© Wuhan University 2023

0 Introduction

Let $m\ge \mathrm{1}$, $p>\mathrm{1}$ and $q\in \left(\mathrm{1,2}\right)$ be three constants, $\mathrm{\Omega }\subset {\mathbb{R}}^N\left(N>p\right)$ be a bounded domain with smooth boundary $\partial \mathrm{\Omega }$, and $u_{\mathrm{0}}\left(x\right)$ be a bounded non-trivial function with ${\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\in W_{\mathrm{0}}^{\mathrm{1},p}\left(\mathrm{\Omega }\right)$. We focus our attention here on dealing with the global existence, extinction and non-extinction phenomenon of the following doubly nonlinear parabolic equation

$\{\begin{array}{l}{u}_t-\mathrm{d}\mathrm{i}\mathrm{v}\left({\left|\nabla \left({\left|u\right|}^{m-\mathrm{1}}u\right)\right|}^{p-\mathrm{2}}\nabla \left({\left|u\right|}^{m-\mathrm{1}}u\right)\right)=f\left(u\right),\left(x,t\right)\in \mathrm{\Omega }\times \left(\mathrm{0},+\mathrm{\infty }\right)\\ u\left(x,t\right)=\mathrm{0},\left(x,t\right)\in \partial \mathrm{\Omega }\times \left(\mathrm{0},+\mathrm{\infty }\right)\\ u\left(x,\mathrm{0}\right)=u_{\mathrm{0}}(x),x\in \mathrm{\Omega }\end{array}$(1)

with

$f\left(u\right)=\{\begin{array}{l}{\left|u\right|}^{q-\mathrm{2}}u\mathrm{l}\mathrm{o}\mathrm{g}\left|u\right|,\mathrm{i}\mathrm{f}u\ne \mathrm{0}\\ \mathrm{0},\mathrm{i}\mathrm{f}u=\mathrm{0}\end{array}$(2)

Nonlinear evolutionary problems with logarithmic nonlinearity like model (1) came from inflation cosmology, super symmetric field theories, quantum mechanics and nuclear physics^[1-6].

In the past few decades, many mathematical researchers have devoted themselves to investigating the doubly nonlinear parabolic equations, and obtained many meaningful results, such as local and global well-posedness, regularity, blow-up in a finite time and extinction singularity ^[7-12]. Especially, the authors ^[13-15] studied the problem

$\{\begin{array}{l}{u}_t-\mathrm{d}\mathrm{i}\mathrm{v}\left({\left|\nabla u^m\right|}^{p-\mathrm{2}}\nabla u^m\right)=\lambda u^q,\left(x,t\right)\in \mathrm{\Omega }\times \left(\mathrm{0},+\mathrm{\infty }\right)\\ u\left(x,t\right)=\mathrm{0},\left(x,t\right)\in \partial \mathrm{\Omega }\times \left(\mathrm{0},+\mathrm{\infty }\right)\\ u\left(x,\mathrm{0}\right)=u_{\mathrm{0}}\left(x\right),x\in \mathrm{\Omega }\end{array}$(3)

where $m>\mathrm{0}$, $p>\mathrm{1}$, $\lambda >\mathrm{0}$, $q>\mathrm{0}$ and $u_{\mathrm{0}}\left(x\right)$ is a non-negative bounded non-trivial function with $u_{\mathrm{0}}^m\left(x\right)\in W_{\mathrm{0}}^{\mathrm{1},p}\left(\mathrm{\Omega }\right)$. For the case $\mathrm{0}<m\left(p-\mathrm{1}\right)<\mathrm{1}$, the authors ^[14,15] proved that the critical blow-up and extinction exponents are $q_{bc}=\mathrm{1}$ and $q_{ec}=m\left(p-\mathrm{1}\right)$, respectively. Compared with the case $\mathrm{0}<m\left(p-\mathrm{1}\right)<\mathrm{1}$, the solution for the case $m\left(p-\mathrm{1}\right)\ge \mathrm{1}$ exhibits completely different properties ^[13]. On the one hand, for any $q>\mathrm{0}$, the non-trivial solution to problem (3) will never become extinct at a finite time. On the other hand, the critical blow-up exponent becomes $q_{bc}^{\mathrm{*}}=m\left(p-\mathrm{1}\right)$.

Recently, Le and Le ^[16,17] considered problem (1) with $m\left(p-\mathrm{1}\right)>\mathrm{1}$ and obtained the existence and non-existence results of the global weak solutions. Precisely speaking, they concluded that if $m\left(p-\mathrm{1}\right)>q-\mathrm{1}$, then for any ${\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\in W_{\mathrm{0}}^{\mathrm{1},p}\left(\mathrm{\Omega }\right)$, problem (1) admits a global solution; if $m\left(p-\mathrm{1}\right)\le q-\mathrm{1}$, then there exists a weak solution to problem (1) which is global provided that $u_{\mathrm{0}}\left(x\right)$ belongs to some specific stable sets, and the weak solution blows up in a finite time provided that $u_{\mathrm{0}}\left(x\right)$ belongs to some specific unstable sets.

According to our knowledge, there is no extinction result of the solution to problem (1) with $\mathrm{0}<m\left(p-\mathrm{1}\right)<\mathrm{1}$. Inspired by the above works, we naturally take the following two questions into consideration. Does the solution to problem (1) with $\mathrm{0}<m\left(p-\mathrm{1}\right)<\mathrm{1}$ exist globally under some certain conditions? If so, is there a critical extinction exponent of the global solutions? In fact, in this paper, we work with the following equivalent formulation of problem (1), obtained by changing variable $v={\left|u\right|}^{m-\mathrm{1}}u$,

$\{\begin{array}{l}{\left({\left|v\right|}^{\frac{\mathrm{1}-m}{m}}v\right)}_t-\mathrm{d}\mathrm{i}\mathrm{v}\left({\left|\nabla v\right|}^{p-\mathrm{2}}\nabla v\right)=f\left(v\right),\left(x,t\right)\in \mathrm{\Omega }\times \left(\mathrm{0},+\mathrm{\infty }\right)\\ v\left(x,t\right)=\mathrm{0},\left(x,t\right)\in \partial \mathrm{\Omega }\times \left(\mathrm{0},+\mathrm{\infty }\right)\\ v\left(x,\mathrm{0}\right)=v_{\mathrm{0}}={\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}},x\in \mathrm{\Omega }\end{array}$(4)

with

$f\left(v\right)=\{\begin{array}{l}\frac{\mathrm{1}}{m}{\left|v\right|}^{\frac{q-\mathrm{1}-m}{m}}v\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|,\mathrm{i}\mathrm{f}v\ne \mathrm{0}\\ \mathrm{0},\mathrm{i}\mathrm{f}v=\mathrm{0}\end{array}$(5)

It is clear that the first equation in problem (4) has degeneracy or singularity at the points where $v\left(x,t\right)=\mathrm{0}$ or $\left|\nabla v\left(x,t\right)\right|=\mathrm{0}$, and hence problem (4) might not have classical solution in general. We introduce the definition of the weak solution to problem (4) as follows.

Definition 1   Let $T>\mathrm{0}$. A measurable function $v\left(x,t\right)$ defined in $\mathrm{\Omega }\times \left[\mathrm{0},T\right)$ is called a weak solution to problem (4) if $v\in L^{\mathrm{\infty }}\left(\mathrm{0},T;W_{\mathrm{0}}^{\mathrm{1},p}\left(\mathrm{\Omega }\right)\right)$, ${\left({\left|v\right|}^{\frac{\mathrm{1}-m}{m}}v\right)}_t\in L^{\mathrm{2}}\left(\mathrm{0},T;L^{\mathrm{2}}\left(\mathrm{\Omega }\right)\right)$, $v\left(x,\mathrm{0}\right)={\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\in W_{\mathrm{0}}^{\mathrm{1},p}\left(\mathrm{\Omega }\right)$ and

$\underset{\mathrm{\Omega }}{\int }\left[{\left({\left|v\right|}^{\frac{\mathrm{1}-m}{m}}v\right)}_t\phi +{\left|\nabla v\right|}^{p-\mathrm{2}}\nabla v\cdot \nabla \phi \right]\mathrm{d}x=\frac{\mathrm{1}}{m}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{q-\mathrm{1}-m}{m}}v\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|\phi \mathrm{d}x$(6)

holds for $\mathrm{a}.\mathrm{e}.t\in \left(\mathrm{0},T\right)$ and any $\phi \in W_{\mathrm{0}}^{\mathrm{1},p}\left(\mathrm{\Omega }\right)$.

Similar to the proof of Theorem 3.3 ^[17], by Faedo-Galerkin method, we can prove the local existence result of the weak solution to problem (4). Now, we state the main results of this paper as follows.

Theorem 1   Assume that $\mathrm{0}<m\left(p-\mathrm{1}\right)<\mathrm{1}$. Then the weak solution $u\left(x,t\right)$ of problem (1) exists globally.

Theorem 2   Assume that $\mathrm{0}<m\left(p-\mathrm{1}\right)<q-\mathrm{1}<\mathrm{1}$. If

$\mathrm{m}\mathrm{a}\mathrm{x}\left\{{\left(\underset{\mathrm{\Omega }}{\int }{\left({\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\right)}_{+}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\theta },{\left(\underset{\mathrm{\Omega }}{\int }{\left({\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\right)}_{-}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\theta }\right\}\le \frac{{\kappa }_{\mathrm{6}}}{\mathrm{2}{\kappa }_{\mathrm{7}}}$(7)

with $\theta =\frac{q-\mathrm{1}-m\left(p-\mathrm{1}\right)+m\beta }{mpa+m+\mathrm{1}}$, $\beta \in \left(\mathrm{0},\frac{\mathrm{2}-q}{m}\right)$ and $a>\mathrm{m}\mathrm{a}\mathrm{x}\left\{-\frac{\mathrm{1}}{p},-\frac{\mathrm{1}}{mp},\frac{\left(N-p\right)\left(m+\mathrm{1}\right)-Npm}{m{p}^{\mathrm{2}}}\right\}$.Then the weak solution $u\left(x,t\right)$ to problem (1) will vanish in finite time, where ${\kappa }_{\mathrm{6}}$ and ${\kappa }_{\mathrm{7}}$ are two positive constants, given by (23) and (24), respectively.

Theorem 3   Assume that $\mathrm{0}<q-\mathrm{1}\le m\left(p-\mathrm{1}\right)<\mathrm{1}$. If

$\{\begin{array}{l}\underset{\mathrm{\Omega }}{\int }{\left|u_{\mathrm{0}}\right|}^{m+\mathrm{1}}\mathrm{d}x>\mathrm{0}\mathrm{a}\mathrm{n}\mathrm{d}E\left({\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\right)\le \mathrm{0},m\left(p-\mathrm{1}\right)=q-\mathrm{1}\\ \underset{\mathrm{\Omega }}{\int }{\left|u_{\mathrm{0}}\right|}^{m+\mathrm{1}}\mathrm{d}x>\mathrm{0}\mathrm{a}\mathrm{n}\mathrm{d}E\left({\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\right)<-\frac{m\left|\mathrm{\Omega }\right|\left[m\left(p-\mathrm{1}\right)-\left(q-\mathrm{1}\right)\right]}{ep{\left(q+m-\mathrm{1}\right)}^{\mathrm{2}}},m\left(p-\mathrm{1}\right)>q-\mathrm{1}\end{array}$(8)

Then the weak solution to problem (1) cannot vanish in finite time, where

$E\left({\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\right)=\frac{\mathrm{1}}{p}{\underset{\mathrm{\Omega }}{\int }\left|\nabla \left({\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\right)\right|}^p\mathrm{d}x-\frac{m}{q+m-\mathrm{1}}{\underset{\mathrm{\Omega }}{\int }\left|u_{\mathrm{0}}\right|}^{q+m-\mathrm{1}}\mathrm{l}\mathrm{o}\mathrm{g}\left|u_{\mathrm{0}}\right|\mathrm{d}x[+mq+m-\mathrm{12}\mathrm{\Omega }u\mathrm{0}q+m-\mathrm{1}dx]$(9)

Remark 1   From Theorems 2 and 3, we know that the critical extinction exponent of the global solutions to problem (1) is $q_{ec}=m\left(p-\mathrm{1}\right)+\mathrm{1}$.

1 Proofs of the Main Results

Proof of Theorem 1   Multiplying both sides of the first equality in (4) by $v\left(x,t\right)$ and integrating over $\mathrm{\Omega }$, one gets

$\frac{\mathrm{1}}{m+\mathrm{1}}\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x+\underset{\mathrm{\Omega }}{\int }{\left|\nabla v\right|}^p\mathrm{d}x=\frac{\mathrm{1}}{m}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+q-\mathrm{1}}{m}}\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|\mathrm{d}x$(10)

Remembering that $q\in \left(\mathrm{1,2}\right)$ and $m\ge \mathrm{1}$, we can select $\beta \in \left(\mathrm{0},\frac{\mathrm{2}-q}{m}\right)$ such that $q+m\beta \in \left(\mathrm{1,2}\right)$. For this chosen $\beta$, we know $\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|\le \frac{\mathrm{1}}{\mathrm{e}\beta }{\left|v\right|}^{\beta }$. Then, from (10), it holds that

$\frac{\mathrm{1}}{m+\mathrm{1}}\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x+\underset{\mathrm{\Omega }}{\int }{\left|\nabla v\right|}^p\mathrm{d}x\le \frac{\mathrm{1}}{\mathrm{e}m\beta }\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+q-\mathrm{1}}{m}+\beta }\mathrm{d}x$(11)

By using Hölder's inequality, (11) leads to

$\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x\le \frac{m+\mathrm{1}}{\mathrm{e}m\beta }\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+q-\mathrm{1}}{m}+\beta }\mathrm{d}x\le {\kappa }_{\mathrm{1}}{\left(\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{m\beta +m+q-\mathrm{1}}{m+\mathrm{1}}}$(12)

where ${\kappa }_{\mathrm{1}}=\frac{m+\mathrm{1}}{\mathrm{e}m\beta }{\left|\mathrm{\Omega }\right|}^{\frac{\mathrm{2}-q-m\beta }{m+\mathrm{1}}}$. By a simple calculation, we get

$\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x\le {\left[{\left(\underset{\mathrm{\Omega }}{\int }{\left|v_{\mathrm{0}}\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{\mathrm{2}-q-m\beta }{m+\mathrm{1}}}+\frac{\left(\mathrm{2}-q-m\beta \right){\kappa }_{\mathrm{1}}}{m+\mathrm{1}}t\right]}^{\frac{m+\mathrm{1}}{\mathrm{2}-q-m\beta }}$(13)

It follows from (13) and Hölder's inequality that

$\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m\beta +m+q-\mathrm{1}}{m}}\mathrm{d}x\le {\left|\mathrm{\Omega }\right|}^{\frac{\mathrm{2}-q-m\beta }{m+\mathrm{1}}}{\left(\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{m\beta +m+q-\mathrm{1}}{m+\mathrm{1}}}\le {\left|\mathrm{\Omega }\right|}^{\frac{\mathrm{2}-q-m\beta }{m+\mathrm{1}}}{\left[{\left(\underset{\mathrm{\Omega }}{\int }{\left|v_{\mathrm{0}}\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{\mathrm{2}-q-m\beta }{m+\mathrm{1}}}+\frac{\left(\mathrm{2}-q-m\beta \right){\kappa }_{\mathrm{1}}}{m+\mathrm{1}}t\right]}^{\frac{m\beta +m+q-\mathrm{1}}{\mathrm{2}-q-m\beta }}$(14)

On the other hand, multiplying both sides of the first equality in (4) by $v_t={\left({\left|u\right|}^{m-\mathrm{1}}u\right)}_t$ and integrating over $\mathrm{\Omega }$, then with the help of Hölder's inequality and Cauchy's inequality with $\epsilon$, we get

$\begin{array}{l}\frac{\mathrm{4}{m}^{\mathrm{2}}}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\underset{\mathrm{\Omega }}{\int }\left[{\left({\left|v\right|}^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_t\right]}^{\mathrm{2}}\mathrm{d}x+\frac{m}{p}\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{\Omega }}{\int }{\left|\nabla v\right|}^p\mathrm{d}x=\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+q-\mathrm{1}}{m}-\mathrm{2}}v\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|v_t\mathrm{d}x\\ =\underset{{\mathrm{\Omega }}_{\mathrm{1}}=\left\{x\in \mathrm{\Omega }:\left|v\right|\ge \mathrm{1}\right\}}{\int }{\left|v\right|}^{\frac{m+q-\mathrm{1}}{m}-\mathrm{2}}v\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|v_t\mathrm{d}x+\underset{{\mathrm{\Omega }}_{\mathrm{2}}=\left\{x\in \mathrm{\Omega }:\left|v\right|<\mathrm{1}\right\}}{\int }{\left|v\right|}^{\frac{m+q-\mathrm{1}}{m}-\mathrm{2}}v\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|v_t\mathrm{d}x\\ \le \frac{\mathrm{2}m}{\mathrm{e}\beta \left(m+\mathrm{1}\right)}\underset{{\mathrm{\Omega }}_{\mathrm{1}}}{\int }{\left|v\right|}^{\frac{\mathrm{2}m\beta +\mathrm{2}q+m-\mathrm{3}}{\mathrm{2}m}}\left|{\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_t\right|\mathrm{d}x+\frac{\mathrm{4}{m}^{\mathrm{2}}}{\mathrm{e}\left(m+\mathrm{1}\right)\left(m+\mathrm{2}q-\mathrm{3}\right)}\underset{{\mathrm{\Omega }}_{\mathrm{2}}}{\int }\left|{\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_t\right|\mathrm{d}x\\ \le \frac{m}{\mathrm{2}{\epsilon }_{\mathrm{1}}\mathrm{e}\beta \left(m+\mathrm{1}\right)}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{\mathrm{2}m\beta +\mathrm{2}q+m-\mathrm{3}}{\mathrm{2}m}}\mathrm{d}x+\frac{\mathrm{2}m{\epsilon }_{\mathrm{1}}}{\mathrm{e}\beta \left(m+\mathrm{1}\right)}{\underset{\mathrm{\Omega }}{\int }\left|{\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_t\right|}^{\mathrm{2}}\mathrm{d}x\\ +\frac{m^{\mathrm{2}}\left|\mathrm{\Omega }\right|}{\mathrm{e}{\epsilon }_{\mathrm{2}}\left(m+\mathrm{1}\right)\left(m+\mathrm{2}q-\mathrm{3}\right)}+\frac{\mathrm{4}{m}^{\mathrm{2}}{\epsilon }_{\mathrm{2}}}{\mathrm{e}\left(m+\mathrm{1}\right)\left(m+\mathrm{2}q-\mathrm{3}\right)}{\underset{\mathrm{\Omega }}{\int }\left|{\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_t\right|}^{\mathrm{2}}\mathrm{d}x\end{array}$(15)

If ${\epsilon }_{\mathrm{1}}$ and ${\epsilon }_{\mathrm{2}}$ are sufficiently small such that $\frac{{\epsilon }_{\mathrm{1}}}{\beta }+\frac{\mathrm{2}m{\epsilon }_{\mathrm{2}}}{m+\mathrm{2}q-\mathrm{3}}\le \frac{\mathrm{2}m\mathrm{e}}{m+\mathrm{1}}$, then from (15), it holds that

$\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{\Omega }}{\int }{\left|\nabla v\right|}^p\mathrm{d}x\le {\kappa }_{\mathrm{3}}+{\kappa }_{\mathrm{2}}{\left(\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m\beta +m+q-\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{\mathrm{2}m\beta +\mathrm{2}q+m-\mathrm{3}}{m\beta +m+q-\mathrm{1}}}$(16)

where ${\kappa }_{\mathrm{2}}=\frac{p}{\mathrm{2}\mathrm{e}\beta {\epsilon }_{\mathrm{1}}\left(m+\mathrm{1}\right)}{\left|\mathrm{\Omega }\right|}^{\frac{\mathrm{2}-q-m\beta }{m\beta +m+q-\mathrm{1}}}$ and ${\kappa }_{\mathrm{3}}=\frac{mp\left|\mathrm{\Omega }\right|}{\mathrm{e}{\epsilon }_{\mathrm{2}}\left(m+\mathrm{1}\right)\left(m+\mathrm{2}q-\mathrm{3}\right)}$. Combining (14) and (16) tells us that

$\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{\Omega }}{\int }{\left|\nabla v\right|}^p\mathrm{d}x\le {\kappa }_{\mathrm{3}}+{\kappa }_{\mathrm{4}}{\left[{\left(\underset{\mathrm{\Omega }}{\int }{\left|v_{\mathrm{0}}\right|}^{\frac{\mathrm{m}+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{\mathrm{2}-q-m\beta }{m+\mathrm{1}}}+\frac{\left(\mathrm{2}-q-m\beta \right){\kappa }_{\mathrm{1}}}{m+\mathrm{1}}t\right]}^{\frac{\mathrm{2}m\beta +\mathrm{2}q+m-\mathrm{3}}{\mathrm{2}-q-m\beta }}$(17)

where ${\kappa }_{\mathrm{4}}={\kappa }_{\mathrm{2}}{\left|\mathrm{\Omega }\right|}^{\frac{\left(\mathrm{2}-q-m\beta \right)\left(\mathrm{2}m\beta +\mathrm{2}q+m-\mathrm{3}\right)}{\left(m+\mathrm{1}\right)\left(m\beta +m+q-\mathrm{1}\right)}}$. Integrating (17), we arrive at

$\begin{array}{l}\underset{\mathrm{\Omega }}{\int }{\left|\nabla v\right|}^p\mathrm{d}x\le \underset{\mathrm{\Omega }}{\int }{\left|\nabla v_{\mathrm{0}}\right|}^p\mathrm{d}x-\frac{\left(m+\mathrm{1}\right){\kappa }_{\mathrm{4}}}{{\kappa }_{\mathrm{1}}\left(m\beta +q+m-\mathrm{1}\right)}{\left(\underset{\mathrm{\Omega }}{\int }{\left|v_{\mathrm{0}}\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{m\beta +q+m-\mathrm{1}}{m+\mathrm{1}}}+{\kappa }_{\mathrm{3}}t\\ +\frac{\left(m+\mathrm{1}\right){\kappa }_{\mathrm{4}}}{{\kappa }_{\mathrm{1}}\left(m\beta +q+m-\mathrm{1}\right)}{\left({\left(\underset{\mathrm{\Omega }}{\int }{\left|v_{\mathrm{0}}\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{\mathrm{2}-q-m\beta }{m+\mathrm{1}}}+\frac{\left(\mathrm{2}-q-m\beta \right){\kappa }_{\mathrm{1}}}{m+\mathrm{1}}t\right)}^{\frac{m\beta +q+m-\mathrm{1}}{\mathrm{2}-q-m\beta }}\end{array}$(18)

which implies that $\underset{\mathrm{\Omega }}{\int }{\left|\nabla v\right|}^p\mathrm{d}x=\underset{\mathrm{\Omega }}{\int }{\left|\nabla \left({\left|u\right|}^{m-\mathrm{1}}u\right)\right|}^p\mathrm{d}x$ is bounded for all $t\in \left[\mathrm{0},+\mathrm{\infty }\right)$. The proof of Theorem 1 is complete.

Proof of Theorem 2   Multiplying both sides of the first equality in (4) by $\phi ={\left|v\right|}^{pa}{v}_{+}=$${\left|u\right|}^{mpa}{\left({\left|u\right|}^{m-\mathrm{1}}u\right)}_{+}$, we find that

$\frac{\mathrm{1}}{mpa+m+\mathrm{1}}\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{\Omega }}{\int }{v}_{+}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x+\frac{pa+\mathrm{1}}{{\left(a+\mathrm{1}\right)}^p}\underset{\mathrm{\Omega }}{\int }{\left|\nabla v_{+}^{a+\mathrm{1}}\right|}^p\mathrm{d}x=\frac{\mathrm{1}}{m}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{q-\mathrm{1}-m}{m}}v{\left|v\right|}^{pa}{v}_{+}\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|\mathrm{d}x\le \frac{\mathrm{1}}{m\mathrm{e}\beta }\underset{\mathrm{\Omega }}{\int }{v}_{+}^{\frac{m\left(pa+\beta +\mathrm{1}\right)+q-\mathrm{1}}{m}}\mathrm{d}x$(19)

By virtue of Hölder's inequality and Sobolev embedding inequality, we have

$\begin{array}{l}\underset{\mathrm{\Omega }}{\int }{v}_{+}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x\le {\left|\mathrm{\Omega }\right|}^{\mathrm{1}-\frac{\left(N-p\right)\left(mpa+m+\mathrm{1}\right)}{Npm\left(a+\mathrm{1}\right)}}{\left(\underset{\mathrm{\Omega }}{\int }{\left(v_{+}^{a+\mathrm{1}}\right)}^{\frac{mpa+m+\mathrm{1}}{m\left(a+\mathrm{1}\right)}·\frac{Npm\left(a+\mathrm{1}\right)}{\left(N-p\right)\left(mpa+m+\mathrm{1}\right)}}\mathrm{d}x\right)}^{\frac{\left(N-p\right)\left(mpa+m+\mathrm{1}\right)}{Npm\left(a+\mathrm{1}\right)}}\\ \le {\left|\mathrm{\Omega }\right|}^{\mathrm{1}-\frac{\left(N-p\right)\left(mpa+m+\mathrm{1}\right)}{Npm\left(a+\mathrm{1}\right)}}{\kappa }_{\mathrm{5}}^{\frac{\left(N-p\right)\left(mpa+m+\mathrm{1}\right)}{Npm\left(a+\mathrm{1}\right)}}{\left(\underset{\mathrm{\Omega }}{\int }{\left|\nabla v_{+}^{a+\mathrm{1}}\right|}^p\mathrm{d}x\right)}^{\frac{mpa+m+\mathrm{1}}{pm\left(a+\mathrm{1}\right)}}\end{array}$(20)

which implies that

$\underset{\mathrm{\Omega }}{\int }{\left|\nabla v_{+}^{a+\mathrm{1}}\right|}^p\mathrm{d}x\ge {\left|\mathrm{\Omega }\right|}^{\frac{\left(N-p\right)\left(m+\mathrm{1}\right)-mp\left(N+pa\right)}{N\left(mpa+m+\mathrm{1}\right)}}{\kappa }_{\mathrm{5}}^{\frac{p-N}{N}}{\left(\underset{\mathrm{\Omega }}{\int }{v}_{+}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{pm\left(a+\mathrm{1}\right)}{mpa+m+\mathrm{1}}}$(21)

where ${\kappa }_{\mathrm{5}}={\kappa }_{\mathrm{5}}\left(p,N\right)$ is the optimal Sobolev embedding constant. Substituting (21) into (19) and using Hölder's inequality, we get

$\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{\Omega }}{\int }{v}_{+}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x+{\kappa }_{\mathrm{6}}{\left(\underset{\mathrm{\Omega }}{\int }{v}_{+}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{pm\left(a+\mathrm{1}\right)}{mpa+m+\mathrm{1}}}\le {\kappa }_{\mathrm{7}}{\left(\underset{\mathrm{\Omega }}{\int }{v}_{+}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{m\left(pa+\beta +\mathrm{1}\right)+q-\mathrm{1}}{mpa+m+\mathrm{1}}}$(22)

where

${\kappa }_{\mathrm{6}}=\frac{\left(mpa+m+\mathrm{1}\right)\left(pa+\mathrm{1}\right)}{{\left(a+\mathrm{1}\right)}^p}{\kappa }_{\mathrm{5}}^{\frac{p-N}{N}}{\left|\mathrm{\Omega }\right|}^{\frac{\left(N-p\right)\left(m+\mathrm{1}\right)-pm\left(N+pa\right)}{N\left(mpa+m+\mathrm{1}\right)}}$(23)

and

${\kappa }_{\mathrm{7}}=\frac{mpa+m+\mathrm{1}}{m\mathrm{e}\beta }{\left|\mathrm{\Omega }\right|}^{\mathrm{1}-\frac{m\left(pa+\beta +\mathrm{1}\right)+q-\mathrm{1}}{mpa+m+\mathrm{1}}}$(24)

Recalling that $\mathrm{0}<m\left(p-\mathrm{1}\right)<q-\mathrm{1}$ and $q+m\beta <\mathrm{2}$, we check that

$\mathrm{0}<\frac{pm\left(a+\mathrm{1}\right)}{mpa+m+\mathrm{1}}<\frac{m\left(pa+\beta +\mathrm{1}\right)+q-\mathrm{1}}{mpa+m+\mathrm{1}}<\mathrm{1}$(25)

On the other hand, our assumption (7) tells us that

$\mathrm{0}<{\kappa }_{\mathrm{7}}<\frac{\mathrm{1}}{\mathrm{2}}{\kappa }_{\mathrm{6}}{\left(\underset{\mathrm{\Omega }}{\int }{v}_{\mathrm{0}+}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{m\left(p-\mathrm{1}\right)-\left(q-\mathrm{1}\right)-m\beta }{mpa+m+\mathrm{1}}}$(26)

Combining (22), (25), (26) and Lemma 1 ^[18], one can claim that there exist two positive constants $\xi$ and $\eta$ such that

$\mathrm{0}\le y\left(t\right):=\underset{\mathrm{\Omega }}{\int }{v}_{+}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x\le \xi {\mathrm{e}}^{-\eta t},t\ge \mathrm{0}$(27)

Choosing

$T_{\mathrm{0}}>\mathrm{m}\mathrm{a}\mathrm{x}\left\{\mathrm{0},\frac{\mathrm{1}}{\eta }\mathrm{l}\mathrm{n}\left[\xi {\left(\frac{\mathrm{2}{\kappa }_{\mathrm{7}}}{k_{\mathrm{6}}}\right)}^{\frac{mpa+m+\mathrm{1}}{q-\mathrm{1}-m\left(p-\mathrm{1}\right)+m\beta }}\right]\right\}$(28)

Then it follows from (22) and (27) that

$\frac{\mathrm{d}}{\mathrm{d}t}y\left(t\right)+\frac{{\kappa }_{\mathrm{6}}}{\mathrm{2}}{\left(y\left(t\right)\right)}^{\frac{pm\left(a+\mathrm{1}\right)}{mpa+m+\mathrm{1}}}\le \mathrm{0},t\ge T_{\mathrm{0}}$(29)

Integrating above inequality, one has

$\mathrm{0}\le y^{\mathrm{1}-\frac{pm\left(a+\mathrm{1}\right)}{mpa+m+\mathrm{1}}}\left(t\right)\le y^{\mathrm{1}-\frac{pm\left(a+\mathrm{1}\right)}{mpa+m+\mathrm{1}}}\left(T_{\mathrm{0}}\right)-\frac{{\kappa }_{\mathrm{6}}\left(m+\mathrm{1}-pm\right)}{\mathrm{2}\left(mpa+m+\mathrm{1}\right)}\left(t-T_{\mathrm{0}}\right)$(30)

which suggests that there exists a

${T_{\mathrm{0}}}^{\mathrm{\text{'}}}\in \left[T_{\mathrm{0}},T_{\mathrm{0}}+\frac{\mathrm{2}\left(mpa+m+\mathrm{1}\right)}{{\kappa }_{\mathrm{6}}\left(m+\mathrm{1}-pm\right)}{y}^{\frac{m+\mathrm{1}-pm}{mpa+m+\mathrm{1}}}\left(T_{\mathrm{0}}\right)\right]$(31)

such that

$\underset{t\to {{T_{\mathrm{0}}}^{\mathrm{\text{'}}}}^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}y\left(t\right)=\underset{t\to {{T_{\mathrm{0}}}^{\mathrm{\text{'}}}}^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}\underset{\mathrm{\Omega }}{\int }{v}_{+}^{\frac{mpa+m+\mathrm{1}}{m}}\left(t\right)\mathrm{d}x=\mathrm{0}$(32)

Moreover, it can be concluded that

$\underset{\mathrm{\Omega }}{\int }{v}_{+}\left(t\right)\mathrm{d}x\le {\left|\mathrm{\Omega }\right|}^{\frac{mpa+\mathrm{1}}{mpa+m+\mathrm{1}}}{\left(\underset{\mathrm{\Omega }}{\int }{v}_{+}^{\frac{mpa+m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{m}{mpa+m+\mathrm{1}}}\to \mathrm{0}\mathrm{a}\mathrm{s}t\to {{T_{\mathrm{0}}}^{\mathrm{\text{'}}}}^{-}$(33)

On the other hand, by a similar way, it can be shown that $\underset{\mathrm{\Omega }}{\int }{v}_{-}\left(t\right)\mathrm{d}x$ will vanish in finite time. Thus $\underset{\mathrm{\Omega }}{\int }\left|v\left(t\right)\right|\mathrm{d}x=\underset{\mathrm{\Omega }}{\int }{v}_{+}\left(t\right)\mathrm{d}x+\underset{\mathrm{\Omega }}{\int }{v}_{-}\left(t\right)\mathrm{d}x$ vanishes in finite time. Recalling that $v={\left|u\right|}^{m-\mathrm{1}}u$, one can claim that the solution $u\left(x,t\right)$ to problem (1) possesses the extinction property. The proof of Theorem 2 is completed.

Proof of Theorem 3   Denoting

$E\left(v\left(t\right)\right)=\frac{\mathrm{1}}{p}\underset{\mathrm{\Omega }}{\int }{\left|\nabla v\right|}^p\mathrm{d}x-\frac{\mathrm{1}}{q+m-\mathrm{1}}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{q+m-\mathrm{1}}{m}}\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|\mathrm{d}x+\frac{m}{{\left(q+m-\mathrm{1}\right)}^{\mathrm{2}}}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{q+m-\mathrm{1}}{m}}\mathrm{d}x$(34)

with $v={\left|u\right|}^{m-\mathrm{1}}u$, then a direct calculation shows that

$\frac{\mathrm{d}E\left(v\left(t\right)\right)}{\mathrm{d}t}=-\frac{\mathrm{4}m}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\underset{\mathrm{\Omega }}{\int }\left[{\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_t\right]}^{\mathrm{2}}\mathrm{d}x$(35)

which implies

$E\left(v\left(t\right)\right)=E\left(v_{\mathrm{0}}\right)-\frac{\mathrm{4}m}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\underset{\mathrm{\Omega }}{\int }\left[{\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_t\right]}^{\mathrm{2}}\mathrm{d}x\mathrm{d}\tau$(36)

Set

$M\left(t\right)=\frac{\mathrm{1}}{m+\mathrm{1}}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x$(37)

A direct calculation tells us that

$\begin{array}{l}{M}^{\text{'}}\left(t\right)=-\underset{\mathrm{\Omega }}{\int }{\left|\nabla v\right|}^p\mathrm{d}x+\frac{\mathrm{1}}{m}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{q+m-\mathrm{1}}{m}}\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|\mathrm{d}x=-pE\left(v\left(t\right)\right)+\frac{q-\mathrm{1}-m\left(p-\mathrm{1}\right)}{m\left(q+m-\mathrm{1}\right)}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{q+m-\mathrm{1}}{m}}\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|\mathrm{d}x\\ +\frac{mp}{{\left(q+m-\mathrm{1}\right)}^{\mathrm{2}}}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{q+m-\mathrm{1}}{m}}\mathrm{d}x\ge -pE\left(v\left(t\right)\right)+\frac{q-\mathrm{1}-m\left(p-\mathrm{1}\right)}{m\left(q+m-\mathrm{1}\right)}\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{q+m-\mathrm{1}}{m}}\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|\mathrm{d}x\end{array}$(38)

By virtue of (36), (38) and Hölder's inequality, one derives

$\begin{array}{l}{M}^{\text{'}}\left(t\right)\ge -pE\left(v_{\mathrm{0}}\right)-\frac{m\left(p-\mathrm{1}\right)-\left(q-\mathrm{1}\right)}{q+m-\mathrm{1}}\underset{{\mathrm{\Omega }}_{\mathrm{1}}=\left\{x\in \mathrm{\Omega }:\left|v\right|\ge \mathrm{1}\right\}\bigcup {\mathrm{\Omega }}_{\mathrm{2}}=\left\{x\in \mathrm{\Omega }:\left|v\right|<\mathrm{1}\right\}}{\int }{\left|v\right|}^{\frac{q+m-\mathrm{1}}{m}}\left|\mathrm{l}\mathrm{o}\mathrm{g}\left|v\right|\right|\mathrm{d}x\\ \ge -pE\left(v_{\mathrm{0}}\right)-\frac{m\left(p-\mathrm{1}\right)-\left(q-\mathrm{1}\right)}{q+m-\mathrm{1}}\left(\frac{\mathrm{1}}{e\beta }\underset{{\mathrm{\Omega }}_{\mathrm{1}}}{\int }{\left|v\right|}^{\frac{m\beta +m+q-\mathrm{1}}{m}}\mathrm{d}x+\frac{m\left|\mathrm{\Omega }\right|}{e\left(m+q-\mathrm{1}\right)}\right)\\ \ge -pE\left(v_{\mathrm{0}}\right)-\frac{m\left(p-\mathrm{1}\right)-\left(q-\mathrm{1}\right)}{e\beta \left(q+m-\mathrm{1}\right)}{\left|\mathrm{\Omega }\right|}^{\frac{\mathrm{2}-q-m\beta }{m+\mathrm{1}}}{\left(\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x\right)}^{\frac{m\beta +m+q-\mathrm{1}}{m+\mathrm{1}}}-\frac{\left[m\left(p-\mathrm{1}\right)-\left(q-\mathrm{1}\right)\right]m\left|\mathrm{\Omega }\right|}{\mathrm{e}{\left(q+m-\mathrm{1}\right)}^{\mathrm{2}}}\end{array}$(39)

If $m\left(p-\mathrm{1}\right)=q-\mathrm{1}$. Then from (39), one can see that, for any $t\ge \mathrm{0}$,

$M\left(t\right)\ge M\left(\mathrm{0}\right)-pE\left(v_{\mathrm{0}}\right)t$(40)

Keeping in mind that

$M\left(\mathrm{0}\right)=\frac{\mathrm{1}}{m+\mathrm{1}}\underset{\mathrm{\Omega }}{\int }{\left|v_{\mathrm{0}}\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x=\frac{\mathrm{1}}{m+\mathrm{1}}\underset{\Omega }{\int }{\left|u_{\mathrm{0}}\right|}^{m+\mathrm{1}}dx>\mathrm{0}$(41)

and

$E\left(v_{\mathrm{0}}\right)=E\left({\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\right)\le \mathrm{0}$(42)

then (40) gives us that $M\left(t\right)>\mathrm{0}$, which means that the solution $u\left(x,t\right)$ to problem (1) cannot vanish in finite time.

If $m\left(p-\mathrm{1}\right)\ge q-\mathrm{1}$. Remembering that

$M\left(\mathrm{0}\right)=\frac{\mathrm{1}}{m+\mathrm{1}}\underset{\mathrm{\Omega }}{\int }{\left|v_{\mathrm{0}}\right|}^{\frac{m+\mathrm{1}}{m}}\mathrm{d}x=\frac{\mathrm{1}}{m+\mathrm{1}}\underset{\Omega }{\int }{\left|u_{\mathrm{0}}\right|}^{m+\mathrm{1}}dx>\mathrm{0}$(43)

and

$E\left(v_{\mathrm{0}}\right)=E\left({\left|u_{\mathrm{0}}\right|}^{m-\mathrm{1}}{u}_{\mathrm{0}}\right)<-\frac{m\left|\mathrm{\Omega }\right|\left[m\left(p-\mathrm{1}\right)-\left(q-\mathrm{1}\right)\right]}{\mathrm{e}p{\left(q+m-\mathrm{1}\right)}^{\mathrm{2}}}$(44)