# Global existence and finite time blowup for a mixed pseudo-parabolic
$p$-Laplacian type equation

Jiazhuo Cheng<EMAIL_ADDRESS>Qiru Wang<EMAIL_ADDRESS>School of Mathematics, Sun Yat-sen University, Guangzhou 510275, Guangdong,
China

###### Abstract

This paper concerns the initial-boundary value problem for a mixed pseudo-
parabolic $p$-Laplacian type equation. By constructing a family of potential
wells, we first present the explicit expression for the depth of potential
well, and then prove the existence, uniqueness and decay estimate of global
solutions and the blowup phenomena of solutions with subcritical initial
energy. Next, we extend parallelly these results to the critical initial
energy. Lastly, the existence, uniqueness and asymptotic behavior of global
solutions with supercritical initial energy are proved by further analyzing
the properties of $\omega$-limits of solutions.

###### keywords:

A mixed pseudo-parabolic $p$-Laplacian type equation; global existence,
uniqueness and decay estimate; finite time blowup; potential wells

###### MSC:

[2020] 35K20, 35K58, 35K91, 35D30.

††journal: Journal

##  1 Introduction

In this paper, we consider the following semilinear pseudo-parabolic equation
with initial-boundary conditions

$\displaystyle\left\\{\begin{split}&u_{t}-\mathrm{div}\left(\left|\nabla
u\right|^{p-2}\nabla u\right)-\triangle
u_{t}=\left|u\right|^{q-1}u-{\mathchoice{{-\mkern-19.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}}_{\Omega}\left|u\right|^{q-1}u\operatorname{d}x,~{}~{}&\mathrm{in}\;\Omega\times\left(0,T\right),\\\
&\frac{\partial
u}{\partial\nu}\left(x,t\right)=0,&\mathrm{on}\;\partial\Omega\times\left(0,T\right),\\\
&u\left(x,0\right)=u_{0}\left(x\right)\not\equiv
0,~{}~{}{\mathchoice{{-\mkern-19.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}}_{\Omega}u_{0}\operatorname{d}x=0,&\mathrm{in}\;\Omega,\end{split}\right.$
(1.1)

where, (1) the first equation of (1.1) is a mixed pseudo-parabolic
$p$-Laplacian type equation,
$p>1,\;\max\left\\{p-1,1\right\\}<q<\frac{np}{n-p}-1$, $T\in(0,\infty]$,
$\Omega\subset\mathbb{R}^{n}\left(n\geq 1\right)$ is a bounded domain with
smooth boundary,
${\mathchoice{{-\mkern-19.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}}_{\Omega}u\operatorname{d}x=\frac{1}{\left|\Omega\right|}\int_{\Omega}u\operatorname{d}x$
and
${\mathchoice{{-\mkern-19.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}}_{\Omega}\left|u\right|^{q-1}u\operatorname{d}x$
is to conserve the spatial integral of the unknown function over time; (2) the
second equation of (1.1) is Neumann boundary condition,
$\frac{\partial}{\partial\nu}$ denotes differentiation with respect to the
outward normal $\nu$ on $\partial\Omega$; (3) the third equation of (1.1) is
initial boundary condition.

Problem (1.1) has attracted much attentions in recent years in the fields of
population dynamics and biological sciences [3, 21, 23, 6], where the total
mass is often conserved or known. Such equations can provide insight into
biological and chemical problems where conservation properties predominate.
Since in mathematics we do not require $u(x,y)$ to be non-negative, we use
$\left|u\right|^{q-1}u$ instead of $u^{q}$ in the problem (1.1) (see [10]). In
addition, when affected by a variety of factors, such as atoms, ions, etc.,
the $\triangle u$ term in the pseudo-parabolic is usually replaced by
$\mathrm{div}\left(\left|\nabla u\right|^{p-2}\nabla u\right)$.

The pseudo-parabolic equation

$u_{t}-u_{xxt}=F\left(x,t,u_{x},u_{xx}\right)$ (1.2)

has been extensively studied by many authors [25, 1, 2, 26, 18]. From 2013 to
2018, the authors in [30, 29, 14, 31] investigated the initial boundary value
problem (1.2) with $F\left(x,t,u_{x},u_{xx}\right)=u_{xx}+u^{p}$. They proved
global existence and asymptotic behavior of solutions with subcritical and
critical initial energy $J(u_{0})\leq d$, further gave the global nonexistence
of solutions under supercritical initial energy $J(u_{0})>d$ through
comparison principle, and the upper bound of the blowup time of supercritical
initial energy is estimated. In 2021, as an expansion of previous studies,
Wang and Xu [27] considered the following semilinear pseudo-parabolic equation
with Neumann boundary condition

$u_{t}-\triangle u-\triangle
u_{t}=\left|u\right|^{p-1}u-{\mathchoice{{-\mkern-19.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}}_{\Omega}\left|u\right|^{p-1}u\operatorname{d}x,$

and proved global existence, uniqueness, asymptotic behavior and blowup of
solutions with subcritical and critical initial energy $J(u_{0})\leq d$.

In 2007, Yin and Jin [32] studied the following equation

$u_{t}-\mathrm{div}\left(\left|\nabla u^{m}\right|^{p-2}\nabla
u^{m}\right)=\lambda\left|u\right|^{q},$ (1.3)

and proved the finite time blowup when $m=1,\;1<p<2,\;q>1,\;\lambda>0$. In
2009, Jin et al. [11] considered problem (1.3) with
$0<m\left(p-1\right)<1,\;q>0,\;\lambda>0$, and the finite time blowup was
proved for $q>m\left(p-1\right)$. In 2014, Qu et al. [20] studied the nonlocal
$p$-Laplace equation

$u_{t}-\mathrm{div}\left(\left|\nabla u\right|^{p-2}\nabla
u\right)=\left|u\right|^{q}-\frac{1}{\left|\Omega\right|}\int_{\Omega}\left|u\right|^{q}\operatorname{d}x$

with Neumann boundary condition, and proved that the solution blows up for
special initial energy, i.e., $J(u_{0})\leq 0$.

Compared with the above studies, there are relatively few studies on the mixed
pseudo parabolic $p$-Laplace equations, see [7, 15, 24, 13, 12, 8]. In 2003,
Liu [13] studied the following nonlinear pseudo parabolic $p$-Laplace equation

$u_{t}-k\triangle u_{t}-\mathrm{div}\left(\left|\nabla u\right|^{p-2}\nabla
u\right)=f\left(u\right),$ (1.4)

and proved the existence and uniqueness of solutions for
$p>2,\;k>0,\;f\left(u\right)=0$. In 2013, Li et al. [12] extended the problem
(1.4) to $p\geq 2,\;0<q<p-1,\;k>0$, and proved the existence and asymptotic
behavior of solutions with $f\left(u\right)=u^{q}$. In 2018, Cao and Liu [4]
considered the equation (1.4) for $1<p<2,\;k\geq
0,\;f\left(u\right)=\left|u\right|^{p-2}u\log\left|u\right|$ and obtained the
global existence and asymptotic behavior of solutions by constructing a family
of potential wells.

In order to overcome the difficulties encountered in the prior estimation of
solutions using Galerkin method, we use the potential well method proposed by
Sattinger [22] in 1968. Liu et al. [16, 17] extended and improved the method
by introducing a family of potential wells and taking the known potential
wells as a special case. Now, it is one of the most useful methods to prove
the global existence and nonexistence of solutions and the vacuum isolation of
solutions for parabolic equations [5, 28].

It is interesting to consider (1.1) due to the role of the corresponding
conservation properties in the real world and its connection to the biological
and chemical model equations (1.4). In addition, keeping balance is an arduous
task between the bad side and the good side of the strong dissipation term
$\triangle u_{t}$, which can not only help the global existence by decaying
energy, but also impede blowup and delay the blowup time. In this paper, it is
important to deal with the blowup situation and maintain this balance, but it
is not easy. Therefore, we combine the theory of potential wells with the
Galerkin method to overcome the existing difficulties, and obtain the global
existence, uniqueness, asymptotic behavior and finite time blow up of
solutions.

The structure of this paper is organized as follows: In Section 2, we give
some notations, definitions and lemmas about the basic properties of the
related functionals and sets. Also, we present the main results of this paper.
In Sections 3 and 4, we are devoted to the subcritical initial energy
$J(u_{0})<d$ and the critical initial energy $J(u_{0})=d$, respectively, and
prove the existence, uniqueness, decay estimate of global solutions and the
blowup phenomena of solutions. In Section 5, we prove the global existence,
uniqueness and asymptotic behavior of solutions for the supercritical initial
energy $J\left(u_{0}\right)>d$ by analyzing the properties of $\omega$-limits
of solutions. The conclusion is made in Section 6.

##  2 Preliminaries and main results

Throughout the whole paper, let $C$ represent generic positive constant, which
may change from line to line. Let $L^{p}\left(\Omega\right)$ be the set of all
measurable functions on $\Omega$ and satisfying
$\int_{\Omega}\left|u\left(x\right)\right|^{p}\operatorname{d}x<\infty$. For
$1\leq p\leq\infty$, denote by ${\left\|\cdot\right\|}_{p}$ the
$L^{p}\left(\Omega\right)$ norm, i.e.,

${\left\|u\right\|}_{p}=\left\\{\begin{array}[]{l}\left(\int_{\Omega}\left|u\left(x\right)\right|^{p}\operatorname{d}x\right)^{\frac{1}{p}},\;\;\;\mathrm{if}\;1\leq
p<\infty;\\\
\mathrm{ess}\;\underset{x\in\Omega}{\mathrm{sup}}\left|u\left(x\right)\right|,\;\;\;\;\;\;\;\mathrm{if}\;p=\infty,\end{array}\right.$

where $\forall u\in L^{p}\left(\Omega\right)$. For $1\leq p\leq+\infty$,
denote

$W^{m,p}\left(\Omega\right)=\left\\{u\in
L^{p}\left(\Omega\right)|D^{\alpha}u\in
L^{p}\left(\Omega\right),\forall\alpha\in Z_{+}^{n},\left|\alpha\right|\leq
m\right\\},$

and

$W_{N}^{{m,p}}\left(\Omega\right)=\left\\{u\in
W^{m,p}\left(\Omega\right)|{\left.\frac{\partial
u}{\partial\nu}\right|}_{\partial\Omega}=0,\int_{\Omega}u\operatorname{d}x=0\right\\},$

where $D^{\alpha}u$ represents the $\alpha$ order weak derivative of $u$.

When $p=2$, we can write
$H^{k}\left(\Omega\right)=W^{k,2}\left(\Omega\right)\left(k=0,1,\cdots\right)$.
Let $\left(\cdot,\cdot\right)$ be the inner product in
$L^{2}\left(\Omega\right)$ and ${\left\|\cdot\right\|}_{H^{1}}$ be the norm of
$H^{1}\left(\Omega\right)$. That is

$\left(u,v\right)=\int_{\Omega}u\left(x\right)v\left(x\right)\operatorname{d}x,\;\;\;\forall
u,v\in L^{2}\left(\Omega\right),$

and

${\left\|u\right\|}_{H^{1}}=\left(\left\|u\right\|_{2}^{2}+\left\|\nabla
u\right\|_{2}^{2}\right)^{\frac{1}{2}},\;\;\;\forall u\in
H^{1}\left(\Omega\right),$

then the norm ${\left\|\cdot\right\|}_{H^{1}}$ is equivalent to the norm
${\left\|\nabla\left(\cdot\right)\right\|}_{2}$.

Next, we define the energy functional $J\left(u\right)$ and the Nehari
functional $I\left(u\right)$ as

$\begin{array}[]{l}J\left(u\right)=\frac{1}{p}\left\|\nabla
u\right\|_{p}^{p}-\frac{1}{q+1}\left\|u\right\|_{q+1}^{q+1},\\\
I\left(u\right)=\left\|\nabla
u\right\|_{p}^{p}-\left\|u\right\|_{q+1}^{q+1},\end{array}$ (2.1)

and define the Nehari manifold as

$\mathcal{N}=\left\\{\left.u\in
W_{N}^{1,p}\left(\Omega\right)\right|I\left(u\right)=0,{\left\|\nabla
u\right\|}_{p}\neq 0\right\\}.$

By $J\left(u\right)$ and $I\left(u\right)$, we set

$\begin{array}[]{l}W=\left\\{\left.u\in
W_{N}^{1,p}\left(\Omega\right)\right|J\left(u\right)<d,I\left(u\right)>0\right\\}\cup\left\\{0\right\\},\\\
V=\left\\{\left.u\in
W_{N}^{1,p}\left(\Omega\right)\right|J\left(u\right)<d,I\left(u\right)<0\right\\},\end{array}$

where

$d=\underset{u\in\mathcal{N}}{\mathrm{inf}}J\left(u\right)$

is the depth of the potential well.

For $\delta>0$, we introduce a family of potential wells as follows

$\begin{array}[]{l}I_{\delta}\left(u\right)=\delta\left\|\nabla
u\right\|_{p}^{p}-\left\|u\right\|_{q+1}^{q+1},\\\
{\mathcal{N}}_{\delta}=\left\\{\left.u\in
W_{N}^{1,p}\left(\Omega\right)\right|I_{\delta}\left(u\right)=0,{\left\|\nabla
u\right\|}_{p}\neq 0\right\\},\\\ W_{\delta}=\left\\{\left.u\in
W_{N}^{1,p}\left(\Omega\right)\right|J\left(u\right)<d\left(\delta\right),I_{\delta}\left(u\right)>0\right\\}\cup\left\\{0\right\\},\\\
V_{\delta}=\left\\{\left.u\in
W_{N}^{1,p}\left(\Omega\right)\right|J\left(u\right)<d\left(\delta\right),I_{\delta}\left(u\right)<0\right\\},\\\
d\left(\delta\right)=\underset{u\in{\mathcal{N}}_{\delta}}{\mathrm{inf}}J\left(u\right).\end{array}$
(2.2)

In what follows, we give some sets and functionals for weak solutions with
high energy levels. Set

$\begin{array}[]{l}{\mathcal{N}}_{+}=\left\\{u\in
W_{N}^{1,p}\left(\Omega\right)\left|I\left(u\right)>0\right.\right\\},\\\
J^{\alpha}=\left\\{u\in
W_{N}^{1,p}\left(\Omega\right)\left|J\left(u\right)\leq\alpha\right.\right\\},\\\
\mathcal{N}^{\alpha}=\mathcal{N}\cap
J^{\alpha}=\left\\{u\in\mathcal{N}\left|J\left(u\right)\leq\alpha\right.\right\\},\;\;\;\forall\alpha>d,\\\
\lambda_{\alpha}=\inf\left\\{\left\|u\right\|_{H^{1}}^{2}\left|u\in\right.\mathcal{N}^{\alpha}\right\\},\;\;\;\forall\alpha>d.\end{array}$

Obviously, $\lambda_{\alpha}$ is non-increasing with respect to $\alpha$.

Integrating the first equation of (1.1) with respect to $x$ over $\Omega$ and
then with respect to $t$ from $0$ to $t$, we obtain
$\int_{\Omega}u\operatorname{d}x=\int_{\Omega}u_{0}\operatorname{d}x=0$. That
is, we have the conservation law for problem (1.1). Before stating the main
theorems, we first give some definitions and lemmas.

As in [9, Page 8], “In general, as we shall see, the conservation law has no
classical solutions, but is well-posed if we allow for properly defined
generalized or weak solutions. This is all to say that we may be forced by the
structure of the particular equation to abandon the search for smooth,
classical solutions. We must instead, while still hoping to achieve the well-
posedness conditions (a)-(c), investigate a wider class of candidates for
solutions. And in fact, even for those PDE which turn out to be classically
solvable, it is often most expedient initially to search for some appropriate
kind of weak solution.”

######  Definition 2.1 (Weak solution [9])

Function $u\left(x,t\right)$ is called a weak solution to problem (1.1) on
$\Omega\times[0,T)$, if $u\in
L^{\infty}\left(0,T;W_{N}^{1,p}\left(\Omega\right)\right)$, $u_{t}\in
L^{2}\left(0,T;W_{N}^{1,2}\left(\Omega\right)\right)$,
$u\left(x,0\right)=u_{0}\left(x\right)\in W_{N}^{1,p}\left(\Omega\right)$ and
satisfies

$\begin{array}[]{l}\int_{0}^{t}\left(\left(u_{\tau},\varphi\right)+\left(\nabla
u_{\tau},\nabla\varphi\right)+\left(\left|\nabla u\right|^{p-2}\nabla
u,\nabla\varphi\right)\right)\operatorname{d}\tau\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\int_{0}^{t}\left(\left|u\right|^{q-1}u-{\mathchoice{{-\mkern-19.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}}_{\Omega}\left|u\right|^{q-1}u\operatorname{d}x,\varphi\right)\operatorname{d}\tau,\end{array}$
(2.3)

for any $\varphi\in W_{N}^{1,p}\left(\Omega\right)\cap
W_{N}^{1,2}\left(\Omega\right)$.

Moreover, the following equality

$\int_{0}^{t}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau+J\left(u\right)=J\left(u_{0}\right)$

holds for $t\in[0,T)$.

######  Definition 2.2 (Maximal existence time)

If $u\left(t\right)$ is a weak solution of problem (1.1), for maximal
existence time $T$ of $u\left(t\right)$, we have the following definition.

_(i)_ For $0\leq t<\infty$, if $u\left(t\right)$ exists, then $T=+\infty$;

_(ii)_ For $0<t_{0}<\infty$, if $0\leq t<t_{0}$, then $u\left(t\right)$
exists, but doesn’t exist at $t=t_{0}$, then $T=t_{0}$.

######  Lemma 2.1 (Relations between $I_{\delta}\left(u\right)$ and
${\left\|\nabla u\right\|}_{p}$)

Let $u\in W_{N}^{1,p}\left(\Omega\right)$.

_(i)_ If $0<{\left\|\nabla u\right\|}_{p}<r\left(\delta\right)$, then
$I_{\delta}\left(u\right)>0$;

_(ii)_ If $I_{\delta}\left(u\right)<0$, then ${\left\|\nabla
u\right\|}_{p}>r\left(\delta\right)$;

_(iii)_ If $I_{\delta}\left(u\right)=0$ and ${\left\|\nabla u\right\|}_{p}\neq
0$, then ${\left\|\nabla u\right\|}_{p}\geq r\left(\delta\right)$,
where
$r\left(\delta\right)=\left(\frac{\delta}{C_{\ast}^{q+1}}\right)^{\frac{1}{q+1-p}}$,
$C_{\ast}$ is the imbedding constant for
$W^{1,p}\left(\Omega\right)\hookrightarrow L^{q+1}\left(\Omega\right)$ and
satisfies

$\frac{1}{C_{\ast}}=\underset{u\in W_{N}^{1,p},u\neq
0}{\inf}\frac{{\left\|\nabla u\right\|}_{p}}{{\left\|u\right\|}_{q+1}}.$ (2.4)

Proof: (i) By $0<{\left\|\nabla u\right\|}_{p}<r\left(\delta\right)$, we
obtain

$\left\|u\right\|_{q+1}^{q+1}\leq C_{\ast}^{q+1}\left\|\nabla
u\right\|_{p}^{q+1}=C_{\ast}^{q+1}\left\|\nabla
u\right\|_{p}^{q+1-p}\left\|\nabla u\right\|_{p}^{p}<\delta\left\|\nabla
u\right\|_{p}^{p},$

then $I_{\delta}\left(u\right)>0$.

(ii) From $I_{\delta}\left(u\right)<0$, we know that ${\left\|\nabla
u\right\|}_{p}\neq 0$. Then

$\delta\left\|\nabla u\right\|_{p}^{p}<\left\|u\right\|_{q+1}^{q+1}\leq
C_{\ast}^{q+1}\left\|\nabla u\right\|_{p}^{q+1}=C_{\ast}^{q+1}\left\|\nabla
u\right\|_{p}^{q+1-p}\left\|\nabla u\right\|_{p}^{p}$

i.e., ${\left\|\nabla u\right\|}_{p}>r\left(\delta\right)$.

(iii) If $I_{\delta}\left(u\right)=0$, ${\left\|\nabla u\right\|}_{p}\neq 0$,
we see that

$\delta\left\|\nabla u\right\|_{p}^{p}=\left\|u\right\|_{q+1}^{q+1}\leq
C_{\ast}^{q+1}\left\|\nabla u\right\|_{p}^{q+1}=C_{\ast}^{q+1}\left\|\nabla
u\right\|_{p}^{q+1-p}\left\|\nabla u\right\|_{p}^{p},$

which implies ${\left\|\nabla u\right\|}_{p}\geq r\left(\delta\right)$.

######  Lemma 2.2 (Properties of $J(\lambda u)$)

Assume $u\in W_{N}^{1,p}\left(\Omega\right)$ and $\left\|u\right\|_{q+1}\neq
0$. Then

_(i)_ $J\left(\lambda u\right)$ is increasing on
$\lambda\in\left[0,\lambda^{\ast}\right]$ and decreasing on
$\lambda\in[\lambda^{\ast},\infty)$, $\lambda=\lambda^{\ast}$ is the maximum
point of $J\left(\lambda u\right)$. Furthermore, $\underset{\lambda\rightarrow
0}{\mathrm{lim}}J\left(\lambda
u\right)=0,\;\underset{\lambda\rightarrow+\infty}{\mathrm{lim}}J\left(\lambda
u\right)=-\infty$;

_(ii)_ $I\left(\lambda u\right)>0$ for $0<\lambda<\lambda^{\ast}$ and
$I\left(\lambda u\right)<0$ for $\lambda^{\ast}<\lambda<\infty$, and
$I\left(\lambda^{\ast}u\right)=0$, where

$\lambda^{\ast}=\left(\frac{\left\|\nabla
u\right\|_{p}^{p}}{\left\|u\right\|_{q+1}^{q+1}}\right)^{\frac{1}{q-p+1}}.$

Proof: From the definition of $J(u)$, we can find that

$J\left(\lambda u\right)=\frac{\lambda^{p}}{p}\left\|\nabla
u\right\|_{p}^{p}-\frac{\lambda^{q+1}}{q+1}\left\|u\right\|_{q+1}^{q+1},$

which gives $\underset{\lambda\rightarrow 0}{\mathrm{lim}}J\left(\lambda
u\right)=0,\;\underset{\lambda\rightarrow+\infty}{\mathrm{lim}}J\left(\lambda
u\right)=-\infty$. On the other hand, we see that

$\frac{dJ\left(\lambda u\right)}{d\lambda}=\lambda^{p-1}\left(\left\|\nabla
u\right\|_{p}^{p}-\lambda^{q-p+1}\left\|u\right\|_{q+1}^{q+1}\right).$ (2.5)

Considering ${\left.\frac{dJ\left(\lambda
u\right)}{d\lambda}\right|}_{\lambda=\lambda^{\ast}}=0$, we have

$\lambda^{\ast}=\left(\frac{\left\|\nabla
u\right\|_{p}^{p}}{\left\|u\right\|_{q+1}^{q+1}}\right)^{\frac{1}{q-p+1}}.$

Taking into account (2.5), we obtain

$\begin{array}[]{l}\frac{dJ\left(\lambda
u\right)}{d\lambda}>0,\;\;\;\mathrm{for}\;0<\lambda<\lambda^{\ast},\\\
\frac{dJ\left(\lambda
u\right)}{d\lambda}<0,\;\;\;\mathrm{for}\;\lambda^{\ast}<\lambda<\infty.\end{array}$

In addition, it is easy to see that

$I\left(\lambda u\right)=\lambda\frac{dJ\left(\lambda
u\right)}{d\lambda}=\left\\{\begin{array}[]{l}>0,\;\;\;0<\lambda<\lambda^{\ast},\\\
=0,\;\;\;\lambda=\lambda^{\ast},\\\
<0,\;\;\;\lambda^{\ast}<\lambda<\infty.\end{array}\right.$

######  Lemma 2.3 (Depth $d$ of potential well)

Let $p$ and $q$ satisfy the conditions given by (1.1). For the depth $d$ of
potential well, we have

$d=\frac{q+1-p}{p\left(q+1\right)}C_{*}^{-\frac{p(q+1)}{q+1-p}}>0.$

Proof: If $u\in\mathcal{N}$, then ${\left\|\nabla u\right\|}_{p}\neq 0$ and
$I\left(u\right)=0$ (or $\left\|\nabla
u\right\|_{p}^{p}-\left\|u\right\|_{q+1}^{q+1}=0$). It follows that

$\displaystyle J\left(u\right)=\frac{1}{p}\left\|\nabla
u\right\|_{p}^{p}-\frac{1}{q+1}\left\|u\right\|_{q+1}^{q+1}=\left(\frac{1}{p}-\frac{1}{q+1}\right)\left\|\nabla
u\right\|_{p}^{p},$

which says $J\left(u\right)>0$.

From Lemma 2.2 (ii), we see that $u\in\mathcal{N}$ must be of the form
$u=\lambda^{\ast}v$ for $v\in W_{N}^{1,p}\left(\Omega\right)$, $v\neq 0$ and
$\lambda^{\ast}=\left(\frac{\left\|\nabla
v\right\|_{p}^{p}}{\left\|v\right\|_{q+1}^{q+1}}\right)^{\frac{1}{q-p+1}}$.
Hence, we have

$\displaystyle
d=\inf_{u\in\mathcal{N}}J(u)=\inf_{u\in\mathcal{N}}\left[\left(\frac{1}{p}-\frac{1}{q+1}\right)\|\nabla
u\|_{p}^{p}\right]$
$\displaystyle\quad=\left(\frac{1}{p}-\frac{1}{q+1}\right)\inf_{u\in\mathcal{N}}\left(\frac{\|\nabla
u\|_{p}^{q+1}}{\|u\|_{q+1}^{q+1}}\right)^{\frac{p}{q+1-p}}$
$\displaystyle\quad=\left(\frac{1}{p}-\frac{1}{q+1}\right)\inf_{v\in
W_{N}^{1,p}\left(\Omega\right),v\neq
0}\left(\frac{\|\nabla\left(\lambda^{\ast}v\right)\|_{p}^{q+1}}{\|\left(\lambda^{\ast}v\right)\|_{q+1}^{q+1}}\right)^{\frac{p}{q+1-p}}$
$\displaystyle\quad=\frac{q+1-p}{p\left(q+1\right)}C_{*}^{-\frac{p(q+1)}{q+1-p}}.$

The proof is complete.

###### Lemma 2.4

For $d(\delta)$ in (2.2), we have

_(i)_
$d(\delta)\geq\frac{1}{p}(1-\delta)r^{p}(\delta)+\frac{q+1-p}{p\left(q+1\right)}\delta
r^{p}(\delta)$. In particular,
$d(1)\geq\frac{q+1-p}{p\left(q+1\right)}\left(\frac{1}{C_{\ast}^{q+1}}\right)^{\frac{p}{q+1-p}}$;

_(ii)_ there is a unique $b$, $b\in(1,\frac{q+1}{p}]$ such that $d(b)=0$, and
$d(\delta)>0$ for $1\leq\delta<b$;

_(iii)_ $d(\delta)$ is increasing on $0<\delta\leq 1$, decreasing on
$1\leq\delta\leq b$, and $\delta=1$ is the maximum point of $d(\delta)$.

Proof: (i) If $u\in N_{\delta}$, that is, $I_{\delta}\left(u\right)=0$ and
${\left\|\nabla u\right\|}_{p}\neq 0$. By Lemma 2.1 (iii), we find

$\begin{array}[]{l}J\left(u\right)=\frac{1}{p}\left(1-\delta\right)\left\|\nabla
u\right\|_{p}^{p}+\frac{\delta}{p}\left\|\nabla
u\right\|_{p}^{p}-\frac{1}{q+1}\left\|u\right\|_{q+1}^{q+1}\\\
\;\;\;\;\;\;\;\;\;\geq\frac{1}{p}\left(1-\delta\right)r^{p}\left(\delta\right)+\frac{q+1-p}{p\left(q+1\right)}\delta
r^{p}\left(\delta\right),\end{array}$

which implies
$d(1)\geq\frac{q+1-p}{p\left(q+1\right)}\left(\frac{1}{C_{\ast}^{q+1}}\right)^{\frac{p}{q+1-p}}$.

(ii) Let $\lambda\left(\delta\right)=\left(\frac{\delta\left\|\nabla
u\right\|_{p}^{p}}{\left\|u\right\|_{q+1}^{q+1}}\right)^{\frac{1}{q+1-p}}$,
then

$\begin{array}[]{l}I_{\delta}\left(\lambda\left(\delta\right)u\right)=\delta\lambda\left(\delta\right)^{p}\left\|\nabla
u\right\|_{p}^{p}-\lambda\left(\delta\right)^{q+1}\left\|u\right\|_{q+1}^{q+1}\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\lambda\left(\delta\right)^{p}\left(\delta\left\|\nabla
u\right\|_{p}^{p}-\lambda\left(\delta\right)^{q+1-p}\left\|u\right\|_{q+1}^{q+1}\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=0,\end{array}$

i.e., $I_{\delta}\left(\lambda\left(\delta\right)u\right)=0$. For
$\lambda\left(\delta\right)u\in N_{\delta}$, we have

$\begin{array}[]{l}d\left(\delta\right)\leq
J\left(\lambda\left(\delta\right)u\right)\leq\frac{1}{p}\lambda\left(\delta\right)^{p}\left\|\nabla
u\right\|_{p}^{p}-\frac{1}{q+1}\lambda\left(\delta\right)^{q+1}\left\|u\right\|_{q+1}^{q+1}\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\left(\frac{1}{p}-\frac{\delta}{q+1}\right)\lambda\left(\delta\right)^{p(x)}\left\|\nabla
u\right\|_{p}^{p}.\end{array}$

Then $d\left(\delta\right)\leq 0$ for $\delta=\frac{q+1}{p}$. On the other
hand, $d\left(1\right)=d>0$. Since $d\left(\delta\right)$ is continuous with
respect to $\delta$, there is a $b\in(1,\frac{q+1}{p}]$ satisfying
$d\left(b\right)=0$.

(iii) If we can prove that
$\underset{N_{\delta^{\prime}}}{\inf}J\left(u\right)=d\left(\delta^{\prime}\right)<d\left(\delta^{\prime\prime}\right)=\underset{N_{\delta^{\prime\prime}}}{\inf}J\left(u\right)$
for $0<\delta^{\prime}<\delta^{\prime\prime}<1$ and
$1<\delta^{\prime\prime}<\delta^{\prime}<b$, respectively, then the lemma is
proved.

In fact, for $0<\delta^{\prime}<\delta^{\prime\prime}<1$, from the definition
of $\lambda\left(\delta\right)$, we see that

$\begin{array}[]{l}J\left(\lambda\left(\delta^{\prime\prime}\right)u\right)-J\left(\lambda\left(\delta^{\prime}\right)u\right)\\\
=\frac{1}{p}\int_{\Omega}(\lambda\left(\delta^{\prime\prime}\right)^{p}\left|\nabla
u\right|^{p}-\lambda\left(\delta^{\prime}\right)^{p}\left|\nabla
u\right|^{p})\operatorname{d}x\\\
\;\;\;\;-\frac{1}{q+1}\int_{\Omega}(\lambda\left(\delta^{\prime\prime}\right)^{q+1}\left|u\right|^{q+1}-\lambda\left(\delta^{\prime}\right)^{q+1}\left|u\right|^{q+1})\operatorname{d}x\\\
=\int_{\Omega}\left|\nabla
u\right|^{p}\int_{\lambda\left(\delta^{\prime}\right)}^{\lambda\left(\delta^{\prime\prime}\right)}\lambda\left(\delta\right)^{p-1}\operatorname{d}\lambda\operatorname{d}x-\int_{\Omega}\left|u\right|^{q+1}\int_{\lambda\left(\delta^{\prime}\right)}^{\lambda\left(\delta^{\prime\prime}\right)}\lambda\left(\delta\right)^{q}\operatorname{d}\lambda\operatorname{d}x\\\
=\int_{\lambda\left(\delta^{\prime}\right)}^{\lambda\left(\delta^{\prime\prime}\right)}\left(\lambda\left(\delta\right)^{p-1}\left\|\nabla
u\right\|_{p}^{p}-\lambda\left(\delta\right)^{q}\left\|u\right\|_{q+1}^{q+1}\right)\operatorname{d}\lambda\\\
=\int_{\lambda\left(\delta^{\prime}\right)}^{\lambda\left(\delta^{\prime\prime}\right)}\lambda\left(\delta\right)^{p-1}\left(1-\delta\right)\left\|\nabla
u\right\|_{p}^{p}\operatorname{d}\lambda.\end{array}$ (2.6)

By the definition of $\lambda\left(\delta\right)$, we know that
$\lambda\left(\delta\right)$ is increasing with respect to $\delta$.
Therefore, from (2.6), we have
$J\left(\lambda\left(\delta^{\prime\prime}\right)u\right)-J\left(\lambda\left(\delta^{\prime}\right)u\right)>0$.

Similarly, for $1<\delta^{\prime\prime}<\delta^{\prime}<b$, we get
$J\left(\lambda\left(\delta^{\prime\prime}\right)u\right)-J\left(\lambda\left(\delta^{\prime}\right)u\right)>0$.

From Lemma 2.4, we can define $d_{0}=\underset{\delta\rightarrow
0^{+}}{\mathrm{lim}}d(\delta)\geq 0$.

###### Lemma 2.5

For $u\in W_{N}^{1,p}$, let $d_{0}<J(u)<d$, and $\delta_{1}<1<\delta_{2}$ be
the two roots of equation $d(\delta)=J(u)$, then the sign of $I_{\delta}(u)$
is invariable in $\delta_{1}<\delta<\delta_{2}$.

Next, we prove that $W_{\delta}$ and $V_{\delta}$ are invariant sets of (1.1)
when $0<J(u_{0})<d$. The following discussion is divided into two parts:
$J(u_{0})$ being in the monotonic interval of $d\left(\delta\right)$ and
$J(u_{0})$ being in the non-monotonic interval of $d\left(\delta\right)$.

######  Proposition 2.1 (Inspired by [4])

Assume that $u$ is a weak solution of (1.1), and the initial value satisfies
$u_{0}\in W_{N}^{1,p}$ and $J\left(u_{0}\right)=\sigma$. Then we get the
following results.

_(i)_ If $0<\sigma\leq d_{0}$, then there is a unique
$\overline{\delta}\in\left(1,b\right)$ satisfying
$d\left(\overline{\delta}\right)=\sigma$, where $b$ is the constant in Lemma
2.4 (ii). Further, if $I\left(u_{0}\right)>0$, then $u\in W_{\delta}$ for any
$1\leq\delta<\overline{\delta}$. Otherwise, if $I\left(u_{0}\right)<0$, then
there is $u\in V_{\delta}$ for any $1\leq\delta<\overline{\delta}$;

_(ii)_ If $d_{0}<\sigma<d$, then $\delta_{1}$ and $\delta_{2}$ satisfy
$\delta_{1}<1<\delta_{2}$ and
$d\left(\delta_{1}\right)=d\left(\delta_{2}\right)=\sigma$. Further, if
$I\left(u_{0}\right)>0$, then there is $u\in W_{\delta}$ for any
$\delta_{1}<\delta<\delta_{2}$. Otherwise, if $I\left(u_{0}\right)<0$, then
$u\in V_{\delta}$ for any $\delta_{1}<\delta<\delta_{2}$.

Proof: Case 1. If $0<J\left(u_{0}\right)=\sigma\leq d_{0}$, that is,
$J\left(u_{0}\right)$ is in the monotonic interval of $d\left(\delta\right)$.
By Lemma 2.4, there is a unique $\overline{\delta}\in\left(1,b\right)$
satisfying $d\left(\overline{\delta}\right)=\sigma$. For any
$1\leq\delta<\overline{\delta}$, we have

$I_{\delta}\left(u_{0}\right)=\left(\delta-1\right)\left\|\nabla
u_{0}\right\|_{p}^{p}+I\left(u_{0}\right)\geq
I\left(u_{0}\right),\;J\left(u_{0}\right)=\sigma=d\left(\overline{\delta}\right)<d\left(\delta\right).$
(2.7)

Multiplying the equation of (1.1) by $u_{t}$, and integrating on
$\Omega\times\left[0,t\right]$, we obtain that

$\int_{0}^{t}\left(\left\|u_{\tau}\right\|_{2}^{2}+\left\|\nabla
u_{\tau}\right\|_{2}^{2}\right)\operatorname{d}\tau+J\left(u\right)=J\left(u_{0}\right)=d\left(\overline{\delta}\right)<d\left(\delta\right),$
(2.8)

for any $t\in\left(0,T\right)$ and any $\delta\in[1,\overline{\delta})$, where
$T$ is the maximal existence time.

If $I\left(u_{0}\right)>0$, it is easy to know from (2.7) that there is
$u_{0}\in W_{\delta}$ for $\delta\in[1,\overline{\delta})$. We assert that if
$t\in\left(0,T\right)$ and $\delta\in[1,\overline{\delta})$, then $u\in
W_{\delta}$. Assuming that the assertion is not tenable, then there exists
$\delta^{\ast}\in[1,\overline{\delta})$ and $t_{0}\in\left(0,T\right)$, such
that $u\in W_{\delta^{\ast}}$ for $t\in\left(0,t_{0}\right)$, but
$u\left(x,t_{0}\right)\in\partial W_{\delta^{\ast}}$, i.e.,

$I_{\delta^{\ast}}\left(u\left(t_{0}\right)\right)=0,\;\left\|\nabla
u\left(t_{0}\right)\right\|_{p}^{p}\neq
0,\;\mathrm{or}\;J\left(u\left(t_{0}\right)\right)=d\left(\delta^{\ast}\right).$

In fact, from (2.8), we can see that $J\left(u\left(t_{0}\right)\right)\leq
J\left(u_{0}\right)<d\left(\delta^{\ast}\right)$, which implies
$I_{\delta^{\ast}}\left(u\left(t_{0}\right)\right)=0$ and $\left\|\nabla
u\left(t_{0}\right)\right\|_{p}^{p}\neq 0$, i.e. $u\left(x,t_{0}\right)\in
N_{\delta^{\ast}}$. Therefore, by the definition of
$d\left(\delta^{\ast}\right)$, we can get
$J\left(u\left(t_{0}\right)\right)\geq d\left(\delta^{\ast}\right)$, which is
a contradiction.

Next, we prove that if $I\left(u_{0}\right)<0$, then $u_{0}\in V_{\delta}$ for
$\delta\in[1,\overline{\delta})$, and $u\in V_{\delta}$ for any
$t\in\left(0,T\right)$ and any $\delta\in[1,\overline{\delta})$. Assuming that
the assertion about $u_{0}$ is not tenable, combined with (2.7), we can know
that $\delta_{\ast}\in[1,\overline{\delta})$ being the first number such that
$u_{0}\in V_{\delta}$ for $\delta\in[1,\delta_{\ast})$ and $u_{0}\in\partial
V_{\delta_{\ast}}$, i.e.,

$I_{\delta_{\ast}}\left(u_{0}\right)=0,\;or\;J\left(u_{0}\right)=d\left(\delta_{\ast}\right).$

Because $J(u_{0})$ is in the strict decreasing interval of $d(\delta)$, then
$J(u_{0})=d(\overline{\delta})<d(\delta_{\ast})$, which implies that
$I_{\delta_{\ast}}(u_{0})=0$. Because $I_{\delta}(u_{0})<0$ for
$\delta\in[1,\delta_{\ast})$, then from Lemma 2.1 ($\mathrm{ii}$) we have
${\left\|\nabla u_{0}\right\|}_{p}>r(\delta)>0$, that is, $u_{0}\in
N_{\delta_{\ast}}$. According to the definition of
$d\left(\delta_{\ast}\right)$, we get $J(u_{0})=d(\overline{\delta})\geq
d(\delta_{\ast})$, which is contradict with the monotonicity of $d(\delta)$.
Assuming that the assertion about $u$ is not tenable, then for
$t\in(0,t_{0})$, there exists $\delta_{\ast}^{\ast}\in[1,\overline{\delta})$
and $t_{0}\in(0,T)$, such that $u\in V_{\delta_{\ast}^{\ast}}$, but
$u(x,t_{0})\in\partial V_{\delta_{\ast}^{\ast}}$, that is,

$I_{\delta_{\ast}^{\ast}}(u(t_{0}))=0,\;\mathrm{or}\;J(u(t_{0}))=d(\delta_{\ast}^{\ast}).$

In fact, $J(u(t_{0}))\leq J(u_{0})<d(\delta_{\ast}^{\ast})$ is obtained from
(2.8), which implies $I_{\delta_{\ast}^{\ast}}(u(t_{0}))=0$. Because
$I_{\delta}(u(t_{0}))<0$ for $\delta\in[1,\delta_{\ast}^{\ast})$, then from
Lemma 2.1 ($\mathrm{ii}$) we get ${\left\|\nabla
u(t_{0})\right\|}_{p}>r(\delta)>0$, that is, $u(x,t_{0})\in
N_{\delta_{\ast}^{\ast}}$. From the definition of $d(\delta_{\ast}^{\ast})$,
we get $J(u(t_{0}))\geq d(\delta_{\ast}^{\ast})$, which is a contradiction.

Case 2. $d_{0}<J(u_{0})=\sigma<d$, that is, $J(u_{0})$ is in the non-monotonic
interval of $d(\delta)$. By Lemma 2.4, it is known that 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 2.5, the
sign of $I_{\delta}(u)$ remains unchanged for $\delta_{1}<\delta<\delta_{2}$.
Thus $I_{\delta}(u_{0})>0$ for $\delta\in(\delta_{1},\delta_{2})$. Therefore,
we have $u_{0}\in W_{\delta}$ for $\delta\in(\delta_{1},\delta_{2})$. The
proof of $u\in W_{\delta}$ is similar to Case 1. If $I(u_{0})<0$, by lemma
2.5, we can still get $I_{\delta}(u_{0})<0$ for
$\delta\in(\delta_{1},\delta_{2})$ and $J(u_{0})<d(\delta)$ 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 Case 1.

By Lagrange mean value theorem, the following lemma can be obtained directly.

######  Lemma 2.6 (Estimate of nonlinear term $\left|u\right|^{q-1}u$)

If $q$ satisfies the conditions given by (1.1),
$\left|u_{1}\right|+\left|u_{2}\right|>0$ and $u_{1}\neq u_{2}$ for any
$u_{1}\left(x,t\right)$, $u_{2}\left(x,t\right)$ with
$\left(x,t\right)\in\Omega\times\left[0,T\right]$, then

$\left|u_{1}\right|^{q-1}u_{1}-\left|u_{2}\right|^{q-1}u_{2}\leq
q\left(\left|u_{1}\right|+\left|u_{2}\right|\right)^{q-1}\left|u_{1}-u_{2}\right|.$

The main results of global existence, uniqueness, decay estimate and blowup
for the case $J\left(u_{0}\right)<d$ are stated in the following two theorems.

###### Theorem 2.1

Assume that $u_{0}\in W_{N}^{1,p}\left(\Omega\right)$, $J\left(u_{0}\right)<d$
and $I\left(u_{0}\right)>0$, then problem (1.1) has a global weak solution
$u\in L^{\infty}\left(0,\infty;W_{N}^{1,p}\left(\Omega\right)\right)$ with
$u_{t}\in L^{2}\left(0,\infty;W_{N}^{1,2}\left(\Omega\right)\right)$, and the
weak solution of problem (1.1) is unique for $p\leq 2$.

Moreover, if $\left\|\nabla u\right\|_{2}^{2}\leq C\left\|\nabla
u\right\|_{p}^{p}$, there exists a constant $\delta>0$ such that
$\left\|u\right\|_{H^{1}}^{2}<\left\|u_{0}\right\|_{H^{1}}^{2}e^{-2\delta t}$
for $t\in[0,\infty)$.

###### Theorem 2.2

Let $u_{0}\in W_{N}^{1,p}\left(\Omega\right)$, $J\left(u_{0}\right)<d$ and
$I\left(u_{0}\right)<0$, then the weak solution $u(t)$ of problem (1.1) blows
up in finite time.

The main results for the case $J\left(u_{0}\right)=d$ are given in the
following two theorems.

###### Theorem 2.3

Assume that $u_{0}\in W_{N}^{1,p}(\Omega)$ and $J\left(u_{0}\right)=d$. If
$I(u_{0})\geq 0$, then problem (1.1) admits a global weak solution $u(t)\in
L^{\infty}(0,\infty;W_{N}^{1,p}(\Omega))$ with $u_{t}\in
L^{2}\left(0,\infty;W_{N}^{1,2}\left(\Omega\right)\right)$, and there exists a
unique weak solution for $p\leq 2$.

Moreover, if $I(u_{0})>0$ and $\left\|\nabla u\right\|_{2}^{2}\leq
C\left\|\nabla u\right\|_{p}^{p}$, there exist constants $t_{1}>0$ and
$\kappa>0$ such that
$\left\|u\right\|_{H^{1}}^{2}<\left\|u\left(t_{1}\right)\right\|_{H^{1}}^{2}e^{-2\kappa\left(t-t_{1}\right)}$
for $t\in(t_{1},+\infty)$.

###### Theorem 2.4

Let $u_{0}\in W_{N}^{1,p}(\Omega)$, $J(u_{0})=d$ and $I(u_{0})<0$, then the
weak solution $u(t)$ of problem (1.1) blows up in finite time.

The following theorem gives the main result for case $J\left(u_{0}\right)>d$.

###### Theorem 2.5

Assume that $u_{0}\in W_{N}^{1,p}\left(\Omega\right)$, $J\left(u_{0}\right)$
is finite and $J\left(u_{0}\right)>d$, $I\left(u_{0}\right)>0$ and
$\left\|u_{0}\right\|_{H^{1}}^{2}\leq\lambda_{J\left(u_{0}\right)}$, then
problem (1.1) has a global weak solution $u\in
L^{\infty}\left(0,\infty;W_{N}^{1,p}(\Omega)\right)$ with $u_{t}\in
L^{2}\left(0,\infty;W_{N}^{1,2}(\Omega)\right)$, and the weak solution is
unique for $p\leq 2$.

Moreover, we have that $u\left(t\right)\rightarrow 0$ as
$t\rightarrow+\infty$.

##  3 Subcritical initial energy
$\boldsymbol{J}\mathbf{(u_{0})}\boldsymbol{<}\boldsymbol{d}$

In this section, we shall consider the global existence, uniqueness and decay
estimate, and blowup of solutions to problem (1.1) for the subcritical initial
energy $J\left(u_{0}\right)<d$.

Proof of Theorem 2.1.

Global existence. Consider a orthonormal basis
$\left\\{\omega_{j}\left(x\right)\right\\}$ on
$W_{N}^{1,p}\left(\Omega\right)$, which is also orthogonal in
$L^{2}\left(\Omega\right)$. Define

$u_{m}\left(x,t\right)=\sum_{j=1}^{m}g_{jm}\left(t\right)\omega_{j}\left(x\right),\;\;\;m=1,2,\cdots,$

where $g_{jm}\left(t\right)$ satisfies the initial value problem of the
following equations

$\left(u_{mt},\omega_{s}\right)+\left(\nabla
u_{mt},\nabla\omega_{s}\right)+\left(\left|\nabla u_{m}\right|^{p-2}\nabla
u_{m},\nabla\omega_{s}\right)=\left(f\left(u_{m}\right),\omega_{s}\right),$
(3.1)
$u_{m}\left(x,0\right)=\sum_{j=1}^{m}a_{jm}\omega_{j}\left(x\right)\rightarrow
u_{0}\left(x\right)\quad{\rm in}\quad W_{N}^{1,p}\left(\Omega\right),$ (3.2)

for $1\leq s\leq m$, in which

$a_{jm}=g_{jm}{(0)}\;\mathrm{and}\;f\left(u\right)=\left|u\right|^{q-1}u-{\mathchoice{{-\mkern-19.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}}_{\Omega}\left|u\right|^{q-1}udx.$

Multiplying the $j^{th}$ equation of (3.1) by $g^{\prime}_{sm}\left(t\right)$,
summing up with respect to $j$, and integrating with respect to the time
variable from 0 to $t$, we readily get the equality

$\int_{0}^{t}\left\|u_{m\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau+J\left(u_{m}\right)=J\left(u_{m}\left(0\right)\right),\;\;\;0\leq
t<\infty.$ (3.3)

From (3.2), we obtain $J\left(u_{m}\left(0\right)\right)\rightarrow
J\left(u_{0}\right)<d$ and $I\left(u_{m}\left(0\right)\right)\rightarrow
I\left(u_{0}\right)>0$, which combining (3.3) implies that

$\int_{0}^{t}\left\|u_{m\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau+J\left(u_{m}\right)<d,\;\;\;0\leq
t<\infty.$ (3.4)

In fact, we obtain $u_{m}\left(t\right)\in W$ for sufficiently large $m$. If
the above is false, then there exists $t_{0}>0$ to satisfy
$u_{m}(x,t_{0})\in\partial W$, i.e.

$I(u_{m}(t_{0}))=0,\;{\left\|\nabla u_{m}(t_{0})\right\|}_{p}\neq
0,\;\mathrm{or}\;J(u_{m}(t_{0}))=d.$

On the one hand, from (3.4), it is easy to know that $J(u_{m}(t_{0}))=d$ is
not tenable. On the other hand, if $I(u_{m}(t_{0}))=0$ and ${\left\|\nabla
u_{m}(t_{0})\right\|}_{p}\neq 0$, it can be seen from the definition of $d$
that $J(u(t_{0}))\geq d$, which is also contradict with (3.4). Thus we get
$u_{m}\left(t\right)\in W$.

With the help of (3.4) and

$J\left(u_{m}\right)=\frac{1}{q+1}I\left(u_{m}\right)+\left(\frac{1}{p}-\frac{1}{q+1}\right)\left\|\nabla
u_{m}\right\|_{p}^{p},$

we have

$\int_{0}^{t}\left\|u_{m\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau+\left(\frac{1}{p}-\frac{1}{q+1}\right)\left\|\nabla
u_{m}\right\|_{p}^{p}<d,\;\;\;0\leq t<\infty,$

where $m$ is sufficiently large, which yields a priori estimate

$\int_{0}^{t}\left\|u_{m\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau<d,\;\;\;0\leq
t<\infty,$ $\left\|\nabla
u_{m}\right\|_{p}^{p}<\frac{p\left(q+1\right)}{q+1-p}d,\;\;\;0\leq t<\infty,$
$\left\|u_{{}_{m}}^{q}\right\|_{\gamma}^{\gamma}=\left\|u_{m}\right\|_{q+1}^{q+1}\leq
C_{\ast}^{q+1}\left(\frac{p\left(q+1\right)}{q+1-p}d\right)^{\frac{q+1}{p}},\;\gamma=\frac{q+1}{q},\;\;0\leq
t<\infty.$

Then, there exists a subsequence of $\left\\{u_{m}\right\\}$, denoted by the
same symbol satisfying

$\left\\{\begin{array}[]{l}u_{m}\rightarrow
u\;\;\mathrm{in}\;L^{\infty}(0,\infty;\;W_{N}^{1,p}(\Omega))\;\mathrm{weak}\;\mathrm{star}\;\mathrm{and}\;\mathrm{a}.\mathrm{e}.\;\mathrm{in}\;Q=\Omega\times(0,\;\infty),\\\
u_{mt}\rightarrow
u_{t}\;\;\mathrm{in}\;L^{2}(0,\infty;\;W_{N}^{1,2}(\Omega))\;\mathrm{weak},\\\
\left|\nabla u_{m}\right|^{p-2}\nabla
u_{m}\rightarrow\chi\;\;\mathrm{in}\;L^{\infty}(0,\infty;\;L^{\frac{p}{p-1}}(\Omega))\;\mathrm{weak}\;\mathrm{star},\\\
\left|u_{m}\right|^{q-1}u_{m}\rightarrow\left|u\right|^{q-1}u\;\;\mathrm{in}\;L^{\infty}(0,\infty;\;L^{\frac{q+1}{q}}(\Omega))\;\mathrm{weak}\;\mathrm{star}.\end{array}\right.$

We fixed $s$ in (3.1) and letting $m\rightarrow\infty$, there hold

$\left(u_{t},\omega_{s}\right)+\left(\nabla
u_{t},\nabla\omega_{s}\right)+\left(\chi,\nabla\omega_{s}\right)=\left(f\left(u\right),\omega_{s}\right)\;\mathrm{for}\;\mathrm{all}\;s,$

and

$\left(u_{t},v\right)+\left(\nabla u_{t},\nabla v\right)+\left(\chi,\nabla
v\right)=\left(f\left(u\right),v\right)\;\mathrm{for}\;\mathrm{all}\;v\in
W_{N}^{1,p}\left(\Omega\right),\;t\in\left(0,\;\infty\right).$ (3.5)

Next, we only need to prove that there is $\chi=\left|\nabla
u\right|^{p-2}\nabla u$ in the weak sense, i.e.

$\left(\chi,\nabla v\right)=\left(\left|\nabla u\right|^{p-2}\nabla u,\nabla
v\right),\;\;\;\forall v\in W_{N}^{1,p}\left(\Omega\right).$ (3.6)

In fact, for any $\psi\in
L^{\infty}(0,T;W_{N}^{1,p}\left(\Omega\right)),\;\varsigma\in
W_{N}^{1,p}\left(\Omega\right),\;0\;\leq\varsigma\leq 1$, we obtain

$\int_{\Omega}\varsigma\left(\left|\nabla u_{m}\right|^{p-2}\nabla
u_{m}-\left|\nabla\psi\right|^{p-2}\nabla\psi\right)\nabla\left(u_{m}-\psi\right)\operatorname{d}x\geq
0,$

i.e.

$\begin{array}[]{l}\int_{\Omega}\varsigma\left|\nabla
u_{m}\right|^{p-2}\left|\nabla
u_{m}\right|^{2}\operatorname{d}x-\int_{\Omega}\varsigma\left|\nabla
u_{m}\right|^{p-2}\nabla u_{m}\nabla\psi\operatorname{d}x\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\int_{\Omega}\varsigma\left|\nabla\psi\right|^{p-2}\nabla\psi\nabla\left(u_{m}-\psi\right)\operatorname{d}x\geq
0.\end{array}$ (3.7)

Noticing that

$\begin{array}[]{l}\int_{\Omega}\varsigma\left|\nabla
u_{m}\right|^{p-2}\left|\nabla u_{m}\right|^{2}\operatorname{d}x\\\
=-\int_{\Omega}\nabla\varsigma\left|\nabla u_{m}\right|^{p-2}\nabla
u_{m}u_{m}\operatorname{d}x-\int_{\Omega}\varsigma\nabla\left(\left|\nabla
u_{m}\right|^{p-2}\nabla u_{m}\right)u_{m}\operatorname{d}x\\\
=-\int_{\Omega}\nabla\varsigma\left|\nabla u_{m}\right|^{p-2}\nabla
u_{m}u_{m}\operatorname{d}x-\int_{\Omega}\varsigma
u_{mt}u_{m}\operatorname{d}x-\int_{\Omega}\varsigma\nabla u_{mt}\nabla
u_{m}\operatorname{d}x\\\ \;\;\;\;-\int_{\Omega}\nabla\varsigma\nabla
u_{mt}u_{m}\operatorname{d}x+\int_{\Omega}\varsigma\left|u\right|_{m}^{q-1}u_{m}u_{m}\operatorname{d}x,\end{array}$
(3.8)

Letting $m\rightarrow\infty$ in (3.7), combined with (3.8),we find

$\begin{array}[]{l}-\int_{\Omega}\nabla\varsigma\chi
u\operatorname{d}x-\int_{\Omega}\varsigma
u_{t}u\operatorname{d}x-\int_{\Omega}\varsigma\nabla u_{t}\nabla
u\operatorname{d}x-\int_{\Omega}\nabla\varsigma\nabla
u_{t}u\operatorname{d}x\\\
+\int_{\Omega}\varsigma\left|u\right|^{q-1}uu\operatorname{d}x-\int_{\Omega}\varsigma\chi\nabla\psi\operatorname{d}x-\int_{\Omega}\varsigma\left|\nabla\psi\right|^{p-2}\nabla\psi\nabla\left(u-\psi\right)\operatorname{d}x\geq
0.\end{array}$ (3.9)

Choosing $v=u\varsigma$ in (3.5), we have

$\begin{array}[]{l}\int_{\Omega}u_{t}u\varsigma\operatorname{d}x+\int_{\Omega}\nabla
u_{t}\nabla u\varsigma\operatorname{d}x+\int_{\Omega}\nabla
u_{t}u\nabla\varsigma\operatorname{d}x+\int_{\Omega}\chi\nabla
u\varsigma\operatorname{d}x+\int_{\Omega}\chi
u\nabla\varsigma\operatorname{d}x\\\
=\int_{\Omega}\left|u\right|^{q-1}uu\varsigma\operatorname{d}x.\end{array}$
(3.10)

Combining (3.9) with (3.10), we get

$\int_{\Omega}\varsigma\left(\chi-\left|\nabla\psi\right|^{p-2}\nabla\psi\right)\nabla\left(u-\psi\right)\operatorname{d}x\geq
0.$

Taking $\psi=u-\lambda v,\;\lambda\geq 0,\;v\in
W_{N}^{1,p}\left(\Omega\right)$ in the above inequality, we obtain

$\int_{\Omega}\varsigma\left(\chi-\left|\nabla\left(u-\lambda
v\right)\right|^{p-2}\nabla\left(u-\lambda v\right)\right)\nabla
v\operatorname{d}x\geq 0.$ (3.11)

Choosing $\lambda\rightarrow 0$ in (3.11), we get

$\int_{\Omega}\varsigma\left(\chi-\left|\nabla u\right|^{p-2}\nabla
u\right)\nabla v\operatorname{d}x\geq 0,\;\forall v\in
W_{N}^{1,p}\left(\Omega\right).$

Obviously, if we choose $\lambda\leq 0$, we can deduce the similar inequality
replacing $\geq$ by $\leq$. Hence, (3.6) holds. Furthermore, (3.2) gives
$u(x,0)=u_{0}(x)$ in $W_{N}^{1,p}(\Omega)$.

Uniqueness. Let $u_{1}$ and $u_{2}$ be the weak solutions of (1.1) with the
same initial data. Define $v=u_{1}-u_{2}$, we have

$v_{\tau}\;+\nabla v_{\tau}+\mathrm{div}\left(|\nabla u_{1}|^{p-2}\nabla
u_{1}\right)-\mathrm{div}\left(|\nabla u_{2}|^{p-2}\nabla
u_{2}\right)=|u_{1}|^{q-1}u_{1}-|u_{2}|^{q-1}u_{2},$ (3.12) $u_{1},u_{2},v\in
L^{\infty}\left(0,\infty;W_{N}^{1,p}\right),\;u_{1t},u_{2t},v_{t}\in
L^{2}\left(0,\infty;W_{N}^{1,2}\right).$

Multiplying (3.12) by $v$, integrating over $\left(0,t\right)\times\Omega$, we
get

$\begin{array}[]{l}\int_{0}^{t}\int_{\Omega}{(v_{\tau}\;v+\nabla
v_{\tau}\nabla v+\left(|\nabla u_{1}|^{p-2}\nabla u_{1}-|\nabla
u_{2}|^{p-2}\nabla u_{2}\right)\nabla
v\;)}\operatorname{d}x\operatorname{d}\tau\\\
=\int_{0}^{t}\int_{\Omega}\left(|u_{1}|^{q-1}u_{1}-|u_{2}|^{q-1}u_{2}\right)v\;\operatorname{d}x\operatorname{d}\tau.\end{array}$
(3.13)

By Lemma 2.6, we get

$\begin{array}[]{l}\int_{0}^{t}\int_{\Omega}\left(\left|u_{1}\right|^{q-1}u_{1}-\left|u_{2}\right|^{q-1}u_{2}\right)v\operatorname{d}x\operatorname{d}\tau\\\
\leq\int_{0}^{t}\int_{\Omega}q\left(\left|u_{1}\right|+\left|u_{2}\right|\right)^{q-1}\left|u_{1}-u_{2}\right|\left|v\right|\operatorname{d}x\operatorname{d}\tau\\\
\leq
C\int_{0}^{t}{\left\|\left(\left|u_{1}\right|+\left|u_{2}\right|\right)^{q-1}\right\|}_{A_{1}}{\left\|u_{1}-u_{2}\right\|}_{A_{2}}{\left\|v\right\|}_{A_{3}}\operatorname{d}\tau\\\
=C\int_{0}^{t}\left\|\left|u_{1}\right|+\left|u_{2}\right|\right\|_{{}^{\left(q-1\right)}A_{1}}^{q-1}{\left\|u_{1}-u_{2}\right\|}_{A_{2}}{\left\|v\right\|}_{A_{3}}\operatorname{d}\tau\\\
\leq
C\int_{0}^{t}{\left\|u_{1}-u_{2}\right\|}_{H^{1}}{\left\|v\right\|}_{H^{1}}\operatorname{d}\tau\\\
\leq
C\int_{0}^{t}\left\|v\right\|_{H^{1}}^{2}\operatorname{d}\tau,\end{array}$
(3.14)

where $(q-1)A_{1}=q+1$ and $A_{2}=A_{3}=q+1<\frac{2n}{n-2}$ by $p\leq 2$. As
$v(x,0)=0$, we obtain

$\begin{array}[]{l}\int_{0}^{t}\int_{\Omega}{(v_{\tau}v+\nabla v_{\tau}\nabla
v)}\operatorname{d}x\operatorname{d}\tau=\frac{1}{\;2}\int_{\Omega}{(\left.v^{2}\left(\tau\right)\right|_{v\left(0\right)}^{v\left(t\right)}+\left.\left|\nabla
v\left(\tau\right)\right|^{2}\right|_{v\left(0\right)}^{v\left(t\right)})}\operatorname{d}x\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{1}{2}\int_{\Omega}{(v^{2}\left(t\right)+\left|\nabla
v\left(t\right)\right|^{2})}\operatorname{d}x\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\frac{1}{2}\left\|v\left(t\right)\right\|_{H^{1}}^{2}.\end{array}$
(3.15)

Let
$\overset{-}{v}=\theta_{1}u_{1}+\left(1-\theta_{1}\right)u_{2},\;\theta_{1}\in\left[0,1\right],$
i.e.,

$\int_{\Omega}\left(\left|\nabla u_{1}\right|^{p-2}\nabla u_{1}-\left|\nabla
u_{2}\right|^{p-2}\nabla u_{2}\right)\nabla
v\operatorname{d}x=\int_{\Omega}\left(p-1\right)\left|\nabla\overset{-}{v}\right|^{p-2}\left|\nabla
v\right|^{2}\operatorname{d}x.$ (3.16)

It follows from (3.13), (3.14), (3.15) and (3.16) that

$\left\|v\right\|_{H^{1}}^{2}\leq
C\int_{0}^{t}\left\|v\right\|_{H^{1}}^{2}\operatorname{d}\tau.$

By Gronwall’s inequality, we obtain $\left\|v\right\|_{H^{1}}^{2}\leq 0$, that
is, $\left\|v\right\|_{H^{1}}^{2}=\left\|u_{1}-u_{2}\right\|_{H^{1}}^{2}=0$.
Thus $u_{1}=u_{2}=0$ a.e. in $\Omega\times\left(0,\infty\right)$.

Asymptotic behavior. Letting $\varphi=u$ in (2.3) and

$\int_{0}^{t}\int_{\Omega}\left({\mathchoice{{-\mkern-19.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}}_{\Omega}\left|u\right|^{q-1}u\operatorname{d}x\right)u\operatorname{d}x\operatorname{d}\tau=\int_{0}^{t}\left({\mathchoice{{-\mkern-19.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}}_{\Omega}\left|u\right|^{q-1}u\operatorname{d}x\right)\int_{\Omega}u\operatorname{d}x\operatorname{d}\tau=0,$

we find

$\frac{1}{2}\left\|u\right\|_{H^{1}}^{2}-\frac{1}{2}\left\|u_{0}\right\|_{H^{1}}^{2}+\;\int_{0}^{t}\left(\left\|\nabla
u\right\|_{p}^{p}-\left\|u\right\|_{q+1}^{q+1}\right)\operatorname{d}s=0,$

then

$\frac{1}{2}\frac{d}{dt}\left\|u\right\|_{H^{1}}^{2}=-I\left(u\right).$ (3.17)

By Proposition 2.1, we know that $u\in W_{\delta}$ for
$1\leq\delta<\overline{\delta}$ or $\delta_{1}<\delta<\delta_{2}$ with
$\delta_{1}<1<\delta_{2}$, and particularly $I(u)>0$. Then, the norm
${\left\|u\right\|}_{H^{1}}$ is equivalent to the norm ${\left\|\nabla
u\right\|}_{2}$ on $H^{1}\left(\Omega\right)$ and combining (2.4), we get

$\begin{array}[]{l}\;\;\frac{1}{2}\frac{d}{dt}\left\|u\right\|_{H^{1}}^{2}=-I\left(u\right)=-\left\|\nabla
u\right\|_{p}^{p}+\left\|u\right\|_{q+1}^{q+1}\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq-\left\|\nabla
u\right\|_{p}^{p}+C_{\ast}^{q+1}\left\|\nabla u\right\|_{p}^{q+1}\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\left(C_{\ast}^{q+1}\left\|\nabla
u\right\|_{p}^{q-p+1}-1\right)\left\|\nabla u\right\|_{p}^{p}\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq C\left(C_{\ast}^{q+1}\left\|\nabla
u\right\|_{p}^{q-p+1}-1\right)\left\|\nabla u\right\|_{2}^{2}\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq C\left(C_{\ast}^{q+1}\left\|\nabla
u\right\|_{p}^{q-p+1}-1\right)\left\|u\right\|_{H^{1}}^{2}.\end{array}$ (3.18)

Now, we will discuss $C_{\ast}^{q+1}\left\|\nabla u\right\|_{p}^{q-p+1}-1<0$.
Taking into account (2.1) and $I(u)>0$ we deduce that

$J(u)>\left(\frac{1}{p}-\frac{1}{q+1}\right)\;\left\|\nabla
u\right\|_{p}^{p},$

that is,

$\left\|\nabla
u\right\|_{p}^{p}<\frac{p\left(q+1\right)}{q+1-p}J\left(u\right)\leq\frac{p\left(q+1\right)}{q+1-p}J\left(u_{0}\right).$
(3.19)

By (3.19), we see that

$C_{\ast}^{q+1}\left\|\nabla
u\right\|_{p}^{q+1-p}<C_{\ast}^{q+1}\left(\frac{p\left(q+1\right)}{q+1-p}J\left(u_{0}\right)\right)^{\frac{q-p+1}{p}}<1.$
(3.20)

From (3.18) it follows that

$\frac{1}{2}\frac{d}{dt}\left\|u\right\|_{H^{1}}^{2}<C\left(C_{\ast}^{q+1}\left(\frac{p\left(q+1\right)}{q+1-p}J\left(u_{0}\right)\right)^{\frac{q-p+1}{p}}-1\right)\left\|u\right\|_{H^{1}}^{2}.$

Considering (3.20), we can know that there exists $\delta>0$ such that

$-\delta:=C\left(C_{\ast}^{q+1}\left(\frac{p\left(q+1\right)}{q+1-p}J\left(u_{0}\right)\right)^{\frac{q-p+1}{p}}-1\right).$

Applying Gronwall’s inequality, we infer for $\delta>0$

$\left\|u\right\|_{H^{1}}^{2}<\left\|u_{0}\right\|_{H^{1}}^{2}e^{-2\delta
t},\;0\leq t<\infty.$

Theorem 2.1 is proved.

Next, in order to prove $u(t)$ blowing up in finite time, we give a
relationship between $\left\|\nabla u\right\|_{p}^{p}$ and the depth $d$ of
potential well with $u_{0}\in V$.

###### Lemma 3.1

Let $u_{0}\in V$, then

$\left\|\nabla u\right\|_{p}^{p}>\frac{p\left(q+1\right)}{q+1-p}d.$

Proof: Assume that $u(t)$ is a weak solution of (1.1) with $J(u_{0})<d$ and
$I(u_{0})<0$, $T$ is the maximal existence time. Recalling Proposition 2.1, it
is easy to see that $u(x,t)\in V_{\delta}$, that is, $I(u)<0$ for $0<t<T$.
From Lemma 2.1, take $\delta=1$, we have

$\left\|\nabla
u\right\|_{p}^{p}>\left(\frac{1}{C_{\ast}^{q+1}}\right)^{\frac{p}{q+1-p}}.$

Coming back to Lemma 2.3, we conclude that

$d<\frac{q+1-p}{p\left(q+1\right)}\left\|\nabla u\right\|_{p}^{p}.$

Proof of Theorem 2.2. We will prove $u(t)$ blowing up in finite time with
$u_{0}\in V$. Arguing by contradiction, assume that the solution is global in
time.

Taking into account $u_{0}\in V$ and Proposition 2.1, we obtain $u\in
V_{\delta}$ for $t\in[0,+\infty)$. Denoting

$G\left(t\right)=\int_{0}^{t}\left\|u\right\|_{H^{1}}^{2}\operatorname{d}\tau+\left(T_{0}-t\right)\left\|u_{0}\right\|_{H^{1}}^{2},\;t\in\left[0,T_{0}\right],$

where $0<T_{0}<+\infty$. Obviously, for any $t\in[0,T_{0}]$ we get $G(t)>0$.
Applying the continuity of $G(t)$ with respect to $t$, we infer that there
exists a constant $\theta>0$, for $t\in[0,T_{0}]$ such that $G(t)\geq\theta$.
Then

$\begin{array}[]{l}G^{\prime}\left(t\right)=\left\|u\right\|_{H^{1}}^{2}-\left\|u_{0}\right\|_{H^{1}}^{2}\\\
\;\;\;\;\;\;\;\;\;=\left\|u\right\|_{2}^{2}+\left\|\nabla
u\right\|_{2}^{2}-\left\|u_{0}\right\|_{2}^{2}-\left\|\nabla
u_{0}\right\|_{2}^{2}\\\
\;\;\;\;\;\;\;\;\;=2\int_{0}^{t}\left(u,u_{\tau}\right)\operatorname{d}\tau+2\int_{0}^{t}\left(\nabla
u,\nabla u_{\tau}\right)\operatorname{d}\tau,\end{array}$ (3.21)

and (3.17) gives

$G^{\prime\prime}\left(t\right)=2\left(u,u_{t}\right)+2\left(\nabla u,\nabla
u_{t}\right)=-2I\left(u\right).$ (3.22)

By (3.21) and Cauchy-Schwarz inequality, we deduce

$\begin{array}[]{l}\left(G^{\prime}\left(t\right)\right)^{2}=4\left(\int_{0}^{t}{\left(u,u_{\tau}\right)}\operatorname{d}\tau+\int_{0}^{t}{\left(\nabla
u,\nabla u_{\tau}\right)}\operatorname{d}\tau\right)^{2}\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;=4\left[\left(\int_{0}^{t}\left(u,u_{\tau}\right)\operatorname{d}\tau\right)^{2}+\left(\int_{0}^{t}\left(\nabla
u,\nabla
u_{\tau}\right)\operatorname{d}\tau\right)^{2}+2\int_{0}^{t}\left(u,u_{\tau}\right)\operatorname{d}\tau\int_{0}^{t}\left(\nabla
u,\nabla u_{\tau}\right)\operatorname{d}\tau\right]\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq
4\Big{[}\int_{0}^{t}\left\|u\right\|_{2}^{2}\operatorname{d}\tau\int_{0}^{t}\left\|u_{\tau}\right\|_{2}^{2}\operatorname{d}\tau+\int_{0}^{t}\left\|\nabla
u\right\|_{2}^{2}\operatorname{d}\tau\int_{0}^{t}\left\|\nabla
u_{\tau}\right\|_{2}^{2}\operatorname{d}\tau\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+2\left(\int_{0}^{t}\left\|u\right\|_{2}^{2}\operatorname{d}\tau\right)^{\frac{1}{2}}\left(\int_{0}^{t}\left\|u_{\tau}\right\|_{2}^{2}\operatorname{d}\tau\right)^{\frac{1}{2}}\left(\int_{0}^{t}\left\|\nabla
u\right\|_{2}^{2}\operatorname{d}\tau\right)^{\frac{1}{2}}\left(\int_{0}^{t}\left\|\nabla
u_{\tau}\right\|_{2}^{2}\operatorname{d}\tau\right)^{\frac{1}{2}}\Big{]}\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq
4\Big{(}\int_{0}^{t}\left\|u\right\|_{2}^{2}\operatorname{d}\tau\int_{0}^{t}\left\|u_{\tau}\right\|_{2}^{2}\operatorname{d}\tau+\int_{0}^{t}\left\|\nabla
u\right\|_{2}^{2}\operatorname{d}\tau\int_{0}^{t}\left\|\nabla
u_{\tau}\right\|_{2}^{2}\operatorname{d}\tau\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\int_{0}^{t}\left\|u\right\|_{2}^{2}\operatorname{d}\tau\int_{0}^{t}\left\|\nabla
u_{\tau}\right\|_{2}^{2}\operatorname{d}\tau+\int_{0}^{t}\left\|\nabla
u\right\|_{2}^{2}\operatorname{d}\tau\int_{0}^{t}\left\|u_{\tau}\right\|_{2}^{2}\operatorname{d}\tau\Big{)}\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq
4\left(\int_{0}^{t}\left\|u\right\|_{2}^{2}\operatorname{d}\tau+\int_{0}^{t}\left\|\nabla
u\right\|_{2}^{2}\operatorname{d}\tau\right)\left(\int_{0}^{t}\left\|u_{\tau}\right\|_{2}^{2}\operatorname{d}\tau+\int_{0}^{t}\left\|\nabla
u_{\tau}\right\|_{2}^{2}\operatorname{d}\tau\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq
4\int_{0}^{t}\left\|u\right\|_{H^{1}}^{2}\operatorname{d}\tau\int_{0}^{t}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau\leq
4G\left(t\right)\int_{0}^{t}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau.\end{array}$
(3.23)

Considering (3.22) and (3.23), it is easy to see that

$\begin{array}[]{l}G^{\prime\prime}\left(t\right)G\left(t\right)-\frac{q+3}{4}\left(G^{\prime}\left(t\right)\right)^{2}\geq
G\left(t\right)\left(G^{\prime\prime}\left(t\right)-\left(q+3\right)\int_{0}^{t}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau\right)\\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=G\left(t\right)\left(-2I\left(u\right)-\left(q+3\right)\int_{0}^{t}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau\right).\end{array}$

Let

$\xi\left(t\right)=-2I\left(u\right)-\left(q+3\right)\int_{0}^{t}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau.$

Recalling (2.1), it follows that

$J\left(u\right)=\left(\frac{1}{p}-\frac{1}{q+1}\right)\left\|\nabla
u\right\|_{p}^{p}+\frac{1}{q+1}I\left(u\right),$

Therefore,

$\xi\left(t\right)=2\left(\frac{q+1}{p}-1\right)\left\|\nabla
u\right\|_{p}^{p}-2\left(q+1\right)J\left(u_{0}\right)+\left(q-1\right)\int_{0}^{t}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau.$

Now, we will discuss in two cases.

(i) If $0<J(u_{0})<d$, from Lemma 3.1 it follows that

$\xi\left(t\right)>2\left(q+1\right)d-2\left(q+1\right)J\left(u_{0}\right)+\left(q-1\right)\int_{0}^{t}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau=\rho>0.$
(3.24)

Then we get

$G^{\prime\prime}\left(t\right)G\left(t\right)-\frac{q+3}{4}\left(G^{\prime}\left(t\right)\right)^{2}\geq\theta\rho>0,\;t\in\left[0,T_{0}\right],$

which yields

$\left(G^{-\vartheta}\left(t\right)\right)^{\prime\prime}=-\frac{\vartheta}{G^{\vartheta+2}\left(t\right)}\left(G^{\prime\prime}\left(t\right)G\left(t\right)-\left(\vartheta+1\right)\left(G^{\prime}\left(t\right)\right)^{2}\right)<0,\;\vartheta=\frac{q-1}{4}.$

Thus, from the proof of [19, Theorem 4.3], there exists a $T>0$ such that

$\lim_{t\rightarrow T}G^{-\vartheta}\left(t\right)=0,$

and

$\lim_{t\rightarrow T}G\left(t\right)=+\infty,$

which is a contradiction with $T=+\infty$.

(ii) If $J(u_{0})\leq 0$, it is easy to get (3.24) directly. The rest is
proved similar to case (i).

The proof of Theorem 2.2 is complete.

##  4 Critical initial energy $J\left(u_{0}\right)=d$

In this section, we shall prove the global existence, uniqueness and decay
estimate, and blowup of solutions to problem (1.1) for the critical initial
energy $J\left(u_{0}\right)=d$.

Proof of Theorem 2.3.

Global existence and uniqueness. From the condition $J\left(u_{0}\right)=d$,
it follows that ${\left\|u_{0}\right\|}_{W^{1,p}}\neq 0$. Denote
$\mu_{s}=1-\frac{1}{s}$ and $u_{s0}=\mu_{s}u_{0}$ for $s=2,3,\cdots$. We
consider the problem (1.1) with the condition

$u\left(x,0\right)=u_{s0}\left(x\right),\;s=2,3,\cdots.$ (4.1)

Coming back to Lemma 2.2 and the initial data $I(u_{0})\geq 0$, we obtain
$\lambda^{\ast}\geq 1$. Further, we deduce $I(u_{s0})=I(\mu_{s}u_{0})>0$ and
$J(u_{s0})=J(\mu_{s}u_{0})<J(u_{0})=d$. Due to Theorem 2.1, the problem (1.1)
with the initial condition (4.1) exists a unique global solution $u_{s}(t)\in
L^{\infty}(0,\infty;W_{N}^{1,p}(\Omega))$ with $u_{st}(t)\in
L^{2}\;(0,\infty;W_{N}^{1,2}(\Omega))$ for each $s=2,3,\cdots$. By Proposition
2.1, we can know that $u_{s}\left(x,t\right)\in W$. As in the proof of Theorem
2.1, we readily get

$\int_{0}^{t}\left\|u_{s\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau<d,\;\;\;0\leq
t<\infty,$

and

$\left\|\nabla
u_{s}\right\|_{p}^{p}<\frac{p\left(q+1\right)}{q+1-p}d,\;\;\;0\leq t<\infty.$

The rest is proved similar to Theorem 2.1.

Asymptotic behavior. The existence of the global solution $u(t)$ of (1.1) is
proved by the above, then we claim that $u\in W$ for $t>0$.

Arguing by contradiction, let $t_{0}>0$ is the first time that
$I(u(t_{0}))=0$. Through the definition of $d$, we can know that
$J(u(t_{0}))\geq d$. But

$0<J(u(t_{0}))=d-\int_{0}^{t_{0}}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau=d_{1}\leq
d$ (4.2)

for any $t_{0}>0$. Then, we have

$J(u(t_{0}))=d.$ (4.3)

Considering (4.2) and (4.3), gives

$\int_{0}^{t_{0}}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau=0,$

that is, $u_{t}\equiv 0$ for $0\leq t\leq t_{0}$, which contradicts
$I(u_{0})>0$ (from (3.22)). Then, we infer $u\in W$ for $0<t<\infty$. Let
$t_{1}>0$ as the initial time, it generates $u(x,t)\in W$ for $t>t_{1}$. Then
by (3.18)-(3.20) we can know that there exists a constant $\varsigma>0$ as

$-\varsigma=C\left(C_{\ast}^{q+1}\left(\frac{p\left(q+1\right)}{q+1-p}J\left(u\left(t_{1}\right)\right)\right)^{\frac{q-p+1}{p}}-1\right)$

such that

$\left\|u\right\|_{H^{1}}^{2}<\left\|u\left(t_{1}\right)\right\|_{H^{1}}^{2}e^{-2\varsigma\left(t-t_{1}\right)},\;t\in\left(t_{1},+\infty\right).$

Theorem 2.3 is proved.

Proof of Theorem 2.4. By the continuity of $I(u)$ and $J(u)$ with respect to
$t$, there exists a sufficiently small $t_{0}>0$ such that $I(u(t_{0}))<0$ and
$J(u(t_{0}))>0$ for $I(u_{0})<0$ and $J(u_{0})=d>0$. Considering (3.22), it is
easy to see that $u_{t}\neq 0$ for $0<t\leq t_{0}$. Then, we have

$J(u(t_{0}))=d-\int_{0}^{t_{0}}\left\|u_{\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau=d_{0}<d.$

Applying Proposition 2.1 and taking $t=t_{0}$ as the initial time, we obtain
$u(x,t)\in V$ for $t>t_{0}$.

The rest is similar to the proof of Theorem 2.2.

##  5 Supercritical initial energy $J\left(u_{0}\right)>d$

In this section, we shall prove the global existence, uniqueness and
asymptotic behavior of solutions to problem (1.1) for the supercritical
initial energy $J\left(u_{0}\right)>d$ by the properties of $\omega$-limits of
solutions.

Proof of Theorem 2.5.

Global existence and uniqueness. Consider the basis $u_{m}\left(x,t\right)$ as
in Theorem 2.1, we claim that $u_{m}\in{\mathcal{N}}_{+}$.

Assuming that the assertion is not tenable, then there exists a $t_{0}>0$ such
that $u_{m}\in{\mathcal{N}}_{+}$ for $t\in\left(0,t_{0}\right)$ and
$u_{m}\left(t_{0}\right)\in{\mathcal{N}}$. By (3.17), we have
$\int_{0}^{t}\left\|u_{m\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau\neq 0$
for $\Omega\times\left(0,t_{0}\right)$. Further, it follows from (3.3) that
$u_{m}(t_{0})\in J^{J\left(u_{m}\left(0\right)\right)}$. Hence,
$u_{m}(t_{0})\in\mathcal{N}^{J\left(u_{m}\left(0\right)\right)}$. Due to the
definition of $\lambda_{J(u_{0})}$, we have

$\left\|u_{m}\left(t_{0}\right)\right\|_{H^{1}}^{2}\geq\lambda_{J(u_{0})}.$
(5.1)

Combining (3.17), by the fact $I\left(u_{m}\right)>0$, gives

$\left\|u_{m}\left(t_{0}\right)\right\|_{H^{1}}^{2}<\left\|u_{m}\left(x,0\right)\right\|_{H^{1}}^{2}\leq\lambda_{J(u_{0})},$

which contradicts with (5.1). Then, for all $t_{0}>0$,
$u_{m}\in{\mathcal{N}}_{+}$.

Further, we can get $u_{m}\left(t\right)\in
J^{J\left(u_{m}\left(0\right)\right)}\cap{\mathcal{N}}_{+}$. On the other
hand, (3.3) yields

$J\left(u_{m}\left(0\right)\right)\geq
J\left(u_{m}\left(t\right)\right)\geq\left(\frac{1}{p}-\frac{1}{q+1}\right)\left\|\nabla
u_{m}\right\|_{p}^{p}+\frac{1}{q+1}I\left(u_{m}\right),$

and

$J\left(u_{m}\left(0\right)\right)\geq\int_{0}^{t}\left\|u_{m\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau,$

which imply boundedness of $\left\|\nabla u_{m}\right\|_{p}^{p}$ and
$\int_{0}^{t}\left\|u_{m\tau}\right\|_{H^{1}}^{2}\operatorname{d}\tau$.

Similar to the proof of Theorem 2.1, then problem (1.1) has a unique global
weak solution $u$.

Asymptotic behavior. We denote by $\omega\left(u_{0}\right)=\underset{t\geq
0}{\cap}\overline{\left\\{u\left(s\right):s\geq t\right\\}}$ the
$\omega$-limit of $u$. By (3.3) and (3.17), we have

$\left\|\omega\right\|_{H^{1}}^{2}<\left\|u_{0}\right\|_{H^{1}}^{2}\leq\lambda_{J\left(u_{0}\right)},\;J\left(\omega\right)\leq
J\left(u_{0}\right),$

for any $\omega\in\omega\left(u_{0}\right)$, which means that
$\omega\not\in\mathcal{N}^{J\left(u_{0}\right)}$ and $\omega\in
J^{J\left(u_{0}\right)}$, i.e., $\omega\not\in\mathcal{N}$. Then,
$\omega\left(u_{0}\right)\cap\mathcal{N}=\varnothing$, which implies
$\omega\left(u_{0}\right)=\left\\{0\right\\}$. Therefore,
$u\left(t\right)\rightarrow 0$ as $t\rightarrow+\infty$.

Theorem 2.5 is proved.

##  6 Conclusions

In this paper, we’ve studied the global existence and finite time blowup of
solutions for a mixed pseudo-parabolic $p$-Laplacian type equation. Compared
with the research in [27], we replace the $\triangle u$ term in the equation
with the $\mathrm{div}\left(\left|\nabla u\right|^{p-2}\nabla u\right)$ term
according to actual physical background. By combining the Galerkin method and
the theory of potential wells, we first present the explicit expression for
the depth $d$ of potential well, and then overcome the difficulties in a
priori estimation of $p$-Laplacian term and nonlinear term
${\mathchoice{{-\mkern-19.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}{{-\mkern-16.0mu\int}}}_{\Omega}\left|u\right|^{q-1}u\operatorname{d}x$
in the equation. In Section 5, we overcome the limitation of the potential
well family method in the $J\left(u_{0}\right)>d$ case by further analyzing
the properties of $\omega$-limits of solutions. However, there exist many
problems to be further studied such as in fractional order equations and
others.

## Availability of data and material

Not applicable.

## Competing interests

The authors declare that they have no competing interests.

## Funding

This research was supported by the National Natural Science Foundation of
China (No. 12071491).

## Author’s contributions

The authors contributed equally to the writing of this paper. All authors read
and approved the final manuscript.

## References

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# A remark on the Whitney Broken Circuit Theorem

Paula M. S. Fialho1, Emanuel Juliano1, Aldo Procacci2. 1Departamento de
Ciência da Computação UFMG, 30161-970 - Belo Horizonte - MG Brazil
2Departamento de Matemática UFMG, 30161-970 - Belo Horizonte - MG Brazil

###### Abstract

In the present note we show, via the connection between chromatic polynomial
and Potts model, that the Whitney Broken circuit theorem is in fact a special
case of a more general identity relating the chromatic polynomial of a graph
${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ to sums over forests of
${\mathbb{G}}$ associated to some partition scheme in ${\mathbb{G}}$.

##  1 Introduction

Along this paper, ${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ represents a
fixed simple graph with vertex set ${\mathbb{V}}$ and edge set ${\mathbb{E}}$.
For any subset $R\subset{\mathbb{V}}$, we denote by ${\mathbb{G}}|_{R}$ the
restriction of ${\mathbb{G}}$ to $R$, i.e.
${\mathbb{G}}|_{R}=(R,{\mathbb{E}}|_{R})$ where
${\mathbb{E}}|_{R}=\\{\\{x,y\\}\subset R:\\{x,y\\}\in{\mathbb{E}}\\}$. A graph
${\mathbb{G}}^{\prime}=({\mathbb{V}}^{\prime},{\mathbb{E}}^{\prime})$ is a
subgraph of ${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ if
${\mathbb{V}}^{\prime}\subseteq{\mathbb{V}}$ and
${\mathbb{E}}^{\prime}\subseteq{\mathbb{E}}$. A subgraph
${\mathbb{G}}^{\prime}$ of ${\mathbb{G}}$ is called spanning if
${\mathbb{V}}^{\prime}={\mathbb{V}}$. A graph
${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ is connected if for any pair $B,C$
of subsets of ${\mathbb{V}}$ such that $B\cup C={\mathbb{V}}$ and $B\cap
C=\emptyset$, there is at least an edge $e\in{\mathbb{E}}$ such that $e\cap
B\neq\emptyset$ and $e\cap C\neq\emptyset$. A tree is a connected graph with
no cycles. A subset $R\subset{\mathbb{V}}$ is said to be connected if
${\mathbb{G}}|_{R}$ is connected; we denote by $\mathrm{C}_{\mathbb{V}}$ the
set of all connected subsets of ${\mathbb{G}}$ with cardinality greater than
one, i.e.

$\mathrm{C}_{\mathbb{V}}=\\{R\subset{\mathbb{V}}:\;\mbox{${\mathbb{G}}|_{R}$
is connected and $|R|\geq 2$}\\}.$ (1.1)

Given $R\in\mathrm{C}_{\mathbb{V}}$, let us represent by
$\mathcal{C}_{{\mathbb{G}}|_{R}}$ the set of all connected spanning subgraphs
of ${{\mathbb{G}}|_{R}}$ and by
$\mathcal{T}_{{\mathbb{G}}|_{R}}\subset\mathcal{C}_{{\mathbb{G}}|_{R}}$ the
set of all spanning trees of ${\mathbb{G}}|_{R}$.

Given $n\in\mathbb{N}$, we set $[n]=\\{1,2,\dots,n\\}$ and given a finite set
$X$, we let $|X|$ denote its cardinality. A coloring of the vertex set
${\mathbb{V}}$ of a graph ${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ using
$q\in\mathbb{N}$ colors is a function $\kappa:{\mathbb{V}}\to[q]$ and we
denote by ${\mathcal{K}}_{\mathbb{V}}(q)$ the set of all colorings of
$\mathbb{V}$ with $q$ colors. Clearly
$\left|{\mathcal{K}}_{\mathbb{V}}(q)\right|=q^{\left|{\mathbb{V}}\right|}$. A
coloring $\kappa\in{\mathcal{K}}_{\mathbb{V}}(q)$ is called proper if for any
edge $\\{x,y\\}\in{\mathbb{E}}$ it holds that $\kappa(x)\neq\kappa(y)$. We
represent by ${\mathcal{K}}^{*}_{\mathbb{V}}(q)$ the set of all proper
colorings of ${\mathbb{V}}$ with $q$ colors. The quantity
$P_{\mathbb{G}}(q):=|{\mathcal{K}}^{*}_{\mathbb{V}}(q)|$, i.e. the number of
proper colorings with $q$ colors of the graph $\mathbb{G}$ is, as a function
of $q$, a polynomial known as the _chromatic polynomial_ of ${\mathbb{G}}$.
The classical representation of the chromatic polynomial $P_{\mathbb{G}}(q)$
of a graph ${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ is as follows.

$P_{\mathbb{G}}(q)=\sum_{E\subset{\mathbb{E}}}(-1)^{|E|}q^{k(E)}$ (1.2)

where $k(E)$ is the number of connected components of the graph
$({\mathbb{V}},E)$ (isolated vertices are considered trivial connected
components, therefore are included).

An alternative representation of $P_{\mathbb{G}}(q)$, known under the name of
Whitney’s broken circuit theorem, was derived by Whitney [15]. To state this
theorem we need to introduce some notations regarding forests of a graph
${\mathbb{G}}$.

A connected component of a graph is said non trivial if the cardinality of its
vertex set is greater than one, i.e., it contains at least one edge. A forest
in ${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ is a subset
$F\subset{\mathbb{E}}$ such that all non trivial connected components of the
graph $({\mathbb{V}},F)$ are trees. We denote by $\mathcal{F}_{\mathbb{G}}$
the set of all forests that are subgraphs of ${\mathbb{G}}$ and we agree that
the empty set belongs to $\mathcal{F}_{\mathbb{G}}$, namely
$({\mathbb{V}},\emptyset)$ is seen as a forest in ${\mathbb{G}}$. Fix now an
ordering on the edge set ${\mathbb{E}}$ of the graph ${\mathbb{G}}$. A circuit
$C$ of ${\mathbb{G}}$ is a sequence of edges $\\{e_{1},\dots,e_{n}\\}$ such
that $e_{n}$ and $e_{1}$ are adjacent, as well as $e_{i}$ and $e_{i+1}$ for
all $i\in[n-1]$. A circuit can also be seen as a subset of ${\mathbb{E}}$. A
circuit $C=\\{e_{1},\dots,e_{n}\\}$ is called simple if it does not contain
any repeated edges or vertices, except for the vertex $v=e_{n}\cap e_{1}$.
Given a simple circuit $C=\\{e_{1},\dots,e_{n}\\}$, the set $C\setminus e$ is
called a broken circuit of ${\mathbb{G}}$ if $e=\max\\{e_{1},\dots,e_{n}\\}$
(where the maximum is in respect to the ordering of ${\mathbb{E}}$ previously
fixed). A forest $F\in\mathcal{F}_{\mathbb{G}}$ is said to be broken circuit
free if it does not contain any broken circuit. Let us represent by
$\mathcal{F}^{*}_{\mathbb{G}}$ the set of all broken circuit free forests of
${\mathbb{G}}$, then the Whitney’s broken circuit theorem can now be stated as
follows.

######  Theorem 1.1 (Whitney [15])

Given a graph ${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ , we have

$P_{\mathbb{G}}(q)=q^{|{\mathbb{V}}|}\sum_{F\in\mathcal{F}^{*}_{\mathbb{G}}}\left(-{1\over
q}\right)^{|F|}.$ (1.3)

Whitney’s broken circuit theorem is relevant because it provides a simple
combinatorial interpretation of the coefficients of the chromatic polynomial.
Many papers are based on this theorem and several generalization (to
hypergraphs, matroids, lattices etc.) have been given (see e.g. [3] and
references therein). In particular, Theorem 1.1 has recently been used in [14]
to prove that the number of list-colorings of a graph
${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ is not less than that of ordinary
$q$-colorings of ${\mathbb{G}}$ provided that $q>1.135|{\mathbb{E}}|$. Even
more recently, the broken circuit theorem has been used in [7] to obtain the
best bounds for the zero-free region of the chromatic polynomial. In [7] the
authors show that for any graph ${\mathbb{G}}$ with maximum degree $\Delta$,
the chromatic polynomial $P_{\mathbb{G}}(q)$ is free of zeros outside the
complex disk $|q|<5.94\Delta$. Their result improves the previous bounds given
in [13] and [5] which were obtained instead via the connection between
$P_{\mathbb{G}}(q)$ and the partition function of the antiferromagnetic Potts
model at zero temperature.

In this note we show that the broken circuit theorem, i.e. Theorem 1.1, is in
fact a special case of a more general representation of $P_{\mathbb{G}}(q)$ in
terms of sum over suitable forest of ${\mathbb{G}}$. To state our theorem, we
need to introduce a so-called partition scheme in ${\mathbb{G}}$ (see e.g.
Sec. 4 of [6] and references therein) according to the following definition.

###### Definition 1.2

Given $R\in\mathrm{C}_{\mathbb{V}}$, a partition scheme in ${\mathbb{G}}|_{R}$
is a map
${\mathrm{m}_{R}}:\mathcal{T}_{{\mathbb{G}}|_{R}}\to\mathcal{C}_{{\mathbb{G}}|_{R}}$
such that for all $\tau\in\mathcal{T}_{{\mathbb{G}}|_{R}}$, it holds that
$\tau\subset{\mathrm{m}_{R}}(\tau)$ and
$\mathcal{C}_{{\mathbb{G}}|_{R}}=\biguplus_{\tau\in\mathcal{T}_{{\mathbb{G}}|_{R}}}[\tau,\mathrm{m}_{R}(\tau)]$,
where $\biguplus$ means disjoint union and
$[\tau,\mathrm{m}_{R}(\tau)]=\\{g\in\mathcal{C}_{{\mathbb{G}}|_{R}}:\tau\subseteq
g\subseteq{\mathrm{m}_{R}(\tau)}\\}$. Having established, for any
$R\subset\mathrm{C}_{\mathbb{V}}$, a partition scheme $\mathrm{m}_{R}$ on
${\cal C}_{{\mathbb{G}}|_{R}}$, the family
${\mathrm{\mathbf{m}}}=\\{\mathrm{m}_{R}\\}_{R\in\mathrm{C}_{\mathbb{V}}}$ is
said to be a partition scheme in ${\mathbb{G}}$ and, for any non trivial
$\tau$ subtree of ${\mathbb{G}}$, we will denote shortly
$\mathrm{\mathbf{m}}(\tau)=\mathrm{m}_{V_{\tau}}(\tau)$ (here $V_{\tau}$
denotes the vertex set of $\tau$).

We are now in the position to state the main result of this note.

###### Theorem 1.3

Let ${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ be a graph and let
$P_{\mathbb{G}}(q)$ be the chromatic polynomial of ${\mathbb{G}}$. Let
$\mathrm{\mathbf{m}}$ be a partition scheme in ${\mathbb{G}}$ according to
Definition 1.2. Then

$P_{\mathbb{G}}(q)=q^{|{\mathbb{V}}|}\sum_{F\in\mathcal{F}^{\bf
m}_{\mathbb{G}}}\left(-{1\over q}\right)^{|F|}$ (1.4)

where $\mathcal{F}^{\bf m}_{\mathbb{G}}$ is the subset of
$\mathcal{F}_{\mathbb{G}}$ formed by the empty set and by all non trivial
forests $F=\\{\tau_{1},\dots,\tau_{k}\\}$ in which
$\mathrm{\mathbf{m}}(\tau_{i})=\tau_{i}$ for each tree $\tau_{i}$, $i\in[k]$.

Remark. It is noteworthy to observe that identity (1.4) holds independently of
the partition scheme ${\bf m}$ in ${\mathbb{G}}$ used. Even more impressive is
that despite the fact that the set
$\mathcal{F}_{\mathbb{G}}^{\mathrm{\mathbf{m}}}$ depends strongly on the
partition scheme $\mathrm{\mathbf{m}}$ , i.e. in general
$\mathcal{F}_{\mathbb{G}}^{\mathrm{\mathbf{m}}}\neq\mathcal{F}_{\mathbb{G}}^{\mathrm{\mathbf{m}}^{\prime}}$
if $\mathrm{\mathbf{m}}\neq\mathrm{\mathbf{m}}^{\prime}$, the coefficients of
1.4, namely
$|\\{F\in\mathcal{F}_{\mathbb{G}}^{\mathrm{\mathbf{m}}}:|F|=k\\}|$, does not
depend on $\mathrm{\mathbf{m}}$. Various partition schemes are available (see
e.g. [6], Sec. 4 and references therein). In particular, in statistical
mechanics, as far as we know, two partition schemes have been used: the first
one is the so called Penrose partition scheme, originally presented in [9] and
used in [4] to improve the Dobrushin criterion [2] for the convergence of the
cluster expansion on the abstract polymer gas; the second one is the minimal-
tree partition scheme, described e.g. in the proof of Lemma 2.2. in [12] and
used in [10] to improve the lower bound of the convergence radius of the Mayer
Series on a gas of classical particles in the continuum interacting via a
stable and regular pair potential. The Penrose partition scheme has also been
used in combinatorics, e.g. in [6] it was a key tool to find zero-free regions
in the complex plane at large $|q|$ for the multivariate Tutte polynomial (see
there Proposition 4.4.and Lemma 4.5). Last but not least, the improvement of
the Lovász local lemma given in [1] also relies crucially on the Penrose
partition scheme.

To conclude the introduction, we show as promised that the broken circuit
theorem, i.e. Theorem 1.1 is in fact a special case of Theorem (1.3). Namely,
the broken circuit theorem is in fact Theorem 1.1 when the partition scheme
$\mathrm{\mathbf{m}}$ is the so-called minimal-tree partition scheme. To see
this, let us construct explicitly the above mentioned partition scheme in
${\mathbb{G}}$.

Minimal-tree partition scheme. We start by choosing a total order $\succ$ in
the set of edges ${\mathbb{E}}$ of ${\mathbb{G}}$. Now, for any
$R\in\mathrm{C}_{\mathbb{V}}$, let
${\mu_{R}}:\mathcal{T}_{{\mathbb{G}}|_{R}}\to\mathcal{C}_{{\mathbb{G}}|_{R}}$
be the map that associates to the tree
$\tau=(R,E_{\tau})\in\mathcal{T}_{{\mathbb{G}}|_{R}}$ the graph
${\mu_{R}}(\tau)\in\mathcal{C}_{{\mathbb{G}}|_{R}}$ whose edge set is
$E_{{\mu_{R}}(\tau)}=E_{\tau}\cup E_{\tau}^{*}$, where $E_{\tau}^{*}$ is
composed by all edges $\\{x,y\\}\subset{\mathbb{E}}|_{R}\setminus E_{\tau}$
such that $\\{x,y\\}\succ\\{z,u\\}$ for every edge $\\{z,u\\}\in E_{\tau}$
belonging to the path from $x$ to $y$ through $\tau$. It is easy to show that
${\mu_{R}}$ is, for any $R\in C_{\mathbb{V}}$, a partition scheme in
$\mathcal{C}_{{\mathbb{G}}|_{R}}$ (see e.g. [10], [11], [12]) and thus the
family ${\bm{\mu}}=\\{\mu_{R}\\}_{R\in\mathrm{C}_{\mathbb{V}}}$ is a partition
scheme in ${\mathbb{G}}$. Then, according to Theorem 1.3, we can write

$P_{\mathbb{G}}(q)=q^{|{\mathbb{V}}|}\sum_{F\in\mathcal{F}^{\bm{\mu}}_{\mathbb{G}}}\left(-{1\over
q}\right)^{|F|},$ (1.5)

where $\bm{\mu}$ is the minimal-tree partition scheme and
$\mathcal{F}^{\bm{\mu}}_{\mathbb{G}}$ is the set of all forests $F$ in
${\mathbb{G}}$ whose non trivial trees $\tau\in F$ are such that
${\bm{\mu}}(\tau)=\tau$. Observe now that for any non trivial tree
$\tau\subset{\mathbb{G}}$ the condition $\bm{\mu}(\tau)=\tau$ simply means
that for any $\\{x,y\\}\in{\mathbb{E}}|_{R}\setminus E_{\tau}$, the path
$p^{\tau}_{xy}$ in $\tau$ from $x$ to $y$ is such that $\\{x,y\\}$ is never
the largest edge in the set $\\{\\{x,y\\}\cup\\{e\\}_{e\in p^{\tau}_{xy}}\\}$.
Therefore, the condition ${\bm{\mu}}(\tau)=\tau$ for any tree in a forest
$F\in\mathcal{F}^{\bm{\mu}}_{{\mathbb{G}}}$ can be rephrased by saying that
$F$ does not contain broken circuits. That is to say,
$\mathcal{F}^{\bm{\mu}}_{\mathbb{G}}=\mathcal{F}^{*}_{\mathbb{G}}$ and the sum
r.h.s. of (1.5) coincides term by term with the sum of the r.h.s. of (1.3).

##  2 Proof of Theorem 1.3

It is long known (see [8] and references therein) that $P_{\mathbb{G}}(q)$
also coincides with the partition function of the antiferromagnetic Potts
model with $q$ states on $\mathbb{G}$ at zero temperature. The Potts model
with $q$ states on a finite graph $\mathbb{G}=(\mathbb{V},\mathbb{E})$ (a
heavily studied spin system in statistical mechanics) is defined by a set of
random variables (a.k.a. “spins”) $\\{\sigma_{x}\\}_{x\in{\mathbb{V}}}$. For
any vertex $x\in{\mathbb{V}}$, the spin $\sigma_{x}$ takes values in the set
of integers $[q]$ and a configuration $\bm{\sigma}$ of the system is a
function $\bm{\sigma}:{\mathbb{V}}\to[q]$. Interpreting $[q]$ as a set of $q$
colors, a spin configuration $\bm{\sigma}$ can be seen as an element of
${\mathcal{K}}_{\mathbb{V}}(q)$. The energy of a spin configuration
$\bm{\sigma}\in{\mathcal{K}}_{\mathbb{V}}(q)$ is defined as

$H_{{\mathbb{G}}}(\bm{\sigma})=-J\sum_{\\{x,y\\}\in\mathbb{E}}\delta_{\sigma_{x}\sigma_{y}}$

where $J\in\mathbb{R}$ and $\delta_{\sigma_{x}\sigma_{y}}$ is the Kronecker
symbol which is equal to one when $\sigma_{x}=\sigma_{y}$ and zero otherwise.
When $J>0$ the model is called ferromagnetic, while if $J<0$ the model is said
to be antiferromagnetic. The probability to find the system in the
configuration $\bm{\sigma}$ is defined as

${\rm Prob}(\bm{\sigma})=\frac{e^{-\beta
H_{{\mathbb{G}}}(\bm{\sigma})}}{Z_{{\mathbb{G}}}(q,\beta)}$ (2.1)

where $\beta\geq 0$ is the inverse temperature. The normalization constant
$Z_{{\mathbb{G}}}(q,\beta)$ in the denominator is called the partition
function and is explicitly given by

$Z_{{\mathbb{G}}}(q,\beta)=\sum_{\bm{\sigma}\in{\mathcal{K}}_{\mathbb{V}}(q)}e^{-\beta
H_{{\mathbb{G}}}(\bm{\sigma})}.$ (2.2)

The case $J<0$ and $\beta=+\infty$ is the antiferromagnetic and zero
temperature Potts model with $q$-states. In this case, when a product
$0\cdot\infty$ appears in the factor $\beta H_{{\mathbb{G}}}(\bm{\sigma})$, it
must be interpreted as zero. With this convention, the only configurations
$\bm{\sigma}$ with non zero probability are those for which
$\delta_{\sigma_{x}\sigma_{y}}=0$ for all $\\{x,y\\}\in{\mathbb{E}}$, i.e.
those configurations in which adjacent spins have different values (or
colors). In other words, the antiferromagnetic Potts model at zero temperature
is such that ${\rm Prob}(\bm{\sigma})\neq 0$ if and only if the configuration
$\bm{\sigma}$ is a proper coloring (i.e.
$\bm{\sigma}\in{\mathcal{K}}^{*}_{\mathbb{V}}(q)$). Clearly, for any
$\bm{\sigma}\in{\mathcal{K}}^{*}_{\mathbb{V}}(q)$ we have that $\exp\\{-\beta
H_{{\mathbb{G}}}(\bm{\sigma}\\}|_{\beta=+\infty,J<0}=1$, so that, according to
(2.1), all colorings in ${\mathcal{K}}^{*}_{\mathbb{V}}(q)$ have the same
probability $1/Z_{\mathbb{G}}(q,\beta)$, and thus the normalization constant
$Z_{\mathbb{G}}(q,\beta)$ coincides with
$|{\mathcal{K}}^{*}_{\mathbb{V}}(q)|$, i.e. we get that

$Z_{\mathbb{G}}(q,\beta)|_{\beta=+\infty,J<0}=P_{\mathbb{G}}(q).$ (2.3)

The connection between chromatic polynomial and antiferromagnetic Potts model
at zero temperature shown by Formula (2.3) has been explored in recent times
to study the location of the zeros of the $P_{\mathbb{G}}(q)$ in the complex
plane (see [13, 5]).

Let us enunciate two rather standard results in statistical mechanics. The
first is basically related to the well known high-temperature expansion for
the partition function of a general discrete spin system while the second is a
long known identity discovered by Oliver Penrose [9] during with the study of
the mathematical properties of continuous particle systems in the high-
temperature and low-density regime.

###### Proposition 2.1

let $Z_{\mathbb{G}}(q,\beta)|_{\beta=+\infty,J<0}$ be the partition function
of the antiferromagnetic Potts model with $q$ state at zero temperature. Then

$Z_{\mathbb{G}}(q,\beta)|_{\beta=+\infty,J<0}=q^{|{\mathbb{V}}|}\Xi_{\mathbb{G}}(q),$
(2.4)

where

$\Xi_{\mathbb{G}}(q)=1+\sum_{k\geq 1}\sum_{\\{R_{1},\dots,R_{k}\\}\atop
R_{i}\in\mathrm{C}_{\mathbb{V}},~{}R_{i}\cap
R_{j}=\emptyset}\prod_{i=1}^{k}z(R_{i},q)$ (2.5)

is the grand canonical partition function of a “polymer gas”, in which the
polymers are connected subsets $R\subset\mathbb{V}$, with cardinality $|R|\geq
2$ (i.e. they are the elements of the set $\mathrm{C}_{\mathbb{V}}$ defined in
(1.1)), subjected to a non-intersection constraint (hard-core interaction) and
with weights (a.k.a. activities) $z(R,q)$ given by

$z(R,q)={1\over
q^{|R|-1}}\sum_{g\in\mathcal{C}_{{\mathbb{G}}|_{R}}}(-1)^{|E_{g}|}$ (2.6)

where $\mathcal{C}_{{\mathbb{G}}|_{R}}$ denotes the set of all connected
spanning subgraphs of ${\mathbb{G}}|_{R}$.

A sketch of the proof of Proposition 2.1 can be found e.g. in [13] and in [6].
However, since this representation of
$Z_{\mathbb{G}}(q,\beta)|_{\beta=+\infty,J<0}$ is crucial for the proof of
Theorem 1.3, we will provide a detailed proof of it in the appendix.

###### Proposition 2.2

Let ${\mathrm{\mathbf{m}}}$ be a partition scheme in
${\mathbb{G}}=({\mathbb{V}},{\mathbb{E}})$ according to Definition 1.2, and
let $\\{w_{e}\\}_{e\in{\mathbb{E}}}$ be any collection of real numbers, then,
for any $R\in\mathrm{C}_{\mathbb{V}}$, we have

$\sum_{g\in\mathcal{C}_{{\mathbb{G}}|_{R}}}\prod_{e\in
E_{g}}w_{e}=\sum_{\tau\in\mathcal{T}_{{\mathbb{G}}|_{R}}}\prod_{e\in
E_{\tau}}w_{e}\prod_{e\in E_{{\mathrm{\mathbf{m}}}(\tau)}\setminus
E_{\tau}}(1+w_{e}),$ (2.7)

where $E_{g}$, $E_{\tau}$, and $E_{{\mathrm{\mathbf{m}}}(\tau)}$ represent the
edge sets of $g$, $\tau$, and ${\mathrm{\mathbf{m}}}(\tau)$ respectively.

The proof of identity (2.7) is straightforward and can be seen e.g. in Sec.
4.1 of [4].

We are now in the position to prove Theorem 1.3. Let us turn initially our
attention to the Penrose identity (2.7) and observe that when $w_{e}=-1$ for
all $e\in{\mathbb{E}}|_{R}$, the l.h.s. of (2.7) becomes
$\sum_{g\in\mathcal{C}_{{\mathbb{G}}|_{R}}}(-1)^{|E_{g}|}$, which, recalling
the definition of $z(q,R)$ given in (2.6) is exactly the sum in the r.h.s. of
(2.6). On the other hand in the r.h.s. of the Penrose identity (2.7), when
$w_{e}=-1$ for all $e\in{\mathbb{E}}|_{R}$, the last product kills the
contribution of any tree $\tau$ such that
$E_{{\mathrm{\mathbf{m}}}(\tau)}\setminus E_{\tau}\neq\emptyset$, so that the
only contributions on the sum over trees in $\mathcal{T}_{{\mathbb{G}}|_{R}}$
come from those trees $\tau$ such that ${\mathrm{\mathbf{m}}}(\tau)=\tau$.
Since for any $\tau\in\mathcal{T}_{{\mathbb{G}}|_{R}}$ we have that
$|\tau|=|R|-1$, as far as the case $w_{e}=-1$ for all $e\in{\mathbb{E}}|_{R}$
is concerned, the Penrose identity (2.7) can be written as

$\sum_{g\in\mathcal{C}_{{\mathbb{G}}|_{R}}}(-1)^{|E_{g}|}=(-1)^{|R|-1}\sum_{\tau\in\mathcal{T}_{{\mathbb{G}}|_{R}}\atop{\mathrm{\mathbf{m}}}(\tau)=\tau}1.$
(2.8)

Hence we can rewrite the activity $z(R,q)$ of a polymer $R$, given in (2.6),
as

$z(R,q)=\sum_{\tau\in\mathcal{T}_{{\mathbb{G}}|_{R}}\atop{\mathrm{\mathbf{m}}}(\tau)=\tau}\left(-{1\over
q}\right)^{|\tau|},$ (2.9)

where ${\mathrm{\mathbf{m}}}$ is any partition scheme on ${\mathbb{G}}$. Then,
recalling (2.5) we get

$\Xi_{\mathbb{G}}(q)=1+\sum_{k\geq 1}\sum_{\\{R_{1},\dots,R_{k}\\}\atop
R_{i}\in\mathrm{C}_{\mathbb{V}},~{}R_{i}\cap
R_{j}=\emptyset}\prod_{i=1}^{k}\sum_{\tau_{i}\in\mathcal{T}_{{\mathbb{G}}|_{R_{i}}}\atop{\mathrm{\mathbf{m}}}(\tau_{i})=\tau_{i}}\left(-{1\over
q}\right)^{|\tau|}=\sum_{F\in\mathcal{F}^{\mathrm{\mathbf{m}}}_{\mathbb{G}}}\left(-{1\over
q}\right)^{|F|},$ (2.10)

where $\mathcal{F}^{\mathrm{\mathbf{m}}}_{\mathbb{G}}$ is the subset of
$\mathcal{F}_{\mathbb{G}}$ formed by the empty set (contributing with the
factor “1” in the sum in the r.h.s. of (2.10)) and by all non trivial forests
$F=\\{\tau_{1},\dots,\tau_{k}\\}$ where each tree $\tau_{i}$, $i\in[k]$, is
such that $\mathrm{\mathbf{m}}(\tau_{i})=\tau_{i}$. Inserting finally (2.10)
into (2.4) and recalling (2.3), we get (1.4).

## Acknowledgements

It is a pleasure to thank Benedetto Scoppola for discussions and comments.
P.M.S.F. and E.J. were supported by FAPEMIG (Fundação de Amparo à Pesquisa do
Estado de Minas Gerais).

## Appendix. Proof of Proposition 2.1

We start by rewriting $Z_{\mathbb{G}}(q)|_{\beta=+\infty,J<0}$ as follows.

$Z_{\mathbb{G}}(q)|_{\beta=+\infty,J<0}=\sum_{\bm{\sigma}\in{\mathcal{K}}_{\mathbb{V}}(q)}\exp\Big{\\{}-\sum_{\\{x,y\\}\subset{\mathbb{V}}}J_{xy}\delta_{\sigma_{x}\sigma_{y}}\Big{\\}}$
(A.1)

where

$J_{xy}=\begin{cases}+\infty&{\rm if}~{}\\{x,y\\}\in\mathbb{E}\\\ 0&{\rm
otherwise}\end{cases}$ (A.2)

reminding that a product $0\cdot\infty$ in the r.h.s. of (A.1) must be
interpreted as zero.

Using the so-called Mayer trick, we can write

$\displaystyle\exp\Big{\\{}-\sum_{\\{x,y\\}\subset{\mathbb{V}}}J_{xy}\delta_{\sigma_{x}\sigma_{y}}\Big{\\}}$
$\displaystyle=\prod_{\\{x,y\\}\subset{\mathbb{V}}}\Big{[}(e^{-J_{xy}\delta_{\sigma_{x}\sigma_{y}}}-1)+1]$
$\displaystyle=\sum_{g\in{\cal G}_{\mathbb{V}}}\prod_{\\{x,y\\}\in
E_{g}}(e^{-J_{xy}\delta_{\sigma_{x}\sigma_{y}}}-1)$
$\displaystyle=1+\sum_{k\geq
1}\sum_{\\{R_{1},\dots,R_{k}\\}\subset{\mathbb{V}}\atop R_{i}\cap
R_{j}=\emptyset,~{}|R_{i}|\geq 2}\prod_{i=1}^{k}\sum_{g\in
G_{R_{i}}}\prod_{\\{x,y\\}\in
E_{g}}(e^{-J_{xy}\delta_{\sigma_{x}\sigma_{y}}}-1)$
$\displaystyle=1+\sum_{k\geq
1}\sum_{\\{R_{1},\dots,R_{k}\\}\subset{\mathbb{V}}\atop R_{i}\cap
R_{j}=\emptyset,~{}|R_{i}|\geq
2}\prod_{i=1}^{k}\rho(R_{i},\bm{\sigma}_{R_{i}})$

where in the second line we have denoted by ${\cal G}_{\mathbb{V}}$ the set of
all graphs whose vertex set is ${\mathbb{V}}$ and, for any $g\in{\cal
G}_{\mathbb{V}}$, $E_{g}$ denotes the edge set of $g$. In the third line, for
any $R\subset{\mathbb{V}}$ such that $|R|\geq 2$, $G_{R}$ denotes the set of
all connected graphs with vertex set $R$. Finally, in the fourth line we have
set, for any $R\subset{\mathbb{V}}$ such that $|R|\geq 2$,

$\rho(R,\bm{\sigma}_{R})=\sum_{g\in G_{R}}\prod_{\\{x,y\\}\in
E_{g}}(e^{-J_{xy}\delta_{\sigma_{x}\sigma_{y}}}-1)$ (A.3)

where for any $R\subset{\mathbb{V}}$, $\bm{\sigma}_{R}$ is the restriction of
$\bm{\sigma}$ to $R$.

Recalling (2.3), we have that the chromatic polynomial $P_{\mathbb{G}}(q)$ can
be written as

$\displaystyle P_{\mathbb{G}}(q)$
$\displaystyle=\sum_{\bm{\sigma}\in{\mathcal{K}}_{\mathbb{V}}(q)}\Bigg{[}1+\sum_{k\geq
1}\sum_{\\{R_{1},\dots,R_{k}\\}\subset{\mathbb{V}}\atop R_{i}\cap
R_{j}=\emptyset,~{}|R_{i}|\geq
2}\prod_{i=1}^{k}\rho(R_{i},\sigma_{R_{i}})\Bigg{]}$
$\displaystyle=q^{|{\mathbb{V}}|}+\sum_{k\geq
1}\sum_{\\{R_{1},\dots,R_{k}\\}\subset{\mathbb{V}}\atop R_{i}\cap
R_{j}=\emptyset,~{}|R_{i}|\geq
2}\prod_{i=1}^{k}\sum_{\bm{\sigma}_{R_{i}}\in{\mathcal{K}}_{R_{i}}(q)}\rho(R_{i},\bm{\sigma}_{R_{i}})q^{|{\mathbb{V}}|-\sum_{i=1}^{k}|R_{i}|}$

According to the last equality above we can rewrite $P_{{\mathbb{G}}}(q)$ as
follows.

$P_{{\mathbb{G}}}(q)=q^{|{\mathbb{V}}|}\Bigg{(}1+\sum_{k\geq
1}\sum_{\\{R_{1},\dots,R_{k}\\}\subset{\mathbb{V}}\atop R_{i}\cap
R_{j}=\emptyset,~{}|R_{i}|\geq 2}\prod_{i=1}^{k}{1\over
q^{|R_{i}|}}\sum_{\bm{\sigma}_{R_{i}}\in{\mathcal{K}}_{R_{i}}(q)}\rho(R_{i},\bm{\sigma}_{R_{i}})\Bigg{)}$
(A.4)

Now observe that, given $R\subset{\mathbb{V}}$ such that $|R|\geq 2$ we have
that

$\sum_{\bm{\sigma}_{R}\in{\mathcal{K}}_{R}(q)}\rho(R,\bm{\sigma}_{R})=\begin{cases}\displaystyle{q\sum\limits_{g\in\mathcal{C}_{{\mathbb{G}}|_{R}}}}(-1)^{|E_{g}|}&\mbox{if
${\mathbb{G}}|_{R}$ is connected}\\\ 0&{\rm otherwise}\end{cases}$ (A.5)

where $\mathcal{C}_{{\mathbb{G}}|_{R}}$ denotes the set of all connected
spanning subgraphs of ${\mathbb{G}}|_{R}$. Indeed, recalling (A.3), we have

$\displaystyle\sum_{\bm{\sigma}_{R}\in{\mathcal{K}}_{R}(q)}\rho(R,\bm{\sigma}_{R})$
$\displaystyle=\sum_{\bm{\sigma}_{R}\in{\mathcal{K}}_{R}(q)}\sum_{g\in
G_{R}}\prod_{\\{x,y\\}\in E_{g}}(e^{-J_{xy}\delta_{\sigma_{x}\sigma_{y}}}-1)$
$\displaystyle=\sum_{\bm{\sigma}_{R}\in{\mathcal{K}}_{R}(q)}\sum_{g\in
G_{R}}\prod_{\\{x,y\\}\in E_{g}}\delta_{\sigma_{x}\sigma_{y}}(e^{-J_{xy}}-1)$
$\displaystyle=\sum_{g\in
G_{R}}\left[\sum\limits_{\bm{\sigma}_{R}\in{\mathcal{K}}_{R}(q)}\prod\limits_{\\{x,y\\}\in
E_{g}}\delta_{\sigma_{x}\sigma_{y}}\right]\prod_{\\{x,y\\}\in
E_{g}}(e^{-J_{xy}}-1)$ $\displaystyle=q\sum_{g\in G_{R}}\prod_{\\{x,y\\}\in
E_{g}}(e^{-J_{xy}}-1)$

where the last equality above holds because, for any $g\in G_{R}$,

$\sum\limits_{\bm{\sigma}_{R}\in{\mathcal{K}}_{R}(q)}\prod\limits_{\\{x,y\\}\in
E_{g}}\delta_{\sigma_{x}\sigma_{y}}=q$

Finally, recalling the definition of $J_{xy}$ given in (A.2), we have that

$\sum_{g\in G_{R}}\prod_{\\{x,y\\}\in
E_{g}}(e^{-J_{xy}}-1)=\begin{cases}\displaystyle{\sum\limits_{g\in\mathcal{C}_{{\mathbb{G}}|_{R}}}}(-1)^{|E_{g}|}&\mbox{if
${\mathbb{G}}|_{R}$ is connected}\\\ 0&{\rm otherwise}\end{cases}$

Therefore inserting (A.5) in (A.4) and recalling that
$\mathrm{C}_{\mathbb{V}}=\\{R\subset{\mathbb{V}}:|R|\geq 2~{}{\rm
and}~{}{\mathbb{G}}|_{R}~{}{\rm is}~{}{\rm connected}\\}$ (i.e.
$\mathrm{C}_{\mathbb{V}}$ is the set of all connected subsets of
${\mathbb{V}}$ with cardinality greater than one), we get

$P_{\mathbb{G}}(q)=q^{|{\mathbb{V}}|}\Xi_{\mathbb{G}}(q)$

where

$\Xi_{\mathbb{G}}(q)=1+\sum_{k\geq 1}\sum_{\\{R_{1},\dots,R_{k}\\}\atop
R_{i}\in\mathrm{C}_{\mathbb{V}},~{}R_{i}\cap
R_{j}=\emptyset}\prod_{i=1}^{k}z(R_{i},q)$

with

$z(R,q)={1\over
q^{|R|-1}}\sum_{g\in\mathcal{C}_{{\mathbb{G}}|_{R}}}(-1)^{|E_{g}|}$

This concludes the proof of Proposition 2.1.

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