Extinction and Nonextinction for the Fast Diffusion Equation

Abstract

This paper deals with the extinction and nonextinction properties of the fast diffusion equation of homogeneous Dirichlet boundary condition in a bounded domain of 𝑅^𝑁 with 𝑁>2. For 0<𝑚<1, under appropriate hypotheses, we show that 𝑚=𝑝 is the critical exponent of extinction for the weak solution. Furthermore, we prove that the solution either extinct or nonextinct in finite time depends strongly on the initial data and the first eigenvalue of −Δ with homogeneous Dirichlet boundary.

1. Introduction

In this paper, we deal with the following fast diffusion equation with gradient absorption terms: 𝑢_𝑡=Δ𝑢^𝑚+𝜆|∇𝑢|^𝑝,()(),𝑢()𝑥,𝑡∈Ω×0,∞𝑥,0=𝑢₀(𝑥)𝑢()(),,𝑥∈Ω,𝑥,𝑡=0,𝑥∈𝜕Ω×0,∞ (1) where 0<𝑚<1, 𝑝>0, 𝜆>0, Ω⊂𝑅^𝑁 with 𝑁>2 is an open bounded domain with smooth boundary, and 𝑢₀∈𝐿^∞(Ω)∩𝑊₀^1,𝑝(Ω) is a nonzero positive function.

Equation (1) appears in a lot of applications to describe the evolution of diffusion processes, in particular, fast diffusion for 0<𝑚<1. In combustion theory, for instance, the function 𝑢(𝑥,𝑡) represents the temperature, the term Δ𝑢^𝑚 represents the thermal diffusion, and 𝜆|∇𝑢|^𝑝 is a source.

Extinction and nonextinction are important properties for solutions of many evolutionary equations, especially for fast diffusion equations. In 1974, Kalashnikov [1] considered the Cauchy problem of equation 𝑢_𝑡=Δ𝑢−𝑢^𝑝 and firstly introduced the definition of extinction for its solution; that is, there exists a finite time 𝑇>0 such that the solution is nontrivial for 0<𝑡<𝑇, but 𝑢(𝑥,𝑡)≡0 for all (𝑥,𝑡)∈Ω×(𝑇,∞). In this case, 𝑇 is called the extinction time.

Since then, many authors became interested in the extinction and nonextinction of all kinds of evolutionary equations. For the following homogeneous Dirichlet boundary value problem: 𝑢_𝑡=Δ𝑢−𝑢^𝑞,()(),𝑢()𝑥,𝑡∈Ω×0,∞𝑥,0=𝑢₀(𝑥)𝑢()().,𝑥∈Ω,𝑥,𝑡=0,𝑥∈𝜕Ω×0,∞ (2) Gu [2] obtained that the nontrivial solutions of the problem (2) vanish identically in a finite time if and only if 0<𝑞<1, which implies that strong absorption will cause extinction in a finite time. More results on the extinction for the problem (2) have also been obtained by many researchers, and we can refer to [3–7] and the references therein. Because of the occurrence of such a phenomenon for a diffusion equation with a different absorption term, that is, the absorption term is a nonnegative function of ∇𝑢 instead of being a nonnegative function of 𝑢, Benachour et al. [8, 9] considered the following Cauchy problem for the viscous Hamilton-Jacobi equation: 𝑢_𝑡=Δ𝑢−|∇𝑢|^𝑞,(𝑥,𝑡)∈𝑅^𝑁×(),𝑢()0,∞𝑥,0=𝑢₀(𝑥),𝑥∈𝑅^𝑁. (3) They proved that the nonnegative classical solutions to the problem (3) are extinct in finite time and have noncompact support if 0<𝑞<1.

Generally, in the problems (2) and (3), there is a comparison between the diffusion term and the absorption term, and the absorption is sufficiently strong to lead any bounded nonnegative solution to zero in finite time. However, while both the source −𝑢^𝑞 in problem (2) and the source −|∇𝑢|^𝑞 in problem (3) are called the “cool source,” the nonlinear source +|∇𝑢|^𝑞 in problem (1) is physically called the “hot source.” As far as we know, the type of “hot source” has complicated influences on the properties of solutions compared with the case of “cool source” [10]. Thus, few works are concerned with extinction property for solutions to the evolutionary equations with “hot source.” In 2005, Li and Wu [10] gave some necessary and sufficient conditions of extinction for the solutions to the following problem with “hot source”: 𝑢_𝑡=Δ𝑢^𝑚+𝜆𝑢^𝑝,()(),𝑢()𝑥,𝑡∈Ω×0,∞𝑥,0=𝑢₀(𝑥)𝑢()(),≥0,𝑥∈Ω,𝑥,𝑡=0,𝑥∈𝜕Ω×0,∞ (4) where 0<𝑚<1. They proved that if 𝑝>𝑚, the solutions to the problem (4) with small initial data vanished in finite time and if 𝑝<𝑚, the maximal solution to the problem (4) is positive for all 𝑡>0. Recently, Tian and Mu [11] studied the following 𝑝-Laplacian equation with nonlinear “hot source”: 𝑢_𝑡=div|∇𝑢|^𝑝−2∇𝑢+𝜆𝑢^𝑝,()(),𝑢()𝑥,𝑡∈Ω×0,∞𝑥,0=𝑢₀(𝑥)𝑢()().,𝑥∈Ω,𝑥,𝑡=0,𝑥∈𝜕Ω×0,∞ (5) They obtained that for 1<𝑝<2, 𝑞=𝑝−1 is the critical exponent of extinction for the weak solution to the problem (5). Also, they show that for 1<𝑝<2 and 𝑞=𝑝−1, the extinction and nonextinction of the solution to the problem (5) depend strongly on the first eigenvalue of the problem −div(|∇𝑢|^𝑝−2∇𝑢)=𝜆|𝑢|^𝑝−2𝑢 in Ω, 𝑢|_𝜕Ω=0. For more results on the extinction properties of equations with “hot source,” we can refer to [12, 13] and the references therein.

By replacing the diffusion term 𝑢^𝑝 with the gradient absorption terms |∇𝑢|^𝑝 in (4), in this paper, we devote to establish the conditions for the extinction of solution to the problem (1), which involves the “hot source” rather than the “cool source.”

2. Preliminaries

In this section, we will give some definitions and lemmas which is useful to the proof of our results in the next section. For our convenience, we first define some sets as follows: Ω_𝑇()𝑢=Ω×0,𝑇,𝑇>0,𝐸=_𝑡∈𝐿²Ω_𝑇∶𝑢∈𝐿^2𝑚Ω_𝑇𝐿²Ω_𝑇;∇𝑢∈𝐿^2𝑝Ω_𝑇𝐸₀=𝜑∈Ω_𝑇∶𝜑_𝑡,Δ𝜑;∇𝜑∈𝐿²Ω_𝑇;𝜑|_𝜕Ω𝑇=0. (6) It is well known that the problem (1) has no classical solution in general. We need to consider its weak solutions which is defined as follows.

Definition 1. For any 𝑇≥0, a function 𝑢(𝑥,𝑡)∈𝐸 is called a weak solution to the problem (1) if the following equalities hold for any 0<𝑡₁<𝑡₂<𝑇 and 0≤𝜑∈𝐸₀: _Ω𝑢𝑥,𝑡₂𝜑𝑥,𝑡₂𝑑𝑥−_Ω𝑢𝑥,𝑡₁𝜑𝑥,𝑡₁=𝑑𝑥_𝑡2𝑡1Ω𝑢𝜑_𝑡+𝑢^𝑚Δ𝜑+|∇𝑢|^𝑝𝑢()𝜑𝑑𝑡𝑑𝑥,𝑥,0=𝑢₀(𝑥),a.e.𝑥∈Ω. (7)

Remark 2. Similarly, to define a subsolution (resp., supersolution) 𝑢(𝑥,𝑡) (resp., 𝑢(𝑥,𝑡)) of the problem (1), we need only to set 𝑢(𝑥,0)<𝑢₀(𝑥) (resp., 𝑢(𝑥,0)>𝑢₀(𝑥)) in Ω, 𝑢(𝑥,𝑡)≤0 (resp.,  𝑢(𝑥,𝑡)≥0) on 𝜕Ω×(0,∞), and the first equality in (7) is replaced by ≤ (resp., ≥) for every 𝜑(𝑥)>0.

As a useful tool in the proof of our main results in the next section, a comparison principle of the following problem is needed: 𝑢_𝑡()(),()(),𝑢()=Δ𝜑𝑥,𝑡,𝑢+𝑓𝑥,𝑡,𝑢𝑥,𝑡∈Ω×0,𝑇𝑥,0=𝑢₀(𝑥)𝑢()().,𝑥∈Ω,𝑥,𝑡=0,𝑥∈𝜕Ω×0,𝑇 (8) Under the following four hypotheses:(H1)𝜑, Δ_𝑥𝜑∈𝐶(Ω_𝑇×𝑅), ∇_𝑥𝜑∈∏^𝑁_𝑖=1𝐶(Ω_𝑇×𝑅), and 𝜑_𝑢∈𝐶(Ω_𝑇×𝑅⧵{0}), s.t. 𝜑(𝑥,𝑡,0)=0 and 𝜑_𝑢(𝑥,𝑡,𝑢)>0 for all (𝑥,𝑡)∈Ω_𝑇 and 𝑢≠0;(H2)𝑓∈𝐶(Ω_𝑇×𝑅) and 𝑓_𝑢∈𝐶(Ω_𝑇×𝑅⧵{0}) with 𝑓(𝑥,𝑡,0)=0 for all (𝑥,𝑡)∈Ω_𝑇;(H3)𝑢₀∈𝐿^∞(Ω) with 𝑢₀>0;(H4) for any constant 𝑎>0, if 𝑢≥0, 𝑣≥𝑎 in Ω_𝑇 and 𝑢, 𝑣∈𝐿^∞(Ω_𝑇), then the functions Φ and 𝐹 defined almost everywhere in Ω_𝑇 by Φ(𝑥,𝑠)≡¹₀𝜑_𝑢(()𝑣)𝐹()≡𝑥,𝑠,𝜃𝑢+1−𝜃𝑑𝜃,𝑥,𝑠¹₀𝜑_𝑓(𝑥,𝑠,𝜃𝑢+(1−𝜃)𝑣)𝑑𝜃 (9) belong to 𝐿^∞(Ω_𝑇) and 𝐿^∞(𝜕Ω_𝑇×[0,𝑇]), respectively, we have the following lemma on the comparison principle of the problem (8).

Lemma 3 (see [14]). Assume that the hypotheses (H1)–(H4) hold. Let 𝑢(𝑥,𝑡;𝑢₀) and 𝑣(𝑥,𝑡;𝑣₀) be nonnegative solutions of (8) on Ω_𝑇 with 𝑢₀≤𝑣₀. Assume further that for any 𝑡₁<𝑇, there is a constant 𝑎(𝑡₁)>0 such that 𝑣(𝑥,𝑡;𝑣₀)≥𝑎(𝑡₁) for all (𝑥,𝑡)∈Ω×[0,𝑡₁]. Then, 𝑢≤𝑣 almost everywhere on Ω_𝑇.

3. Main Results

In this section, by applying the energy method introduced in [11] and the comparison result in Lemma 3 of the problem (8), we give the main results on the extinction and nonextinction of the solution to the problem (1).

3.1. Extinction of the Solution

In this subsection, we consider the extinction of the solution to the problem (1) and give the conditions for the extinction of the solution to the problem (1).

Theorem 4. Assume that 0<𝑚<1, and let 𝑢(𝑥,𝑡) be a weak solution of the problem (1). If 𝑚<𝑝≤2/(3−𝑚), then, for sufficiently small initial data, there exists a finite time 𝑇, such that 𝑢(𝑥,𝑡)≡0 (10) for all (𝑥,𝑡)∈Ω×(𝑇,+∞).

Proof. First of all, multiplying the first equation of the problem (1) by 𝑢^𝑠−1  (𝑠>1) and integrating over Ω, we can obtain the following crucial equation to our proof: 1𝑠𝑑𝑑𝑡_Ω𝑢^𝑠()𝑑𝑥+4𝑚𝑠−1(𝑚+𝑠−1)^2Ω∇𝑢^{(𝑚+𝑠−1)/22}𝑑𝑥=𝜆𝑚+𝑠−12^−𝑝_Ω∇𝑢^{(𝑚+𝑠−1)/2𝑝}𝑢^{((3−𝑠−𝑚)/2)𝑝+𝑠−1}𝑑𝑥. (11) Then, we prove the theorem by the following two cases.
First, we consider the case of (𝑁−2)/(𝑁+2)≤𝑚<1. Let 𝑠=1+𝑚 in (11), and we can get by the Hölder inequality and Poincare inequality that ‖𝑢(⋅,𝑡)‖^𝑚_1+𝑚≤|Ω|^{(𝑚/(1+𝑚))−((𝑁−2)/2𝑁)}𝑢^𝑚(⋅,𝑡)_{2𝑁/(𝑁−2)}≤𝐶₀|Ω|^{(𝑚/(1+𝑚))−((𝑁−2)/2𝑁)}∇𝑢^𝑚(⋅,𝑡)₂, (12) where 𝐶₀ is the Sobolev embedding constant depending only on 𝑝 and 𝑁.
Since 𝑝≤2/(3−𝑚), 0<𝑚<1, it follows from the Young’s inequality for any 𝜀>0 that _Ω∇𝑢^𝑚(⋅,𝑡)^𝑝𝑢^{𝑝(1−𝑚)+𝑚}𝑑𝑥≤𝜀∇𝑢^𝑚(⋅,𝑡)²₂+𝑐(𝜀)‖𝑢‖^{2(𝑝−𝑝𝑚+𝑚)/(2−𝑝)}_{2(𝑝−𝑝𝑚+𝑚)/(2−𝑝)}. (13) Also, since the assumptions 𝑝≤2/(3−𝑚) and 0<𝑚<1 imply that 2(𝑝−𝑝𝑚+𝑚)/(2−𝑝)≤1+𝑚, we can obtain by the Hölder inequality that ‖𝑢‖₂^{2(𝑝−𝑝𝑚+𝑚)/(2−𝑝)}_{𝑝−𝑝𝑚+𝑚/(2−𝑝)}≤𝐶₁|Ω|^{2(𝑝−𝑝𝑚+𝑚)/((1+𝑚)(2−𝑝))}‖𝑢‖^{2(𝑝−𝑝𝑚+𝑚)/(2−𝑝)}_1+𝑚. (14) Therefore, from the previous inequalities (11)–(14), we can obtain the following differential inequality: 1𝑑1+𝑚‖𝑢‖𝑑𝑡^1+𝑚1+𝑚+1−𝜆𝑚^−𝑝𝜀𝐶₀⁻²×|Ω|^{(𝑚/(1+𝑚))−((𝑁−2)/2𝑁)}‖𝑢‖^2𝑚_1+𝑚≤𝜆𝑚^−𝑝𝑐(𝜀)𝐶₁|Ω|^{(2+𝑚𝑝−3𝑝)/((2−𝑝)(1+𝑚))}×‖𝑢‖^{2(𝑝−𝑝𝑚+𝑚)/(2−𝑝)}_1+𝑚. (15) Choose 𝜀 sufficiently small such that 1−𝜆𝑚^−𝑝𝜀>0 and initial data ‖𝑢₀‖ sufficiently small such that 𝑢₀^{2(𝑝−𝑚)/(2−𝑝)}_1+𝑚<𝜆⁻¹𝑚^𝑝𝑐(𝜀)⁻¹𝐶₁⁻¹1−𝜆𝑚^−𝑝𝜀𝐶₀⁻²×|Ω|^{(𝑚/(1+𝑚))−((𝑁−2)/2𝑁)−((2+𝑚𝑝−3𝑝)/((2−𝑝)(1+𝑚)))}. (16) Thus, we can obtain that 1𝑑1+𝑚‖𝑢‖𝑑𝑡^1+𝑚1+𝑚+𝐶₃‖𝑢‖^2𝑚_1+𝑚≤0, (17) where 𝐶₂=(1−𝜆𝑚^−𝑝𝜀)𝐶₀⁻²|Ω|^{((𝑛−2)/2𝑛)−(𝑚/(1+𝑚))}−𝜆𝑚^−𝑝𝑐(𝜀)𝐶₁|Ω|^{((2+𝑚𝑝−3𝑝)/((2−𝑝)(1+𝑚)))}‖𝑢₀‖^{2(𝑝−𝑚)/(2−𝑝)}_1+𝑚>0.
By integrating the inequality (17), we can obtain that there exists a finite time 𝑇=‖‖𝑢₀‖^1−𝑚_1+𝑚/(1−𝑚)𝐶₂ such that ‖𝑢‖_1+𝑚≤𝑢₀^1−𝑚_1+𝑚−(1−𝑚)𝐶₂𝑡^1/(1−𝑚)(],‖𝑢‖,𝑡∈0,𝑇_1+𝑚=0,𝑡∈[𝑇,+∞), (18) which implies that 𝑢(𝑥,𝑡) vanishes in finite time 𝑇.
Next, we consider the second case of 0<𝑚<(𝑁−2)/(𝑁+2)<(𝑁−2)/𝑁. Let 𝑠=(𝑁/2)(1−𝑚)>1 in (11), and we can get by the Poincare inequality that ‖𝑢(⋅,𝑡)‖_𝑠^{(𝑚+𝑠−1)/2}=𝑢^{(𝑚+𝑠−1)/2}(⋅,𝑡)_{2𝑁/(𝑁−2)}≤𝐶₃∇𝑢^{(𝑚+𝑠−1)/2}(⋅,𝑡)₂, (19) where 𝐶₃ is the Sobolev embedding constant depending only on 𝑁, 𝑚, and 𝑠. Also, by the similar calculations as the proof in the first case, we can get from the Young’s inequality for any 𝜂>0 that _Ω∇𝑢^{(𝑚+𝑠−1)/2𝑝}𝑢^{((3−𝑠−𝑚)/2)𝑝+𝑠−1}𝑑𝑥≤𝜂∇𝑢^{(𝑚+𝑠−1)/22}₂+𝑐𝜂|Ω|^{(2+𝑝𝑚−3𝑝)/(2−𝑝)𝑠}×‖𝑢‖_𝑠^{((3−𝑠−𝑚)𝑝+2𝑠−2)/(2−𝑝)}. (20) Choose 𝜂 sufficiently small such that 4𝑚(𝑠−1)/(𝑚+𝑠−1)²−𝜆(((1+𝑚)((𝑁/2)−1))/2)^−𝑝𝜂>0 and initial data ‖𝑢₀‖ sufficiently small such that 𝑢_0𝑠^{2(𝑝−𝑚)/(2−𝑝)}<𝜆⁻¹𝑐𝜂⁻¹𝐶₃⁻²(1+𝑚)((𝑁/2)−1)2^−𝑝×|Ω|^{(2+𝑝𝑚−3𝑝)/(2−𝑝)𝑠}×4𝑚(𝑠−1)(𝑚+𝑠−1)²())−𝜆1+𝑚)((𝑁/2−12^−𝑝𝜂. (21) Then, it follows from the inequalities (11) and (19)–(21) that 1𝑠𝑑‖𝑢‖𝑑𝑡^𝑠𝑠+𝐶₄‖𝑢‖_𝑠^{𝑚+𝑠−1}≤0, (22) where 𝐶₄=𝐶₃⁻²((4𝑚(𝑠−1)/(𝑚+𝑠−1)²) − 𝜆(((1+𝑚)((𝑁/2)−1))/2)^−𝑝𝜂)−𝜆𝑐(𝜂)|Ω|^{(2+𝑝𝑚−3𝑝)/(𝑝−2)𝑠}‖𝑢₀‖_𝑠^{2(𝑝−𝑚)/(2−𝑝)}.
By integrating the inequality (22), we can obtain that there exists a finite time 𝑇=‖𝑢₀‖_𝑠^1−𝑚/(1−𝑚)𝐶₄ such that ‖𝑢‖_𝑠≤𝑢_0𝑠^1−𝑚−(1−𝑚)𝐶₄𝑡^1/(1−𝑚)(],‖𝑢‖,𝑡∈0,𝑇_𝑠=0,𝑡∈[𝑇,+∞), (23) which implies that 𝑢(𝑥,𝑡) vanishes in finite time 𝑇. This completes the proof of Theorem 4.

Theorem 5. If 𝑚=𝑝≤2/(3−𝑚), the solution to the problem (1) vanishes in finite time for 𝜆 sufficiently small.

Proof. When 𝑝=𝑚, we note that the left sides of the inequalities (16) and (21) always equal 1. Thus, by a similar argument in the proof of Theorem 4, we can choose 𝜆 sufficiently small such that the inequalities (16) and (21) hold, which imply that there exist 𝑇<+∞ such that 𝑢(𝑥,𝑡) vanishes identically for all (𝑥,𝑡)∈Ω_𝑇. This completes the proof of Theorem 5.

3.2. Nonextinction of the Solution

In this subsection, we investigate the conditions under which the solution 𝑢(𝑥,𝑡) of the problem (1) cannot become extinct.

Theorem 6. If 𝑝<𝑚, the weak solution 𝑢(𝑥,𝑡) of (1) cannot vanish in finite time for any nonnegative initial date 𝑢₀ with 𝜆 being sufficiently large.

Proof. In order to prove Theorem 6, we first define two useful functions as follows. The first function 𝜙(𝑥)∈𝐻¹₀(Ω) satisfying max_𝑥∈Ω𝜙(𝑥)=1 is the associated eigenfunction with the principal eigenvalue 𝜆₁ of the following problem: −Δ𝜙=𝜆₁𝜙,𝑥∈Ω,𝜙|_𝜕Ω=0. (24) For 𝑝<𝑚, we define the second useful function as follows: 𝑔(𝑡)=𝑎𝑏^{1/(𝑚−𝑝)}1−exp(−𝐶𝑡)^1/(1−𝑝), (25) where 𝑎<𝑏 and 𝐶∈(0,(𝑚−𝑝)(𝑎^1−𝑝/𝑏^1−𝑚)^1/(1−𝑝)). It follows from [11] that the function 𝑔(𝑡) satisfies the following properties: 𝑔^′(𝑡)≤−𝑎𝑔^𝑚(𝑡)+𝑏𝑔^𝑝(𝑡),𝑔(0)(𝑡)=0,0<𝑔<1,for𝑡>0. (26) Now, let 𝑣(𝑥,𝑡)=𝑔(𝑡)𝜙(𝑥)^1/𝑚 be a function Ω×(0,∞). Next, we will show that 𝑣(𝑥,𝑡) is a subsolution of the problem (1). In fact, from the definitions of functions 𝑔(𝑡), 𝑣(𝑥,𝑡) and the properties (26) of the function 𝑔(𝑡), we have that 𝐿(𝑣(=𝑥,𝑡))_𝑡0Ω𝑣_𝑡()𝜑()+𝑥,𝑠𝑥,𝑠𝑑𝑥𝑑𝑠_𝑡0Ω∇𝑢^𝑚⋅∇𝜑−𝜆|∇𝑣|^𝑝𝜑()=𝑥,𝑠𝑑𝑥𝑑𝑠_𝑡0Ω𝑣_𝑡(𝑥,𝑠)−Δ𝑣^𝑚−|∇𝑣|^𝑝𝜑()=𝑥,𝑠𝑑𝑥𝑑𝑠_𝑡0Ω𝑔^′(𝑡)𝜙(𝑥)^1/𝑚+𝜆₁𝑔^𝑚(𝑡)𝜙(𝑥)−𝜆𝑔^𝑝(𝑡)𝑚^−𝑝𝜙(𝑥)^{((1−𝑚)𝑝)/𝑚}∇𝜙^𝑝()≤×𝜑𝑥,𝑠𝑑𝑥𝑑𝑠_𝑡0Ω−𝑎𝑔^𝑚(𝑡)𝜙(𝑥)^1/𝑚−𝜆𝑔^𝑝(𝑡)×𝑚^−𝑝𝜙(𝑥)^{((1−𝑚)𝑝)/𝑚}∇𝜙^𝑝−𝑏𝜆𝜙(𝑥)^1/𝑚−𝜆₁𝜆𝑔^𝑚−𝑝(𝑡)𝜙(𝑥()≤)×𝜑𝑥,𝑠𝑑𝑥𝑑𝑠_𝑡0Ω−𝜆𝑔^𝑝(𝑡)𝑚^−𝑝𝜙(𝑥)^{((1−𝑚)𝑝)/𝑚}∇𝜙^𝑝−𝑏𝜆𝜙(𝑥)^1/𝑚−𝜆₁𝜆𝑔^𝑚−𝑝(𝑡)𝜙(𝑥()≤)×𝜑𝑥,𝑠𝑑𝑥𝑑𝑠_𝑡0Ω−𝜆𝑔^𝑝(𝑡)𝜙(𝑥)^{((1−𝑚)𝑝)/𝑚}×𝑚^−𝑝∇𝜙^𝑝−𝑏𝜆𝜙(𝑥)^{(1−𝑝+𝑝𝑚)/𝑚}−𝜆₁𝜆𝑔^𝑚−𝑝(𝑡)𝜙(𝑥)^{(𝑚−𝑝+𝑝𝑚)/𝑚}𝜑(𝑥,𝑠)𝑑𝑥𝑑𝑠. (27) In order to prove that 𝐿(𝑣(𝑥,𝑡))<0 which implies that 𝑣(𝑥,𝑡) is a subsolution of the problem (1), we only show that _Ω𝑚^−𝑝∇𝜙^𝑝−𝑏𝜆𝜙(𝑥)^{(1−𝑝+𝑝𝑚)/𝑚}−𝜆₁𝜆𝑔^𝑚−𝑝(𝑡)𝜙(𝑥)^{(𝑚−𝑝+𝑝𝑚)/𝑚}𝑑𝑥≥0. (28) Since 0<𝑔(𝑡)<1 and 𝑚>𝑝, we have that 𝑔^𝑚−𝑝(𝑡)𝜙(𝑥)<1,_{(1−𝑝+𝑝𝑚)/𝑚}<𝜙(𝑥)^{(𝑚−𝑝+𝑝𝑚)/𝑚}. (29) By choosing 𝜆≥𝑚^𝑝(𝑏+𝜆₁)(‖𝜙‖^{(𝑚−𝑝+𝑝𝑚)/𝑚}_{(𝑚−𝑝+𝑝𝑚)/𝑚}/‖∇𝜙‖^𝑝_𝑝), we can get that _Ω𝑚^−𝑝∇𝜙^𝑝−𝑏+𝜆₁𝜆𝜙(𝑥)^{(𝑚−𝑝+𝑝𝑚)/𝑚}𝑑𝑥≥0, (30) which together with (29) implies that (28) holds. Therefore, 𝑣(𝑥,𝑡) is a subsolution of the problem (1). Moreover, since 𝑣(𝑥,0)=𝑔(0)𝜙(𝑥)=0≤𝑢₀ in Ω and 𝑣|_(𝜕Ω)𝑡=0, we can obtain by the comparison principle that 𝑢(𝑥,𝑡)≥𝑣(𝑥,𝑡)>0 in Ω×(0,+∞), which implies that the weak solution 𝑢(𝑥,𝑡) of (1) cannot vanish in finite time. This completes the proof of Theorem 6.

Remark 7. From Theorems 4–6, we observe that 𝑞=𝑚 is the critical exponent of extinction for the solution to the problem (1).

Acknowledgments

This work is supported in part by NSF of China (11101069, 11071266) and in part by the Natural Science Foundation Project of CQ CSTC (2010BB9218).