# Global existence and blow-up of solutions to the nonlinear porous medium
equation

Bolys Sabitbek  Bolys Sabitbek: School of Mathematical Sciences Queen Mary
University of London United Kingdom and Al-Farabi Kazakh National University
Almaty, Kazakhstan E-mail address<EMAIL_ADDRESS>and  Berikbol Torebek
Berikbol Torebek: Department of Mathematics: Analysis, Logic and Discrete
Mathematics Ghent University Belgium and Institute of Mathematics and
Mathematical Modeling Almaty, Kazakhstan E-mail address
<EMAIL_ADDRESS>
###### Abstract.

In this paper, a nonlinear porous medium equation under a new condition on
nonlinearity is considered in a bounded domain. Firstly, we present the blow-
up of the positive solution to the considered problem for the negative initial
energy. By introducing a family of potential wells, we establish the invariant
sets and vacuum isolating solutions of nonlinear porous medium equation. For
the subcritical and critical initial energy cases, we obtain a global
existence, asymptotic behavior and blow-up phenomena in a finite time of the
positive solution to the nonlinear porous medium equation.

###### Key words and phrases:

Blow-up, $p$-Laplacian, porous medium equation, global solution

###### 1991 Mathematics Subject Classification:

35K92; 35B44, 35A01.

The first author was supported by EPSRC grant EP/R003025/2. The second author
was supported by FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial
Differential Equations. No new data was collected or generated during the
course of this research.

###### Contents

  1. 1 Introduction
    1. 1.1 Setting problem
    2. 1.2 Literature overview
    3. 1.3 Main objective of the paper
    4. 1.4 Plan of the paper
  2. 2 Blow-up results for $J(u_{0})<0$
  3. 3 Family of Potential Wells
  4. 4 Low initial energy $J(u_{0})<d$
    1. 4.1 Invariant sets
    2. 4.2 Global existence of weak solution
    3. 4.3 Asymptotic behavior of solutions
    4. 4.4 Finite time blow-up for $J(u_{0})<d$
  5. 5 Critical initial energy $J(u_{0})=d$
    1. 5.1 Global existence for $J(u_{0})=d$
    2. 5.2 Asymptotic behavior of solutions
    3. 5.3 Finite time blow-up for $J(u_{0})=d$

##  1\. Introduction

###  1.1. Setting problem

The main purpose of this paper is to investigate a global existence and blow-
up of the positive solutions to a nonlinear porous medium problem

$\displaystyle\begin{cases}u_{t}(x,t)-\nabla\cdot(|\nabla u^{m}|^{p-2}\nabla
u^{m})=f(u(x,t)),\,\,\,&(x,t)\in\Omega\times(0,+\infty),\\\
u(x,t)=0,\,\,\,&(x,t)\in\partial\Omega\times[0,+\infty),\\\
u(x,0)=u_{0}(x)\geq 0,\,\,\,&x\in\overline{\Omega},\end{cases}$ (1.1)

where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain with a smooth boundary
$\partial\Omega$, $m\geq 1$ and $2\leq p<n$. Let $f(u)$ satisfy the following
assumptions:

  * (i)

$f(u)$ is a locally Lipschitz continuous, $f(0)=0$ and $f(u)>0$ for $u>0$.

  * (ii)

$f(u)$ is convex for $u>0$.

  * (iii)

For $p<\alpha\leq\gamma<\frac{pn}{n-p}$, we have

$u^{m}f(u)\leq\gamma(F(u)-\sigma),$ (1.2)

and

$\alpha F(u)\leq u^{m}f(u)+\beta u^{pm}+\alpha\sigma,$ (1.3)

where $F(u)=\int_{0}^{u}s^{m-1}f(s)ds$, $\sigma>0$, and
$\beta<\frac{\lambda_{1,p}(\alpha-m-1)}{m+1}$ with $\lambda_{1,p}$ is the
first eigenvalue of $p$-Laplacian.

For our convenience, these assumptions on the nonlinearity $f(u)$ denoted as
$(H)$.

###  1.2. Literature overview

Recently, Chung-Choi [3] introduced a new condition

$\alpha\int_{0}^{u}f(s)ds\leq uf(u)+\beta u^{p}+\alpha\sigma,\,\,u>0,$ (1.4)

for blow-up solutions to $p$-Laplace parabolic equation that can be recovered
from (1.1) for $m=1$. Their proof of blow-up phenomena is based on the
concavity method, which was introduced in the abstract form by Levine [13],
and developed in the following works [14] and [20]. This work [3] motivated to
extend this new condition on $f(u)$ to the porous medium equation and also
establish the global existence and blow-up of the positive solutions to
problem (1.1) by the potential wells method.

As far as we know, the potential wells method was first proposed by Sattinger
[23] to study the properties of solutions of semilinear hyperbolic equation
and was developed by many authors ([16, 17, 19, 21, 25, 27, 28, 31, 32]). It
is worth noting the paper of Payne and Sattinger [19], devoted to a more
detailed description of the potential wells method for the nonlinear wave and
heat equations. In 2006, Liu-Zhao [17] generalized and improved the results of
Payne-Sattinger [19], by obtaining the invariant sets and vacuum isolation of
solutions. Later, Xu [31] complemented the result of Liu-Zhao by considering
the critical initial energy $E(0)=d.$ The applications of the potential wells
method to the study of various nonlinear evolution equations is still
relevant. This is confirmed by the recently published papers devoted to this
direction (see for example [5, 11, 15, 18, 26, 30, 33] and references
therein).

The porous medium equation is one of the important examples of the nonlinear
parabolic equations. The physical applications of the porous medium equation
describe widely processes involving fluid flow, heat transfer or diffusion,
and its other applications in different fields such as mathematical biology,
lubrication, boundary layer theory, and etc. There is a huge literature
dealing with an existence and nonexistence of solutions to problem (1.1) for
the reaction term $u^{p}$ on the bounded and unbounded domains, for example,
[1, 2, 4, 6, 8, 9, 7, 10, 14, 22]. By using the concavity method, Schaefer
[24] established a condition on the initial data of a Dirichlet type initial-
boundary value problem for the porous medium equation with a power function
reaction term when blow-up of the solution in finite time occurs and a global
existence of the solution holds. We refer more details to Vazquez’s book [29]
which provides a systematic presentation of the mathematical theory of the
porous medium equation.

###  1.3. Main objective of the paper

In this paper, we will be interested in answering the following two questions
related to the nonlinear porous medium equation (1.1) with $f(u)$ satisfies
the conditions $(H)$:

(Question 1):

Is it possible to obtain the finite time blow-up results to problem (1.1) in
case

$\displaystyle J(u_{0}):=\frac{1}{pm}\int_{\Omega}|\nabla
u^{m}_{0}(x)|^{p}dx-\int_{\Omega}(F(u_{0}(x))-\sigma)dx<0$

and $m>1$?

If $m=1,$ then the finite time blow-up of solutions was obtained by Chung and
Choi in [3].

(Question 2):

Can one extend the potential wells method to the nonlinear porous medium
equation (1.1) in the subcritical initial energy case $0<J(u_{0})\leq d$ with

$d:=\inf\\{J(u):u^{m}\in W_{0}^{1,p}(\Omega)\backslash\\{0\\},I(u)=0\\},$

and prove the global existence, blow-up and asymptotic behavior of the
solution to problem (1.1) using potential wells theory?

As far as the authors know, the potential wells method has not yet been used
to study the porous medium equations with the nonlinear source term.

###  1.4. Plan of the paper

The main aim of this paper are to answer above two questions.

This paper is organised as follows:

Section 2:

We present the blow-up of the positive solution to the nonlinear porous medium
equation (1.1) in the case $J(u_{0})<0$. This generalizes the blow-up results
to the semilinear heat equation with $p$-Laplacian, obtained by Chung and Choi
in [3].

Section 3:

We shall introduce a family of potential wells for nonlinear porous medium
equation (1.1) and give a series of their properties.

Section 4:

This section is devoted to the use of the family of potential wells to obtain
invariant sets and vacuum isolating solutions of problem (1.1). Also, we will
prove the global existence, asymptotic behavior and blow-up of solutions for
problem (1.1) with $J(u_{0})<d$. That is, when $J(u_{0})<d$ and

$I(u_{0}):=||\nabla
u_{0}^{m}||^{p}_{L^{p}(\Omega)}-\int_{\Omega}u_{0}^{m}f(u)dx>0,$

we have proved that there is a global solution to problem (1.1). Moreover, in
case $pm>m+1$ it decreases polynomially, and in case $pm=m+1$ it decreases
exponentially. In the case $J(u_{0})<d$ and $I(u_{0})<0$ we show that the
solution of problem (1.1) blows up in finite time.

Section 5:

We establish the global existence, asymptotic behavior and blow-up of
solutions for problem (1.1) with $J(u_{0})=d$. That is, when $J(u_{0})=d$ and
$I(u_{0})>0$ we have proved that, there is a global solution to problem (1.1).
Moreover, in case $pm>m+1$ it decreases polynomially, and in case $pm=m+1$ it
decreases exponentially. In the case $J(u_{0})=d$ and $I(u_{0})<0$ we show
that the solution of problem (1.1) blows up in finite time.

Section 6:

Here we provide a final discussion and some open questions related to problem
(1.1).

##  2\. Blow-up results for $J(u_{0})<0$

In this section, we present the blow-up of the positive solution to the
nonlinear porous medium equation (1.1) in the case $J(u_{0})<0$. Here we
provide the lemma that will be useful in the proof.

######  Lemma 2.1 (Theorem 1.1, [12]).

There exists $\lambda_{1,p}>0$ and $0<w\in W^{1,p}_{0}(\Omega)$ in
$\Omega\subset\mathbb{R}^{n}$ such that

$\displaystyle\begin{cases}\nabla\cdot(|\nabla w(x)|^{p-2}\nabla
w(x))+\lambda_{1,p}w(x)=0,\,\,\,&x\in\Omega,\\\
w(x)=0,\,\,\,&x\in\partial\Omega,\end{cases}$ (2.1)

where $\lambda_{1,p}$ is given by

$\lambda_{1,p}:=\inf_{v\in W_{0}^{1,p}(\Omega)}\frac{\int_{\Omega}|\nabla
v|^{p}dx}{\int_{\Omega}|v|^{p}dx}>0.$

Recall that $\lambda_{1,p}$ is the first eigenvalue of $p$-Laplace operator
and $w$ is a corresponding eigenfunction.

Denote

$\displaystyle J(u):=\frac{1}{pm}\int_{\Omega}|\nabla
u^{m}(x,t)|^{p}dx-\int_{\Omega}(F(u(x,t))-\sigma)dx,$

and

$\displaystyle J(u_{0}):=\frac{1}{pm}\int_{\Omega}|\nabla
u^{m}_{0}(x)|^{p}dx-\int_{\Omega}(F(u_{0}(x))-\sigma)dx.$

We know that

$J(u)=J(u_{0})+\int_{0}^{t}J^{\prime}(u)d\tau,$ (2.2)

where

$\displaystyle\int_{0}^{t}J^{\prime}(u)d\tau$
$\displaystyle=\frac{1}{pm}\int_{0}^{t}\int_{\Omega}\frac{d}{d\tau}|\nabla
u^{m}(x,\tau)|^{p}dxd\tau-\int_{0}^{t}\int_{\Omega}\frac{d}{d\tau}(F(u(x,\tau))+\sigma)dxd\tau$
$\displaystyle=\frac{1}{m}\int_{0}^{t}\int_{\Omega}|\nabla
u^{m}(x,\tau)|^{p-2}\nabla u^{m}\cdot\nabla(u^{m}(x,\tau))_{\tau}dxd\tau$
$\displaystyle-\int_{0}^{t}\int_{\Omega}F_{u}(u(x,\tau))u_{\tau}(x,\tau)dxd\tau$
$\displaystyle=-\frac{1}{m}\int_{0}^{t}\int_{\Omega}[\Delta_{p}u^{m}+f(u)](u^{m}(x,\tau))_{\tau}dxd\tau$
$\displaystyle=-\int_{0}^{t}\int_{\Omega}u^{m-1}(x,\tau)u_{\tau}^{2}(x,\tau)dxd\tau.$

This yields

$J(u)+\int_{0}^{t}\int_{\Omega}u^{m-1}(x,\tau)u_{\tau}^{2}(x,\tau)dxd\tau=J(u_{0}).$

Below we give a main result of this section.

######  Theorem 2.2.

Let $\Omega$ be a bounded domain of $\mathbb{R}^{n}$ with smooth boundary
$\partial\Omega$. Let a function $f$ satisfy

$\alpha F(u)\leq u^{m}f(u)+\beta u^{pm}+\alpha\sigma,\,\,\,u>0,$ (2.3)

where $\alpha>pm$ with $m\geq 1$ and $\sigma>0$,

$F(u)=\int_{0}^{u}s^{m-1}f(s)ds,$ (2.4)

and $0<\beta\leq\frac{\lambda_{1,p}(\alpha-pm)}{m+1}$ with $m\geq 1$. If
$u_{0}^{m}\in L^{\infty}(\Omega)\cap W_{0}^{1,p}(\Omega)$ satisfies the
inequality

$J(u_{0})=\frac{1}{pm}\int_{\Omega}|\nabla
u_{0}^{m}(x)|^{p}dx-\int_{\Omega}\left(F(u_{0}(x))-\sigma\right)dx<0,$ (2.5)

then there cannot exist a positive solution $u$ of (1.1) existing for all
times $T^{*}$ such that

$0<T^{*}\leq\frac{M}{\sigma\int_{\Omega}u_{0}^{m+1}(x)dx},$ (2.6)

that is,

$\lim_{t\rightarrow
T^{*}}\int_{0}^{t}\int_{\Omega}u^{m+1}(x,\tau)dxd\tau=+\infty.$ (2.7)

###### Proof of Theorem 2.2.

Let

$E_{p}(t):=\int_{0}^{t}\int_{\Omega}u^{m+1}(x,\tau)dxd\tau+M,\,\,t\geq 0,$
(2.8)

with the positive constant $M$. Then we have

$E^{\prime}_{p}(t)=\int_{\Omega}u^{m+1}(x,t)dx=(m+1)\int_{\Omega}\int_{0}^{t}u^{m}(x,\tau)u_{\tau}(x,\tau)d\tau
dx+\int_{\Omega}u^{m+1}_{0}(x)dx.$

By using the condition (2.3), Lemma 2.1 and
$0<\beta\leq\frac{\lambda_{1,p}(\alpha-mp)}{pm}$, we estimate the second
derivative of $E_{p}(t)$ with respect to time

$\displaystyle E^{\prime\prime}_{p}(t)$
$\displaystyle=(m+1)\int_{\Omega}u^{m}(x,t)u_{t}(x,t)dx$
$\displaystyle=(m+1)\int_{\Omega}u^{m}(x,t)\Delta_{p}u^{m}(x,t)+(m+1)\int_{\Omega}u^{m}(x,t)f(u(x,t))dx$
$\displaystyle=-(m+1)\int_{\Omega}|\nabla
u^{m}(x,t)|^{p}dx+(m+1)\int_{\Omega}u^{m}(x,t)f(u(x,t))dx$
$\displaystyle\geq-(m+1)\int_{\Omega}|\nabla
u^{m}(x,t)|^{p}dx+(m+1)\int_{\Omega}\left[\alpha F(u(x,t))-\beta
u^{pm}(x,t)-\alpha\sigma\right]dx$
$\displaystyle=-\alpha(m+1)\left[\frac{1}{pm}\int_{\Omega}|\nabla
u^{m}(x,t)|^{p}dx-\int_{\Omega}(F(u(x,t))-\sigma)dx\right]$
$\displaystyle+\left(\frac{\alpha(m+1)}{pm}-m-1\right)\int_{\Omega}|\nabla
u^{m}(x,t)|^{p}dx-\beta(m+1)\int_{\Omega}u^{pm}(x,t)dx$
$\displaystyle\geq-\alpha(m+1)J(u)+\left[\lambda_{1,p}\frac{(\alpha-
pm)(m+1)}{pm}\ -\beta(m+1)\right]\int_{\Omega}u^{pm}(x,t)dx$
$\displaystyle\geq-\alpha(m+1)J(u).$

Therefore, $E^{\prime\prime}_{p}(t)$ can be rewritten in the following form

$\displaystyle E^{\prime\prime}_{p}(t)\geq-\alpha(m+1)J(u_{0})+\alpha
pm\int_{0}^{t}\int_{\Omega}u^{m-1}(x,\tau)u_{\tau}^{2}(x,\tau)dxd\tau,$

and

$\displaystyle(E^{\prime}_{p}(t))^{2}$
$\displaystyle\leq(m+1)^{2}(1+\delta)\left(\int_{0}^{t}\int_{\Omega}u^{m+1}dxd\tau\right)\left(\int_{0}^{t}\int_{\Omega}u^{m-1}u_{\tau}^{2}(x,\tau)dxd\tau\right)$
$\displaystyle+\left(1+\frac{1}{\delta}\right)\left(\int_{\Omega}u_{0}^{m+1}(x)dx\right)^{2}.$

Then by taking $\varepsilon=\delta=\frac{\sqrt{pm\alpha}}{m+1}-1>0$, we
discover

$\displaystyle
E^{\prime\prime}_{p}(t)E_{p}(t)-(1+\varepsilon)(E^{\prime}_{p}(t))^{2}$
$\displaystyle\geq-\alpha M(m+1)J(u_{0})$
$\displaystyle+pm\alpha\left(\int_{0}^{t}\int_{\Omega}u^{m+1}(x,\tau)dxd\tau\right)\left(\int_{0}^{t}\int_{\Omega}u_{\tau}^{2}(x,\tau)u^{m-1}(x,\tau)dxd\tau\right)$
$\displaystyle-(m+1)^{2}(1+\varepsilon)(1+\delta)\left(\int_{0}^{t}\int_{\Omega}u^{m+1}dxd\tau\right)\left(\int_{0}^{t}\int_{\Omega}u^{m-1}u_{\tau}^{2}(x,\tau)dxd\tau\right)$
$\displaystyle-(1+\varepsilon)\left(1+\frac{1}{\delta}\right)\left(\int_{\Omega}u_{0}^{m+1}(x)dx\right)^{2}$
$\displaystyle\geq-\alpha
M(m+1)J(u_{0})-(1+\sigma)\left(1+\frac{1}{\delta}\right)\left(\int_{\Omega}u_{0}^{m+1}(x)dx\right)^{2}.$

By assumption $J(u_{0})<0$, thus if we select $M$ sufficiently large we have

$E^{\prime\prime}_{p}(t)E_{p}(t)-(1+\varepsilon)(E^{\prime}_{p}(t))^{2}>0.$
(2.9)

We can see that the above expression for $t\geq 0$ implies

$\frac{d}{dt}\left[\frac{E^{\prime}_{p}(t)}{E^{\varepsilon+1}_{p}(t)}\right]>0,$

hence

$\begin{cases}E^{\prime}_{p}\geq\left[\frac{E^{\prime}_{p}(0)}{E^{\varepsilon+1}_{p}(0)}\right]E^{1+\varepsilon}_{p}(t),\\\
E_{p}(0)=M.\end{cases}$

Then for $\varepsilon=\frac{\sqrt{pm\alpha}}{m+1}-1>0$, we arrive at

$E_{p}(t)\geq\left(\frac{1}{M^{\sigma}}-\frac{\varepsilon\int_{\Omega}u^{m+1}_{0}(x)dx}{M^{\varepsilon+1}}t\right)^{-\frac{1}{\varepsilon}}.$

Then the blow-up time $T^{*}$ satisfies

$0<T^{*}\leq\frac{M}{\varepsilon\int_{\Omega}u_{0}^{m+1}dx}.$

That completes the proof. ∎

##  3\. Family of Potential Wells

Define

$\displaystyle J(u)$ $\displaystyle=\frac{1}{pm}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}-\int_{\Omega}[F(u)-\sigma]dx,$

where $u^{m}\in W_{0}^{1,p}(\Omega)$, $1<p<n$, and $m\geq 1$. We denote the
Nehari functional by

$I(u)=||\nabla u^{m}||^{p}_{L^{p}(\Omega)}-\int_{\Omega}u^{m}f(u)dx.$

The depth of potential well is defined by

$d:=\inf\\{J(u):u^{m}\in W_{0}^{1,p}(\Omega)\backslash\\{0\\},I(u)=0\\}.$

Also the quantity $d$ can be understood as a mountain pass energy. For
instance, there is the valley. If we pour water into the valley containing a
local minimum at the origin, as the water level rises in the direction of the
potential energy and passes at a local maximum $d$.

######  Lemma 3.1.

Let $f(u)$ satisfy $(H)$, then we have

  * $(a)$

$wf^{\prime}(w)-m(p-1)f(w)\geq 0$ holds for $u\geq 0$;

  * $(b)$

$|F(u)-\sigma|\leq A|u|^{\gamma}$ for some $A>0$ and all $u\in R$;

  * $(c)$

$F(u)-\sigma\geq B|u|^{\lambda}$ for $|u|\geq 1$ with
$B=\frac{F(\widetilde{u})}{\widetilde{u}^{\lambda}}>0$ and
$\lambda=\frac{\alpha}{1+\beta}>pm$.

###### Proof of Lemma 3.1.

$(a)$ is proved by the variational method. Let

$i(\tau)=J(w+\tau v)=\frac{1}{pm}\int_{\Omega}|\nabla(w+\tau
v)^{m}|^{p}dx-\int_{\Omega}F(w+\tau v)dx.$

Let us compute the second variation of $i(\tau)$. The first derivative yields

$\displaystyle i^{\prime}(\tau)$ $\displaystyle=\int_{\Omega}|\nabla(w+\tau
v)^{m}|^{p-2}\nabla(w+\tau v)^{m}\cdot\nabla[(w+\tau v)^{m-1}v]dx$
$\displaystyle-\int_{\Omega}(w+\tau v)^{m-1}f(w+\tau v)vdx,$

and

$\displaystyle i^{\prime\prime}(\tau)=$ $\displaystyle
m(p-2)\int_{\Omega}|\nabla(w+\tau v)^{m}|^{p-4}|\nabla(w+\tau
v)^{m}\cdot\nabla[(w+\tau v)^{m-1}v]|^{2}dx$
$\displaystyle+m\int_{\Omega}|\nabla(w+\tau v)^{m}|^{p-2}|\nabla[(w+\tau
v)^{m-1}v]|^{2}dx$ $\displaystyle+(m-1)\int_{\Omega}|\nabla(w+\tau
v)^{m}|^{p-2}\nabla(w+\tau v)^{m}\cdot\nabla[(w+\tau v)^{m-2}v^{2}]dx$
$\displaystyle-(m-1)\int_{\Omega}(w+\tau v)^{m-2}v^{2}f(w+\tau
v)dx-\int_{\Omega}(w+\tau v)^{m-1}v^{2}f^{\prime}(w+\tau v)dx.$

Then the second variation of $J$ at a critical point $w$ is

$\displaystyle i^{\prime\prime}(0)=$ $\displaystyle m(p-2)\int_{\Omega}|\nabla
w^{m}|^{p-4}|\nabla w^{m}\cdot\nabla[w^{m-1}v]|^{2}dx+m\int_{\Omega}|\nabla
w^{m}|^{p-2}|\nabla[w^{m-1}v]|^{2}dx$ $\displaystyle+(m-1)\int_{\Omega}|\nabla
w^{m}|^{p-2}\nabla w^{m}\cdot\nabla[w^{m-2}v^{2}]dx$
$\displaystyle-(m-1)\int_{\Omega}w^{m-2}v^{2}f(w)dx-\int_{\Omega}w^{m-1}v^{2}f^{\prime}(w)dx.$

If $w$ is a non-trivial critical point, then $i^{\prime\prime}(0)\leq 0$. Let
$w$ be an admissible test function, then we compute

$\displaystyle i^{\prime\prime}(0)=$ $\displaystyle(mp-1)\int_{\Omega}|\nabla
w^{m}|^{p}dx-(m-1)\int_{\Omega}w^{m}f(w)dx-\int_{\Omega}w^{m+1}f^{\prime}(w)dx$
$\displaystyle=$
$\displaystyle\int_{\Omega}w^{m}\left[(mp-1)(-\Delta_{p}w^{m})-(m-1)f(w)-wf^{\prime}(w)\right]dx$
$\displaystyle=$
$\displaystyle-\int_{\Omega}w^{m}\left[wf^{\prime}(w)-m(p-1)f(w)\right]dx.$

Since $i^{\prime\prime}(0)\leq 0$, we have $wf^{\prime}(w)-m(p-1)f(w)\geq 0$.

$(b)$ From $(H)$, the growth condition $|u^{m}f(u)|\leq\gamma|F(u)-\sigma|$
implies

$\displaystyle\frac{|u^{m-1}f(u)|}{|F(u)-\sigma|}\leq\frac{\gamma}{|u|}\,\,\,\text{
for }\,\,\,u\neq 0.$

Taking any known fixed $\widetilde{u}\neq 0$, we integrate above inequality
from $\widetilde{u}$ to $u$, which gives

$\displaystyle\int_{\widetilde{u}}^{u}\frac{|s^{m-1}f(s)|}{|F(s)-\sigma|}ds$
$\displaystyle\leq\int_{\widetilde{u}}^{u}\frac{\gamma}{|s|}ds,$

and by using $F^{\prime}(u)=u^{m-1}f(u)$, we have

$\displaystyle\ln\frac{|F(u)-\sigma|}{|F(\widetilde{u})-\sigma|}\leq\ln\frac{|u|^{\gamma}}{|\widetilde{u}|^{\gamma}},$

that is $|F(u)-\sigma|\leq A|u|^{\gamma},$ where
$A=\frac{|F(\widetilde{u})-\sigma|}{|\widetilde{u}|^{\gamma}}$.

$(c)$ From $(H)$ we have $\alpha F(u)\leq u^{m}f(u)+\beta
u^{pm}+\alpha\sigma$. For $|u|\geq 1$, we write

$\displaystyle\frac{\alpha}{u}\leq\frac{u^{m-1}f(u)}{F(u)-\sigma}+\frac{\beta
u^{pm-1}}{F(u)-\sigma}\leq(1+\beta)\frac{u^{m-1}f(u)}{F(u)-\sigma},$

then we get

$\frac{u^{m-1}f(u)}{F(u)-\sigma}\geq\frac{\lambda}{u},$ (3.1)

where $\lambda=\frac{\alpha}{1+\beta}$. As the previous approach on $(b)$, we
get $F(u)-\sigma\geq B|u|^{\lambda}$ where
$B=\frac{F(\widetilde{u})}{\widetilde{u}^{\lambda}}$. ∎

######  Lemma 3.2.

Let $f(u)$ satisfy $(H)$. Let $u^{m}\in W_{0}^{1,p}(\Omega)$, $||\nabla
u^{m}||_{L^{p}(\Omega)}\neq 0$, and

$\displaystyle\phi(\epsilon)=\frac{1}{\epsilon^{m(p-1)}}\int_{\Omega}u^{m}f(\epsilon
u)dx.$ (3.2)

Then we have the following properties of function $\phi$:

  * •

$\phi(\epsilon)$ is increasing on $0<\epsilon<\infty$.

  * •

$\lim_{\epsilon\rightarrow 0}\phi(\epsilon)=0$ and
$\lim_{\epsilon\rightarrow+\infty}\phi(\epsilon)=+\infty$.

###### Proof of Lemma 3.2.

Let

$\displaystyle\frac{d\phi(\epsilon)}{d\epsilon}$
$\displaystyle=\frac{1}{\epsilon^{m(p-1)+1}}\int_{\Omega}[\epsilon
u^{m+1}f^{\prime}(\epsilon u)-m(p-1)u^{m}f(\epsilon u)]dx$
$\displaystyle=\frac{1}{\epsilon^{m(p-1)+1}}\int_{\Omega}u^{m}[\epsilon
uf^{\prime}(\epsilon u)-m(p-1)f(\epsilon u)]dx>0.$

By $(a)$ in Lemma 3.1, we see that $\phi(\epsilon)$ is increasing on
$0<\epsilon<\infty$.

The property $\lim_{\epsilon\rightarrow 0}\phi(\epsilon)=0$ comes from the
fact that

$\displaystyle|\phi(\epsilon)|$
$\displaystyle\leq\frac{1}{\epsilon^{pm}}\int_{\Omega}|\epsilon^{m}u^{m}f(\epsilon
u)|dx\leq\frac{\gamma A}{\epsilon^{pm}}\int_{\Omega}|\epsilon u|^{\gamma}dx$
$\displaystyle=\gamma\epsilon^{\gamma-
pm}A||u||^{\gamma}_{L^{\gamma}(\Omega)}.$

The property $\lim_{\epsilon\rightarrow+\infty}\phi(\epsilon)=+\infty$ can be
shown by using the fact $u^{m}f(u)\geq 0$ and $(c)$ in Lemma 3.1 as follows

$\displaystyle\phi(\epsilon)$
$\displaystyle=\frac{1}{\epsilon^{pm}}\int_{\Omega}\epsilon^{m}u^{m}f(\epsilon
u)dx$
$\displaystyle\geq\frac{1}{\epsilon^{pm}}\int_{\Omega_{\epsilon}}\epsilon^{m}u^{m}f(\epsilon
u)dx$ $\displaystyle\geq\frac{1}{\epsilon^{pm}}\int_{\Omega_{\epsilon}}[\alpha
F(\epsilon u)-\beta\epsilon^{pm}u^{pm}-\alpha\sigma]dx$
$\displaystyle\geq\alpha B\epsilon^{\lambda-
pm}\int_{\Omega_{\epsilon}}|u|^{\lambda}dx-\beta\int_{\Omega_{\epsilon}}u^{pm}dx,$

where $\Omega_{\epsilon}=\\{x\in\Omega\,\,|\,\,u(x)\geq\frac{1}{\epsilon}\\}$
and

$\lim_{\epsilon\rightarrow+\infty}\int_{\Omega_{\epsilon}}|u|^{\lambda}dx=||u||^{\lambda}_{L^{\lambda}(\Omega)}>0.$

This completes the proof. ∎

When nonlinearity term $f(u)$ satisfies condition $(H)$, then all non-trivial
critical points are a priori unstable equilibria for porous equation (1.1).
The next lemma will provide the local minimum of $J$, a critical point
$\epsilon^{*}$ and positiveness of a depth of potential well.

######  Lemma 3.3.

Let $f(u)$ satisfy $(H)$, $u^{m}\in W_{0}^{1,p}(\Omega)$, and $||\nabla
u^{m}||_{L^{p}(\Omega)}\neq 0$. Then

  * (i)

$\lim_{\epsilon\rightarrow 0}J(\epsilon u)=0$.

  * (ii)

$\frac{d}{d\epsilon}J(\epsilon u)=\frac{1}{\epsilon}I(\epsilon u)$.

  * (iii)

$\lim_{\epsilon\rightarrow+\infty}J(\epsilon u)=-\infty$.

  * (iv)

There exists a unique $\epsilon^{*}=\epsilon^{*}(u)>0$ such that

$\frac{d}{d\epsilon}J(\epsilon u)|_{\epsilon=\epsilon^{*}}=0.$ (3.3)

  * (v)

$J(\epsilon u)$ is increasing on $0\leq\epsilon\leq\epsilon^{*}$, decreasing
on $\epsilon^{*}\leq\epsilon<\infty$ and takes the maximum at
$\epsilon=\epsilon^{*}$.

  * (vi)

$I(\epsilon u)>0$ for $0<\epsilon<\epsilon^{*}$, $I(\epsilon u)<0$ for
$\epsilon^{*}<\epsilon<\infty$ and $I(\epsilon^{*}u)=0$.

###### Proof of Lemma 3.3.

The proof of (i) follows from $|F(u)-\sigma|\leq A|u|^{\gamma}$ and

$J(\epsilon u)=\frac{\epsilon^{pm}}{pm}\int_{\Omega}|\nabla
u^{m}|^{p}dx-\int_{\Omega}[F(\epsilon u)-\sigma]dx.$

The proof of (ii) follows from

$\displaystyle\frac{d}{d\epsilon}J(\epsilon u)$
$\displaystyle=\epsilon^{pm-1}\int_{\Omega}|\nabla
u^{m}|^{p}dx-\epsilon^{m-1}\int_{\Omega}u^{m}f(\epsilon u)dx$
$\displaystyle=\frac{1}{\epsilon}I(\epsilon u).$

Let

$\displaystyle J(\epsilon u)$
$\displaystyle=\frac{\epsilon^{pm}}{pm}\int_{\Omega}|\nabla
u^{m}|^{p}dx-\int_{\Omega}[F(\epsilon u)-\sigma]dx$
$\displaystyle\leq\frac{\epsilon^{pm}}{pm}\int_{\Omega}|\nabla
u^{m}|^{p}dx-B\epsilon^{\lambda}\int_{\Omega_{\epsilon}}|u|^{\lambda}dx.$

Since $\lambda>pm$, $J(\epsilon u)$ goes to $-\infty$ for
$\epsilon\rightarrow+\infty$. This guaranties the existence of a critical
point $\epsilon^{*}$.

The case (iv), the uniqueness of critical point, can be proved by supposing
that there two roots of equation $\frac{dJ(\epsilon u)}{d\epsilon}=0$ as
$\epsilon_{1}<\epsilon_{2}$. Then we have

$\epsilon_{1}^{pm-m}\int_{\Omega}|\nabla
u^{m}|^{p}-\int_{\Omega}u^{m}f(\epsilon_{1}u)dx=0,$

and

$\epsilon_{2}^{pm-m}\int_{\Omega}|\nabla
u^{m}|^{p}-\int_{\Omega}u^{m}f(\epsilon_{2}u)dx=0.$

Elimination of $\int_{\Omega}|\nabla u^{m}|^{p}$ from above equations gives

$\int_{\Omega}u^{m}\left[\frac{f(\epsilon_{2}u)}{\epsilon_{2}^{pm-m}}-\frac{f(\epsilon_{1}u)}{\epsilon_{1}^{pm-m}}\right]dx=0.$

This expression can be rewritten by the following substitution
$w=\epsilon_{1}u$ and $\epsilon=\epsilon_{2}/\epsilon_{1}>1$ as

$\frac{1}{\epsilon^{pm-m}}\int_{\Omega}w^{m}f(\epsilon
w)dx=\int_{\Omega}w^{m}f(w)dx.$ (3.4)

It is easy to see that equation (3.4) does not hold.

The case (v) and (vi) follow from Lemma 3.2 and

$\frac{d}{d\epsilon}J(\epsilon u)=\frac{1}{\epsilon}I(\epsilon
u)=\epsilon^{pm-1}\left[\int_{\Omega}|\nabla
u^{m}|^{p}dx-\phi(\epsilon)\right].$

The proof is completed. ∎

In order to introduce the family of potential wells $\\{W_{\delta}\\}$ and
$\\{V_{\delta}\\}$, we define

$\displaystyle I_{\delta}(u)=\delta\int_{\Omega}|\nabla
u^{m}|^{p}-\int_{\Omega}u^{m}f(u)dx,\,\,\,\delta>0,$

and

$d(\delta):=\inf\\{J(u):u^{m}\in
W_{0}^{1,p}(\Omega)\backslash\\{0\\},I_{\delta}(u)=0\\}.$

######  Lemma 3.4.

Let $f(u)$ satisfy $(H)$. Suppose that $u^{m}\in W_{0}^{1,p}(\Omega)$ and

$\displaystyle
r(\delta)=\left(\frac{\delta}{aC^{\gamma}_{*}}\right)^{\frac{1}{\gamma-p}}\,\,\,\text{and}\,\,\,a=\sup\frac{u^{m}f(u)}{|u^{m}|^{\gamma}},$

where $C_{*}$ is the embedding constant form $W_{0}^{1,p}(\Omega)$ into
$L^{\gamma}(\Omega)$. Then we have

  * (i)

If $0<||\nabla u^{m}||_{L^{p}(\Omega)}<r(\delta)$, then $I_{\delta}(u)>0$. In
the case $\delta=1$, if $0<||\nabla u^{m}||_{L^{p}(\Omega)}<r(1)$, then
$I(u)>0$.

  * (ii)

If $I_{\delta}(u)<0$, then $||\nabla u^{m}||_{L^{p}(\Omega)}>r(\delta)$. In
the case $\delta=1$, if $I(u)<0$, then $||\nabla
u^{m}||_{L^{p}(\Omega)}>r(1)$.

  * (iii)

If $I_{\delta}(u)=0$, then $||\nabla u^{m}||_{L^{p}(\Omega)}\geq r(\delta)$.
In the case $\delta=1$, if $I(u)=0$, then $||\nabla
u^{m}||_{L^{p}(\Omega)}\geq r(1)$.

######  Lemma 3.5 (Gagliardo-Nirenberg-Sobolev inequality).

Let $\Omega$ be a bounded, open subset of $\mathbb{R}^{n}$, and suppose
$\partial\Omega$ is $C^{1}$. Assume $1\leq p<n$, $p^{*}=\frac{np}{n-p}$, and
$f\in W^{1,p}(\Omega)$. Then $f\in L^{p^{*}}(\Omega)$, with estimate

$||f||_{L^{p^{*}}(\Omega)}\leq C||f||_{W^{1,p}(\Omega)},$

the constant $C$ depending only on $p$, $n$, and $\Omega$.

###### Proof of Lemma 3.4.

(i) From $0<||\nabla u^{m}||_{L^{p}(\Omega)}<r{(\delta)}$ we have

$\displaystyle\int_{\Omega}u^{m}f(u)dx$
$\displaystyle\leq\int_{\Omega}|u^{m}f(u)|dx\leq
a\int_{\Omega}|u^{m}|^{\gamma}dx=a||u^{m}||^{\gamma}_{L^{\gamma}(\Omega)}$
$\displaystyle\leq aC_{*}^{\gamma}||\nabla
u^{m}||^{\gamma-p}_{L^{p}(\Omega)}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}<\delta||\nabla u^{m}||^{p}_{L^{p}(\Omega)},$

which gives $I_{\delta}(u)>0$.

(ii) From $I_{\delta}(u)<0$ we get

$\displaystyle\delta||\nabla u^{m}||^{p}_{L^{p}(\Omega)}$
$\displaystyle<\int_{\Omega}u^{m}f(u)dx\leq
a||u^{m}||^{\gamma}_{L^{\gamma}(\Omega)}$ $\displaystyle\leq
aC_{*}^{\gamma}||\nabla u^{m}||^{\gamma-p}_{L^{p}(\Omega)}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)},$

which implies $||\nabla u^{m}||_{L^{p}(\Omega)}>r(\delta)$.

(iii) From $I_{\delta}(u)=0$ we obtain

$\displaystyle\delta||\nabla u^{m}||^{p}_{L^{p}(\Omega)}$
$\displaystyle\leq\int_{\Omega}u^{m}f(u)dx\leq
a||u^{m}||^{\gamma}_{L^{\gamma}(\Omega)}$ $\displaystyle\leq
aC_{*}^{\gamma}||\nabla u^{m}||^{\gamma-p}_{L^{p}(\Omega)}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)},$

which implies $||\nabla u^{m}||_{L^{p}(\Omega)}\geq r(\delta)$. ∎

Next lemma describes the properties of $d(\delta)$ for a family of potential
wells.

######  Lemma 3.6.

Let $f(u)$ satisfy $(H)$. Then

  * (i)

$d(\delta)>a(\delta)r^{p}(\delta)$ for
$0<\delta<\frac{\alpha}{pm}-\frac{\beta}{\lambda_{1,p}}$, where
$a(\delta)=\frac{1}{pm}-\frac{\delta}{\alpha}-\frac{\beta}{\lambda_{1,p}\alpha}$.

  * (ii)

$\lim_{\delta\rightarrow 0}d(\delta)=0$ and there exists a unique $b$,
$\frac{\alpha}{pm}-\frac{\beta}{\lambda_{1,p}}\leq b\leq\frac{\gamma}{pm}$
such that $d(b)=0$ and $d(\delta)>0$ for $0<\delta<b$.

  * (iii)

$d(\delta)$ is strictly increasing on $0\leq\delta\leq 1$, strictly decreasing
on $1\leq\delta\leq\infty$ and takes the maximum $d=d(1)$ at $\delta=1$.

###### Proof of Lemma 3.6.

(i) If $I_{\delta}(u)=0$ and $||\nabla u||_{L^{p}(\Omega)}\neq 0$, then we
have $||\nabla u||_{L^{p}(\Omega)}\geq r(\delta)$ from Lemma 3.4. By applying
$I_{\delta}(u)=0$, Lemma 2.1 and 3.4, we have

$\displaystyle J(u)$ $\displaystyle=\frac{1}{pm}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}-\int_{\Omega}[F(u)-\sigma]dx$
$\displaystyle\geq\frac{1}{pm}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}-\frac{1}{\alpha}\int_{\Omega}u^{m}f(u)dx-\frac{\beta}{\alpha}||u^{m}||^{p}_{L^{p}(\Omega)}$
$\displaystyle\geq\left[\frac{1}{pm}-\frac{\delta}{\alpha}-\frac{\beta}{\lambda_{1,p}\alpha}\right]||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}$ $\displaystyle\geq a(\delta)r^{p}(\delta),$

where $0<\delta<\frac{\alpha}{pm}-\frac{\beta}{\lambda_{1,p}}$.

$(ii)$ Lemma 3.2 implies that for $u^{m}\in W_{0}^{1,p}(\Omega)$, $||\nabla
u^{m}||_{L^{p}(\Omega)}\neq 0$ and any $\delta>0$ we can define a unique
$\epsilon(\delta)=\phi^{-1}(\delta||\nabla u^{m}||^{p}_{L^{p}(\Omega)})$ such
that

$\epsilon^{pm}\phi(\epsilon)=\int_{\Omega}(\epsilon u)^{m}f(\epsilon
u)dx=\delta||\nabla(\epsilon u)^{m}||^{p}_{L^{p}(\Omega)}.$ (3.5)

Then for $I_{\delta}(\epsilon u)=0$ we get

$\lim_{\delta\rightarrow 0}\epsilon(\delta)=0\,\,\text{ and
}\,\,\lim_{\delta\rightarrow+\infty}\epsilon(\delta)=+\infty.$

From Lemma 3.3 it follows that

$\displaystyle\lim_{\delta\rightarrow 0}J(\epsilon
u)=\lim_{\epsilon\rightarrow 0}J(\epsilon u)=0\,\,$ $\displaystyle\text{ and
}\,\,\lim_{\delta\rightarrow 0}d(\delta)=0,$
$\displaystyle\lim_{\delta\rightarrow+\infty}J(\epsilon
u)=\lim_{\epsilon\rightarrow+\infty}J(\epsilon u)=-\infty\,\,$
$\displaystyle\text{ and
}\,\,\lim_{\delta\rightarrow+\infty}d(\delta)=-\infty.$

Existence of $d(b)=0$ for $b\geq\alpha/pm-\beta/\lambda_{1,p}$ and
$d(\delta)>0$ for $0<\delta<b$ follows from the part $(i)$ and above
expressions. Upper bound of $b\leq\gamma/pm$ follows from the facts
$|u^{m}f(u)|\leq\gamma|F(u)-\sigma|$ and $I_{\delta}(u)=0$, then

$\displaystyle J(u)$ $\displaystyle=\frac{1}{pm}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}-\int_{\Omega}[F(u)-\sigma]dx$
$\displaystyle\geq\frac{1}{pm}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}-\gamma^{-1}\int_{\Omega}u^{m}f(u)dx$
$\displaystyle=\left(\frac{1}{pm}-\frac{\delta}{\gamma}\right)||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}<0,\,\,\,\text{ for }\,\,\,\delta>\gamma/pm.$

For the case $(iii)$, it is enough to show that
$d(\delta^{\prime})<d(\delta^{\prime\prime})$ for any
$0<\delta^{\prime}<\delta^{\prime\prime}<1$ or
$1<\delta^{\prime\prime}<\delta^{\prime}<b$. In other word, we need to prove
that for some $c(\delta^{\prime},\delta^{\prime\prime})>0$, we have

$J(v)>J(u)-c(\delta^{\prime},\delta^{\prime\prime}),$

for any $u^{m}\in W_{0}^{1,p}(\Omega)$ and $v^{m}\in W_{0}^{1,p}(\Omega)$ with
$I_{\delta^{\prime\prime}}(u)=0$, $||\nabla^{m}u||_{L^{p}(\Omega)}\neq 0$ and
$I_{\delta^{\prime}}(u)=0$, $||\nabla v^{m}||_{L^{p}(\Omega)}\neq 0$,
respectively.

First, we define $\epsilon(\delta)$ for $u$ as (3.5), then
$I_{\delta}(\epsilon(\delta)u)=0$ and $\epsilon(\delta^{\prime\prime})=1$. Let
us have $g(\epsilon)=J(\epsilon u)$, then

$\displaystyle\frac{d}{d\epsilon}g(\epsilon)$
$\displaystyle=\frac{1}{\epsilon}\left(||\nabla(\epsilon
u)^{m}||^{p}_{L^{p}(\Omega)}-\int_{\Omega}\epsilon^{m}u^{m}f(\epsilon
u)dx\right)$ $\displaystyle=(1-\delta)\epsilon^{pm-1}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)},$

where we use the fact $\delta||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}=\int_{\Omega}u^{m}f(u)dx$.

Now we choose $v=\epsilon(\delta^{\prime})u$, then $I_{\delta^{\prime}}(v)=0$
and $||\nabla v^{m}||_{L^{p}(\Omega)}\neq 0$. For
$0<\delta^{\prime}<\delta^{\prime\prime}<1$, we get

$\displaystyle J(u)-J(v)$
$\displaystyle=J(\epsilon(\delta^{\prime\prime})u)-J(\epsilon(\delta^{\prime})u)$
$\displaystyle=g(1)-g(\epsilon(\delta^{\prime}))$
$\displaystyle=(1-\delta^{\prime})\epsilon(\delta^{\prime})||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}(1-\epsilon(\delta^{\prime}))$
$\displaystyle>(1-\delta^{\prime\prime})\epsilon(\delta^{\prime})r^{p}(\delta^{\prime\prime})(1-\epsilon(\delta^{\prime}))\equiv
c(\delta^{\prime},\delta^{\prime\prime}).$

For $1<\delta^{\prime\prime}<\delta^{\prime}<b$, we have

$\displaystyle J(u)-J(v)$
$\displaystyle=J(\epsilon(\delta^{\prime\prime})u)-J(\epsilon(\delta^{\prime})u)$
$\displaystyle=g(1)-g(\epsilon(\delta^{\prime}))$
$\displaystyle>(\delta^{\prime\prime}-1)r^{p}(\delta^{\prime\prime})\epsilon(\delta^{\prime\prime})(\epsilon(\delta^{\prime\prime})-1).$

This completes the proof. ∎

######  Lemma 3.7.

Let $f(u)$ satisfy $(H)$ and
$0<\delta<\frac{\alpha}{pm}-\frac{\beta}{\lambda_{1,p}}$. Then the following
properties hold

  * (i)

Suppose that $J(u)\leq d(\delta)$ and $I_{\delta}(u)>0$, then

$0<||\nabla u^{m}||^{p}_{L^{p}(\Omega)}<\frac{d(\delta)}{a(\delta)}.$

In the case $\delta=1$, if $J(u)\leq d$ and $I(u)>0$, then

$0<||\nabla u^{m}||^{p}_{L^{p}(\Omega)}<\frac{d}{a(1)}.$

  * (ii)

Suppose that $J(u)\leq d(\delta)$ and $I_{\delta}(u)=0$, then

$0<||\nabla u^{m}||^{p}_{L^{p}(\Omega)}<\frac{d(\delta)}{a(\delta)}.$

In the case $\delta=1$, if $J(u)\leq d$ and $I(u)=0$, then

$0<||\nabla u^{m}||^{p}_{L^{p}(\Omega)}<\frac{d}{a(1)}.$

  * (iii)

Suppose that $J(u)\leq d(\delta)$ and $||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}>d(\delta)/a(\delta)$, then $I_{\delta}(u)<0$.
In the case $\delta=1$, if $J(u)\leq d$ and $||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}>{d}/{a(1)}$, then $I(u)<0$.

###### Proof of Lemma 3.7.

The case $(i)$ follows from

$\displaystyle J(u)$ $\displaystyle=\frac{1}{pm}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}-\int_{\Omega}[F(u)-\sigma]dx$
$\displaystyle\geq\frac{1}{pm}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}-\frac{1}{\alpha}\int_{\Omega}u^{m}f(u)dx-\frac{\beta}{\alpha}\int_{\Omega}u^{pm}dx$
$\displaystyle>\left[\frac{1}{pm}-\frac{\delta}{\alpha}-\frac{\beta}{\lambda_{1,p}\alpha}\right]||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}$ $\displaystyle=a(\delta)||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}.$

The proof of cases $(ii)$ and $(iii)$ follows from the similar argument. ∎

Then a family of potential wells can be defined for $0<\delta<b$ as follows

$\displaystyle W_{\delta}$ $\displaystyle:=\\{u^{m}\in
W_{0}^{1,p}(\Omega)\,\,|\,\,I_{\delta}(u)>0,J(u)<d(\delta)\\}\cup\\{0\\},$
$\displaystyle\overline{W}_{\delta}$ $\displaystyle:=W_{\delta}\cup\partial
W_{\delta}=\\{u^{m}\in W_{0}^{1,p}(\Omega)\,\,|\,\,I_{\delta}(u)\geq
0,J(u)\leq d(\delta)\\},$

In the case $\delta=1$, we have

$\displaystyle W$ $\displaystyle:=\\{u^{m}\in
W_{0}^{1,p}(\Omega)\,\,|\,\,I(u)>0,J(u)<d\\}\cup\\{0\\},$
$\displaystyle\overline{W}$ $\displaystyle:=\\{u^{m}\in
W_{0}^{1,p}(\Omega)\,\,|\,\,I(u)\geq 0,J(u)\leq d\\}.$

In addition, we define

$\displaystyle V_{\delta}$ $\displaystyle:=\\{u^{m}\in
W_{0}^{1,p}(\Omega)\,\,|\,\,I_{\delta}(u)<0,J(u)<d(\delta)\\},$ $\displaystyle
V$ $\displaystyle:=\\{u^{m}\in W_{0}^{1,p}(\Omega)\,\,|\,\,I(u)<0,J(u)<d\\}.$

From the definition of $W_{\delta}$, $V_{\delta}$ and Lemma 3.6, we derive the
following properties:

######  Lemma 3.8.

If $0<\delta^{\prime}<\delta^{\prime\prime}\leq 1$ and
$1\leq\delta^{\prime\prime}<\delta^{\prime}<b$, then
$W_{\delta^{\prime}}\subset W_{\delta^{\prime\prime}}$ and
$V_{\delta^{\prime}}\subset V_{\delta^{\prime\prime}}$, respectively.

######  Lemma 3.9.

Let $0<J(u)<d$ for some $u^{m}\in W_{0}^{1,p}(\Omega)$. Let
$\delta_{1}<\delta_{2}$ are two roots of equation $J(u)=d(\delta)$. Then the
sign of $I_{\delta}(u)$ is unchangeable for $\delta_{1}<\delta<\delta_{2}$.

###### Proof of Lemma 3.9.

The proof follows from contradiction argument. $||\nabla
u^{m}||_{L^{p}(\Omega)}\neq 0$ comes from $J(u)>0$. If sign of $I_{\delta}(u)$
is changeable for $\delta_{1}<\delta<\delta_{2}$, then there exists
$\delta^{*}\in(\delta_{1},\delta_{2})$ such that $I_{\delta^{*}}(u)=0$. Thus
by the definition of $d(\delta)$ we have $J(u)\geq d(\delta^{*})$ that
contradicts

$J(u)=d(\delta_{1})=d(\delta_{2})<d(\delta^{*}),$

this completes the proof. ∎

##  4\. Low initial energy $J(u_{0})<d$

In this section, we study a global existence of problem (1.1).

###  4.1. Invariant sets

######  Theorem 4.1.

Let $f(u)$ satisfy $(H)$, $u_{0}^{m}(x)\in W_{0}^{1,p}(\Omega)$. Assume that
$0<e<d$, then equation $d(\delta)=e$ has two roots $\delta_{1}<\delta_{2}$.
Therefore, we formulate the following properties

  * $(i)$

All solutions of problem (1.1) with $J(u_{0})=e$ belong to $W_{\delta}$ for
$\delta_{1}<\delta<\delta_{2}$, provided $I(u_{0})>0$.

  * $(ii)$

All solutions of problem (1.1) with $J(u_{0})=e$ belong to $V_{\delta}$ for
$\delta_{1}<\delta<\delta_{2}$, provided $I(u_{0})<0$.

###### Proof of Theorem 4.1.

In the case $(i)$, we suppose that $u(x,t)$ is any solution of problem (1.1)
with $J(u_{0})=e$ and $I(u_{0})>0$. If $I(u_{0})>0$, then we have

$\int_{0}^{t}\int_{\Omega}u^{m}u_{\tau}^{2}dxd\tau+J(u)\leq
J(u_{0})=d(\delta_{1})=d(\delta_{2})<d(\delta),\,\,\,\delta_{1}<\delta<\delta_{2}.$
(4.1)

From (4.1) and the definition of $d(\delta)$ we get $I_{\delta}(u_{0})>0$ and
$J(u_{0})<d(\delta)$ that is $u_{0}^{m}(x)\in W_{\delta}$ for $0<\delta<b$.

Now we prove $u^{m}(t)\in W_{\delta}$ for $\delta_{1}<\delta<\delta_{2}$ and
$0<t<T$, where $T$ is the maximal existence time of $u^{m}(t)$. Arguing by
contradiction, there must exist $t_{0}\in(0,T)$ such that
$u^{m}(t_{0})\in\partial W_{\delta_{0}}$ for some
$\delta_{0}\in(\delta_{1},\delta_{2})$, and

$I_{\delta_{0}}(u(t_{0}))=0\,\,\,\,\,\text{or}\,\,\,\,\,J(u(t_{0}))=d(\delta_{0}).$

From (4.1) it is easy to see that $J(u(t_{0}))=d(\delta_{0})$ is impossible.
If $I_{\delta_{0}}(u(t_{0}))=0$, then by the definition of $d(\delta_{0})$ we
have $J(u(t_{0}))\geq d(\delta_{0})$ that contradicts (4.1).

In the case $(ii)$, we assume again that $u(x,t)$ is any solution of problem
(1.1) with

$J(u_{0})=e\,\,\,\,\text{and}\,\,\,\,I(u_{0})<0.$

As a previous case, $u_{0}^{m}(x)\in V_{\delta}$ can be established by Lemma
3.9 and (4.1).

The proof of $u^{m}(t)\in V_{\delta}$ for $\delta\in(\delta_{1},\delta_{2})$
follows from arguing by contradiction. Let $t_{0}\in(0,T)$ be the first time
such that $u^{m}(t)\in V_{\delta}$ for $t\in[0,t_{0})$ and
$u^{m}(t_{0})\in\partial V_{\delta_{0}}$, i.e.

$I_{\delta_{0}}(u(t_{0}))=0\,\,\,\,\text{or}\,\,\,\,J(u(t_{0}))=d(\delta_{0}),$

for some $\delta_{0}\in(\delta_{1},\delta_{2})$. From (4.1) it follows that
$J(u(t_{0}))\neq d(\delta_{0})$. If $I_{\delta_{0}}(u(t_{0}))=0$, then
$I_{\delta_{0}}(u(t))<0$ for $t\in(0,t_{0})$ and Lemma 3.4 yield

$||\nabla
u^{m}(t)||_{L^{p}(\Omega)}>r(\delta_{0})\,\,\,\,\text{and}\,\,\,\,||\nabla
u^{m}(t_{0})||_{L^{p}(\Omega)}\geq r(\delta_{0}).$

Hence by the definition of $d(\delta_{0})$ we have $J(u(t_{0}))\geq
d(\delta_{0})$ which contradicts (4.1). ∎

###  4.2. Global existence of weak solution

######  Definition 4.2.

We say a function

$u^{m}\in L^{\infty}(0,T;W_{0}^{1,p}(\Omega)),\,\text{ with }u_{t}\in
L^{\infty}(0,T;L^{2}(\Omega)),$

is a weak solution of the initial-boundary problem (1.1) provided

$(u_{t},v)+(|\nabla u^{m}|^{p-2}\nabla u^{m},\nabla v)=(f(u),v)$ (4.2)

for each $v\in W^{1,p}_{0}(\Omega)$ and a.e. time $0\leq t\leq T$ with

$u^{m}(x,0)=u_{0}^{m}(x)\,\,\,\text{ in }\,\,W_{0}^{1,p}(\Omega).$

Now we construct a global weak solution solution of the initial-boundary
problem (1.1) by the Galerkin approximation method.

######  Theorem 4.3.

(Global existence for $J(u_{0})<d$). Let $f$ satisfy $(H)$, $u_{0}^{m}\in
W_{0}^{1,p}(\Omega)$. Assume that $J(u_{0})<d$ and $I(u_{0})>0$. The initial-
boundary problem (1.1) admits a global weak solution $u(t)\in
L^{\infty}(0,\infty;W_{0}^{1,p}(\Omega))$ along with $u_{t}(t)\in
L^{\infty}(0,\infty;L^{2}(\Omega))$ and $u^{m}(t)\in W$ for $t\in[0,\infty)$.

###### Proof of Theorem 4.3.

We employ the Galerkin method by selecting smooth functions
$\omega_{j}=(\omega_{j}(x))$ $(j=1,\ldots)$ such that for a fixed positive
integer $k$

$u_{k}(t):=\sum_{j=1}^{k}g_{jk}(t)\omega_{j}(x),\,\,\,k=1,2,\ldots$

satisfying

$(u_{k}^{\prime},\omega_{s}^{m})+(|\nabla u^{m}_{k}|^{p-2}\nabla
u^{m}_{k},\nabla\omega_{s}^{m})=(f(u_{k}),\omega_{s}^{m})\,\,\,(0\leq
t<\infty,\,\,s=1,\ldots,k).$ (4.3)
$u_{k}^{m}(x,0)=\sum_{j=1}^{k}a_{jk}^{m}\omega_{j}^{m}(x)\rightarrow
u_{0}^{m}(x)\,\,\,\text{ in }\,\,W^{1,p}_{0}(\Omega).$ (4.4)

Multiplying (4.3) by $(g_{sk}^{m})^{\prime}(t)$ and integrating with respect
to time from $0$ to $t$, which gives

$\displaystyle
m\int_{0}^{t}(u_{k}^{\prime},g_{sk}^{m-1}g_{sk}^{\prime}w_{s}^{m})d\tau$
$\displaystyle+\int_{0}^{t}(|\nabla u^{m}_{k}|^{p-2}\nabla
u^{m}_{k},(g_{sk}^{m})^{\prime}\nabla\omega_{s}^{m})d\tau$
$\displaystyle=\int_{0}^{t}(f(u_{k}),(g_{sk}^{m})^{\prime}\omega_{s}^{m})d\tau,$

for $(0\leq t<\infty,\,\,s=1,\ldots,k)$, then summing over $s$, we have

$\displaystyle\int_{0}^{t}([u_{k}^{\prime}(\tau)]^{2},u^{m-1}_{k}(\tau))d\tau$
$\displaystyle+\frac{1}{pm}\int_{0}^{t}\frac{d}{d\tau}\int_{\Omega}|\nabla
u^{m}_{k}(\tau)|^{p}dxd\tau$
$\displaystyle=\frac{1}{pm}\int_{0}^{t}\frac{d}{d\tau}\int_{\Omega}F(u_{k}(\tau))dxd\tau.$

That yields

$\displaystyle\int_{0}^{t}([u_{k}^{\prime}(\tau)]^{2},u^{m-1}_{k}(\tau))d\tau+J(u_{k}(t))=J(u_{k}(0)),\,\,\,t\in[0,\infty).$

From (4.4) we get $J(u_{k}(0))\rightarrow J(u_{0})$, then for sufficiently
large $k$, we have

$\displaystyle\int_{0}^{t}([u_{k}^{\prime}(\tau)]^{2},u^{m-1}_{k}(\tau))d\tau+J(u_{k}(t))<d,\,\,\,t\in[0,\infty).$
(4.5)

Combining (4.5) and Theorem 4.1, we obtain $u_{k}^{m}(t)\in W$ for
$t\in[0,\infty)$ and sufficiently large $k$. From (4.5) and

$\displaystyle J(u)$ $\displaystyle=\frac{1}{pm}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}-\int_{\Omega}[F(u)-\sigma]dx$
$\displaystyle\geq\frac{1}{pm}||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}-\frac{1}{\alpha}\int_{\Omega}u^{m}f(u)dx-\frac{\beta}{\alpha}\int_{\Omega}u^{pm}dx$
$\displaystyle\geq\left[\frac{1}{pm}-\frac{1}{\alpha}-\frac{\beta}{\lambda_{1,p}\alpha}\right]||\nabla
u^{m}||^{p}_{L^{p}(\Omega)}+\frac{1}{\alpha}I(u),$

we establish

$\displaystyle\int_{0}^{t}([u_{k}^{\prime}(\tau)]^{2},u^{m-1}_{k}(\tau))d\tau+\left[\frac{1}{pm}-\frac{1}{\alpha}-\frac{\beta}{\lambda_{1,p}\alpha}\right]||\nabla
u^{m}_{k}||^{p}_{L^{p}(\Omega)}<d,\,\,\,t\in[0,\infty)$ (4.6)

for sufficiently large $k$. Then for $0\leq t<\infty$ (4.6) implies the
following estimates

$\displaystyle||\nabla
u^{m}_{k}||^{p}_{L^{p}(\Omega)}<\left[\frac{1}{pm}-\frac{1}{\alpha}-\frac{\beta}{\lambda_{1,p}\alpha}\right]^{-1}d,$
$\displaystyle||u^{m}_{k}||^{\gamma}_{L^{\gamma}(\Omega)}\leq
C_{*}^{\gamma}||\nabla
u^{m}_{k}||_{L^{p}(\Omega)}^{\gamma}<C_{*}^{\gamma}\left[\frac{1}{pm}-\frac{1}{\alpha}-\frac{\beta}{\lambda_{1,p}\alpha}\right]^{-\frac{\gamma}{p}}d^{\frac{\gamma}{p}},$
$\displaystyle\int_{0}^{t}([u_{k}^{\prime}(\tau)]^{2},u^{m-1}_{k}(\tau))d\tau<d,$

and

$\displaystyle||f(u_{k})||^{q}_{L^{q}(\Omega)}$
$\displaystyle\leq\int_{\Omega}[\gamma
A|u_{k}|^{\gamma-m}]^{q}dx=\gamma^{q}A^{q}||u_{k}^{m}||^{\gamma}_{L^{\gamma}(\Omega)}$
$\displaystyle<\gamma^{q}A^{q}C_{*}^{\gamma}\left[\frac{1}{pm}-\frac{1}{\alpha}-\frac{\beta}{\lambda_{1,p}\alpha}\right]^{-\frac{\gamma}{p}}d^{\frac{\gamma}{p}},$

where $q=\frac{m\gamma}{\gamma-m}$.

According the above energy estimates, we see that sequence
$\\{u_{k}^{m}\\}_{k=1}^{\infty}$ is bounded in
$L^{\infty}(0,\infty;W_{0}^{1,p}(\Omega))$ and
$\\{u_{k}^{\prime}\\}_{k=1}^{\infty}$ is bounded in
$L^{\infty}(0,\infty;L^{2}(\Omega))$.

Consequently, there exists a subsequence
$\\{u_{k_{l}}\\}_{l=1}^{\infty}\subset\\{u_{k}\\}_{k=1}^{\infty}$ and the
functions $u(t)\in L^{\infty}(0,\infty;W_{0}^{1,p}(\Omega))$, with
$u^{\prime}(t)\in L^{\infty}(0,\infty;L^{2}(\Omega))$ and $f(u)\in
L^{\infty}(0,\infty;L^{q}(\Omega))$, such that

$\displaystyle u_{k_{l}}^{m}$ $\displaystyle\rightarrow u^{m}\,\,\text{ weakly
star in }L^{\infty}(0,\infty;W_{0}^{1,p}(\Omega))\,\,\text{ and a.e.
}Q=\Omega\times[0,\infty)$ $\displaystyle u_{k_{l}}^{\prime}$
$\displaystyle\rightarrow u_{t}\,\,\text{ weakly in
}L^{\infty}(0,\infty;L^{2}(\Omega)),$ $\displaystyle f(u_{k_{l}})$
$\displaystyle\rightarrow f(u)\,\,\text{ weakly star in
}L^{\infty}(0,\infty;L^{q}(\Omega))\,\,\,\text{ and a.e.
}Q=\Omega\times[0,\infty).$

For fixed $s$ in (4.3) and taking $k=k_{l}\rightarrow\infty$ we get

$(u_{t},\omega_{s}^{m})+(|\nabla u^{m}|^{p-2}\nabla
u^{m},\nabla\omega_{s}^{m})\rangle dx=(f(u),\omega_{s}^{m}),\,\,\text{ for all
}\,\,s,$

and

$\int_{\Omega}u_{t}vdx+\int_{\Omega}|\nabla u^{m}|^{p-2}\nabla
u^{m}\cdot\nabla vdx=\int_{\Omega}f(u)vdx,\,\,\,\forall v\in
W_{0}^{1,p}(\Omega),\,\,t\in[0,T),$ $u^{m}(x,0)=u_{0}^{m}(x)\,\,\,\text{ in
}\,\,W_{0}^{1,p}(\Omega).$

Therefore $u(x,t)$ is a global weak solution of problem (1.1). ∎

###  4.3. Asymptotic behavior of solutions

Since Theorem 4.3 provides the existence of global weak solutions of initial-
boundary problem (1.1), we shall consider the asymptotic behavior of solutions
for problem (1.1).

######  Theorem 4.4.

Let $f(u)$ satisfy $(H)$ and $u_{0}^{m}(x)\in W_{0}^{1,p}(\Omega)$. Suppose
that $0<J(u_{0})<d$ and $I(u_{0})>0$.

(i):

If $pm>m+1,$ then for a global weak solution $u(x,t)$ of problem (1.1), there
exists a positive constant $\lambda$ such that

$||u(t)||_{L^{m+1}(\Omega)}\leq(||u_{0}||_{L^{m+1}(\Omega)}^{m+1-pm}+\lambda
t)^{-\frac{1}{pm-m-1}},\,0\leq t<\infty.$

(ii):

If $pm=m+1,$ then the global weak solution $u(x,t)$ of problem (1.1) satisfies
the estimate

$||u(t)||_{L^{m+1}(\Omega)}\leq
e^{-(m+1)(1-\delta_{1})Ct}||u_{0}||_{L^{m+1}(\Omega)},\,C>0,\,0\leq t<\infty.$

###### Proof of Theorem 4.4.

(i) Let $u(t)$ be any global weak solution of initial-boundary problem (1.1)
with the initial conditions $J(u_{0})<d$ and $I(u_{0})>0$. Let us multiply
(4.2) by $d(t)\in[0,\infty)$, then we have

$\displaystyle(u_{t},d(t)v)dx+(|\nabla u^{m}|^{p-2}\nabla u^{m},\nabla
d(t)v)=(f(u),d(t)v),$

for all $v(x)\in W_{0}^{1,p}(\Omega)$ and $d(t)\in[0,\infty)$ and write

$\int_{\Omega}u_{t}wdx+\int_{\Omega}|\nabla u^{m}|^{p-2}\nabla
u^{m}\cdot\nabla wdx=\int_{\Omega}f(u)wdx,\,\,\,\forall w\in
L^{\infty}(0,\infty;W_{0}^{1,p}(\Omega)).$

Setting $w=u^{m}$, we arrive at

$\int_{\Omega}u_{t}u^{m}dx+\int_{\Omega}|\nabla
u^{m}|^{p}dx=\int_{\Omega}u^{m}f(u)dx,$

and it can be rewritten such as

$\frac{1}{m+1}\frac{d}{dt}\int_{\Omega}u^{m+1}dx+I(u)=0.$

Theorem 4.1 along with

$0<J(u_{0})<d\,\,\,\,\text{and}\,\,\,\,I(u_{0})>0$

imply that $u(t)\in W_{\delta}$ for $\delta\in(\delta_{1},\delta_{2})$ and
$0\leq t<\infty$, where $\delta_{1}<\delta_{2}$ are the two roots of equation
$d(\delta)=J(u_{0})$. Hence, we have

$I_{\delta}(u)\geq 0\,\,\,\text{for}\,\,\,\delta\in(\delta_{1},\delta_{2})$

and

$I_{\delta_{1}}(u)\geq 0\,\,\,\text{for}\,\,\,0\leq t<\infty.$

Then we have

$\displaystyle\frac{1}{m+1}\frac{d}{dt}\int_{\Omega}u^{m+1}dx+(1-\delta_{1})\int_{\Omega}|\nabla
u^{m}|^{p}dx+I_{\delta_{1}}(u)=0,$

and

$\displaystyle\frac{1}{m+1}\frac{d}{dt}\int_{\Omega}u^{m+1}dx+(1-\delta_{1})\int_{\Omega}|\nabla
u^{m}|^{p}dx\leq 0.$

By the Sobolev inequality

$\int_{\Omega}|\nabla u^{m}|^{p}dx\geq
C_{*}^{p}\left(\int_{\Omega}|u^{m}|^{p^{*}}dx\right)^{\frac{pm}{p^{*}m}},\,p^{*}=\frac{np}{n-p},$

and embedding for $mp^{*}>m+1$

$||u||_{L^{mp^{*}}(\Omega)}\geq C||u||_{L^{m+1}(\Omega)},$

we obtain the following estimate

$\int_{\Omega}|\nabla u^{m}|^{p}dx\geq
C_{*}^{p}C\left(\int_{\Omega}u^{m+1}dx\right)^{\frac{pm}{m+1}}.$

That gives

$\frac{1}{m+1}\frac{d}{dt}\int_{\Omega}u^{m+1}dx+(1-\delta_{1})C_{*}^{p}C\left(\int_{\Omega}u^{m+1}dx\right)^{\frac{pm}{m+1}}\leq
0.$ (4.7)

Therefore, there exists a constant $\lambda>0$ such that

$\int_{\Omega}u^{m+1}(t)dx\leq\left(\left[\int_{\Omega}u^{m+1}_{0}dx\right]^{1-s}+\lambda
t\right)^{-\frac{1}{s-1}},$

where $s=\frac{pm}{m+1}$ with $pm>m+1$ and

$\lambda:=(m+1)(s-1)(1-\delta_{1})C_{*}^{p}C>0.$

The case (ii) immediately follows from the inequality (4.7), if we replace
$pm$ by $m+1.$

This completes the proof. ∎

###  4.4. Finite time blow-up for $J(u_{0})<d$

######  Theorem 4.5.

Let $f(u)$ satisfy $(H)$ and $u_{0}^{m}(x)\in W_{0}^{1,p}(\Omega)$. Suppose
that $J(u_{0})<d$ and $I(u_{0})<0$. Then the solution of problem (1.1) blows
up in a finite time, there exists time $T^{*}>0$ such that

$\lim_{t\rightarrow T^{*}}\int_{0}^{t}\int_{\Omega}u^{m+1}dxd\tau=+\infty.$
(4.8)

###### Proof of Theorem 4.5.

The strategy of proof is to show the differential inequality

$M^{\prime\prime}(t)M(t)-(1+\delta)(M^{\prime}(t))^{2}>0,$ (4.9)

where

$M(t)=\int_{0}^{t}\int_{\Omega}u^{m+1}dxd\tau.$ (4.10)

We can see that the above expression for $t\geq 0$ implies

$\frac{d}{dt}\left[\frac{M^{\prime}(t)}{M^{\delta+1}(t)}\right]>0,$

hence

$\begin{cases}M^{\prime}(t)\geq\left[\frac{M^{\prime}(0)}{M^{\delta+1}(0)}\right]M^{1+\delta}(t),\\\
M(0)>0.\end{cases}$

Then we arrive at

$M(t)\geq\left(\frac{1}{M^{\delta}(0)}-\frac{\alpha\int_{\Omega}u^{m+1}_{0}(x)dx}{M^{\delta+1}(0)}t\right)^{-\frac{1}{\delta}}.$

Then the blow-up time $T^{*}$ satisfies

$0<T^{*}\leq\frac{M(0)}{\delta\int_{\Omega}u_{0}^{m+1}dx}.$

In order to prove (4.9), we need the following computations

$\displaystyle M^{\prime}(t)$ $\displaystyle=\int_{\Omega}u^{m+1}dx$

and

$\displaystyle M^{\prime\prime}(t)$
$\displaystyle=(m+1)\int_{\Omega}[u^{m}f(u)-|\nabla u^{m}|^{p}]dx=-(m+1)I(u).$

Also, we have

$\displaystyle M^{\prime\prime}(t)$
$\displaystyle=(m+1)\int_{\Omega}u^{m}(x,t)u_{t}(x,t)dx$
$\displaystyle=(m+1)\int_{\Omega}u^{m}(x,t)\Delta_{p}u^{m}(x,t)+(m+1)\int_{\Omega}u^{m}(x,t)f(u(x,t))dx$
$\displaystyle=-(m+1)\int_{\Omega}|\nabla
u^{m}(x,t)|^{p}dx+(m+1)\int_{\Omega}u^{m}(x,t)f(u(x,t))dx$
$\displaystyle\geq-(m+1)\int_{\Omega}|\nabla
u^{m}(x,t)|^{p}dx+(m+1)\int_{\Omega}\left[\alpha F(u(x,t))-\beta
u^{pm}(x,t)-\alpha\sigma\right]dx$
$\displaystyle=-\alpha(m+1)\left[\frac{1}{pm}\int_{\Omega}|\nabla
u^{m}(x,t)|^{p}dx-\int_{\Omega}(F(u(x,t))-\sigma)dx\right]$
$\displaystyle+\left(\frac{\alpha(m+1)}{pm}-m-1\right)\int_{\Omega}|\nabla
u^{m}(x,t)|^{p}dx-\beta(m+1)\int_{\Omega}u^{pm}(x,t)dx$
$\displaystyle\geq-\alpha(m+1)J(u)+\left[\lambda_{1,p}\frac{(\alpha-
pm)(m+1)}{pm}\ -\beta(m+1)\right]\int_{\Omega}u^{pm}(x,t)dx$
$\displaystyle\geq-\alpha(m+1)J(u_{0})+\alpha(m+1)\int_{0}^{t}\int_{\Omega}u^{m-1}(u_{\tau})^{2}dxd\tau$
$\displaystyle+\left[\lambda_{1,p}\frac{(\alpha-pm)(m+1)}{pm}\
-\beta(m+1)\right]\int_{\Omega}u^{pm}dx,$

and

$\displaystyle(M^{\prime}(t))^{2}$
$\displaystyle\leq(m+1)^{2}(1+\delta)\left(\int_{0}^{t}\int_{\Omega}u^{m+1}dxd\tau\right)\left(\int_{0}^{t}\int_{\Omega}u^{m-1}u_{\tau}^{2}(x,\tau)dxd\tau\right)$
$\displaystyle+\left(1+\frac{1}{\delta}\right)\left(\int_{\Omega}u_{0}^{m+1}(x)dx\right)^{2}\,\,\,\text{
for }\delta>0.$

Then we write

$\displaystyle
M^{\prime\prime}(t)M(t)-(1+\delta)(M^{\prime}(t))^{2}\geq-\alpha(m+1)M(t)J(u_{0})+C_{1}M(t)\int_{\Omega}u^{pm}dx$
$\displaystyle+\alpha(m+1)\int_{0}^{t}\int_{\Omega}u^{m-1}(u_{\tau})^{2}dxd\tau\int_{0}^{t}\int_{\Omega}u^{m+1}dxd\tau$
$\displaystyle-(m+1)^{2}(1+\delta)^{2}\left(\int_{0}^{t}\int_{\Omega}u^{m+1}dxd\tau\right)\left(\int_{0}^{t}\int_{\Omega}u^{m-1}u_{\tau}^{2}(x,\tau)dxd\tau\right)$
$\displaystyle-(1+\alpha)\left(1+\frac{1}{\delta}\right)\left(\int_{\Omega}u_{0}^{m+1}(x)dx\right)^{2},$

where $C_{1}:=\lambda_{1,p}\frac{(\alpha-pm)(m+1)}{pm}\ -\beta(m+1)$.
Furthermore, by taking $\delta=\sqrt{\frac{\alpha}{m+1}}-1>0$ the above
expression yields

$\displaystyle M^{\prime\prime}(t)M(t)$
$\displaystyle-(1+\alpha)(M^{\prime}(t))^{2}\geq
M(t)\left[\frac{C_{1}}{2}\int_{\Omega}u^{pm}dx-\alpha(m+1)J(u_{0})\right]$
$\displaystyle+\frac{C_{1}}{2}M(t)\int_{\Omega}u^{pm}dx-(1+\alpha)\left(1+\frac{1}{\delta}\right)\left(\int_{\Omega}u_{0}^{m+1}(x)dx\right)^{2}.$

Notice that $\int_{\Omega}u^{pm}(t)dx$ is a convex function of $t$, it follows
that if there exists a time $t_{1}$ such that
$pm\int_{\Omega}u^{pm-1}(t_{1})u^{\prime}(t_{1})dx>0$, then
$\int_{\Omega}u^{pm}(t)dx$ is increasing for all $t>t_{1}$. For sufficiently
large time $t$ the following quantity

$\frac{C}{2}\int_{\Omega}u^{pm}dx>\alpha(m+1)J(u_{0})$

becomes positive for $J(u_{0})<0$ and $0<J(u_{0})<d$. Also, for sufficiently
large time $t$ we have

$\frac{C_{1}}{2}M(t)\int_{\Omega}u^{pm}dx>(1+\alpha)\left(1+\frac{1}{\delta}\right)\left(\int_{\Omega}u_{0}^{m+1}(x)dx\right)^{2}.$

∎

##  5\. Critical initial energy $J(u_{0})=d$

###  5.1. Global existence for $J(u_{0})=d$

######  Theorem 5.1.

(Global existence for $J(u_{0})=d$). Let $f$ satisfy $(H)$, $u_{0}^{m}\in
W_{0}^{1,p}(\Omega)$. Assume that $J(u_{0})=d$ and $I(u_{0})\geq 0$. The
initial-boundary problem (1.1) admits a global weak solution $u^{m}(t)\in
L^{\infty}(0,\infty;W_{0}^{1,p}(\Omega))$ along with $u_{t}(t)\in
L^{\infty}(0,\infty;L^{2}(\Omega))$ and $u(t)\in\overline{W}=W\cup\partial W$
for $t\in[0,\infty)$.

###### Proof of Theorem 5.1.

Note that $J(u_{0})=d$ implies $||\nabla u_{0}^{m}||_{L^{p}(\Omega)}\neq 0$.
Let us pick a sequence $\\{\epsilon_{k}\\}_{k=1}^{\infty}$ such that
$0<\epsilon_{k}<1$ for $k=1,\ldots$ and $\epsilon_{k}\rightarrow 1$ as
$k\rightarrow\infty$. We consider the following initial-boundary problem

$\displaystyle\begin{cases}u_{t}-\nabla\cdot(|\nabla u^{m}|^{p-2}\nabla
u^{m})=f(u),\,\,\,&(x,t)\in\Omega\times(0,+\infty),\\\
u(x,t)=0,\,\,\,&(x,t)\in\partial\Omega\times[0,+\infty),\\\
u(x,0)=u_{0k}(x)\geq 0,\,\,\,&x\in\overline{\Omega},\end{cases}$ (5.1)

where $u_{0k}(x)=\epsilon_{k}u_{0}(x)$ for $k=1,2,\ldots$. From $I(u_{0})\geq
0$ and Lemma 3.3, we have $\epsilon^{*}=\epsilon^{*}(u_{0})\geq 1$. This gives

$I(u_{0k})=I(\epsilon_{k}u_{0})>0$

and

$J(u_{0k})=J(\epsilon_{k}u_{0})<J(u_{0})=d.$

From Theorem 4.3 we know that for each $k$ problem 5.1 admits a global weak
solution $u_{k}(t)\in L^{\infty}(0,\infty;W_{0}^{1,p}(\Omega))$ with
$u_{k}^{\prime}(t)\in L^{\infty}(0,\infty;L^{2}(\Omega))$ and $u_{k}(t)\in W$
for $t\in[0,\infty)$ satisfying

$(u_{k}\,v)+(|\nabla u^{m}_{k}|^{p-2}\nabla u_{k}^{m},\nabla
v)=(f(u_{k}),v),\,\,\,\,\forall v\in W_{0}^{1,p}(\Omega),t>0,$

and

$\int_{0}^{t}([u_{k}^{\prime}(\tau)]^{2},u^{m-1}_{k}(\tau))d\tau+J(u_{k})\leq
J(u_{0k})<d,\,\,\,t\in[0,\infty).$

Then we establish

$\displaystyle\int_{0}^{t}([u_{k}^{\prime}(\tau)]^{2},u^{m-1}_{k}(\tau))d\tau+\left[\frac{1}{pm}-\frac{1}{\alpha}-\frac{\beta}{\lambda_{1,p}\alpha}\right]||\nabla
u^{m}_{k}||^{p}_{L^{p}(\Omega)}<d,\,\,\,t\in[0,\infty)$ (5.2)

for sufficiently large $k$. The remaining steps of the proof is the same way
as in Theorem 4.3. ∎

###  5.2. Asymptotic behavior of solutions

Since Theorem 5.1 provides the existence of global weak solutions of initial-
boundary problem (1.1) for critical energy cindition $J(u_{0})=d$, we shall
consider the asymptotic behavior of solutions for problem (1.1).

######  Theorem 5.2.

Let $f(u)$ satisfy $(H)$ and $u_{0}^{m}(x)\in W_{0}^{1,p}(\Omega)$. Suppose
that $J(u_{0})=d$ and $I(u_{0})\geq 0$.

(i):

If $pm>m+1,$ then for a global weak solution $u(x,t)$ of problem (1.1), there
exists a positive constant $\lambda$ such that

$||u(t)||_{L^{m+1}(\Omega)}\leq(||u_{0}||_{L^{m+1}(\Omega)}^{m+1-pm}+\lambda
t)^{-\frac{1}{pm-m-1}},\,t_{1}\leq t<\infty,\,t_{1}>0.$

(ii):

If $pm=m+1,$ then the global weak solution $u(x,t)$ of problem (1.1) satisfies
the estimate

$||u(t)||_{L^{m+1}(\Omega)}\leq
e^{-(m+1)(1-\delta_{1})Ct}||u_{0}||_{L^{m+1}(\Omega)},\,C>0,\,t_{1}\leq
t<\infty.$

###### Proof of Theorem 5.2.

Recall that the existence of a global weak solution of problem (1.1) is given
in Theorem 5.1 with $J(u_{0})=d$ and $I(u_{0})>0$, which implies $I(u)\geq 0$
for $0\leq t<\infty$. Now we consider the following two cases:

$(a)$ Assume that $I(u)>0$ for $0\leq t<\infty$. From $(u_{t},u^{m})=-I(u)<0$
we have $\int_{\Omega}u_{t}^{2}u^{m-1}dx>0$, it follows that
$\int_{0}^{t}(u^{2}_{\tau},u^{m-1})d\tau$ is increasing for $0\leq t<\infty$.
Picking any $t_{1}>0$ and letting

$d_{1}=d-\int_{0}^{t_{1}}(u^{2}_{\tau},u^{m-1})d\tau.$

Since we have

$\displaystyle J(u)\leq
J(u_{0})-\int_{0}^{t_{1}}(u^{2}_{\tau},u^{m-1})d\tau=d_{1}<d,$

and $u^{m}(t)\in W_{\delta}$ for $\delta_{1}<\delta<\delta_{2}$ and $t_{1}\leq
t<\infty$, where $\delta_{1}<\delta_{2}$ are two roots of equation
$d(\delta)=d_{1}$. Then we have $I_{\delta_{1}}(u)\geq 0$ for $t\geq t_{1}$
which gives

$\displaystyle\frac{1}{m+1}\frac{d}{dt}\int_{\Omega}u^{m+1}dx+(1-\delta_{1})\int_{\Omega}|\nabla
u^{m}|^{p}dx+I_{\delta_{1}}(u)=0,$

and

$\displaystyle\frac{1}{m+1}\frac{d}{dt}\int_{\Omega}u^{m+1}dx+(1-\delta_{1})\int_{\Omega}|\nabla
u^{m}|^{p}dx\leq 0.$

for $t_{1}\leq t<\infty$. By making use of following estimate

$\int_{\Omega}|\nabla u^{m}|^{p}dx\geq
C_{*}^{p}C\left(\int_{\Omega}u^{m+1}dx\right)^{\frac{pm}{m+1}},$

we have

$\displaystyle\frac{1}{m+1}\frac{d}{dt}\int_{\Omega}u^{m+1}dx+(1-\delta_{1})C_{*}^{p}C\left(\int_{\Omega}u^{m+1}dx\right)^{\frac{pm}{m+1}}\leq
0.$

Therefore, for $t_{1}\leq t<\infty$, there exists a constant $\lambda>0$ such
that

$\int_{\Omega}u^{m+1}(t)dx\leq\left(\left[\int_{\Omega}u^{m+1}_{0}dx\right]^{1-s}+\lambda
t\right)^{-\frac{1}{s-1}},$

where $s=\frac{pm}{m+1}$ with $pm>m+1$ and

$\lambda:=(m+1)(s-1)(1-\delta_{1})C_{*}^{p}C>0.$

$(b)$ Assume that there exists $t_{1}>0$ such that $I(u(t_{1}))=0$ and
$I(u(t))>0$ for $0\leq t<t_{1}$. Then
$\int_{0}^{t}(u^{2}_{\tau},u^{m-1})d\tau$ is increasing for $0\leq t<t_{1}$.
Then we have

$J(u(t_{1}))\leq d-\int_{0}^{t_{1}}(u^{2}_{\tau},u^{m-1})d\tau<d,$

it follows that $||\nabla u^{m}||_{L^{p}(\Omega)}=0$. Then we get

$\displaystyle\frac{1}{m+1}\frac{d}{dt}\int_{\Omega}u^{m+1}dx+(1-\delta_{1})\int_{\Omega}|\nabla
u^{m}|^{p}dx\leq 0,$

for $t_{1}\leq t<\infty$. The rest of proof is same as the previous case. The
case (ii) can be proved similarly. This completes the proof of Theorem 5.2. ∎

###  5.3. Finite time blow-up for $J(u_{0})=d$

######  Theorem 5.3.

Let $f(u)$ satisfy $(H)$ and $u_{0}^{m}(x)\in W_{0}^{1,p}(\Omega)$. Suppose
that $J(u_{0})=d$ and $I(u_{0})<0$. Then the solution of problem (1.1) blows
up in a finite time, there exists time $T>0$ such that

$\lim_{t\rightarrow T}\int_{0}^{t}\int_{\Omega}u^{m+1}dxd\tau=+\infty.$ (5.3)

###### Proof of Theorem 5.3.

Let

$M(t)=\int_{0}^{t}\int_{\Omega}u^{m+1}dxd\tau.$ (5.4)

Then as in the case $J(u_{0})<d$ we have

$\displaystyle M^{\prime\prime}(t)M(t)$
$\displaystyle-(1+\alpha)(M^{\prime}(t))^{2}\geq
M(t)\left[\frac{C_{1}}{2}\int_{\Omega}u^{pm}dx-\alpha(m+1)J(u_{0})\right]$
$\displaystyle+\frac{C_{1}}{2}M(t)\int_{\Omega}u^{pm}dx-(1+\alpha)\left(1+\frac{1}{\delta}\right)\left(\int_{\Omega}u_{0}^{m+1}(x)dx\right)^{2},$

where

$C_{1}:=\lambda_{1,p}\frac{(\alpha-pm)(m+1)}{pm}\ -\beta(m+1)$

and

$\delta=\sqrt{\frac{\alpha}{m+1}}-1>0.$

Since $\int_{\Omega}u^{pm}(t)dx$ is a convex function of $t$, it follows that
if there exists a time $t_{1}$ such that

$pm\int_{\Omega}u^{pm-1}(t_{1})u^{\prime}(t_{1})dx>0,$

then $\int_{\Omega}u^{pm}(t)dx$ is increasing for all $t>t_{1}$. For
sufficiently large time $t$ the following quantity

$\frac{C}{2}\int_{\Omega}u^{pm}dx>\alpha(m+1)d,$

and

$\frac{C_{1}}{2}M(t)\int_{\Omega}u^{pm}dx>(1+\alpha)\left(1+\frac{1}{\delta}\right)\left(\int_{\Omega}u_{0}^{m+1}(x)dx\right)^{2}.$

That gives

$M^{\prime\prime}(t)M(t)-(1+\alpha)(M^{\prime}(t))^{2}>0.$

The rest of proof is well-known. We finish the proof of Theorem 5.3. ∎

## Conclusion and open questions

In this paper, we obtained some results on global existence, asymptotic
behavior and blow-up of solutions for the porous medium equation (1.1) under
condition $0<J(u_{0})\leq d$ or $J(u_{0})<0$. For convenience, these cases are
given in the table below

| ”+” global existence | ”-” blow-up
---|---|---
$J(u_{0})<0$ | - | -
$0<J(u_{0})<d$ | + | -
$J(u_{0})=d$ | + | -
$d<J(u_{0})$ | ? | ?
| $I(u_{0})\geq 0$ | $I(u_{0})<0$

It is easy to notice from the above table that in supercritical initial energy
case $d<J(u_{0})$ the problem (1.1) has not yet been investigated. It should
be noted that such cases were studied in [32] for the pseudoparabolic equation
with power nonlinearity.

All results for this paper were obtained for the porous medium equation (1.1)
with $m\geq 1$. That is, the fast diffusion equation case with $0<m<1$ is
still open.

The above reasoning allows us to state with confidence about the future
perspective of the problem (1.1).

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1963; Furth & Rosenbluth, 1964).

Using the expressions for $B_{2}$ at the bottom of the mirror, Eqs. (81), one
may show that the maximum-$J$ condition is equivalent to

$\displaystyle X_{1c}^{\prime\prime}X_{1c}>0,$ (86a)
$\displaystyle\left(\frac{X_{1c}^{\prime}}{X_{1c}^{2}}\right)^{2}+(X_{1s}^{\prime})^{2}-\frac{B_{0}^{\prime\prime}}{2B_{0}X_{1c}^{2}}>\frac{X_{1c}X_{1c}^{\prime\prime}}{2}\left[\frac{1}{X_{1c}^{4}}+\left(\frac{X_{1s}^{\prime\prime}}{X_{1c}^{\prime\prime}}\right)^{2}\right].$
(86b)

These two conditions may qualitatively be interpreted as

$\displaystyle\text{Elongation~{}increase}\neq 0,$
$\displaystyle\mathrm{Twist}^{2}~{}-~{}\frac{\mathrm{Mirror~{}ratio}}{\mathrm{Well~{}width}^{2}}>\mathrm{Elongation~{}increase}.$

Eq. (86a) is a statement about the elongation of flux surfaces having to grow
away from the middle of the mirror, where $X_{1c}^{2}$ is the elongation of
the elliptical cross-section in the $x$-direction. We are free to take
$X_{1c}>1$ to make this direction coincide with the major axis of the ellipse.
Then $X_{1c}^{\prime\prime}>0$ can be interpreted as a condition for growing
elongation away from the minimum (in both directions). This property indeed
constitutes a typical feature of optimised magnetic mirrors (see Fig. 11).

Equation (86b) is more interesting than Eq. (86a), as it involves more aspects
of the field. Maximum-$J$ behaviour can only be attained if there is a
sufficiently large ‘twist’ of field lines across the bottom of the magnetic
well. By ‘twist’ we refer to the contribution from $X_{1c}^{\prime}$ and
$X_{1s}^{\prime}$, which describes a left-right symmetry breaking of the
field. This ‘local shearing’ of field lines and lack of symmetry about the
minimum of the configuration is common in mirror designs. The twist of the
field must be strong enough to overcome the effect of elongation and the
natural tendency for a negative radial magnetic field derivative, indicated by
the $B_{0}^{\prime\prime}$ term.

All in all, strong twist and large elongation are helpful for maximum-$J$
behaviour, but they must be balanced in a careful way. Making the minimum flat
(reducing $B_{0}^{\prime\prime}$) is also helpful. These features are visible
in typically shaped mirrors such as the example shown in Fig. 11.

###  C.4 Omnigeneity in a magnetic mirror

Magnetic mirrors can be made to be omnigenous by careful tailoring of the
quadrupole magnetic field (Hall & McNamara, 1975; Catto & Hazeltine, 1981).
What this means in an open ended device is that the surfaces of constant
$\psi$ correspond to precession surfaces for all trapped species in the sense
introduced by Hall & McNamara (1975). In order to characterise this property,
we measure of the ‘radial’ drift (normal to constant $\psi$ surfaces) by the
quantity $Y_{\mathrm{omn}}=\nabla\Phi\times\nabla
B\cdot\nabla\psi/\mathbf{B}\cdot\nabla B$, which plays a key role in the
analysis of stellarator omnigeneity and in our coordinates $(\psi,\alpha,z)$
can be written as

$Y_{\mathrm{omn}}=\partial_{\alpha}\Phi-\partial_{z}\Phi\frac{\partial_{\alpha}B}{\partial_{z}B}.$
(88)

We follow the approach of Rodríguez & Plunk (2023) to obtain a near-axis
expansion of the condition for omnigeneity, which requires that
$Y_{\mathrm{omn}}$ is equal and opposite at bounce points.

Expanding this condition in powers of $r$, we obtain
$Y_{\mathrm{omn}}^{(1)}=0$ to first order by virtue of the straight axis:
there is no leading curvature drift. It is only at the next order that there
is a non-zero radial drift, so we may write
$Y_{\mathrm{omn}}^{(2)}=Y_{\mathrm{omn,c}}^{(2)}\cos
2\alpha+Y_{\mathrm{omn,s}}^{(2)}\sin 2\alpha$. The expansion gives

$\displaystyle Y_{\mathrm{omn,c}}^{(2)}$
$\displaystyle=2\left(\Phi_{2s}-\frac{B_{0}}{B_{0}^{\prime}}B_{2s}\right)=-2\frac{B_{0}^{2}}{B_{0}^{\prime}}\left[\left(\frac{\Phi_{2s}}{B_{0}}\right)^{\prime}+V_{2}\right],$
(89a) $\displaystyle Y_{\mathrm{omn,s}}^{(2)}$
$\displaystyle=-2\left(\Phi_{2c}-\frac{B_{0}}{B_{0}^{\prime}}B_{2c}\right)=2\frac{B_{0}^{2}}{B_{0}^{\prime}}\left[\left(\frac{\Phi_{2c}}{B_{0}}\right)^{\prime}+V_{1}\right],$
(89b)

where,

$\displaystyle
V_{1}=\frac{1}{4}\left[(Y_{1s}^{\prime})^{2}+(X_{1s}^{\prime})^{2}-(Y_{1c}^{\prime})^{2}-(X_{1c}^{\prime})^{2}\right],$
(90a) $\displaystyle
V_{2}=-\frac{1}{2}\left(X_{1c}^{\prime}X_{1s}^{\prime}+Y_{1c}^{\prime}Y_{1s}^{\prime}\right),$
(90b)

and we have made use of Eq. (80). For omnigeneity, the expressions for
$Y_{\mathrm{omn,c}}^{(2)}$ and $Y_{\mathrm{omn,s}}^{(2)}$ must be even in $z$
(provided $B_{0}$ is symmetric about its minimum). This imposes constraints on
the choice of $X_{1c}$ and $X_{1s}$.

It is natural to ask whether this condition is at all compatible with the
maximum-$J$ requirements devived above. In other words, is it possible to
design an omnigenous, maximum-$J$ magnetic mirror? Let us consider omnigeneity
at the bottom of the mirror, in which the condition reduces to that of locally
vanishing radial drift. Evaluating the square brackets in Eqs. (89) at the
minimum, and requiring them to vanish (assuming a first order zero of
$B_{0}^{\prime\prime}\neq 0$),

$\displaystyle X_{1s}^{\prime\prime}=0,$ (91a)
$\displaystyle\left(\frac{X_{1c}^{\prime}}{X_{1c}^{2}}\right)^{2}+(X_{1s}^{\prime})^{2}=\frac{B_{0}^{\prime\prime}}{2B_{0}X_{1c}^{2}}+\frac{X_{1c}X_{1c}^{\prime\prime}}{2}\left(1+\frac{1}{X_{1c}^{4}}\right),$
(91b)

which, incidentally, implies $B_{20}/B_{0}=X_{1c}^{\prime\prime}X_{1c}/2$ and
$B_{2c}=0=B_{2s}$ at the minimum. If we choose $X_{1c}^{\prime\prime}>0$ to
comply by the first of the two maximum-$J$ conditions, Eq. (86a), the second
maximum-$J$ condition, Eq. (86b), is satisfied by satisfying the condition of
omnigeneity, Eq. (91b). This suggests that it is possible to construct a
mirror that is omnigeneous and satisfies the maximum-$J$ condition
simultaneously, at least at the minimum. Indeed, Catto & Hazeltine (1981) have
shown that it is possible in a finite region around the minimum.

###  C.5 Example of approximately omnigeneneous and maximum-$J$ mirror

To venture beyond the bottom of the well, we present a numerical example in
which the maximum-$J$ and omnigeneity conditions are both satisfied over a
non-zero interval in $z$. To construct such an example, we choose for
simplicity a symmetric solution with well defined parity (so that $X_{1s}$ and
$X_{1c}$ are odd and even respectively). This guarantees the correct behaviour
of Eq. (89b), but leaves us free to find inputs such that
$V_{1}+(\Phi_{2c}/B_{0})^{\prime}=0$. This task can be formulated as an
optimisation problem in which $X_{1s}$ and $X_{1c}$ are the degrees of freedom
(fixing $B_{0}$). We initialise the search with a configuration that satisfies
the omnigeneity requirement at the bottom of the well (as we know in closed
form what this choice should be). The result of the optimisation (with 6
scalar degrees of freedome) is presented in Fig. 11.

Figure 11: Example of an optimised ‘symmetric’ magnetic mirror. The figure
shows a 3D rendition of a near-axis magnetic mirror optimised for omnigeneity
and the maximum-$J$ property. The bottom plots illustrate the elongation of
flux surfaces, $\mathcal{E}$, and the normalised precession frequency (with
$\hat{\omega}_{\alpha}>0$ indicating maximum-$J$ behaviour) as a function of
$z$, where the black line represents the average over the field-line label
$\alpha$. The mirror was constructed assuming symmetry of $X_{1c}$ and
$X_{1s}$, and employs a simple quadratic magnetic field.

##  Appendix D Details of near-axis optimisation for a vacuum QI field with
the maximum-$J$ property

In this Appendix, we document details of the optimisation approach to the
near-axis field presented in Figure 8 of the main text. A more in-depth
discussion will be presented elsewhere.

The main goal of our optimisation effort is to construct a proof-of-concept
near-axis field that exhibits the maximum-$J$ property in the omnigenous
portion of the trapped-particle population. In that regard, we do not optimise
for the particles in the so-called buffer-region (Plunk et al., 2019), the
part of the domain where omnigeneity is violated in order to enforce
periodicity of the magnetic field. In general, narrow buffer regions lead to
large shaping gradients, and these regions are therefore often made relatively
wide although omnigeneity then suffers. As far as the maximum-$J$ property is
concerned, this will increase the number of particles with large ($\sim 1/r$)
detrimental precession. We shall however not be concerned with this detail but
argue that we could, in principle, shrink the buffer region to become
arbitrarily small.

Focusing, then, on the other contributions to the precession, we need to
introduce a measure for maximum-$J$ behaviour that serves as a target function
in the numerical optimisation. In order to exclude contributions from the
buffer regions proportional to $1/r$, we evaluate the precession frequency as
the sum of Eqs. (47) and (52), and construct the target function as the sum

$g_{\omega_{\alpha}}=\sum_{k}\hat{\omega}_{\alpha}^{2}(k),$

taken over values of $k$ such that $\hat{\omega}_{\alpha}<0$. The required
integrals are evaluated using the help of the BAD library in python
(Mackenbach et al., 2023b).

A second ingredient in the optimisation is the condition of omnigeneity
(outside the buffer regions). Given first-order inputs, second order choices
necessary to enforce the correct form for $B_{2c}$. Accordingly, we construct
$X_{2c}$ and $X_{2s}$ following Eqs. (61) and the equivalent for $X_{2s}$,
assuming that $B_{2c}$ has the correct form for QI. Writing
$X_{2c}=\tilde{X}_{2c}c_{2\alpha_{1}}+\tilde{X}_{2s}s_{2\alpha_{1}}$ and
$X_{2s}=\tilde{X}_{2c}s_{2\alpha_{1}}-\tilde{X}_{2s}c_{2\alpha_{1}}$, we find

$\displaystyle\tilde{X}_{2c}=\frac{1}{\kappa
B_{0}}\left[\hat{\mathcal{T}}_{c}c_{2\alpha_{1}}+\hat{\mathcal{T}}_{s}s_{2\alpha_{1}}+\frac{B_{0}^{2}}{4}\left(\frac{d^{2}}{B_{0}^{\prime}}\right)^{\prime}\right],$
(92a) $\displaystyle\tilde{X}_{2s}=\frac{1}{\kappa
B_{0}}\left[\hat{\mathcal{T}}_{c}s_{2\alpha_{1}}-\hat{\mathcal{T}}_{s}c_{2\alpha_{1}}-B_{2s}^{\mathrm{QI}}\right].$
(92b)

where $c_{2\alpha_{1}}=\cos 2\alpha_{1}$ and $s_{2\alpha_{1}}=\sin
2\alpha_{1}$, and

$\displaystyle\hat{\mathcal{T}}_{c}=\frac{B_{0}}{l^{\prime}}\left[Z_{2c}^{\prime}+2\iota_{0}Z_{2s}+\frac{q_{c}^{2}-q_{s}^{2}+r_{c}^{2}-r_{s}^{2}}{4l^{\prime}}\right],$
(93a)
$\displaystyle\hat{\mathcal{T}}_{s}=\frac{B_{0}}{l^{\prime}}\left[Z_{2s}^{\prime}-2\iota_{0}Z_{2c}+\frac{q_{c}q_{s}+r_{c}r_{s}}{2l^{\prime}}\right],$
(93b)

and $\alpha_{1}=\nu-\iota\varphi$ in the notation of Appendix A, following
Garren & Boozer (1991). This construction guarantees QI behaviour at second
order for any odd $B_{2s}^{\mathrm{QI}}$, as can be checked by substitution.
The only exceptions are points where the curvature vanishes, where we learnt
in Appendix B, Eq. (64), that the QI condition reduces to a condition on
lower-order coefficients. An indication of the latter is that the construction
will generally diverge wherever $\kappa=0$, from which more general
expressions than those of Eq. (64) can be obtained which do not assume QI at
first order (this is important for the tops of the well in practice). Thus,
one must add to the optimisation a measure of the deviation from QI at the
points of minimum and maximum field strength, which the optimiser should
attempt to reduce. We denote the sum of this ‘residual’ at all extrema along
the field by $\check{g}_{\mathrm{QI}}$. To simplify the available choices, we
further take $p_{2}=0$ and $\tilde{X}_{2s}=0$, which is not the most general
choice, and most likely not even the best, but suffices for our purposes.
Although this approach enforces QI at second order exactly, it does not appear
to be the best one in practice, as it leads to highly shaped configurations,
and unnecessarily enforces QI at second order in regions where it is already
violated at first order. However, it suffices for our purposes of proof-of-
principle construction, and we leave further refinements for future work.

The target function to be numerically is thus
$g=g_{\omega_{\alpha}}+\check{g}_{\mathrm{QI}}$, and we proceed to discuss the
degrees of freedom over which the optimisation is to be carried out. At first
order, there are many free functions, some of which will be fixed for
simplicity. The magnetic field strength is taken to vary along the axis as
$B_{0}=1+0.16\cos\varphi$, but the shape of the magnetic axis and the
elongation measure $\bar{d}$ are allowed to change. For the former, we vary
three even Fourier harmonics (three for each $Z$ and $R$ describing the axis
in cylindrical coordinates), a fourth one being chosen consistently to make
the curvature vanish at $\phi=0,~{}\pi$ (Rodriguez et al., 2022; Camacho Mata
et al., 2022). This restricts us to first order zeroes of curvature (Rodriguez
et al., 2022; Mata & Plunk, 2023). The function $\bar{d}$ is allowed to have 7
degrees of freedom through collocation points of a symmetric spline, symmetry
being necessary to preserve stellarator symmetry.

With the optimisation procedure defined this way, we finally need to specify a
starting point, which we take to be a generic QI axis with $N=1$ and
$\bar{d}=1$. Other details, such as the buffer region size etc. can be found
in the supplementary material, in which all the numerical tools are provided.
The optimisation is performed using the near-axis code pyQIC (Jorge et al.,
2023) with appropriate upgrades (use of splines, second order choice, etc.),
and the optimisation libraries of scipy (the algorithms Nelder-Mead and BFGS
are used, the latter as a refinement of the former). Broadly speaking, the
optimisation proceeds in two stages: (i) Nelder-Mead optimisation for QI and
maximum-$J$ behaviour at the minima and maxima of $B_{0}$; (ii) full
optimisation (on a few field lines) of the function $g$. Step (i) has the
benefit of only requiring a first-order near-axis solution, thus being quite
fast, while (ii) is slower as it needs to compute $\hat{\omega}_{\alpha}$.
Additional refinements are of course possible.

The results of the optimisation are presented in Figure 8, and all the
pertinent files and scripts are included in the Zenodo repository associated
with this paper.

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