© Wuhan University 2024

0 Introduction

In the present article, our attention is focused on the blow-up behavior of the following porous medium equation with superline source and local linear boundary dissipation

$\{\begin{array}{ll}{u}_t=\mathrm{\Delta }{u}^m+{\left|u\right|}^{p-\mathrm{2}}u,& x\in \mathrm{\Omega }, t>\mathrm{0},\\ u(x,t)=\mathrm{0},& x\in {\mathrm{\Gamma }}_{\mathrm{0}}, t>\mathrm{0},\\ \frac{\partial u^m}{\partial \mathit{n}}=-u_t,& x\in {\mathrm{\Gamma }}_{\mathrm{1}}, t>\mathrm{0},\\ u(x,\mathrm{0})=u_{\mathrm{0}}(x),& x\in \mathrm{\Omega },\end{array}$(1)

where $m>\mathrm{1}$, $p>m+\mathrm{1}$, $\mathrm{\Omega }$ is a bounded open subset of ${\mathbb{R}}^n$ with $C^{\mathrm{1}}$ boundary $\partial \mathrm{\Omega }$ and $\mathit{n}$ represents the unit outer normal vector to $\partial \mathrm{\Omega }$. $\left\{{\mathrm{\Gamma }}_{\mathrm{0}},{\mathrm{\Gamma }}_{\mathrm{1}}\right\}$ is a partition of the boundary $\partial \mathrm{\Omega }$ satisfying $\partial \mathrm{\Omega }={\mathrm{\Gamma }}_{\mathrm{0}}\bigcup {\mathrm{\Gamma }}_{\mathrm{1}},\overline{{\mathrm{\Gamma }}_{\mathrm{0}}}\bigcap \overline{{\mathrm{\Gamma }}_{\mathrm{1}}}=\mathrm{\varnothing }.$

Moreover, ${\mathrm{\Gamma }}_{\mathrm{0}}$ and ${\mathrm{\Gamma }}_{\mathrm{1}}$ are measurable over $\partial \mathrm{\Omega }$, endowed with $(n-\mathrm{1})$-dimensional surface measure $\sigma$ and $\sigma ({\mathrm{\Gamma }}_{\mathrm{0}})>\mathrm{0}$.

Problem (1) with $m=\mathrm{1}$ can be regarded as a mathematical model to depict a heat reaction-diffusion process that occurs inside a solid body $\mathrm{\Omega }$ surrounded by a fluid, with contact ${\mathrm{\Gamma }}_{\mathrm{1}}$ and having an internal cavity with a contact boundary ${\mathrm{\Gamma }}_{\mathrm{0}}.$ In this physics background, the quantity of heat produced by the reaction is proportional to a superlinear power of the temperature. To avoid an internal explosion in $\mathrm{\Omega }$, a refrigeration system is installed in the fluid. The operational mechanism of this refrigeration system lies in the fact that the heat absorbed from the fluid is proportional to the power of the rate of change of the temperature, which can be expressed as: $\frac{\partial u}{\partial \mathit{n}}=-|u_t{|}^{q-\mathrm{2}}{u}_t, x\in {\mathrm{\Gamma }}_{\mathrm{1}},$ where $\frac{\partial u}{\partial \mathit{n}}$ stands for the heat flux from $\mathrm{\Omega }$ to the fluid.

Evolution equations ^[1-8] with boundary damping have attracted the attention of mathematicians in the past period. For instance, Fiscella and Vitillaro ^[9] studied the following problem with local nonlinear boundary dissipation

$\{\begin{array}{ll}{u}_t-\mathrm{\Delta }u=|u{|}^{p-\mathrm{2}}u,& x\in \mathrm{\Omega }, t>\mathrm{0},\\ u(x,t)=\mathrm{0}& x\in {\mathrm{\Gamma }}_{\mathrm{0}}, t>\mathrm{0},\\ \frac{\partial u}{\partial \mathit{n}}=-|u_t{|}^{q-\mathrm{2}}{u}_t,& x\in {\mathrm{\Gamma }}_{\mathrm{1}}, t>\mathrm{0},\\ u(x,\mathrm{0})=u_{\mathrm{0}}(x),& x\in \overline{\mathrm{\Omega }},\end{array}$(2)

where $\mathrm{2}\le p\le \mathrm{1}+\frac{{\mathrm{2}}^{\mathrm{*}}}{\mathrm{2}}$ and ${\mathrm{2}}^{\mathrm{*}}$ denotes the Sobolev conjugate of 2. Using the monotonicity method of Lions^[9] and a contraction argument, they proved the local well-posedness in the Hadamard sense. Moreover, in the case of a superlinear source, i.e. $p>\mathrm{2}$, under the condition

$J(u_{\mathrm{0}})=\frac{\mathrm{1}}{\mathrm{2}}\underset{\Omega }{\int }|\nabla u_{\mathrm{0}}(x){|}^{\mathrm{2}}\mathrm{d}x-\frac{\mathrm{1}}{p}\int |u_{\mathrm{0}}(x){|}^p\mathrm{d}x<d:=\underset{u\in H_{{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{1}}\left(\mathrm{\Omega }\right)\setminus \{\mathrm{0}\}}{\mathrm{i}\mathrm{n}\mathrm{f}}\underset{\lambda >\mathrm{0}}{\mathrm{s}\mathrm{u}\mathrm{p}}J\left(\lambda u\right),$

they gave the global existence and finite time blow-up results. To be precise, if

$K(u_{\mathrm{0}})=\underset{\Omega }{\int }|\nabla u_{\mathrm{0}}(x){|}^{\mathrm{2}}\mathrm{d}x-\underset{\Omega }{\int }|u_{\mathrm{0}}(x){|}^p\mathrm{d}x\ge \mathrm{0},$

then the weak solution is global, while if

$K(u_{\mathrm{0}})=\underset{\Omega }{\int }|\nabla u_{\mathrm{0}}(x){|}^{\mathrm{2}}\mathrm{d}x-\underset{\Omega }{\int }|u_{\mathrm{0}}(x){|}^p\mathrm{d}x\le \mathrm{0}$

and $q<q_{\mathrm{0}}(p):=\frac{\mathrm{2}(n+\mathrm{1})p-\mathrm{4}(n-\mathrm{1})}{n(p-\mathrm{2})+\mathrm{4}},$ then the weak solution will blow up in some finite time. In a recent work, the authors ^[10] considered problem (1) with $m=\mathrm{1}$, and obtained the finite time blow-up result for arbitrary high initial energy. Moreover, under some additional conditions, the authors also gave estimates of the blow-up time. In addition, using some differential inequality techniques, the authors ^{[11, 12]} considered the lower bounds for the blow-up time of blow-up solutions to some porous medium equations with null Dirichlet boundary conditions or homogeneous Neumann boundary conditions.

To the best of our knowledge, there is no previous work on the blow-up behavior of the solution to the problem (1). Building on the aforementioned work, we will analyze the effects of the nonlinear diffusion and the local linear boundary dissipation on the blow-up phenomenon of problem (1). In order to deal with the difficulties caused by the nonlinear diffusion term $\mathrm{\Delta }{u}^m$ better and more effectively, throughout this paper, we work with the following equivalent formulation of the problem (1) obtained by changing variables $u^m=v$,

$\{\begin{array}{ll}{\left(v^{\frac{\mathrm{1}}{m}}\right)}_t=\mathrm{\Delta }v+{\left|v\right|}^{\frac{p-\mathrm{2}}{m}}{v}^{\frac{\mathrm{1}}{m}},& x\in \mathrm{\Omega }, t>\mathrm{0},\\ v(x,t)=\mathrm{0},& x\in {\mathrm{\Gamma }}_{\mathrm{0}}, t>\mathrm{0},\\ \frac{\partial v}{\partial \mathit{n}}=-{\left(v^{\frac{\mathrm{1}}{m}}\right)}_t,& x\in {\mathrm{\Gamma }}_{\mathrm{1}}, t>\mathrm{0},\\ v(x,\mathrm{0})=v_{\mathrm{0}}(x)=u_{\mathrm{0}}^m(x),& x\in \mathrm{\Omega }.\end{array}$(3)

First, we obtain the finite time blow-up criterion of the solution to the problem (3) by using a modified concavity method (Theorem 1). Second, for any $a\in \mathbb{R},$we prove that there exists a $v_{\mathrm{0}}\in H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}\left(\mathrm{\Omega }\right)$ with initial energy $J\left(v_{\mathrm{0}}\right)=a$ that leads to a finite time blow-up solution (Corollary 1). Finally, the lower bounds of the blow-up time are derived by combining the interpolation inequality for $L^q$-norms, the Sobolev embedding theorem, with some differential inequality techniques (Theorem 2).

The article is organized as follows: In Section 1, we introduce some notations and state some useful lemmas. In Section 2, we give the finite time blow-up criterion and the lower and upper estimates of the blow-up time.

1 Preliminaries

In this section, we first introduce some notations, definitions, and some known results. Throughout this article, we denote ${\Vert \cdot \Vert }_q={\Vert \cdot \Vert }_{L^q\left(\mathrm{\Omega }\right)}$, ${\Vert \cdot \Vert }_{q,{\mathrm{\Gamma }}_{\mathrm{1}}}={\Vert \cdot \Vert }_{L^q\left({\mathrm{\Gamma }}_{\mathrm{1}}\right)}$ for some $q\in \left[\mathrm{1},+\mathrm{\infty }\right]$, and the Hilbert space

$H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}\left(\mathrm{\Omega }\right)=\left\{v\in H^{\mathrm{1}}\left(\mathrm{\Omega }\right): v=\mathrm{0} \mathrm{f}\mathrm{o}\mathrm{r} x\in {\mathrm{\Gamma }}_{\mathrm{0}}\right\}.$

$\left(\cdot ,\cdot \right)$ and ${\left(\cdot ,\cdot \right)}_{{\mathrm{\Gamma }}_{\mathrm{1}}}$ stand for the inner products on the Hilbert spaces $L^{\mathrm{2}}\left(\mathrm{\Omega }\right)$ and $L^{\mathrm{2}}\left({\mathrm{\Gamma }}_{\mathrm{1}}\right)$ , respectively. From the trace theorem, one knows that there exists a continuous trace mapping $H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}\left(\mathrm{\Omega }\right)$$\hookrightarrow$$L^{\mathrm{2}}\left(\partial \mathrm{\Omega }\right)$. Moreover, since $\sigma \left({\mathrm{\Gamma }}_{\mathrm{0}}\right)>\mathrm{0},$ then, Theorem 6.7-5 ^[13] tells us that a Poincaré-type inequality holds. Therefore, ${\Vert \nabla v\Vert }_{\mathrm{2}}$ is equivalent to the norm ${\Vert v\Vert }_{H_{{\mathrm{\Gamma }}_{\mathrm{0}}}}^{\mathrm{1}}={\left({\Vert v\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert \nabla v\Vert }_{\mathrm{2}}^{\mathrm{2}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}$ in the space $H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}\left(\mathrm{\Omega }\right).$ On the other hand, since $m>\mathrm{1}$, one can define the following positive optimal constants

$S_{\mathrm{1}}=\underset{\underset{v\ne \mathrm{0}}{v\in H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}}\left(\mathrm{\Omega }\right)}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{{\Vert v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}}{{\Vert \nabla v\Vert }_{\mathrm{2}}^{\mathrm{2}}}, S_{\mathrm{2}}=\underset{\underset{v\ne \mathrm{0}}{v\in H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}}\left(\mathrm{\Omega }\right)}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{{\Vert v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\Vert }_{\mathrm{2}}^{\mathrm{2}}}{{\Vert \nabla v\Vert }_{\mathrm{2}}^{\mathrm{2}}}.$(4)

The definition of the weak solution to the problem (3) is given as follows.

${\mathrm{2}}^{\mathrm{*}}=\{\begin{array}{cc}\mathrm{\infty },& \mathrm{i}\mathrm{f}\mathrm{ }n\le \mathrm{2};\\ \frac{\mathrm{2}n}{n-\mathrm{2}},& \mathrm{i}\mathrm{f}\mathrm{ }n>\mathrm{2}.\end{array}$

A function $v:=v\left(x,t\right)\in L^{\mathrm{\infty }}\left(\mathrm{0},T;H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}(\mathrm{\Omega })\right)$ with ${\left(v^{\frac{\mathrm{1}}{m}}\right)}_t\in L^{\mathrm{2}}\left(\left(\mathrm{0},T\right)\times \mathrm{\Omega }\right)$ is called a weak solution of problem (3) on $[\mathrm{0},T)\times \mathrm{\Omega }$ if $v(x,\mathrm{0})=v_{\mathrm{0}}=u_{\mathrm{0}}^m$, and

$\left({\left(v^{\frac{\mathrm{1}}{m}}\right)}_t,\varphi \right)+\left(\nabla v\left(t\right),\nabla \varphi \right)+{\left({\left(v^{\frac{\mathrm{1}}{m}}\right)}_t,\varphi \right)}_{{\mathrm{\Gamma }}_{\mathrm{1}}}=\underset{\mathrm{\Omega }}{\int }{\left|v\right|}^{\frac{p-\mathrm{2}}{m}}{v}^{\frac{\mathrm{1}}{m}}\varphi \mathrm{d}x$(5)

holds for a.e. $t\in \left(\mathrm{0},T\right)$ and any $\varphi \in H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}\left(\mathrm{\Omega }\right).$ Moreover, the spatial trace of $v$ has a distributional time derivative $v_t$ on $\left(\mathrm{0},T\right)\times \partial \mathrm{\Omega }$, belonging to $L^{\mathrm{2}}\left(\left(\mathrm{0},T\right)\times \partial \mathrm{\Omega }\right)$.

$\underset{t\to T^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}\rho \left(t\right)=\underset{t\to T^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathrm{1}}{m+\mathrm{1}}\left({\Vert v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\left(t\right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\left(t\right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)=+\mathrm{\infty }.$

In what follows, we introduce the energy-functional $J\left(v\right)=\frac{\mathrm{1}}{\mathrm{2}}{\Vert \nabla v\Vert }_{\mathrm{2}}^{\mathrm{2}}-\frac{m}{m+p-\mathrm{1}}{\Vert v\Vert }_{\frac{m+p-\mathrm{1}}{m}}^{\frac{m+p-\mathrm{1}}{m}},$ and Nehari functional

$K\left(v\right)={\Vert \nabla v\Vert }_{\mathrm{2}}^{\mathrm{2}}-{\Vert v\Vert }_{\frac{m+p-\mathrm{1}}{m}}^{\frac{m+p-\mathrm{1}}{m}}$

in $H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}(\mathrm{\Omega })$ related to problem (3). Evidently, one knows that $J\left(v\right)$ and $K\left(v\right)$ are continuous on $H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}(\mathrm{\Omega })$, and for a.e. $t\in \left(\mathrm{0},T\right),$

$\frac{\mathrm{d}}{\mathrm{d}t}J(v(t))=-\frac{\mathrm{4}m}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}\left({\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_t\left(t\right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_t\left(t\right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)\le \mathrm{0}$(6)

which implies that

$J\left(v\left(t\right)\right)=J\left(v_{\mathrm{0}}\right)-\frac{\mathrm{4}m}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left({\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)\mathrm{d}\tau$(7)

Now, we introduce the definition of the potential well-depth $d=\underset{v\in {N}}{\mathrm{i}\mathrm{n}\mathrm{f}}J(v)=\underset{v\in H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}(\mathrm{\Omega })\backslash \{\mathrm{0}\}}{\mathrm{i}\mathrm{n}\mathrm{f}}\underset{\lambda >\mathrm{0}}{\mathrm{s}\mathrm{u}\mathrm{p}}J(\lambda v),$ where ${N}$ is the Nehari manifold ${N}=\left\{v\in H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}(\mathrm{\Omega })|K(v)=\mathrm{0}\right\}\backslash \left\{\mathrm{0}\right\}.$ Indeed, $d$ also can be characterized as

$d=\frac{p-m-\mathrm{1}}{\mathrm{2}\left(m+p-\mathrm{1}\right)}{\left(\underset{\underset{v\ne \mathrm{0}}{v\in H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}}\left(\mathrm{\Omega }\right)}{\mathrm{s}\mathrm{u}\mathrm{p}}\frac{{\Vert v\Vert }_{\frac{m+p-\mathrm{1}}{m}}}{{\Vert \nabla v\Vert }_{\mathrm{2}}}\right)}^{-\frac{\mathrm{2}(m+p-\mathrm{1})}{p-m-\mathrm{1}}}.$

We are now able to give some lemmas, which will play a key role in our proof of the main results.

${\lambda }^{\mathrm{*}}={\left(\frac{{\Vert \nabla v\Vert }_{\mathrm{2}}^{\mathrm{2}}}{{\Vert v\Vert }_{\frac{m+p-\mathrm{1}}{m}}^{\frac{m+p-\mathrm{1}}{m}}}\right)}^{\frac{m}{p-m-\mathrm{1}}}\in \left(\mathrm{0,1}\right).$

Thereupon, one has ${\lambda }^{\mathrm{*}}v\in {N}$. Furthermore, according to the definition of the potential well depth $d$, one can arrive at

$\begin{array}{l}d=\underset{v\in {N}}{\mathrm{i}\mathrm{n}\mathrm{f}}J(v)\le J({\lambda }^{\mathrm{*}}v)=\frac{p-m-\mathrm{1}}{\mathrm{2}(p+m-\mathrm{1})}{\Vert {\lambda }^{\mathrm{*}}v\Vert }_{\frac{m+p-\mathrm{1}}{m}}^{\frac{m+p-\mathrm{1}}{m}}+\frac{\mathrm{1}}{\mathrm{2}}K\left({\lambda }^{\mathrm{*}}v\right)\\ =\frac{p-m-\mathrm{1}}{\mathrm{2}(p+m-\mathrm{1})}({\lambda }^{\mathrm{*}}{)}^{\frac{m+p-\mathrm{1}}{m}}{\Vert v\Vert }_{\frac{m+p-\mathrm{1}}{m}}^{\frac{m+p-\mathrm{1}}{m}}<\frac{p-m-\mathrm{1}}{\mathrm{2}\left(m+p-\mathrm{1}\right)}{\Vert v\Vert }_{\frac{m+p-\mathrm{1}}{m}}^{\frac{m+p-\mathrm{1}}{m}}\end{array}$(8)

which results in ${\Vert v\Vert }_{\frac{m+p-\mathrm{1}}{m}}^{\frac{m+p-\mathrm{1}}{m}}>\frac{\mathrm{2}\left(m+p-\mathrm{1}\right)}{\left(p-\mathrm{1}-m\right)}d.$

The proof of the Lemma 1 is completed.

$J\left(v_{\mathrm{0}}\right)\ge J\left(v\left(t\right)\right)=\frac{p-m-\mathrm{1}}{\mathrm{2}(m+p-\mathrm{1})}{\Vert \nabla v\left(t\right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+\frac{m}{m+p-\mathrm{1}}K\left(v\left(t\right)\right)\ge \frac{p-m-\mathrm{1}}{\mathrm{2}(m+p-\mathrm{1})}{\Vert \nabla v\left(t\right)\Vert }_{\mathrm{2}}^{\mathrm{2}},$

which contradicts the assumption that $v$ is a finite-time blow-up weak solution. The proof of Lemma 2 is completed.

Lemma 3 ^[14] Suppose that a positive function $F$ on $\left[\mathrm{0},T\right]$ satisfies the following conditions:$\left(\mathrm{i}\right)$$F$ is differentiable on $\left[\mathrm{0},T\right]$ and $F^{\text{'}}$ is absolutely continuous on $\left[\mathrm{0},T\right]$ with $F^{\text{'}}\left(\mathrm{0}\right)>\mathrm{0}$; $\left(\mathrm{i}\mathrm{i}\right)$ there exists a positive constant $\alpha >\mathrm{0}$ such that

$F\left(t\right)F^{″}\left(t\right)-\left(\mathrm{1}+\alpha \right){\left(F^{\text{'}}\left(t\right)\right)}^{\mathrm{2}}\ge \mathrm{0}$

holds for a.e. $t\in \left[\mathrm{0},T\right]$. Then $T\le \frac{F\left(\mathrm{0}\right)}{\alpha F^{\text{'}}\left(\mathrm{0}\right)}.$

2 The Finite Time Blow-up Results

Recalling that $\rho \left(t\right)=\frac{\mathrm{1}}{m+\mathrm{1}}\left({\Vert v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\left(t\right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\left(t\right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right),$ one can easily check that

$\frac{\mathrm{d}}{\mathrm{d}t}\rho \left(t\right)=\frac{\mathrm{1}}{m}\left[\left(v^{\frac{\mathrm{1}}{m}}\left(t\right),v_t\left(t\right)\right)+{\left(v^{\frac{\mathrm{1}}{m}}\left(t\right),v_t\left(t\right)\right)}_{{\mathrm{\Gamma }}_{\mathrm{1}}}\right]=-K\left(v\left(t\right)\right)$(9)

For any $T_{\mathrm{1}}\in \left(\mathrm{0},T\right)$, we define an auxiliary function as the form

$F(t)={\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\rho (\tau )\mathrm{d}\tau +\left(T_{\mathrm{1}}-t\right)\rho (\mathrm{0})+\frac{m\beta }{m+\mathrm{1}}{\left(t+\sigma \right)}^{\mathrm{2}}, t\in \left[\mathrm{0},T_{\mathrm{1}}\right]$(10)

where $\beta$ and $\sigma$ are two positive parameters which will be determined later. It is clear that $F$ is positive on $\left[\mathrm{0},T_{\mathrm{1}}\right]$. By a simple calculation, one has

$F^{\text{'}}\left(t\right)=\rho \left(t\right)-\rho \left(\mathrm{0}\right)+\frac{\mathrm{2}m\beta }{m+\mathrm{1}}\left(t+\sigma \right)={\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\frac{\mathrm{d}}{\mathrm{d}t}\rho (\tau )\mathrm{d}\tau +\frac{\mathrm{2}m\beta }{m+\mathrm{1}}(t+\sigma )=\frac{\mathrm{1}}{m}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left[\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)+{\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)}_{{\mathrm{\Gamma }}_{\mathrm{1}}}\right]\mathrm{d}\tau +\frac{\mathrm{2}m\beta }{m+\mathrm{1}}(t+\sigma )$(11)

and $F^{\text{'}}\left(\mathrm{0}\right)=\frac{\mathrm{2}m\beta \sigma }{m+\mathrm{1}}>\mathrm{0}$. Moreover, from (6) and (7), it follows that

$\begin{array}{l}{F}^{″}\left(t\right)=\frac{\mathrm{d}}{\mathrm{d}t}\rho (t)+\frac{\mathrm{2}m\beta }{m+\mathrm{1}}=-K\left(v\left(t\right)\right)+\frac{\mathrm{2}m\beta }{m+\mathrm{1}}=-\left(\frac{m+p-\mathrm{1}}{m}J(v(t)-\frac{p-m-\mathrm{1}}{\mathrm{2}m}{\Vert \nabla v\left(t\right)\Vert }_{\mathrm{2}}^{\mathrm{2}}\right)+\frac{\mathrm{2}m\beta }{m+\mathrm{1}}\\ =\frac{p-m-\mathrm{1}}{\mathrm{2}m}{\Vert \nabla v\left(t\right)\Vert }_{\mathrm{2}}^{\mathrm{2}}-\frac{m+p-\mathrm{1}}{m}J(v_{\mathrm{0}})+\frac{\mathrm{2}m\beta }{m+\mathrm{1}}+\frac{\mathrm{4}\left(m+p-\mathrm{1}\right)}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left({\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)\mathrm{d}\tau \end{array}$(12)

On the other hand, employing the Young's inequality and Cauchy-Schwartz inequality, we obtain:

$\begin{array}{l}\xi \left(t\right)=\left[\frac{\mathrm{1}}{m}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left({\Vert v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\left(\tau \right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\left(\tau \right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)\mathrm{d}\tau +\beta {(t+\sigma )}^{\mathrm{2}}\right]\cdot \left[\frac{\mathrm{4}m}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left({\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)\mathrm{d}\tau +\frac{\mathrm{4}{m}^{\mathrm{2}}\beta }{{(m+\mathrm{1})}^{\mathrm{2}}}\right]\\ -{\left[\frac{\mathrm{1}}{m}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left[\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)+{\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)}_{{\mathrm{\Gamma }}_{\mathrm{1}}}\right]\mathrm{d}\tau +\frac{\mathrm{2}m\beta }{m+\mathrm{1}}(t+\sigma )\right]}^{\mathrm{2}}\end{array}$

$\begin{array}{l}\ge {\left\{\frac{\mathrm{2}}{m+\mathrm{1}}\left({\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left[\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}},{\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\right)+{\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}},{\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\right)}_{{\mathrm{\Gamma }}_{\mathrm{1}}}\right]\mathrm{d}\tau \right)+\frac{\mathrm{2}m\beta }{m+\mathrm{1}}\left(t+\sigma \right)\right\}}^{\mathrm{2}}-{\left[\frac{\mathrm{1}}{m}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left[\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)+{\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)}_{{\mathrm{\Gamma }}_{\mathrm{1}}}\right]\mathrm{d}\tau +\frac{\mathrm{2}m\beta }{m+\mathrm{1}}(t+\sigma )\right]}^{\mathrm{2}}\\ ={\left[\frac{\mathrm{1}}{m}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left[\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)+{\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)}_{{\mathrm{\Gamma }}_{\mathrm{1}}}\right]\mathrm{d}\tau +\frac{\mathrm{2}m\beta }{m+\mathrm{1}}(t+\sigma )\right]}^{\mathrm{2}}-{\left[\frac{\mathrm{1}}{m}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left[\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)+{\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)}_{{\mathrm{\Gamma }}_{\mathrm{1}}}\right]\mathrm{d}\tau +\frac{\mathrm{2}m\beta }{m+\mathrm{1}}(t+\sigma )\right]}^{\mathrm{2}}=\mathrm{0}\end{array}$(13)

Now, we are in the position to estimate $F{F}^{″}-\lambda {\left(F^{\text{'}}\right)}^{\mathrm{2}}$with $\lambda =\frac{m+p-\mathrm{1}}{m+\mathrm{1}}>\mathrm{1}$. Combining (10), (11), and (12), one can arrive at:

$\begin{array}{l}F{F}^{″}-\lambda {\left(F^{\text{'}}\right)}^{\mathrm{2}}=F{F}^{″}-\lambda {\left[\frac{\mathrm{1}}{m}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left[\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)+{\left(v^{\frac{\mathrm{1}}{m}},v_{\tau }\right)}_{{\mathrm{\Gamma }}_{\mathrm{1}}}\right]\mathrm{d}\tau +\frac{\mathrm{2}m\beta }{m+\mathrm{1}}(t+\sigma )\right]}^{\mathrm{2}}\\ =F{F}^{″}+\lambda \left[\xi \left(t\right)-\frac{m+\mathrm{1}}{m}\left(F\left(t\right)-\left(T-t\right)\rho \left(\mathrm{0}\right)\right)\cdot \left(\frac{\mathrm{4}m}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left({\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)\mathrm{d}\tau +\frac{\mathrm{4}{m}^{\mathrm{2}}\beta }{{(m+\mathrm{1})}^{\mathrm{2}}}\right)\right]\\ \ge F{F}^{″}-\frac{\mathrm{4}m\lambda \beta }{m+\mathrm{1}}F(t)-\frac{\mathrm{4}\lambda }{m+\mathrm{1}}F(t){\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left({\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)\mathrm{d}\tau \\ =F[\frac{p-m-\mathrm{1}}{\mathrm{2}m}{\Vert \nabla v\left(t\right)\Vert }_{\mathrm{2}}^{\mathrm{2}}-\frac{m+p-\mathrm{1}}{m}J(v_{\mathrm{0}})+\frac{\mathrm{2}m\beta }{m+\mathrm{1}}+\frac{\mathrm{4}\left(m+p-\mathrm{1}\right)}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left({\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)\mathrm{d}\tau \\ -\frac{\mathrm{4}\lambda }{m+\mathrm{1}}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left({\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)\mathrm{d}\tau -\frac{\mathrm{4}m\lambda \beta }{m+\mathrm{1}}]\\ =F[\left(\frac{\mathrm{4}\left(m+p-\mathrm{1}\right)}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}-\frac{\mathrm{4}\lambda }{m+\mathrm{1}}\right){\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}}\left({\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert {\left(v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\right)}_{\tau }\left(\tau \right)\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)\mathrm{d}\tau \end{array}$(14)

Noticing that, $\lambda =\frac{m+p-\mathrm{1}}{m+\mathrm{1}}$, then (14) leads to

$F{F}^{″}-\frac{m+p-\mathrm{1}}{m+\mathrm{1}}{\left(F^{\text{'}}\right)}^{\mathrm{2}}\ge F\cdot \left[\frac{p-m-\mathrm{1}}{\mathrm{2}m}{\Vert \nabla v\left(t\right)\Vert }_{\mathrm{2}}^{\mathrm{2}}-\frac{m+p-\mathrm{1}}{m}J\left(v_{\mathrm{0}}\right)-\frac{\mathrm{2}m\beta \left(\mathrm{2}p+m-\mathrm{3}\right)}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}\right]$(15)

Up to now, from the above discussion, one can summarize the following lemma.

$\frac{p-m-\mathrm{1}}{\mathrm{2}m}{\Vert \nabla v\left(t\right)\Vert }_{\mathrm{2}}^{\mathrm{2}}-\frac{m+p-\mathrm{1}}{m}J\left(v_{\mathrm{0}}\right)-\frac{\mathrm{2}m\beta \left(\mathrm{2}p+m-\mathrm{3}\right)}{{\left(m+\mathrm{1}\right)}^{\mathrm{2}}}\ge \mathrm{0}$

holds for any $t\in (\mathrm{0},T_{\mathrm{1}}]$. Then $\mathrm{0}<T_{\mathrm{1}}\le \frac{{\left(m+\mathrm{1}\right)}^{\mathrm{3}}\rho \left(\mathrm{0}\right)}{m{\left(p-\mathrm{2}\right)}^{\mathrm{2}}\beta }.$

$T_{\mathrm{1}}<\frac{F\left(\mathrm{0}\right)}{\frac{p-\mathrm{2}}{m+\mathrm{1}}{F}^{\text{'}}(\mathrm{0})}=\frac{T_{\mathrm{1}}\rho \left(\mathrm{0}\right)+\frac{m\beta {\sigma }^{\mathrm{2}}}{m+\mathrm{1}}}{\frac{p-\mathrm{2}}{m+\mathrm{1}}\cdot \frac{\mathrm{2}m\beta \sigma }{m+\mathrm{1}}}=\frac{\rho \left(\mathrm{0}\right){\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\mathrm{2}m\left(p-\mathrm{2}\right)\beta \sigma }{T}_{\mathrm{1}}+\frac{m+\mathrm{1}}{\mathrm{2}\left(p-\mathrm{2}\right)}\sigma$(16)

Choosing $\sigma \in \left(\frac{\rho \left(\mathrm{0}\right){\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\mathrm{2}m\left(p-\mathrm{2}\right)\beta },+\mathrm{\infty }\right)$ to guarantee $\frac{\rho \left(\mathrm{0}\right){\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\mathrm{2}m\left(p-\mathrm{2}\right)\beta \sigma }<\mathrm{1}$, then (16) tells us that,

$T_{\mathrm{1}}\le {\left(\mathrm{1}-\frac{\rho \left(\mathrm{0}\right){\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\mathrm{2}m\left(p-\mathrm{2}\right)\beta \sigma }\right)}^{-\mathrm{1}}\frac{m+\mathrm{1}}{\mathrm{2}\left(p-\mathrm{2}\right)}\sigma$(17)

By a series of calculations, one can verify that the right side of (17) takes its minimum at the point

$\sigma ={\sigma }_{\beta }=\frac{\rho \left(\mathrm{0}\right){\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{m\left(p-\mathrm{2}\right)\beta }\in \left(\frac{\rho \left(\mathrm{0}\right){\left(m+\mathrm{1}\right)}^{\mathrm{2}}}{\mathrm{2}m\left(p-\mathrm{2}\right)\beta },+\mathrm{\infty }\right).$

In other words, one has $T_{\mathrm{1}}\le \frac{{\left(m+\mathrm{1}\right)}^{\mathrm{3}}\rho \left(\mathrm{0}\right)}{m{\left(p-\mathrm{2}\right)}^{\mathrm{2}}\beta }.$

The proof of Lemma 4 is completed.

Now, we give the finite time blow-up criteria as follows.

${\mathrm{ℬ}}_{\mathrm{1}}=\left\{v\in H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}:J\left(v\right)<d, K\left(v\right)<\mathrm{0}\right\},{\mathrm{ℬ}}_{\mathrm{2}}=\left\{v\in H_{{\mathrm{\Gamma }}_{\mathrm{0}}}^{\mathrm{1}}:J\left(v\right)<\frac{\left(p-m-\mathrm{1}\right)\left({\Vert v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert v^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)}{\mathrm{2}\left(m+p-\mathrm{1}\right)\left(S_{\mathrm{1}}+S_{\mathrm{2}}\right)}\right\}.$

Then, the weak solution $v$ to the problem (3) blows up in a finite time. Moreover, one has

$T\le \frac{\mathrm{2}m\left(\mathrm{2}p+m-\mathrm{3}\right)}{{\left(p-\mathrm{2}\right)}^{\mathrm{2}}\left(m+p-\mathrm{1}\right)}\cdot \frac{{\Vert v_{\mathrm{0}}^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert v_{\mathrm{0}}^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}}{d-J\left(v_{\mathrm{0}}\right)}$

for $v_{\mathrm{0}}\in {\mathrm{ℬ}}_{\mathrm{1}}$ and

$T\le \frac{\mathrm{2}m\left(\mathrm{2}p+m-\mathrm{3}\right)}{{\left(p-\mathrm{2}\right)}^{\mathrm{2}}\left(m+p-\mathrm{1}\right)}\cdot \frac{{\Vert v_{\mathrm{0}}^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert v_{\mathrm{0}}^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}}{\frac{p-m-\mathrm{1}}{\mathrm{2}\left(m+p-\mathrm{1}\right)\left(S_{\mathrm{1}}+S_{\mathrm{2}}\right)}\left({\Vert v_{\mathrm{0}}^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\Vert }_{\mathrm{2}}^{\mathrm{2}}+{\Vert v_{\mathrm{0}}^{\frac{m+\mathrm{1}}{\mathrm{2}m}}\Vert }_{\mathrm{2},{\mathrm{\Gamma }}_{\mathrm{1}}}^{\mathrm{2}}\right)-J\left(v_{\mathrm{0}}\right)}$