Blow-Up Phenomena for a Non-Newton Filtration Equation with Local Linear Boundary Dissipation

Abstract

In this article, we consider the finite time blow-up phenomenon for a class of non-Newton filtration equations with local linear boundary dissipation. Using a modified concavity method, the sufficient condition for the finite time blow-up is given. Furthermore, we obtain the result of the blow-up solution with arbitrary high initial energy. Meanwhile, the upper and lower estimates of the blow-up time are discussed.

1. Introduction

In this article, our interest is mainly focused on the blow-up phenomena of the following q-Laplace equation with superlinear source and local linear boundary dissipation

$$\left\{\begin{array}{cc}{u}_t-\mathrm{div}\left({|\nabla u|}^{q-2}\nabla u\right)={|u|}^{p-2}u,\hfill & (x,t)\in \Omega \times (0,+\infty ),\hfill \\ u\left(x,t\right)=0,\hfill & (x,t)\in {\Gamma }_0\times (0,+\infty ),\hfill \\ \frac{\partial u}{\partial \nu }=-u_t,\hfill & (x,t)\in {\Gamma }_1\times (0,+\infty ),\hfill \\ u\left(x,0\right)=u_0\left(x\right),\hfill & x\in \overline{\Omega },\hfill \end{array}\right.$$

(1)

where p and q are two positive constants that fulfill the condition $2\le q<p<q^{*}$, $q^{*}$ is the Sobolev conjugate of q, i.e., $q^{*}=\frac{Nq}{N-q}$ when $N>q$, while $q^{*}=+\infty$ when $N\le q$. $\Omega \subset {\mathbb{R}}^N$ is a bounded open domain with $C^1$ boundary $\partial \Omega$, $\partial \Omega ={\Gamma }_0\cup {\Gamma }_1$, ${\Gamma }_0\cap {\Gamma }_1=\varnothing$ (that is, ${\Gamma }_0\cap {\Gamma }_1$ equals an empty set), ${\Gamma }_0$ and ${\Gamma }_1$ are measurable over $\partial \Omega$, endowed with $(n-1)$-dimensional surface measure $\sigma$ and $\sigma \left({\Gamma }_0\right)>0$, and $\nu$ represents the unit outward normal vector to $\partial \Omega$.

Problem (1) with $q=2$ can be viewed as a model to describe a process of heat reaction–diffusion occurring inside a solid body $\Omega$ surrounded by a fluid, which contacts ${\Gamma }_1$ and has an internal cavity with contact boundary ${\Gamma }_0$. In this physics content, the quantity of heat produced by the reaction is proportional to the superlinear power of the temperature. To avoid the internal explosion in $\Omega$, a refrigeration system is installed in the fluid. The working mechanism of this refrigeration system is that the heat absorbed from the fluid is proportional to a power of the rate of change in the temperature, which can be expressed as

$$\frac{\partial u}{\partial \nu }=-{|u_t|}^{m-2}{u}_t,(x,t)\in {\Gamma }_1\times (0,+\infty ),$$

where $\frac{\partial u}{\partial \nu }$ stands for the heat flux from $\Omega$ to the fluid. One can see [1,2] for more details of the physics background.

In the past, various evolution equations with dynamic boundary conditions have attracted the attention of many mathematicians (see [3,4,5,6,7,8,9,10]). In particular, Fiscella et al. in [1] studied the following problem with local nonlinear boundary dissipation

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t-\Delta u={|u|}^{p-2}u,\hfill & (x,t)\in \Omega \times (0,+\infty ),\hfill \\ u(x,t)=0,\hfill & (x,t)\in {\Gamma }_0\times (0,+\infty ),\hfill \\ \frac{\partial u}{\partial \nu }=-{|u_t|}^{m-2}{u}_t,\hfill & (x,t)\in {\Gamma }_1\times (0,+\infty ),\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in \overline{\Omega },\hfill \end{array}\right.\end{array}$$

(2)

where $m>1$, $2\le p\le 1+\frac{2^{*}}{2}$, $2^{*}$ denotes the critical exponent of Sobolev embedding $H^1\left(\Omega \right)\hookrightarrow L^p\left(\Omega \right)$, i.e., $2^{*}=\frac{2N}{N-2}$ when $N\ge 3$, while $2^{*}=+\infty$ when $N=1,2$. Using the monotonicity method of J. L. lions and a contraction argument, they proved the local well posedness in the Hadamard sense. Moreover, for the case of the superlinear source, i.e., $p>2$, under the assumption that the initial energy is smaller than the mountain pass level, they showed the global existence and finite time blow-up results. To be exact, if

$$K\left(u_0\right)=\underset{\Omega }{\int }|\nabla u_0\left(x\right){|}^2\mathrm{d}x-\underset{\Omega }{\int }{|u_0\left(x\right)|}^p\mathrm{d}x\ge 0,$$

then the weak solution is global, while if

$$K\left(u_0\right)=\underset{\Omega }{\int }|\nabla u_0\left(x\right){|}^2\mathrm{d}x-\underset{\Omega }{\int }{|u_0\left(x\right)|}^p\mathrm{d}x\le 0,$$

and

$$m<\frac{2p(N+1)-4(N-1)}{N(p-2)+4},$$

then the solution will blow up in a finite time. Very recently, Sun et al. in [2] considered problem (2) with $m=2$ and obtained the finite time blow-up result for arbitrary high initial energy. Moreover, for some special cases, they obtained the estimates of the blow-up time. Recently, Yang et al. in [11] extended the results in [2] to the following porous medium equation

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t-\Delta u^r={|u|}^{p-2}u,\hfill & (x,t)\in \Omega \times (0,+\infty ),\hfill \\ u(x,t)=0,\hfill & (x,t)\in {\Gamma }_0\times (0,+\infty ),\hfill \\ \frac{\partial u^r}{\partial \nu }=-u_t,\hfill & (x,t)\in {\Gamma }_1\times (0,+\infty ),\hfill \\ u(x,0)=u_0\left(x\right),\hfill & x\in \overline{\Omega },\hfill \end{array}\right.\end{array}$$

(3)

where $r>1$ and $r+1<p<r\left(2^{*}-1\right)+1$. Under some appropriate assumptions on the initial datum $u_0\left(x\right)$, the blow-up criterion and the estimate of the blow-up time are obtained.

On the other hand, there are a lot of studies in the literature focused on the blow-up behaviors of the solutions to some non-Newton filtration equations with other boundary conditions. For example, Tan in [12] considered the following problem

$$\left\{\begin{array}{cc}\frac{u_t}{{\left|x\right|}^2}-\mathrm{div}\left({|\nabla u|}^{q-2}\nabla u\right)=u^p,\hfill & (x,t)\in \Omega \times (0,+\infty ),\hfill \\ u\left(x,t\right)=0,\hfill & (x,t)\in \partial \Omega \times (0,+\infty ),\hfill \\ u\left(x,0\right)=u_0\left(x\right),\hfill & x\in \overline{\Omega },\hfill \end{array}\right.$$

(4)

where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$ with $N\ge 3$, and $\Omega$ contains coordinate origin, $2<q<N$ and $q<p+1<\frac{Nq}{N-q}$, $\left|x\right|=\sqrt{x_1^2+\cdots +x_N^2}$ for $x=\left(x_1,\cdots ,x_N\right)\in \Omega$, $u_0\left(x\right)$ is a non-negative non-trivial function. When the initial energy is subcritical, by using the energy method and Hardy inequality, Tan obtained the sufficient conditions for the solutions to exist globally or to blow up in finite time. Moreover, the asymptotic estimate of the global solution is deduced. When problem (4) is at a critical initial energy level, Liu in [13] showed the global existence and finite time blow-up results of the solutions. When the initial energy is supercritical, Han in [14] proved that problem (4) admits blow-up solutions and obtained the estimates of the blow-up time from both above and below.

As far as we know, there is no result yet on the blow-up solutions of problem (1). Inspired by the work mentioned above, we will evaluate the impact of the q-Laplace operator, superlinear source, and local linear boundary dissipation term on the blow-up singularity of the solution. From semilinear to quasilinear, from the null Dirichlet boundary condition to the dynamic boundary condition, the structures and properties of the corresponding energy functionals and Nehari manifolds will change fundamentally. Hence, the methods used in previous works [1,2,12,13,14] cannot be directly used to handle our problem (1). More skills are needed to deal with the difficulties caused by the simultaneous occurrence of the q-Laplace operator and boundary dissipation term.

Since the first equation in problem (1) has degeneracy at the points where $\left|\nabla u\right|=0$, problem (1) might not have a classical solution in general. In order to state the definition of the weak solution and the main results of this article, we first introduce some notations. For given $q\in \left[1,+\infty \right]$, denote the norms of $L^q(\Omega )$ and $L^q\left({\Gamma }_1\right)$ by ${\parallel ·\parallel }_q$ and ${\parallel ·\parallel }_{q,{\Gamma }_1}$, respectively. $(·,·)$ and ${(·,·)}_{{\Gamma }_1}$ stand for the inner products on the Hilbert spaces $L^2(\Omega )$ and $L^2\left({\Gamma }_1\right)$, respectively. We introduce the Hilbert space

$$W_{{\Gamma }_0}^{1,q}(\Omega )=\left\{u\in W^{1,q}(\Omega ):u{|}_{{\Gamma }_0}=0\right\},$$

endowed with the norm

$${\parallel u\parallel }_{W_{{\Gamma }_0}^{1,q}}={\left({\parallel u\parallel }_q^q+{\parallel \nabla u\parallel }_q^q\right)}^{\frac{1}{q}},$$

where $u{|}_{{\Gamma }_0}$ represents the restriction of the trace of u on $\partial \Omega$ to ${\Gamma }_0$.

Definition 1. 

Suppose that $u_0\in W_{{\Gamma }_0}^{1,q}(\Omega )$ and $2\le q<p<q^{*}.$ A function $u:=u\left(x,t\right)\in L^{\infty }\left(0,T;W_{{\Gamma }_0}^{1,q}\left(\Omega \right)\right)$ with $u_t\in L^2\left(0,T;L^2\left(\Omega \right)\right)$ is called a weak solution of problem (1) on $\Omega \times [0,T)$ if $u\left(x,0\right)=u_0$, and

$$\begin{array}{c}\hfill \left(u_t,\varphi \right)+\left({\left|\nabla u\right|}^{q-2}\nabla u,\nabla \varphi \right)+{\left(u_t,\varphi \right)}_{{\Gamma }_1}=\underset{\Omega }{\int }{|u\left(t\right)|}^{p-2}u\varphi \mathrm{d}x,\end{array}$$

(5)

holds for a.e. $t\in (0,T)$ and any test function $\varphi \in W_{{\Gamma }_0}^{1,q}\left(\Omega \right)$. Moreover, the spatial trace of u on $\partial \Omega \times (0,T)$ has a distributional time derivative $u_t\in L^2\left(\partial \Omega \times \left(0,T\right)\right)$.

We define the energy functional (see [12])

$$J\left(u\left(t\right)\right)=\frac{1}{q}\parallel \nabla u\left(t\right){\parallel }_q^q-\frac{1}{p}{\parallel u\left(t\right)\parallel }_p^p,$$

the Nehari functional (see [12])

$$K\left(u\left(t\right)\right)=\parallel \nabla u\left(t\right){\parallel }_q^q-{\parallel u\left(t\right)\parallel }_p^p,$$

in $W_{{\Gamma }_0}^{1,q}\left(\Omega \right)$ related to problem (1), and the depth of potential well

$$d=inf\left\{\underset{\lambda >0}{sup}J\left(\lambda u\right):u\in W_{{\Gamma }_0}^{1,q}(\Omega )\setminus \left\{0\right\}\right\}.$$

We also define the Nehari manifold

$$\mathcal{N}=\{u\in W_{{\Gamma }_0}^{1,q}(\Omega )\mid K\left(u\right)=0\}\setminus \left\{0\right\},$$

and

$${\mathcal{N}}_{-}=\{u\in W_{{\Gamma }_0}^{1,q}(\Omega )\mid K\left(u\right)<0\}.$$

Then, d can also be characterized as

$$d=\underset{u\in \mathcal{N}}{inf}J\left(u\right)=\frac{p-q}{pq}{B}_1^{-\frac{pq}{p-q}}>0,$$

where

$$B_1=\underset{u\in W_{{\Gamma }_0}^{1,q}(\Omega )\setminus \left\{0\right\}}{sup}\frac{{\parallel u\parallel }_p}{{\parallel \nabla u\parallel }_q}.$$

Definition 2. 

Suppose that u is a weak solution of problem (1). We say that u blows up in finite time T if

$$\underset{t\to T^{-}}{lim}\rho \left(u\left(t\right)\right)=\underset{t\to T^{-}}{lim}\frac{1}{2}\left({\parallel u\parallel }_2^2+{\parallel u\parallel }_{2,{\Gamma }_1}^2\right)=+\infty .$$

The main results of this article are stated as follows.

Theorem 1. 

Suppose that $2\le q<p<q^{*}$ and the initial data $u_0$ is in one of the following two sets:

$${\mathcal{B}}_1=\left\{u\in W_{{\Gamma }_0}^{1,q}(\Omega ):J\left(u\right)<d\mathit{and}K\left(u\right)<0\right\},$$

$${\mathcal{B}}_2=\left\{u\in W_{{\Gamma }_0}^{1,q}(\Omega ):J\left(u\right)<\frac{p-q}{pq(S_1+S_2)}\left({\parallel u\parallel }_2^2+{\parallel w\parallel }_{2,{\Gamma }_1}^2\right)\right\},$$

where $S_1$ and $S_2$ are defined in (6). Then, the weak solution u of problem (1) will blow up in finite time T. In addition, one has

$$T\le \frac{4(p-1)}{p{(p-2)}^2}·\frac{\parallel u_0{\parallel }_2^2+{\parallel u_0\parallel }_{2,{\Gamma }_1}^2}{d-J\left(u_0\right)},$$

for $u_0\in {\mathcal{B}}_1$ and

$$T\le \frac{4(p-1)}{{(p-2)}^2}·\frac{\parallel u_0{\parallel }_2^2+{\parallel u_0\parallel }_{2,{\Gamma }_1}^2}{\frac{p-q}{q(S_1+S_2)}\left(\parallel u_0{\parallel }_2^2+{\parallel u_0\parallel }_{2,{\Gamma }_1}^2\right)-J\left(u_0\right)},$$

for $u_0\in {\mathcal{B}}_2$.

Corollary 1. 

For any $a\in \mathbb{R},$ denote the energy level set by

$$J^a:=\{u\in W_{{\Gamma }_0}^{1,q}(\Omega )|J\left(u\right)=a\}.$$

Then, $J^a\cap {\mathcal{B}}_2\ne \varnothing .$

Theorem 2. 

Suppose that

$$2\le q<p\le min\left\{q^{*},q\left(1+\frac{2}{N}\right)\right\},$$

(one can check that $min\left\{q^{*},q\left(1+\frac{2}{N}\right)\right\}=q\left(1+\frac{2}{N}\right)$. In fact, $q^{*}-q\left(1+\frac{2}{N}\right)=\frac{2q+N(q-2)}{N\left(N-q\right)}>0$ for $N>q$ and $q^{*}=+\infty >q\left(1+\frac{2}{N}\right)$ for $N\le q$). The weak solution u of problem (1) blows up in finite time T and $u\left(t\right)\in {\mathcal{N}}_{-}$ for any $t\in (0,T)$. Then,

$$T\ge \frac{\tilde{C}}{(\parallel u_0{\parallel }_2^2+\parallel u_0{{\parallel }_{2,{\Gamma }_1})}^{\frac{q(p-2)}{2q-N(p-q)}}},$$

where $\tilde{C}$ is a positive constant given by (26).

This paper is organized as follows. In Section 2, some definitions and notations are given, and some useful auxiliary lemmas are collected. In Section 3, the conditions on the occurrence of blow-up singularity are discussed, and the bounds of the blow-up time are deduced. Finally, we give a conclusion in Section 4.

2. Preliminaries

From the trace theorem, it follows that there exists a continuous trace mapping $W_{{\Gamma }_0}^{1,q}(\Omega )\hookrightarrow L^q(\partial \Omega )$. On the other hand, since $\sigma \left({\Gamma }_0\right)>0$, a Poincaré-type inequality holds (see [15]), and hence ${\parallel \nabla u\parallel }_q$ can be viewed as an equivalent norm to the norm

$${\parallel u\parallel }_{W_{{\Gamma }_0}^{1,q}}={\left({\parallel u\parallel }_q^q+{\parallel \nabla u\parallel }_q^q\right)}^{\frac{1}{q}},$$

in the space $W_{{\Gamma }_0}^{1,q}(\Omega )$. Throughout this article, we denote

$$\begin{array}{c}\hfill S_1=\underset{u\in W_{{\Gamma }_0}^{1,q}(\Omega )\setminus \left\{0\right\}}{sup}\frac{{\parallel u\parallel }_{2,{\Gamma }_1}^2}{{\parallel \nabla u\parallel }_q^q},S_2=\underset{u\in W_{{\Gamma }_0}^{1,q}(\Omega )\setminus \left\{0\right\}}{sup}\frac{{\parallel u\parallel }_2^2}{{\parallel \nabla u\parallel }_q^q}.\end{array}$$

(6)

From the definitions of $J\left(u\right(t\left)\right)$, $K\left(u\right(t\left)\right)$, and $\rho \left(u\right(t\left)\right)$, one knows that they are continuous on $W_{{\Gamma }_0}^{1,q}\left(\Omega \right)$. Furthermore, one has

$$\frac{\mathrm{d}}{\mathrm{d}t}\rho \left(u\left(t\right)\right)=\left(u\left(t\right),u_t\left(t\right)\right)+{\left(u\left(t\right),u_t\left(t\right)\right)}_{{\Gamma }_1}=-K\left(u\left(t\right)\right),$$

(7)

and

$$\begin{array}{c}\hfill \frac{\mathrm{d}}{\mathrm{d}t}J\left(u\left(t\right)\right)=-\left(\parallel u_t{\left(t\right)\parallel }_{2,{\Gamma }_1}^2+{\parallel u_t\left(t\right)\parallel }_2^2\right)\le 0,\end{array}$$

(8)

which implies that

$$\begin{array}{c}\hfill J\left(u\left(t\right)\right)-J\left(u_0\right)=-\underset{0}{\overset{t}{\int }}\left(\parallel u_{\tau }{\left(\tau \right)\parallel }_2^2+{\parallel u_{\tau }\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau .\end{array}$$

(9)

We now give some lemmas, which play a key role in our proofs of the main results.

Lemma 1. 

Suppose that $2\le q<p<q^{*}$ and $u\in {\mathcal{N}}_{-}$. Then,

$${\parallel u\parallel }_p^p>\frac{pq}{p-q}d.$$

Proof. 

Noticing that ${\parallel \nabla u\parallel }_q^q-{\parallel u\parallel }_p^p=K\left(u\right)<0,$ one has $u\ne 0$. On the other hand, by a simple calculation, one has $K\left({\lambda }^{*}u\right)=0$ with ${\lambda }^{*}={\left(\frac{{\parallel \nabla u\parallel }_q^q}{{\parallel u\parallel }_p^p}\right)}^{\frac{1}{p-q}}\in (0,1)$. That is to say, ${\lambda }^{*}u\in \mathcal{N}$. With the help of the definition of d, one can find that

$$\begin{array}{cc}\hfill d=\underset{u\in \mathcal{N}}{inf}J\left(u\right)\le & J\left({\lambda }^{*}u\right)\hfill \\ \hfill =& \frac{p-q}{pq}{∥{\lambda }^{*}u∥}_p^p+\frac{1}{q}K\left({\lambda }^{*}u\right)\hfill \\ \hfill =& \frac{p-q}{pq}{\left({\lambda }^{*}\right)}^p{\parallel u\parallel }_p^p\hfill \\ \hfill <& \frac{p-q}{pq}{\parallel u\parallel }_p^p,\hfill \end{array}$$

which leads to the desired result. The proof of Lemma 1 is complete. □

Lemma 2. 

Suppose that $2\le q<p<q^{*}$ and the weak solution u of problem (1) blows up in finite time T. Then, there is a $t^{*}\in [0,T)$, such that $u\left(t^{*}\right)\in {\mathcal{N}}_{-}.$

Proof. 

By contradiction, suppose that $K\left(u\right(t\left)\right)\ge 0$ for any $t\in [0,T).$ Then, it follows from (9) that

$$\begin{array}{cc}\hfill J\left(u_0\right)\ge J\left(u\left(t\right)\right)& =\frac{1}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q-\frac{1}{p}{\parallel u\left(t\right)\parallel }_p^p\hfill \\ \hfill & =\frac{p-q}{pq}{\parallel \nabla u\parallel }_q^q+\frac{1}{p}K\left(u\left(t\right)\right)\hfill \\ \hfill & \ge \frac{p-q}{pq}{\parallel \nabla u\left(t\right)\parallel }_q^q,\hfill \end{array}$$

which contradicts the assumption that u is a finite time blow-up weak solution. The proof of Lemma 2 is complete. □

Our blow-up criterion is based on the following Levine’s concavity method (see [16]).

Lemma 3. 

(see [16]). Suppose that a positive function F on $[0,T]$ fulfills the following conditions:

(i) F is differentiable on $[0,T]$, and $F^{\prime }$ is absolutely continuous on $[0,T]$ with $F^{\prime }\left(0\right)>0$;

(ii) there is a positive constant $\alpha >0$ such that, for a.e. $t\in [0,T]$,

$$F\left(t\right)F^{″}\left(t\right)-\left(1+\alpha \right){\left(F^{\prime }\left(t\right)\right)}^2\ge 0.$$

Then,

$$T\le \frac{F\left(0\right)}{\alpha F^{\prime }\left(0\right)}.$$

3. The Finite Time Blow-Up Results

For arbitrary $T_1\in \left(0,T\right)$, define

$$\begin{array}{c}\hfill F\left(t\right)=\underset{0}{\overset{t}{\int }}\rho \left(u\left(\tau \right)\right)\mathrm{d}\tau +(T_1-t)\rho \left(u_0\right)+\frac{1}{2}\beta {(t+\sigma )}^2,t\in [0,T_1],\end{array}$$

(10)

where $\beta$ and $\sigma$ are two positive parameters, which will be determined later. From (7), one has

$$\begin{array}{cc}\hfill F^{\prime }\left(t\right)& =\rho \left(u\left(t\right)\right)-\rho \left(u_0\right)+\beta (t+\sigma )\hfill \\ \hfill & =\underset{0}{\overset{t}{\int }}\frac{\mathrm{d}}{\mathrm{d}\tau }\rho \left(u\left(\tau \right)\right)\mathrm{d}\tau +\beta (t+\sigma )\hfill \\ \hfill & =\underset{0}{\overset{t}{\int }}\left[(u,u_{\tau })+{(u,u_{\tau })}_{{\Gamma }_1}\right]\mathrm{d}\tau +\beta (t+\sigma ).\hfill \end{array}$$

(11)

Combining (9) with the definitions of $K\left(u\left(t\right)\right)$ and $J\left(u\left(t\right)\right)$ yields

$$\begin{array}{cc}\hfill F^{″}\left(t\right)& =\frac{\mathrm{d}}{\mathrm{d}t}\rho \left(u\left(t\right)\right)+\beta =-K\left(u\left(t\right)\right)+\beta =-\left(pJ\left(u\left(t\right)\right)-\frac{p-q}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q\right)+\beta \hfill \\ \hfill & =\frac{p-q}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q+p\left[\underset{0}{\overset{t}{\int }}\left(\parallel u_{\tau }{\left(\tau \right)\parallel }_2^2+{\parallel u_{\tau }\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau -J\left(u_0\right)\right]+\beta .\hfill \end{array}$$

(12)

Moreover, it is easy to verify that F is positive on $[0,T_1]$ and $F^{\prime }\left(0\right)=\frac{1}{2}\beta \sigma >0.$ Now, we are in the position to estimate $F{F}^{″}-\lambda {\left(F^{\prime }\right)}^2$, where $\lambda >1$ is a constant which will be given later. Using Cauchy–Schwartz inequality and Young’s inequality, one can prove that

$$\begin{array}{ccc}\xi \left(t\right)& :=& \left[\underset{0}{\overset{t}{\int }}\left({\parallel u\left(\tau \right)\parallel }_2^2+{\parallel u\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau +\beta {(t+\sigma )}^2\right]·\left[\underset{0}{\overset{t}{\int }}\left(\parallel u_{\tau }{\left(\tau \right)\parallel }_2^2+{\parallel u_{\tau }\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau +\beta \right]\\ \hfill & \hfill & -{\left[\underset{0}{\overset{t}{\int }}(u,u_{\tau })\mathrm{d}\tau +\underset{0}{\overset{t}{\int }}{(u,u_{\tau })}_{{\Gamma }_1}\mathrm{d}\tau +\beta (t+\sigma )\right]}^2\hfill \\ \hfill & \hfill =& \left[\underset{0}{\overset{t}{\int }}{\parallel u\left(\tau \right)\parallel }_2^2\mathrm{d}\tau ·\underset{0}{\overset{t}{\int }}{\parallel u_{\tau }\left(\tau \right)\parallel }_2^2-{\left(\underset{0}{\overset{t}{\int }}(u,u_{\tau })\mathrm{d}\tau \right)}^2\right]\hfill \\ \hfill & \hfill & +\left[\underset{0}{\overset{t}{\int }}{\parallel u\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\mathrm{d}\tau ·\underset{0}{\overset{t}{\int }}{\parallel u_{\tau }\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\mathrm{d}\tau -{\left(\underset{0}{\overset{t}{\int }}{(u,u_{\tau })}_{{\Gamma }_1}\mathrm{d}\tau \right)}^2\right]\hfill \\ \hfill & \hfill & +[\underset{0}{\overset{t}{\int }}{\parallel u\left(\tau \right)\parallel }_2^2\mathrm{d}\tau ·\underset{0}{\overset{t}{\int }}\parallel u_{\tau }{\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\mathrm{d}\tau +\underset{0}{\overset{t}{\int }}{\parallel u\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\mathrm{d}\tau ·\underset{0}{\overset{t}{\int }}{\parallel u_{\tau }\left(\tau \right)\parallel }_2^2\mathrm{d}\tau \hfill \\ \hfill & \hfill & -2\underset{0}{\overset{t}{\int }}(u,u_{\tau })\mathrm{d}\tau ·\underset{0}{\overset{t}{\int }}{(u,u_{\tau })}_{{\Gamma }_1}\mathrm{d}\tau ]\hfill \\ \hfill & \hfill & +\left[\beta {(t+\sigma )}^2{\int }_0^t\parallel u_{\tau }{\left(\tau \right)\parallel }_2^2\mathrm{d}\tau +\beta {\int }_0^t{\parallel u\left(\tau \right)\parallel }_2^2\mathrm{d}\tau -2\beta (t+\sigma ){\int }_0^t(u,u_{\tau })\mathrm{d}\tau \right]\hfill \\ \hfill & \hfill & +\left[\beta {(t+\sigma )}^2\underset{0}{\overset{t}{\int }}\parallel u_{\tau }{\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\mathrm{d}\tau +\beta \underset{0}{\overset{t}{\int }}{\parallel u\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\mathrm{d}\tau -2\beta (t+\sigma )\underset{0}{\overset{t}{\int }}{(u,u_{\tau })}_{{\Gamma }_1}\mathrm{d}\tau \right]\hfill \\ \hfill & \hfill \ge & 0.\hfill \end{array}$$

Therefore, in slight of (10), (11), and (12), one obtains

$$\begin{array}{c}F{F}^{″}-\lambda {\left(F^{\prime }\right)}^2\stackrel{\left(11\right)}{=}F{F}^{″}-\lambda {\left\{\underset{0}{\overset{t}{\int }}\left[(u,u_{\tau })+{(u,u_{\tau })}_{{\Gamma }_1}\right]\mathrm{d}\tau +\beta (t+\sigma )\right\}}^2\hfill \\ =F{F}^{″}+\lambda \left\{\xi \left(t\right)-\left[\underset{0}{\overset{t}{\int }}\left({∥u\left(\tau \right)∥}_2^2+{∥u\left(\tau \right)∥}_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau +\beta {(t+\sigma )}^2\right]\right.\hfill \\ ·\left.\left[\underset{0}{\overset{t}{\int }}\left({∥u_{\tau }\left(\tau \right)∥}_2^2+{∥u_{\tau }\left(\tau \right)∥}_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau +\beta \right]\right\}\hfill \\ \stackrel{\left(10\right)}{=}F{F}^{″}+\lambda \left\{\xi \left(t\right)-2\left[F\left(t\right)-(T-t)\rho \left(u_0\right)\right]·\left[\underset{0}{\overset{t}{\int }}\left(\parallel u_{\tau }{\left(\tau \right)\parallel }_2^2+{\parallel u_{\tau }\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau +\beta \right]\right\}\hfill \\ \ge F{F}^{″}-2\lambda F\left(t\right)·\left[\underset{0}{\overset{t}{\int }}\left(\parallel u_{\tau }{\left(\tau \right)\parallel }_2^2+{\parallel u_{\tau }\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau +\beta \right]\hfill \\ \stackrel{\left(12\right)}{=}F\left(t\right)·\left\{\left[\frac{p-q}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q+p\underset{0}{\overset{t}{\int }}\left(\parallel u_{\tau }{\left(\tau \right)\parallel }_2^2+{\parallel u_{\tau }\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau -pJ\left(u_0\right)+\beta \right]\right.\hfill \\ \left.-2\lambda \left[\underset{0}{\overset{t}{\int }}\left(\parallel u_{\tau }{\left(\tau \right)\parallel }_2^2+{\parallel u_{\tau }\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau +\beta \right]\right\}\hfill \\ =F\left(t\right)\hfill \\ ·\left[\frac{p-q}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q+(p-2\lambda )\underset{0}{\overset{t}{\int }}\left(\parallel u_{\tau }{\left(\tau \right)\parallel }_2^2+{\parallel u_{\tau }\left(\tau \right)\parallel }_{2,T_1}^2\right)\mathrm{d}\tau -pJ\left(u_0\right)+(1-2\lambda )\beta \right].\hfill \end{array}$$

Choosing $\lambda =\frac{p}{2}$, then from the above inequality, one has, for a.e. $t\in [0,T_1]$,

$$\begin{array}{c}\hfill F{F}^{″}-\frac{p}{2}{\left(F^{″}\right)}^2\ge F·\left[\frac{p-q}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q-pJ\left(u_0\right)-\beta (p-1)\right].\end{array}$$

(13)

So far, based on the above analysis, one can immediately summarize the following lemma.

Lemma 4. 

Suppose that $\rho \left(u_0\right)>0$ and there exists some $\beta >0$ such that

$$\frac{p-q}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q-pJ\left(u_0\right)-(p-1)\beta \ge 0,$$

holds for any $t\in (0,T_1].$ Then,

$$0<T_1\le \frac{8\rho \left(u_0\right)}{\beta {(p-2)}^2}.$$

Proof. 

An application of Lemma 3 and (13) gives us

$$\begin{array}{c}\hfill T_1\le \frac{F\left(0\right)}{\frac{p-2}{2}·F^{\prime }\left(0\right)}=\frac{T_1\rho \left(u_0\right)+\frac{1}{2}\beta {\sigma }^2}{\frac{p-2}{2}·\sigma \beta }=\frac{2\rho \left(u_0\right)}{\sigma \beta (p-2)}{T}_1+\frac{\sigma }{p-2}.\end{array}$$

(14)

Picking $\sigma \in \left(\frac{2\rho \left(u_0\right)}{\beta (p-2)},+\infty \right)$ such that

$$\frac{2\rho \left(u_0\right)}{\sigma \beta (p-2)}<1,$$

thereupon, (14) is applicable to produce

$$T_1\le \frac{\sigma }{p-2}{\left(1-\frac{2\rho \left(u_0\right)}{\sigma \beta (p-2)}\right)}^{-1}.$$

(15)

A straightforward calculation tells us that the right side of (15) takes its minimum at

$$\sigma ={\sigma }_{\beta }=\frac{4\rho \left(u_0\right)}{\beta (p-2)}\in \left(\frac{2\rho \left(u_0\right)}{\beta (p-2)},+\infty \right).$$

Due to $T_1$ being independent of $\sigma$, it yields that

$$T_1\le \frac{{\sigma }_{\beta }}{p-2}{\left(1-\frac{2\rho \left(u_0\right)}{{\sigma }_{\beta }\beta (p-2)}\right)}^{-1}=\frac{8\rho \left(u_0\right)}{\beta {(p-2)}^2}.$$

The proof of Lemma 4 is complete. □

Up to now, the proof of Theorem 1 can be given as follows.

Proof of Theorem 1. 

For the case $u_0\in {\mathcal{B}}_1$, we first prove that $u\left(t\right)\in {\mathcal{B}}_1$ for any $t\in [0,T)$. In view of (9), one has

$$\begin{array}{c}\hfill J\left(u\left(t\right)\right)=J\left(u_0\right)-\underset{0}{\overset{t}{\int }}\left(\parallel u_{\tau }{\left(\tau \right)\parallel }_2^2+{\parallel u_{\tau }\left(\tau \right)\parallel }_{2,{\Gamma }_1}^2\right)\mathrm{d}\tau \le J\left(u_0\right)<d.\end{array}$$

(16)

Now, our goal is to show that $K\left(u\right(t\left)\right)<0$ for any $t\in [0,T)$. To do this, we suppose, on the contrary, that there is a $t_1\in [0,T)$, such that $K\left(u\left(t_1\right)\right)=0$ and $K\left(u\right(t\left)\right)<0$ in $[0,t_1)$. By applying Lemma 1 and the continuity of the mapping $t\mapsto {\parallel \nabla u\left(t\right)\parallel }_q$, one obtains

$$\parallel \nabla u\left(t_1\right){\parallel }_q^q=\underset{t\to t_1^{-}}{lim}{\parallel \nabla u\left(t\right)\parallel }_q^q\ge \frac{pq}{p-q}d>0.$$

Hence, $u\left(t_1\right)\in \mathcal{N}.$ But, we infer from the definition of the potential well depth d that

$$d=\underset{u\in \mathcal{N}}{inf}J\left(u\right)\le J\left(u\left(t_1\right)\right),$$

which contradicts (16).

Furthermore, it is clear that $\rho \left(u_0\right)>0.$ Then, making use of Lemma 1 again, one can claim that

$$\frac{p-q}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q-pJ\left(u_0\right)-(p-1)\beta >pd-pJ\left(u_0\right)-(p-1)\beta \ge 0,$$

which holds for $t\in (0,T)$, and

$$\beta \in \left(0,\frac{p}{p-1}\left(d-J\left(u_0\right)\right)\right].$$

Up to now, Lemma 4 can be used to deduce that

$$0<T\le \frac{8(p-1)\rho \left(u_0\right)}{p{(p-2)}^2\left(d-J\left(u_0\right)\right)}=\frac{4(p-1)}{p{(p-2)}^2}·\frac{\parallel u_0{\parallel }_2^2+{\parallel u_0\parallel }_{2,{\Gamma }_1}^2}{d-J\left(u_0\right)}<\infty .$$

For the case $u_0\in {\mathcal{B}}_2.$ By (6) and (7), one has, for a.e. $t\in (0,T)$,

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}t}\rho \left(u\left(t\right)\right)=-K\left(u\left(t\right)\right)& =\frac{p-q}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q-pJ\left(u\left(t\right)\right)\hfill \\ \hfill & \ge \frac{p-q}{q}·\frac{2}{S_1+S_2}\rho \left(u\left(t\right)\right)-pJ\left(u\left(t\right)\right)\hfill \\ \hfill & =\frac{p}{A}\left(\rho \left(u\right(t\left)\right)-AJ\left(u\right(t\left)\right)\right),\hfill \end{array}$$

(17)

where

$$A=\frac{pq(S_1+S_2)}{2(p-q)}>0.$$

Set $H\left(t\right)=\rho \left(u\right(t\left)\right)-AJ\left(u\right(t\left)\right)$. With (8) and (17) in mind, one has, for a.e. $t\in (0,T)$,

$$\frac{\mathrm{d}}{\mathrm{d}t}H\left(t\right)=\frac{\mathrm{d}}{\mathrm{d}t}\rho \left(u\left(t\right)\right)-A\frac{\mathrm{d}}{\mathrm{d}t}J\left(u\left(t\right)\right)\ge \frac{\mathrm{d}}{\mathrm{d}t}\rho \left(u\left(t\right)\right)\ge \frac{p}{A}H\left(t\right),$$

together with Gronwall’s inequality (see [17]), this leads to

$$H\left(t\right)\ge e^{\frac{p}{A}t}H\left(0\right).$$

(18)

On the other hand, the hypothesis $u_0\in {\mathcal{B}}_2$ tells us that

$$H\left(0\right)=\rho \left(u_0\right)-AJ\left(u_0\right)=\frac{1}{2}\left(\parallel u_0{\parallel }_2^2+{\parallel u_0\parallel }_{2,T_1}^2\right)-\frac{p(S_1+S_2)}{p-q}J\left(u_0\right)>0.$$

(19)

Combining (17) and (18) with (19) results in

$$\frac{\mathrm{d}}{\mathrm{d}t}\rho \left(u\left(t\right)\right)\ge \frac{p}{A}H\left(t\right)\ge \frac{p}{A}{e}^{\frac{p}{A}t}H\left(0\right)>0\mathrm{for}\mathrm{a}.\mathrm{e}.t\in (0,T),$$

which implies that $\rho \left(u\right(t\left)\right)$ is nondecreasing over $[0,T)$. Furthermore, bearing (6) in mind, one can claim that

$$\begin{array}{cc}\hfill & \frac{p-q}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q-pJ\left(u_0\right)-(p-1)\beta \hfill \\ \hfill & \ge \frac{p-q}{q}·\frac{2}{S_1+S_2}\rho \left(u\left(t\right)\right)-pJ\left(u_0\right)-(p-1)\beta \hfill \\ \hfill & \ge \frac{p-q}{q}·\frac{2}{S_1+S_2}\rho \left(u_0\right)-pJ\left(u_0\right)-(p-1)\beta \hfill \\ \hfill & =\frac{p}{A}H\left(0\right)-(p-1)\beta \hfill \\ \hfill & \ge 0,\hfill \end{array}$$

provided that $\beta \in \left(0,\frac{pH\left(0\right)}{A(p-1)}\right].$ A simple application of Lemma 4 yields

$$0<T\le \frac{8A(p-1)\rho \left(u_0\right)}{p{(p-2)}^{2}H\left(0\right)},$$

namely,

$$T\le \frac{4(p-1)}{{(p-2)}^2}·\frac{\parallel u_0{\parallel }_2^2+{\parallel u_0\parallel }_{2,{\Gamma }_1}^2}{\frac{p-q}{q(S_1+S_2)}\left(\parallel u_0{\parallel }_2^2+{\parallel u_0\parallel }_{2,{\Gamma }_1}^2\right)-J\left(u_0\right)}.$$

The proof of Theorem 1 is complete. □

With the help of the proof of Theorem 1, one can understand that the sets ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ are invariant under the semi-flow related to problem (1). Roughly speaking, if the initial data $u_0\in {\mathcal{B}}_1$, then for any $t\in (0,T)$, $u\left(t\right)$ still belongs to ${\mathcal{B}}_1$, and if the initial data $u_0\in {\mathcal{B}}_2$, then $u\left(t\right)\in {\mathcal{B}}_2$ also holds for any $t\in (0,T)$. On the other hand, when $u\left(t\right)\in {\mathcal{B}}_2$, a simple application of (6) results in

$$\begin{array}{cc}\hfill K\left(u\left(t\right)\right)& =\parallel \nabla u\left(t\right){\parallel }_q^q-\parallel u\left(t\right){\parallel }_p^p=pJ\left(u\left(t\right)\right)-\frac{p-q}{q}{\parallel \nabla u\left(t\right)\parallel }_q^q\hfill \\ \hfill & \le p\left[J\left(u\left(t\right)\right)-\frac{p-q}{pq(S_1+S_2)}\left(\parallel u\left(t\right){\parallel }_2^2+{\parallel u\left(t\right)\parallel }_{2,{\Gamma }_1}^2\right)\right]<0.\hfill \end{array}$$

Thereupon, if the initial data $u_0\in {\mathcal{B}}_1\cup {\mathcal{B}}_2,$ then one has, for any $t\in (0,T)$,

$$u\left(t\right)\in {\mathcal{N}}_{-}=\{u\in W_{{\Gamma }_0}^{1,q}(\Omega )\mid K\left(u\right)<0\}.$$

Based on the above discussion, a question naturally arises in mind, that is, whether the assumption $u_0\in {\mathcal{N}}_{-}$ is sufficient enough to ensure the occurrence of the finite time blow-up phenomenon or not. Giving an answer to this equation is not an easy task, and we have little idea of how to handle it at present. Here we only refer readers to [18] for a similar study.

In addition, it is necessary to point out that both ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ are non-empty. The following proof of Corollary 1 tells us that, for any $a\in \mathbb{R},$ there is an initial datum $u_0\in W_{{\Gamma }_0}^{1,q}(\Omega )$ with initial energy $J\left(u_0\right)=a$, which leads to the finite time blow-up solution.

Proof of Corollary 1. 

We shall use the technique introduced by Sun et al. in [2] to complete the result. Suppose that ${\Omega }_1$ and ${\Omega }_2$ are two disjoint open subdomains of $\Omega$, which fulfill

$$\mathrm{dist}\left({\overline{\Omega }}_1,\partial \Omega \right)>0,\mathrm{dist}\left({\overline{\Omega }}_2,\partial \Omega \right)>0\mathrm{and}\mathrm{dist}\left({\overline{\Omega }}_1,{\overline{\Omega }}_2\right)>0.$$

A direct application of the proof of Theorem 3.7 in [19] tells us that there is a sequence $\left\{v_k\right\}\subset W_0^{1,q}\left({\Omega }_1\right)$, such that

$$\frac{1}{q}\underset{{\Omega }_1}{\int }|\nabla v_k{\left(x\right)|}^q\mathrm{d}x-\frac{1}{p}\underset{{\Omega }_1}{\int }{|v_k\left(x\right)|}^p\mathrm{d}x\to +\infty \mathrm{as}k\to +\infty .$$

(20)

On the other hand, by taking a nonzero function $w\in C_0^{\infty }(\Omega )$ satisfying $\mathrm{supp}w\subset {\Omega }_2,$ one can conclude that, for any $a\in \mathbb{R},$

$$\begin{array}{c}\hfill a-\left(\frac{1}{q}\underset{{\Omega }_2}{\int }{|\nabla \left(rw\left(x\right)\right)|}^q\mathrm{d}x-\frac{1}{p}\underset{{\Omega }_2}{\int }{|rw\left(x\right)|}^p\mathrm{d}x\right)\to +\infty \mathrm{as}r\to +\infty ,\end{array}$$

(21)

and there is $r_0>0$, such that, for any $r>r_0$,

$$\begin{array}{c}\hfill \frac{p-q}{p(S_1+S_2)}\underset{{\Omega }_2}{\int }{|rw\left(x\right)|}^2\mathrm{d}x=r^2·\frac{p-q}{p(S_1+S_2)}\underset{{\Omega }_2}{\int }{|w\left(x\right)|}^2\mathrm{d}x>a.\end{array}$$

(22)

With (20) and (21) in mind, one can claim that there are $k_0\in {\mathbb{N}}_{+}=\left\{1,2,\cdots \right\}$ and $r_1>r_0$, such that

$$\begin{array}{c}\hfill \frac{1}{q}\underset{{\Omega }_1}{\int }|\nabla v_{k_0}{\left(x\right)|}^q\mathrm{d}x-\frac{1}{p}\underset{{\Omega }_1}{\int }{|v_{k_0}\left(x\right)|}^p\mathrm{d}x=a-\left(\frac{1}{q}\underset{{\Omega }_2}{\int }|\nabla \left(r_{1}w\left(x\right)\right){|}^q\mathrm{d}x-\frac{1}{p}\underset{{\Omega }_2}{\int }{|r_{1}w\left(x\right)|}^p\mathrm{d}x\right).\end{array}$$

(23)

Put $u_0=\tilde{v}+r_{1}w,$ where

$$\begin{array}{cc}\hfill \tilde{v}\left(x\right)& =\left\{\begin{array}{cc}0,\hfill & x\in \Omega \setminus {\Omega }_1,\hfill \\ v_{k_0}\left(x\right),\hfill & x\in {\Omega }_1.\hfill \end{array}\right.\hfill \\ \hfill \end{array}$$

It is clear that $u_0\in W_{{\Gamma }_0}^{1,q}(\Omega )$ and $u_0\left(x\right)=0$ in $\Omega \setminus ({\Omega }_1\cup {\Omega }_2)$. Keeping (22) and (23) in mind leads to

$$\begin{array}{cc}\hfill J\left(u_0\right)& =\frac{1}{q}\left(\underset{{\Omega }_1}{\int }+\underset{{\Omega }_2}{\int }\right)|\nabla u_0{\left(x\right)|}^q\mathrm{d}x-\frac{1}{p}\left(\underset{{\Omega }_1}{\int }+\underset{{\Omega }_2}{\int }\right){|u_0\left(x\right)|}^p\mathrm{d}x\hfill \\ \hfill & =\left(\frac{1}{q}\underset{{\Omega }_1}{\int }|\nabla v_{k_0}{\left(x\right)|}^q\mathrm{d}x-\frac{1}{p}\underset{{\Omega }_1}{\int }{|v_{k_0}\left(x\right)|}^p\mathrm{d}x\right)\hfill \\ \hfill & +\left(\frac{1}{q}\underset{{\Omega }_2}{\int }|\nabla \left(r_{1}w\left(x\right)\right){|}^q\mathrm{d}x-\frac{1}{p}\underset{{\Omega }_2}{\int }{|r_{1}w\left(x\right)|}^p\mathrm{d}x\right)\hfill \\ \hfill & \stackrel{\left(23\right)}{=}a\hfill \\ \hfill & \stackrel{\left(22\right)}{<}\frac{p-q}{pq(S_1+S_2)}\underset{{\Omega }_2}{\int }{\left|r_{1}w\left(x\right)\right|}^2\mathrm{d}x\hfill \\ \hfill & =\frac{p-q}{pq(S_1+S_2)}\underset{{\Omega }_2}{\int }{|u_0\left(x\right)|}^2\mathrm{d}x\hfill \\ \hfill & \le \frac{p-q}{pq(S_1+S_2)}\left(\parallel u_0{\parallel }_2^2+{\parallel u_0\parallel }_{2,{\Gamma }_1}^2\right),\hfill \end{array}$$

which means that $u_0\in J^a\cap {\mathcal{B}}_2.$ The proof of Corollary 1 is complete. □

Proof of Theorem 2. 

Noticing that $u\in {\mathcal{N}}_{-}$ for any $t\in (0,T),$ then Lemma 1 and the Sobolev embedding theorem lead to ${\parallel \nabla u\left(t\right)\parallel }_q>0$, ${\parallel u\left(t\right)\parallel }_p>0$ and

$$\rho \left(u\left(t\right)\right)=\frac{1}{2}{\parallel u\left(t\right)\parallel }_2^2+\frac{1}{2}{\parallel u\left(t\right)\parallel }_{2,{\Gamma }_1}^2\ge \frac{1}{2}{\parallel u\left(t\right)\parallel }_2^2>0.$$

A simple application of Gagliardo–Nirenberg interpolation inequality (see [17]) results in

$$\begin{array}{c}\hfill {\parallel \nabla u\left(t\right)\parallel }_q^q<{\parallel u\left(t\right)\parallel }_p^p\le S_3{\parallel u\left(t\right)\parallel }_2^{p(1-\sigma )}{\parallel \nabla u\left(t\right)\parallel }_q^{p\sigma },\end{array}$$

(24)

where $S_3$ is a positive constant and $\sigma =\frac{Nq(p-2)}{p(2q-2N+Nq)}.$ Keeping $p<q\left(1+\frac{2}{N}\right)$ in mind, one can easily check that $q-p\sigma >0$. Hence, one can infer from (24) that

$${\parallel \nabla u\left(t\right)\parallel }_q\le S_3^{\frac{1}{q-p\sigma }}{\parallel u\left(t\right)\parallel }_2^{\frac{p-p\sigma }{q-p\sigma }}.$$

Combining (7) with (24) tells us that

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}t}\rho \left(u\left(t\right)\right)& =-K\left(u\left(t\right)\right)={\parallel u\left(t\right)\parallel }_p^p-{\parallel \nabla u\left(t\right)\parallel }_q^q<{\parallel u\left(t\right)\parallel }_p^p\hfill \\ \hfill & \le S_3{\parallel u\left(t\right)\parallel }_2^{p(1-\sigma )}{\parallel \nabla u\left(t\right)\parallel }_q^{p\sigma }\hfill \\ \hfill & \le S_3{\parallel u\left(t\right)\parallel }_2^{p(1-\sigma )}·S_3^{\frac{p\sigma }{q-p\sigma }}{\parallel u\left(t\right)\parallel }_2^{\frac{p\sigma (p-p\sigma )}{q-p\sigma }}=S_3^{\frac{q}{q-p\sigma }}{\parallel u\left(t\right)\parallel }_2^{2·\frac{pq(1-\sigma )}{q-p\sigma }}\hfill \\ \hfill & \le S_3^{\frac{q}{q-p\sigma }}{\left[2\rho \left(u\right(t\left)\right)\right]}^{\frac{pq(1-\sigma )}{q-p\sigma }}=S_4{\left[\rho \left(t\right)\right]}^{\frac{pq(1-\sigma )}{q-p\sigma }},\hfill \end{array}$$

where

$$S_4=S_3^{\frac{q}{q-p\sigma }}·2^{\frac{pq(1-\sigma )}{q-p\sigma }}>0.$$

Recalling that $\frac{p-p\sigma }{q-p\sigma }>1,$ integrating the differential inequality

$$\frac{\mathrm{d}}{\mathrm{d}t}\rho \left(u\left(t\right)\right)<S_4{\left[\rho \left(u\left(t\right)\right)\right]}^{\frac{pq(1-\sigma )}{q-p\sigma }},$$

from 0 to t gives us that

$$\begin{array}{c}\hfill {\rho }^{-\frac{pq(1-\sigma )-2(q-p\sigma )}{2(q-p\sigma )}}\left(u\left(t\right)\right)-{\rho }^{-\frac{pq(1-\sigma )-2(q-p\sigma )}{2(q-p\sigma )}}\left(u_0\right)>-S_4·\frac{pq(1-\sigma )-2(q-p\sigma )}{2(q-p\sigma )}t.\end{array}$$

(25)

Thanks to the weak solution u that blows up in finite time, one has $\underset{t\to T^{-}}{lim}\rho \left(u\left(t\right)\right)=+\infty .$ Letting $t\to T^{-}$ in (25) yields

$$-{\rho }^{-\frac{pq(1-\sigma )-2(q-p\sigma )}{2(q-p\sigma )}}\left(u_0\right)\ge -S_4·\frac{pq(1-\sigma )-2(q-p\sigma )}{2(q-p\sigma )}T,$$

which is equivalent to

$$T\ge \frac{2(q-p\sigma )}{S_4[pq(1-\sigma )-2(q-p\sigma )]}{\rho }^{-\frac{pq(1-\sigma )-2(q-p\sigma )}{2(q-p\sigma )}}\left(u_0\right)=\frac{\tilde{C}}{(\parallel u_0{\parallel }_2^2+\parallel u_0{{\parallel }_2)}^{\frac{q(p-2)}{2q-N(p-q)}}},$$

where

$$\tilde{C}=\frac{q-p\sigma }{S_4[pq(1-\sigma )-2(q-p\sigma )]}·2^{\frac{pq(1-\sigma )}{2(q-p\sigma )}}>0.$$

(26)

The proof of Theorem 2 is complete. □

4. Conclusions

In this article, we deal with the blow-up phenomena of the solutions to a non-Newton filtration equation with local linear boundary dissipation. By analyzing the effect of the local linear boundary dissipation on the blow-up behaviors of the solutions, along with the modified concavity method, sufficient conditions on the occurrence of the blow-up singularity are obtained. Moreover, by combining the differential inequality approach and Gagliardo–Nirenberg interpolation inequality, the explicit upper and lower bounds of the blow-up time are deduced when blow-up singularity occurs. However, the present methods cannot be directly used to handle the blow-up properties of the solutions when the linear boundary dissipation is replaced by nonlinear boundary dissipation. We hope to find some new and interesting results for the nonlinear boundary dissipation case in the near future.