, where $Q_T={\mathbb{E}}\times[0,T]$.

(2) Critical initial energy case($E(0)=d$ and $I(u)>0$):
First $E(0)=d$ implies that 
$\|\nabla_{\mathbb{E}}u_0\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}\neq 0$. 
Pick a sequence ${\lambda_m}$ such that $0<\lambda_m<1$, $m=1,2,\ldots$ and 
$\lambda_m\to 1$ as $m\to\infty$. Let $u_{0m}(x)=\lambda_m u_0(x)$, we 
consider the initial conditions
\begin{equation}\label{3.3}
u(x,0)=u_{0m}(x)\quad  \text{in } \operatorname{int}(\mathbb{E})
\end{equation}
and the corresponding problem \eqref{1.1} with \eqref{3.3}. 
From $I(u_0)\geq 0$ and Lemma \ref{le3.2}, we have $\lambda^*=\lambda^*(u_0)\geq 1$. 
Thus, we get $I(u_{0m})=I(\lambda_mu_0)>0$ and $J(u_{0m}=J(\lambda_m u_0)<J(u_0)=d$. 
From the low initial case, it follows that for each $m$, 
problem \eqref{1.1} with \eqref{3.3} admits a global weak solution 
$u_m\in L^{\infty}(0,T;\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}))$
 with $\partial_tu_m\in L^{\infty}(0,T;\mathcal{L}_2^{\frac{n+1}{2}}
({\mathbb{E}}))\cap \mathcal{L}_2^{\frac{n+1}{2}}(Q_T)$, where 
$Q_T={\mathbb{E}}\times[0,T]$ and $u_m(t)\in W$ for $0\leq t<\infty$ satisfying
\begin{align*}
&\int_{\mathbb{E}}\omega^h\partial_{tt}u_m\cdot v\,d\sigma
 +\int_{\mathbb{E}}\omega^h\nabla_{\mathbb{E}}u_m\cdot 
\nabla_{\mathbb{E}} v\,d\sigma
+\int_{\mathbb{E}}\omega^h\partial_tu_m\cdot v\,d\sigma
+\int_{\mathbb{E}}\omega^h Vu_m\cdot v\,d\sigma \\
&=\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u_m|^{p_k}u\cdot v \,d\sigma
-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u_m|^{q_j}u\cdot v\,d\sigma, \quad
 \forall v\in\mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}), 
\end{align*}
for $t>0$,
and
\begin{equation}\label{3.4}
\int_0^t\|\partial_tu_m(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}\,d\tau
+J(u_m)+\frac{1}{2}\int_{\mathbb{E}}\omega^h|\partial_tu_m|^2 \,d\sigma
\leq E(u_{0m})<d, 
\end{equation}
for $ 0\leq t<\infty$.
From \eqref{3.4} and
\begin{align*}
J(u_m)&\geq\big(\frac{1}{2}-\frac{1}{p+1}\big)
\Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u_m|^2\,d\sigma
+\int_{\mathbb{E}}\omega^h V|u_m|^2\,d\sigma\Big)-\frac{1}{p+1}I(u_m)\\
&\geq\frac{p-1}{2(p+1)}(1+{C^*}^2)
\|\nabla_{\mathbb{E}}u_m\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2,
\end{align*}
we get
\begin{equation*}
\frac{p-1}{2(p+1)}(1+{C^*}^2)\|\nabla_{\mathbb{E}}u_m
\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2+\int_0^t\|\partial_tu(\tau)
 \|_{\mathcal{L}_2^{\frac{n+1}{2}}({\mathbb{E}})}^2\,d\tau<d, \quad 0\leq t<\infty.
\end{equation*}
The remainder of the proof is similar to the low initial case, then the proof 
is complete.
\end{proof}

\begin{theorem}\label{th1.2}
Let $u_0\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$, $u_0\in W$ 
and $I(u)>0$, then there exist two positive constants $\widehat{C}$ and
 $\zeta$ independent of $t$ such that $0<E(t)\leq \widehat{C} e^{-\zeta t}$ 
for all $t\geq 0$.
\end{theorem}

To prove Theorem \ref{th1.2}, we need the following lemma:

\begin{lemma}\label{le2.6}
Assume that $ 1<p<\frac{N+2}{N-2}$ and $N=n+h+1\geq3$, $u$ is the solution of
problem \eqref{1.1} with
\begin{equation*}
u_0\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}),\quad
u_1\in \mathcal{L}_2^{\frac{n+1}{2}}({\mathbb{E}})\cap 
\mathcal{L}_2^{\frac{n+1}{2}}(Q_T),\quad Q_T={\mathbb{E}}\times[0,T].
\end{equation*}
If
\begin{equation}\label{3.1}
E(0)\leq \frac{2(p+1)}{p-1} \Big(\sum_{k=1}^la_kC_k^{p_1+1}\Big)^{-\frac{2}{p_1-1}}
(1+{C^*}^2),
\end{equation}
then $u\in W$ on $[0,T]$.
\end{lemma}

\begin{proof}
Assume that there exists some time $T^*>0$ such that $u(t)\in W$, where 
$0\leq t<T^*$ and $u(T^*)\in\partial W$, we can obtain $I(u(T^*))=0$ and 
$u(T^*)\neq0$. At the time $I(u(T^*))=0$ and $u(T^*)\neq0$, notice that
\begin{align*}
J(u)
&\geq  (\frac{1}{2}-\frac{1}{p+1})
 \Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\Big)-\frac{1}{p+1}I(u)\\
&\geq \frac{p-1}{2(p+1)}(1+{C^*}^2)\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2
 ^\frac{n+1}{2}({\mathbb{E}})}^2.
\end{align*}
Then, by the energy identity 
$ \int_0^t \|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}\,d\tau+J(u)+\frac{1}{2}\int_{\mathbb{E}}\omega^h|\partial_tu|^2
\,d\sigma\leq E(0)$, we have
\begin{equation}\label{3.7}
\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2  
 \leq \frac{2(p+1)}{p-1}(1+{C^*}^2)^{-1}J(u)\\
   \leq\frac{2(p+1)}{p-1}(1+{C^*}^2)^{-1}E(0).
\end{equation}
It follows from the Sobolev-poincar\'e's inequality, \eqref{3.1} and 
\eqref{3.7}, that
\begin{equation}\label{3.10}
\begin{aligned}
 & \sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 -\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma \\
&\leq\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma\\
&\leq \sum_{k=1}^la_kC_k^{p_k+1}\|\nabla_{\mathbb{E}}
 u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^{p_k+1}\\
&\leq \sum_{k=1}^la_kC_k^{p_1+1}(\frac{2(p+1)}{p-1}(1+{C^*}^2)^{-1}
 E(0))^{\frac{p_1-1}{2}}\|\nabla_{\mathbb{E}}
 u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2,
\end{aligned}
\end{equation}
which implies that $I(u(T^*))>0$ for $t=T^*$. This completes the proof. 
\end{proof}

\begin{proof}[Proof of Theorem \ref{th1.2}]
From
\begin{align*}
E(t)=&\frac{1}{2}\int_{\mathbb{E}}\omega^h|\partial_tu|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
 +\frac{1}{2}\int_{\mathbb{E}}\omega^hV|u|^2\,d\sigma \\
&-\sum_{k=1}^l\frac{a_k}{p_k+1}\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 +\sum_{j=1}^s\frac{b_j}{q_j+1}\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\\
\geq&\frac{1}{2}\int_{\mathbb{E}}\omega^h|\partial_tu|^2\,d\sigma
+(\frac{1}{2}-\frac{1}{p+1})
\Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^hV|u|^2\,d\sigma\Big) \\
&+\frac{1}{p+1}I(u),
\end{align*}
we have $0<E(t)$ for all $t\geq 0$.

The proof of other inequality relies on the construction of a Lyapunov functional. 
And the Lyapunov functional is performed by a suitable modification of energy. 
For this purpose, let $\theta>0$, which will be chosen later, we define
\begin{equation}\label{3.11}
L(t)=E(t)+\theta \int_{\mathbb{E}}\omega^h u\partial_tu\,d\sigma.
\end{equation}
It is straightforward to see that $L(t)$ and $E(t)$ are equivalent in the sense. 
So there exist two positive constants $\beta_1$ and $\beta_2$ depending on 
$\theta$ such that for $t\geq0$, we have
\begin{equation}\label{3.12}
\beta_1E(t)\leq L(t)\leq \beta_2E(t).
\end{equation}
By taking the time derivative of the function $L(t)$ defined in
  \eqref{3.11}, using equation \eqref{1.1} and performing several 
integration by parts, we get
\begin{equation}\label{3.13}
\begin{aligned}
\frac{dL(t)}{dt}  
& =  \frac{d}{dt}E(t)+\theta\int_{\mathbb{E}}\omega^h|\partial_tu|^2\,d\sigma
 +\theta\int_{\mathbb{E}}\omega^h|u\partial_{tt}u|\,d\sigma\\
&  =-\|\partial_tu\|_{\mathcal{L}_2^\frac{n+1}{2}(\mathbb{E})}^2
 +\theta\|\partial_tu\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2\\
  &  \quad +\theta(-\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
-\int_{\mathbb{E}}\omega^h|u\partial_tu|\,d\sigma
 -\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma)\\
  &  \quad +\theta(\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma).
\end{aligned}
\end{equation}
Using Young inequality and Sobolev equality, for any $r>0$, we obtain
\begin{equation}\label{3.14}
\begin{aligned}
\int_{\mathbb{E}}\omega^h|u\partial_tu|\,d\sigma
&\leq\frac{1}{4r}\int_{\mathbb{E}}\omega^h|\partial_tu|^2\,d\sigma
 +r\int_{\mathbb{E}}\omega^h|u|^2\,d\sigma \\
&\leq\frac{1}{4r}\int_{\mathbb{E}}\omega^h|\partial_tu|^2\,d\sigma
 +C^2r\|\nabla _{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2,
\end{aligned}
\end{equation}
where $C$ is the Sobolev constant. By Lemma \ref{le2.6}, we have
\begin{align*}
&\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
 -\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\\
&\leq \sum_{k=1}^la_kC_k^{p_k+1}\|\nabla_{\mathbb{E}}u
 \|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^{p+1}\\
&\leq \sum_{k=1}^la_kC_k^{p_1+1}\Big(\frac{2(p+1)}{p-1}(1+{C^*}^2)^{-1}
E(0)\Big)^{\frac{p_1-1}{2}}\|\nabla_{\mathbb{E}}
 u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2.
\end{align*}
Inserting \eqref{3.14} into \eqref{3.13}, we arrive at
\begin{align*}
&\frac{dL(t)}{dt} \\
&\leq (\theta+\frac{\theta}{4r}-1)\|\partial_tu\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2
+\big(\theta \sum_{k=1}^la_kC_k^{p_1+1}\Big(\frac{2(p+1)}{p-1}
(1+ {C^*}^2)^{-1}E(0)\Big)^{\frac{p_1-1}{2}}\\
&\quad -\theta+\theta C^2r -\theta C^2\Big)
\|\nabla _{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2.
\end{align*}
By Lemma \ref{le2.6} we know 
$ \sum_{k=1}^la_kC_k^{p_1+1}\Big(\frac{2(p+1)}{p-1}(1+ {C^*}^2)^{-1}E(0)
\Big)^{\frac{p_1-1}{2}}\leq 1$. Then we choose $r<1$ such that
\begin{equation*}
\theta \sum_{k=1}^la_kC_k^{p_1+1}
\Big(\frac{2(p+1)}{p-1}(1+ {C^*}^2)^{-1}E(0)\Big)^{\frac{p_1-1}{2}}
-\theta+\theta C^2r-\theta C^2<0.
\end{equation*}
From this inequality we may find $\phi>0$, which depends only on $r$ such that
\begin{equation*}
\frac{dL(t)}{dt}\leq(\theta+\frac{\theta}{4r}-1)\|\partial_t
u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
-\theta\phi\|\nabla _{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2.
\end{equation*}
Consequently using the definition of the energy $E(t)$, for any suitable 
positive constant $M$, we  obtain
\begin{equation}\label{3.15}
\frac{dL(t)}{dt}\leq- M\theta E(t)
+\Big(\theta+\frac{\theta}{4r}-1+\frac{\theta M}{2}\Big)
\|\partial_tu\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
+\theta(\frac{M}{2}-\phi)
\|\nabla _{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2.
\end{equation}
Then, choosing $M<2\phi$, and $\theta$ small enough such that
\begin{equation*}
\theta+\frac{\theta}{4r}-1+\frac{\theta M}{2}<0,
\end{equation*}
inequality \eqref{3.15} becomes
\begin{equation}\label{3.16}
\frac{dL(t)}{dt}\leq- M\theta E(t) \quad  \text{for all } t\geq0.
\end{equation}
On the other hand, by \eqref{3.12}, setting $ \zeta=\frac{M\theta}{\beta_2}$, 
inequality \eqref{3.16} becomes
\begin{equation*}
\frac{dL(t)}{dt}\leq- \zeta L(t) \quad \text{for all } t\geq0.
\end{equation*}
Integrating this differential inequality between $0$ and $t$ gives the 
following estimate for the function $L(t)$
\begin{equation*}
L(t)\leq\dot{C} e^{-\zeta t} \quad \text{for all}  t\geq0.
\end{equation*}
Consequently, by using \eqref{3.12} once again, we conclude that
\begin{equation*}
E(t)\leq \widehat{C} e^{-\zeta t} \quad  \text{for all } t\geq0.
\end{equation*}
The proof is complete.
\end{proof}

\section{Finite time blow-up of solution}\label{se4}

In this section, we discuss the blow-up in finite time and seek a lower bound 
for the blow-up time $T^*$ for the solution of \eqref{1.1}.

\begin{theorem}\label{th1.3}
Assume that $u$ is a local solution of problem \eqref{1.1}, and 
$u_0\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}})$. 
If $I(u)<0$ and $E(0)\leq d$, 
and $\int_\mathbb{E} \omega^h u_0u_1 d\sigma >0$ when $0\le E(0)<d$, where 
$ d<\frac{p-1}{2(p+1)}(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^hV|u|^2\,d\sigma)$ then the solution $u$ blows up 
in a finite time.
\end{theorem}
The following three lemmas will be used to prove Theorem \ref{th1.3}.

\begin{lemma}\label{le2.2}
Let $u$ be the unique local solution of problem \eqref{1.1} and assume that $u$ satisfy $E(0)\leq d$, one has $I(u)<0$ and
\begin{align*}
d<&\frac{p-1}{2(p+1)}\Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2
\,d\sigma+\int_{\mathbb{E}}\omega^hV|u|^2\,d\sigma\Big)\\
<&\frac{p-1}{2(p+1)}\Big(\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma\Big),
\end{align*}
for all $t\in [0,T_{Max})$.
\end{lemma}

\begin{proof}
Since $I(u)<0$ for all $0\leq t<T^*$, it follows that
\begin{equation}\label{2.2}
\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^hV|u|^2\,d\sigma
<\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma.
\end{equation}
For all $0\leq t<T^*$, using the definition of $d$, we get
\begin{equation}\label{2.3}
d<\frac{p-1}{2(p+1)}\Big(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2
\,d\sigma+\int_{\mathbb{E}}\omega^hV|u|^2\,d\sigma\Big).
\end{equation}
Then \eqref{2.2} and \eqref{2.3} imply 
\[
 \sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma
>\frac{2(p+1)}{p-1}d>0. 
 \]
By the continuity of 
$ t\mapsto \sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma$ we get
 $u(T^*)\neq0$. From the definition of $J(u)$, we obtain
\begin{equation*}
d\leq\frac{p-1}{2(p+1)}\Big(\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u(T^*)|^{p_k+1}
\,d\sigma-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u(T^*)|^{q_j+1}\,d\sigma\Big)
=J(u(T^*)),
\end{equation*}
which contradicts to $J(u(T^*)\leq E(T^*)<d$. Then, we obtain
\begin{align*}
d<&\frac{p-1}{2(p+1)}(\int_{\mathbb{E}}\omega^h|\nabla _{\mathbb{E}}u|^2\,d\sigma
+\int_{\mathbb{E}}\omega^hV|u|^2\,d\sigma)\\
<&\frac{p-1}{2(p+1)}(\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma
-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma).
\end{align*}
\end{proof}

\begin{lemma}\label{le2.4}
Assume $M(t)>0$ is a twice differentiable function, then the inequality
\begin{equation*}
M(t)M''(t)-(1+\alpha )M'(t)^2\geq 0
\end{equation*}
holds for $t>0$ and $\alpha>0$. If $M(0)>0$ and $M'(0)>0$, then there 
exists a time $ T^*\leq\frac{M(0)}{\alpha M'(0)}$ such that 
$\lim_{t\to T^{*-}}M(t)=\infty$.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{th1.3}]
We consider $M:[0,T]\to R_+$ defined by
\begin{equation*}
M(t)=\|u(t)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
 +\int_0^t\|u(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2\,d\tau
+(T-t)\|u_0\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2+\kappa(t+T_0)^2,
\end{equation*}
where $T$ and $T_0$ are positive constants. Furthermore,
\begin{equation}\label{2.10}
\begin{aligned}
M'(t)  
& =2\int_{\mathbb{E}}u\partial_tudx+\|u(t)\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2-\|u_0\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
 +2\kappa(t+T_0)\\
& =2\int_{\mathbb{E}}u\partial_tudx+2\int_0^1(u(\tau),\partial_tu(\tau))_2\,d\tau
 +2\kappa(t+T_0),
\end{aligned}
\end{equation}
and consequently,
\begin{align*}
M''(t)
&=2(\partial_{tt}u,u)+2\|\partial_t u\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2+2(\partial_tu,u)+2\kappa\\
&=2\Big[-\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2-\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma
 +\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k+1}\,d\sigma\Big]\\
&-2\Big[\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma
 -\|\partial_t u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2-\kappa\Big].
\end{align*}
For almost every $t\in[0,T]$, we get
\begin{align*}
&M(t)M''(t)-\frac{p+3}{4}M'(t)^2\\
&=M(t)M''(t) +(p+3)\Big[\eta(t)-\Big(M(t)-(T-t)\| u_0\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2\Big) \\
&\quad\times \Big(\|\partial_t u\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2+\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2\,d\tau+\kappa\Big)\Big],\\
\end{align*}
where $\eta:[0,T]\to R_+$ is the function defined by
\begin{align*}
\eta(t)
&=\Big(\| u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
+\int_0^t\|u(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2\,d\tau
+\kappa(t+T_0)^2\Big) \\
&\quad\times \Big(\| \partial_tu\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
+\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2\,d\tau
+\kappa\Big)\\
&\quad -\Big(\int_{\mathbb{E}}\omega^hu\partial_t u\,d\sigma+\int_0^t(u,\partial_t u)_2\,d\tau
+\kappa(t+T_0)\Big)^2 
\geq0.
\end{align*}
As a consequence, we have differential equality
\begin{equation}\label{2.11}
M(t)M''(t)-\frac{p+3}{4}M'(t)^2\geq M(t)\xi(t).
\end{equation}
For almost every $t\in[0,T]$, $\xi:[0,T]\to \mathbb{R}_+$ is the map defined by
\begin{align*}
\xi(t)
=&M''(t)+\Big(\| \partial_tu\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
+\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2\,d\tau
+\kappa\Big)(p+3)\\
=&2\Big[-\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
 -\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma+\sum_{k=1}^la_k\int_{\mathbb{E}}
 \omega^h|u|^{p_k+1}\,d\sigma\Big]\\
&-2\Big[\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j+1}\,d\sigma
 -\|\partial_t u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2-\kappa\Big]\\
&-(p+3)\Big(\|\partial_tu\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
 +\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2\,d\tau
 +\kappa\Big)\\
\geq&-2(p+1)E(t)+(p-1)\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2+(p-1)\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\\
&-(p+3)\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2\,d\tau-(p+1)\kappa\\
=&-2(p+1)E(0)+(p-1)\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2+(p-1)\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\\
&+(p-1)\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2\,d\tau-(p+1)\kappa.
\end{align*}
\smallskip

\noindent\textbf{Case 1.}
If $E(0)<0$, then
\begin{align*}
\xi(t)
=&-2(p+1)E(0)+(p-1)\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2+(p-1)\int_{\mathbb{E}}\omega^h V|u|^2\,d\sigma\\
&+(p-1)\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
\,d\tau-(p+1)\kappa\\
\geq&-2(p+1)E(0)+(p-1)(1+{C^*}^2)\|\nabla_{\mathbb{E}}
 u\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2\\
&+(p-1)\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2\,d\tau-(p+1)\kappa.
\end{align*}
Choosing $\kappa$ satisfying $\kappa\leq-2E(0)$, we have
\begin{equation}\label{2.12}
\xi(t)\geq (p-1)(1+{C^*}^2)\|\nabla_Eu\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2+(p-1)\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2\,d\tau\geq 0.
\end{equation}
\smallskip

\noindent\textbf{Case 2.}
If $0\leq E(0)<d$, then by Lemma \ref{le2.2}, we have
\begin{align*}
\xi(t)=&-2(p+1)E(0)+(p-1)\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2+(p-1)\int_{\mathbb{E}}\omega^hV|u|^2\,d\sigma\\
&+(p-1)\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}})}^2
 \,d\tau-(p+1)\kappa\\
=&2(p+1)(d-E(0))+(p-1)\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2\,d\tau-(p+1)\kappa\\
\geq &(p-1)\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
 ({\mathbb{E}})}^2\,d\tau-(p+1)\kappa.
\end{align*}
Choosing $\kappa=0$, we have 
$ (p-1)\int_0^t\|\partial_tu(\tau)\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2\,d\tau\geq0$. Then we obtain
\begin{equation}\label{2.13}
\xi(t)\geq 0.
\end{equation}

Therefore, from \eqref{2.11} \eqref{2.12} and \eqref{2.13}, we obtain
\begin{equation*}
M(t)M''(t)-\frac{p+3}{4}M'(t)^2\geq 0.
\end{equation*}
By Lemma \ref{le2.4} and \eqref{2.10}, for every $t\in[0,T]$, if $E(0)<0$, 
we then choose $T_0$ sufficiently large such that
 $ M'(0)=2\int_{\mathbb{E}}\omega^hu_0 u_1\,d\sigma+2bT_0>0$; 
if $0\leq E(0)<d$, the condition $ \int_{\mathbb{E}}\omega^hu_0u_1\,d\sigma>0$ 
also ensure that $M'(0)>0$. As $ \frac{p+3}{4}>1$, letting $ \alpha=\frac{p-1}{4}$, 
we get $\lim_{t\to T^{*-}}M(t)=\infty$ by the convexity argument, which implies 
that $\lim_{t\to T^{*-}}\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^2=\infty$.
\end{proof}

\begin{theorem}\label{th2.1}
Assume that
\begin{equation*}
1<p\leq\frac{N}{N-2},\quad N\geq 3,\quad
 u_0\in \mathcal{H}_{2,0}^{1,\frac{n+1}{2}}({\mathbb{E}}),\quad
 u_1\in\mathcal{L}_2^\frac{n+1}{2}({\mathbb{E}}),\quad E(0)<0,
\end{equation*}
Then the solution blows up at time $T^*$.
\end{theorem}

\begin{proof}
We define the auxiliary function
\begin{equation*}
\Phi(t):=\int_{\mathbb{E}}\omega^h(u_t^2+|\nabla_{\mathbb{E}}u|^2+Vu^2)\,d\sigma.
\end{equation*}
Differentiating $\Phi(t)$ and making use of the divergence theorem, we obtain
\begin{equation}\label{53}
\begin{aligned}
\Phi'(t)  
&  =2\int_{\mathbb{E}}\omega^h(u_tu_{tt}+|\nabla_{\mathbb{E}}u|
|\nabla_{\mathbb{E}}u_t|+Vuu_t)\,d\sigma\\
  &  =2\int_{\mathbb{E}}\omega^hu_t(u_{tt}+|\Delta_{\mathbb{E}}u|+Vu)\,d\sigma\\
  &  =2\Big(\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k}u_t\,d\sigma
-\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j}u_t\,d\sigma
-\int_{\mathbb{E}}\omega^h|u_t|^2\,d\sigma\Big).
\end{aligned}
\end{equation}
The two terms on the right-hand side of \eqref{53} can be estimated as follows
\begin{align*}
\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k}\,d\sigma-\sum_{j=1}^sb_j
\int_{\mathbb{E}}\omega^h|u|^{q_j}\,d\sigma 
&\leq\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k}\,d\sigma \\
&\leq \sum_{k=1}^la_kC_k^{p_k}\|\nabla_{\mathbb{E}}u\|_{\mathcal{L}_2^\frac{n+1}{2}
({\mathbb{E}})}^{p_k} \\
&\leq C\Big(\int_{\mathbb{E}}\omega^h|\nabla_{\mathbb{E}}u|^2\,d\sigma\Big)^{p_1}.
\end{align*}
Then, we have
\begin{align*}
&2\Big(\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k}u_t\,d\sigma
 -\sum_{j=1}^sb_j\int_{\mathbb{E}}\omega^h|u|^{q_j}u_t\,d\sigma\Big)\\
&\leq 2\sum_{k=1}^la_k\int_{\mathbb{E}}\omega^h|u|^{p_k}u_t\,d\sigma\\
& \leq 2\Big(\int_{\mathbb{E}}\omega^hu_t^2\,d\sigma\sum_{k=1}^la_k
 \int_{\mathbb{E}}\omega^h|u|^{2p_k}\,d\sigma\Big)^{1/2} \\
&\leq 2\sqrt{C}\Big(\int_{\mathbb{E}}\omega^hu_t^2\,d\sigma\Big)^{1/2}
\Big(\int_{\mathbb{E}}\omega^h|\nabla_{\mathbb{E}}u|^2\,d\sigma\Big)^{p_1/2}\\
&\leq \sqrt{C}\Big\{\int_{\mathbb{E}}\omega^hu_t^2\,d\sigma
+\Big(\int_{\mathbb{E}}\omega^h|\nabla_{\mathbb{E}}u|^2\,d\sigma\Big)^{p_1}
\Big\}
\leq\sqrt{C}(\Phi+\Phi^{p_1}),
\end{align*}
which leads to the differential inequality
\begin{equation*}
\Phi'(t)\leq\alpha\Phi+\beta\Phi^{p_1},
\end{equation*}
where $\alpha=\beta=\sqrt{C}$. This differential inequality may be reduced to a linear differential inequality for $v(t):=\Phi^{1-p_1}$. Integrating this inequality, we obtain
\begin{equation}\label{54}
(\Phi(t))^{1-p_1}\geq\{(\Phi(0))^{1-p_1}+1\}
e^{-(p_1-1)\sqrt{C}t}-1.
\end{equation}
It follows from \eqref{54}, that $\Phi(t)$ remains bounded for $t\in[0,T)$ with
\begin{equation*}
T^*=\frac{1}{(p_1-1)\sqrt{C}}\log\{1+(\Phi(0))^{1-p_1}\}.
\end{equation*}
It gives that $T^*$ is a lower bound for the blow-up time.
\end{proof}

\subsection*{Acknowledgments}
This work was supported by the National Natural Science Foundation of 
China (grant number 11771214), and by the 
initial funding for talents of Nanjing University of Information Science 
and Technology.



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\end{document}