Blow-Up of Solutions for a Class of Sixth Order Nonlinear Strongly Damped Wave Equation

Abstract

We consider the blow-up phenomenon of sixth order nonlinear strongly damped wave equation. By using the concavity method, we prove a finite time blow-up result under assumptions on the nonlinear term and the initial data.

1. Introduction

It is well known that nonlinear strongly damped wave equation is proposed to describe all kinds of viscous vibration system. The global well-posedness of third order nonlinear strongly damped wave equation 𝑢_𝑡𝑡−𝛼Δ𝑢_𝑡−Δ𝑢=𝑓(𝑢),𝛼>0,𝑥∈Ω,𝑡>0, (1) was studied by Webb [1] firstly. He gave the existence and asymptotic behavior of strong solutions for the problem (1). Then this result was improved by Y. C. Liu and D. C. Liu [2]. The existence and uniqueness of strong solutions were proved under the hypothesis of the weaker conditions. For two classes of strongly damped nonlinear wave equation, the finite time blow-up of solutions was proved by Shang [3]. A number of authors (Chen et al. [4], Zhou [5], and Al’shin et al. [6]) have shown the existence of the global weak solutions and the global attractors for third order nonlinear strongly damped wave equation.

For the fourth order nonlinear strongly damped wave equation, there are also some results about initial boundary value problem or Cauchy problem [7–9]. In [7], Shang studied the initial boundary value problem of the following equation: 𝑢_𝑡𝑡−Δ𝑢−Δ𝑢_𝑡−Δ𝑢_𝑡𝑡=𝑓(𝑢),𝑥∈Ω,𝑡>0. (2) Under some assumptions on 𝑓 and 𝑛=1,2,3, he investigated the existence, uniqueness, asymptotic behavior, and blow-up phenomenon of the solutions.

In [8], Xu et al. considered the initial boundary value problem of fourth order wave equation with viscous damping term 𝑢_𝑡𝑡−𝛼𝑢_𝑥𝑥𝑡+𝑢_{𝑥𝑥𝑥𝑥}=𝑓𝑢_𝑥𝑥,𝑥∈Ω,𝑡>0. (3) They proved the global existence and nonexistence of the solution by argument related to the potential well-convexity method.

In order to investigate the water wave problem with surface tension, Schneider and Wayne [10] studied a class of Boussinesq equation as follows: 𝑢_𝑡𝑡=𝑢_𝑥𝑥+𝑢_{𝑥𝑥𝑡𝑡}+𝜇𝑢_{𝑥𝑥𝑥𝑥}−𝑢_{𝑥𝑥𝑥𝑥𝑡𝑡}+𝑢²_𝑥𝑥, (4) where 𝑥,𝑡,𝜇∈𝐑. This type of equations can be formally derived from the 2D water wave problem and models the water wave problem with surface tension. They proved that the long wave limit can be described approximately by two decoupled Kawahara equations. A more natural model seems to be an extension from the classical Boussinesq equation as follows (see [11]): 𝑢_𝑡𝑡=𝑢_𝑥𝑥+𝜇+1𝑢_{𝑥𝑥𝑥𝑥}−𝑢_{𝑥𝑥𝑥𝑥𝑥𝑥}+𝑢²_𝑥𝑥. (5) Wang and Mu [12] studied the Cauchy problem of the equation 𝑢_𝑡𝑡=𝑢_𝑥𝑥+𝑢_{𝑥𝑥𝑡𝑡}−𝑢_{𝑥𝑥𝑥𝑥}−𝑢_{𝑥𝑥𝑥𝑥𝑡𝑡}+𝑓(𝑢)_𝑥𝑥. (6) They obtained the existence and uniqueness of the local solutions and proved the blow-up of solutions to the problem (6). Esfahani et al. [13] studied the solutions of 𝑢_𝑡𝑡=𝑢_𝑥𝑥+𝛽𝑢_{𝑥𝑥𝑥𝑥}+𝑢_{𝑥𝑥𝑥𝑥𝑥𝑥}+|𝑢|^𝛼𝑢_𝑥𝑥, (7) where 𝛽=±1 and 𝛼>0. They proved the local well-posedness in 𝐿²(𝑅) and 𝐻¹(𝑅) and gave finite time blow-up results to the problem (7).

For the sixth order nonlinear wave equation with strong damping term 𝑢_𝑡𝑡−Δ𝑢−𝜈Δ𝑢_𝑡+Δ²𝑢+Δ²𝑢_𝑡𝑡=Δ𝑓(𝑢), (8) H. W. Wang and S. B. Wang [14] established a global existence result of small amplitude solutions of the Cauchy problem (8) for all space dimensions 𝑛>1. When 𝜈=0, H. W. Wang and S. B. Wang [15] studied the long-time behavior of small solutions of the Cauchy problem for a Rosenau equation. The decay and scattering for small amplitude solution are established.

In this paper, we study a class of sixth order nonlinear strongly damped wave equation: 𝑢_𝑡𝑡−Δ𝑢_𝑡=𝐝𝐢𝐯𝜎^→(∇𝑢)+Δ𝑢+𝜇Δ²𝑢+Δ³𝑢()𝑢,𝑥∈Ω,𝑡>0,𝑥,0=𝑢₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥)ℬ,𝑥∈Ω,_𝑖𝑢(𝑥,𝑡)=0,𝑥∈𝜕Ω,𝑡>0, (9) where 𝜇>0, Ω is a bounded domain of 𝑅^𝑛(𝑛≥1) with a smooth boundary 𝜕Ω, and ℬ_𝑖, 𝑖=0,1,2, are homogeneous boundary condition: 𝐷=𝑢∈𝐶⁶Ω∣ℬ_𝑖𝑢=∇^𝑖𝑢≡0,𝑖=0,1,2,𝑥∈𝜕Ω. (10) By using the ideas of the concavity theory introduced by Levine [17], we prove the finite time blow-up results under assumption on the nonlinear term 𝐝𝐢𝐯𝜎^→(∇𝑢) and the initial data 𝑢₀,𝑢₁.

2. Preliminaries and Main Results

In this section, we introduce some notations, basic ideas, and important lemmas which will be needed in the course of the paper.

Let 𝐻=ℒ²(Ω) be a Hilbert space which is equipped with the scalar product (𝑢,𝑣)=∫_Ω𝑢(𝑥)𝑣(𝑥)𝑑𝑥.

Now, we define 𝐴𝑢=−Δ𝑢−𝜇Δ²𝑢−Δ³𝑢, (11) where 𝐴∶𝐷→𝐻 is a symmetric linear operator and satisfies (𝑢,𝐴𝑣)=(𝐴𝑢,𝑣) for all 𝑢,𝑣∈𝐷⊆𝐻.

For the nonlinear term 𝐝𝐢𝐯𝜎^→(∇𝑢) of the problem (9), 𝜎^→=(𝜎₁,𝜎₂,…,𝜎_𝑛)∶𝐷^𝑛→𝐻^𝑛 is a vector function which satisfies the following conditions.(a)Assume that the Fréchet derivative 𝜎^→_𝑥 is a symmetric, bounded, linear operator on 𝐻^𝑛 and that 𝑥→𝜎^→_𝑥 is a continuous map from 𝐷^𝑛 to 𝔏(𝐻)^𝑛.(b)The scalar valued function 𝐺∶𝐷→R is defined by 𝐺(𝑢)=¹₀𝐝𝐢𝐯𝜎^→=𝜌∇𝑢,𝑢𝑑𝜌_Ω10𝐝𝐢𝐯𝜎^→𝜌∇𝑢𝑢𝑑𝜌𝑑𝑥=_Ω¹₀𝑑𝑑𝜌₀^𝜌𝑢𝐝𝐢𝐯𝜎^→()=∇𝑧𝑑𝑧𝑑𝜌𝑑𝑥_Ω𝑢0𝐝𝐢𝐯𝜎^→(∇𝑧)𝑑𝑧𝑑𝑥, (12) where 𝐺(𝑢) denotes the potential associated with 𝐝𝐢𝐯𝜎^→(∇𝑢). The Fréchet derivative of 𝐺(𝑢) is 𝐺_𝑢 which can be shown to act as follows: 𝐺_𝑢𝑣=𝐝𝐢𝐯𝜎^→(∇𝑢),𝑣, (13) for all 𝑢, 𝑣∈𝐷.(c)Assume that for some 𝛼>0𝐝𝐢𝐯𝜎^→(∇𝑢),𝑢≥2(2𝛼+1)𝐺(𝑢), (14) for all 𝑢∈𝐷.

To obtain the finite time blow-up result, we need the following interpolation inequality of Evance [16] for function in 𝑊₀^𝑘,𝑝(Ω).

Lemma 1. For all 𝑢∈𝑊₀^𝑘,𝑝(Ω), if 1≤𝑝<∞, 𝑗 and 𝑘 are integers, and 1≤𝑗≤𝑘, (𝑗/𝑘)≤𝛼≤1, then 𝐷^𝑗𝑢_𝐿𝑞(Ω)≤𝐶𝐷^𝑘𝑢^𝛼_𝐿𝑝(Ω)‖𝐷𝑢‖_𝐿^1−𝛼𝑟_(Ω),∀𝑢∈𝑊₀^𝑘,𝑝(Ω), (15) where (1/𝑞)−(𝑗/𝑛)=𝛼((1/𝑝)−(𝑘/𝑛))+(1−𝛼)((1/𝑟)−(1/𝑛)), with the constant 𝐶 depending only on (𝑝, 𝑟, 𝑗, 𝑘, 𝛼, 𝑛) and Ω.
In particular, if 𝑗=2, 𝑘=3, and 𝑝=𝑞=𝑟=2, one has 𝐷²𝑢_𝐿2(Ω)≤𝐶𝐷³𝑢^𝛼_𝐿2(Ω)‖𝐷𝑢‖_𝐿^1−𝛼_2(Ω),∀𝑢∈𝑊₀^3,2(Ω). (16)

Lemma 2. Assume that 𝑢∈𝑊₀^3,2(Ω) and 0<𝜇≤(1/𝐶₁)(with 𝐶₁ depending on the constant 𝐶 of Sobolev’s interpolation inequality); then (𝐴𝑢,𝑢)≥0.

Proof. By Lemma 1, we see that 𝐷²𝑢_𝐿2(Ω)≤𝐶𝐷³𝑢^𝛼_𝐿2(Ω)‖𝐷𝑢‖_𝐿^1−𝛼_2(Ω),∀𝑢∈𝑊₀^3,2(Ω). (17) Using Young’s inequality, we have 𝐷²𝑢²_𝐿2(Ω)≤𝐶𝐷³𝑢_𝐿^2𝛼_2(Ω)‖𝐷𝑢‖_𝐿^2(1−𝛼)_2(Ω)≤𝐶𝛼𝐷³𝑢²_𝐿2(Ω)+(1−𝛼)‖𝐷𝑢‖²_𝐿2(Ω)≤𝐶₁𝐷³𝑢²_𝐿2(Ω)+‖𝐷𝑢‖²_𝐿2(Ω). (18) For the operator 𝐴, using integration of parts, we have (𝐴𝑢,𝑢)=−Δ𝑢−𝜇Δ²𝑢−Δ³=‖‖𝑢,𝑢∇𝑢²_𝐿2(Ω)+∇³𝑢²_𝐿2(Ω)−𝜇∇²𝑢²_𝐿2(Ω)≥1𝐶₁∇²𝑢²_𝐿2(Ω)−𝜇∇²𝑢²_𝐿2(Ω)=1𝐶₁−𝜇∇²𝑢²_𝐿2(Ω), (19) where 0<𝜇≤(1/𝐶₁).
The verification of the action of 𝐺_𝑥 can be proved from the definition. The details, not being germane to this paper, are omitted here. But a formula will be useful in the sequel as follows.

Lemma 3. Let 𝐺(𝑢)=∫¹₀(𝐝𝐢𝐯𝜎(∇(𝜌𝑢)),𝑢)𝑑𝜌; then one has 𝐺(𝑢(𝑡))−𝐺(𝑢(0))=^𝑡₀𝐝𝐢𝐯𝜎∇𝑢𝜂,𝑢_𝜂𝜂𝑑𝜂, (20) for 𝑢∶[0,𝑇)→𝐷 with a strongly continuous derivative 𝑢_𝑡.

Proof. By the chain rule and the action of 𝐺_𝑥, we have 𝑑𝐺(𝑢)=𝑑𝑑𝑡𝑑𝑡¹₀𝐝𝐢𝐯𝜎^→∇𝑑𝜌𝑢,𝑢𝑑𝜌=−𝑑𝑡¹₀^→𝜎∇𝜌𝑢,∇𝑢𝑑𝜌=−¹₀𝜌𝜎^→_𝜌∇𝑢⋅∇𝑢_𝑡,∇𝑢+𝜎^→𝜌∇𝑢,∇𝑢_𝑡𝑑𝜌=−¹₀𝜌𝜎^→_𝜌∇𝑢⋅∇𝑢,∇𝑢_𝑡+𝜎^→𝜌∇𝑢,∇𝑢_𝑡𝑑𝜌=−¹₀𝜌𝑑𝑑𝜌^→𝜎𝜌∇𝑢,∇𝑢_𝑡+𝜎^→𝜌∇𝑢,∇𝑢_𝑡𝑑𝜌=−¹₀𝑑𝜌𝑑𝜌^→𝜎𝜌∇𝑢,∇𝑢_𝑡=𝑑𝜌𝐝𝐢𝐯𝜎^→(∇𝑢),𝑢_𝑡, (21) where we have used the symmetry of 𝜎^→_𝑥 in the fourth line.

The following lemma contributing to the result of this paper is analogous to Corollary 1.1 of [17] with slight modification.

Lemma 4. Assume that 𝜎^→ is homogenous of degree 1+𝛾 for some 𝛾>0 (i.e., 𝜎^→(𝑠ℎ)=𝑠^→1+𝛾𝜎(ℎ) for all 𝑠>0 and for all ℎ∈𝐷). Let (ℎ₀,𝐝𝐢𝐯𝜎^→(∇ℎ₀))>0 for some ℎ₀∈𝐷. Then there are infinitely many vectors 𝑢₀ such that 𝐺𝑢₀>12𝑢₀,𝐴𝑢₀=12∇³𝑢₀²_𝐿2(Ω)+∇𝑢₀²_𝐿2(Ω)−𝜇∇²𝑢₀²_𝐿2(Ω). (22)

Proof. Let 𝑢₀=𝑠ℎ₀, where 𝑠 is large enough so that 𝑠^𝛾𝐺ℎ₀=𝑠^𝛾1₀𝐝𝐢𝐯𝜎^→𝜌∇ℎ₀,ℎ₀>1𝑑𝜌2ℎ₀,𝐴ℎ₀=12∇³ℎ₀²_𝐿2(Ω)+∇ℎ₀²_𝐿2(Ω)−𝜇∇²ℎ₀²_𝐿2(Ω). (23) Then for all 𝑠>𝑠₀=^𝛾2+𝛾∇³ℎ₀²_𝐿2(Ω)+∇ℎ₀²_𝐿2(Ω)−𝜇∇²ℎ₀²_𝐿2(Ω)2𝐝𝐢𝐯𝜎^→∇ℎ₀,ℎ₀, (24) we have 𝐺𝑢₀=¹₀𝐝𝐢𝐯𝜎^→𝜌𝑠∇ℎ₀,𝑠ℎ₀𝑑𝜌=−𝜎¹₀^→𝜌𝑠∇ℎ₀,𝑠∇ℎ₀𝑑𝜌=𝑠^2+𝛾𝐺ℎ₀>12𝑢₀,𝐴ℎ₀. (25)
The local existence of solution for the problem (9) can be obtained by the standard Faedo-Galerkin approximation methods. The interested reader is referred to Lions [18] or Robinson [19] for details.

Next, we are ready to state the blow-up result of this paper.

Theorem 5. Let 𝑢∶[0,𝑇)→𝐷 be a strongly continuously differentiable solution of (9) in the 𝐷 norm. Suppose that 0<𝜇≤(1/𝐶₁) with (𝐶₁ depending on the constant 𝐶 of Sobolev’s interpolation inequality) (𝐝𝐢𝐯𝜎^→(∇𝑢),𝑢)≥2(2𝛼+1)𝐺(𝑢), where (𝐺(𝑢)=∫¹₀(𝐝𝐢𝐯𝜎^→(𝜌∇𝑢),𝑢)𝑑𝜌). Finally let 𝑢₀ satisfy 𝐺𝑢₀>12𝑢₀,𝐴𝑢₀=12∇³𝑢₀²_𝐿2(Ω)+∇𝑢₀²_𝐿2(Ω)−𝜇∇²𝑢₀²_𝐿2(Ω). (26) Then the solution u can only exist on a bounded interval [0,𝑇), and in fact 𝑇≤4𝛼²𝛽𝑢₀,𝑢₀+∇𝑢₀,∇𝑢₀−2𝛼𝑢₀,𝑢₁+∇𝑢₀,∇𝑢₀−2𝛼𝑢₀,𝑢₁²+4𝛼²𝛽𝑢₀,𝑢₀²×4𝛼²𝛽∇𝑢₀,∇𝑢₀−2𝛼𝑢₀,𝑢₁²+4𝛼²𝛽𝑢₀,𝑢₀^1/2−1, (27) while also lim_{𝑡→𝑇−}‖𝑢‖²_𝐿2(Ω)+^𝑡₀‖∇𝑢‖²_𝐿2(Ω)𝑑𝜂=+∞, (28) and consequently limsup_{𝑡→𝑇−}‖𝑢‖²_𝐻1=+∞. (29)

Proof. For arbitrary 𝑇₀, 𝛽, 𝜏>0 and 𝑡∈[0,𝑇₀), let 𝐹(𝑡)=(𝑢,𝑢)+^𝑡₀()+𝑇𝑢,−Δ𝑢𝑑𝜂₀−𝑡𝑢₀,−Δ𝑢₀+𝛽(𝑡+𝜏)². (30) A direct computation yields 𝐹^′(𝑡)=2𝑢,𝑢_𝑡+(𝑢,−Δ𝑢)−𝑢₀,−Δ𝑢₀+2𝛽(𝑡+𝜏)=2𝑢,𝑢_𝑡+(∇𝑢,∇𝑢)−∇𝑢₀,∇𝑢₀+2𝛽(𝑡+𝜏)=2𝑢,𝑢_𝑡+2^𝑡₀∇𝑢,∇𝑢_𝜂𝑑𝜂+2𝛽(𝑡+𝜏). (31) Suppressing the argument 𝑡, we see that 𝐹^′′(𝑡)=2𝑢,𝑢_𝑡𝑡+2𝑢_𝑡,𝑢_𝑡+2^𝑡₀∇𝑢,∇𝑢_𝜂𝜂𝑑𝜂+2∇𝑢₁,∇𝑢₀+2𝛽. (32) Hence, from (30), (31), and (32), we find after some algebra that 𝐹^′′𝐹−(𝛼+1)𝐹^′(𝑡)²=4(𝛼+1)𝑆²+2𝐹𝑢,𝑢_𝑡𝑡−(2𝛼+1)𝑢_𝑡,𝑢_𝑡+2𝐹^𝑡₀∇𝑢,∇𝑢_𝜂𝜂𝑑𝜂−2(𝛼+1)^𝑡₀∇𝑢_𝜂,∇𝑢_𝜂(𝑇𝑑𝜂+4𝛼+1)₀−𝑡∇𝑢₀,∇𝑢₀×𝑢_𝑡,𝑢_𝑡+^𝑡₀∇𝑢_𝜂,∇𝑢_𝜂𝑑𝜂+𝛽+2𝐹∇𝑢₁,∇𝑢₀−(2𝛼+1)𝛽, (33) where 𝑆²=−𝑢,𝑢_𝑡+^𝑡₀∇𝑢,∇𝑢_𝜂𝑑𝜂+𝛽(𝑡+𝜏)²+(𝑢,𝑢)+^𝑡₀(𝑢,−Δ𝑢)𝑑𝜂+𝛽(𝑡+𝜏)²×𝑢_𝑡,𝑢_𝑡+^𝑡₀∇𝑢_𝜂,∇𝑢_𝜂𝑑𝜂+𝛽. (34) Using Schwarz’s inequality, we have (𝑢,𝑢)𝑢_𝑡,𝑢_𝑡≥𝑢,𝑢_𝑡²,_𝑡0(∇𝑢,∇𝑢)𝑑𝜂^𝑡₀∇𝑢_𝜂,∇𝑢_𝜂𝑑𝜂≥^𝑡₀∇𝑢,∇𝑢_𝜂𝑑𝜂²,(𝑢,𝑢)^𝑡₀∇𝑢_𝜂,∇𝑢_𝜂𝑑𝜂+𝑢,𝑢_𝑡^𝑡₀(∇𝑢,∇𝑢)𝑑𝜂≥2(𝑢,𝑢)𝑢_𝑡,𝑢_𝑡^𝑡₀(∇𝑢,∇𝑢)𝑑𝜂^𝑡₀∇𝑢_𝜂,∇𝑢_𝜂𝑑𝜂≥2𝑢,𝑢_𝑡^𝑡₀∇𝑢,∇𝑢_𝜂()𝑢𝑑𝜂,𝑢,𝑢𝛽+_𝑡,𝑢_𝑡𝛽(𝑡+𝜏)²≥2(𝑢,𝑢)𝑢_𝑡,𝑢_𝑡𝛽²(𝑡+𝜏)²≥2𝑢,𝑢_𝑡𝛽(𝑡+𝜏). (35) By (35), we have 𝑆²≥0. Let 𝐻(𝑡)=𝑢,𝑢_𝑡𝑡−(2𝛼+1)𝑢_𝑡,𝑢_𝑡+∇𝑢₁,∇𝑢₀−()𝛽−2𝛼+1_𝑡0Δ𝑢_𝜂,𝑢_𝜂𝑑𝜂+2(𝛼+1)^𝑡₀Δ𝑢_𝜂,𝑢_𝜂=𝑑𝜂𝑢,𝑢_𝑡𝑡−(2𝛼+1)𝑢_𝑡,𝑢_𝑡+∇𝑢₁,∇𝑢₀−()𝛽−2𝛼+1_𝑡0𝑢_𝜂𝜂,𝑢_𝜂𝑑𝜂+2(𝛼+1)^𝑡₀𝑢_𝜂𝜂,𝑢_𝜂−𝑑𝜂_𝑡0(𝐴𝑢,𝑢)_𝜂−2(𝛼+1)𝐴𝑢,𝑢_𝜂+𝑑𝜂_𝑡0𝐝𝐢𝐯𝜎^→(∇𝑢),𝑢_𝜂−2(𝛼+1)𝐝𝐢𝐯𝜎^→(∇𝑢),𝑢_𝜂𝑑𝜂. (36) Thus 𝐻^′(𝑡)=−2𝛼𝑢_𝑡𝑡,𝑢_𝑡+2𝛼𝑢_𝑡+,𝐴𝑢𝐝𝐢𝐯𝜎^→(∇𝑢),𝑢_𝑡()𝑑−2𝛼+1𝐺(𝑢)𝑑𝑡=−2𝛼Δ𝑢_𝑡,𝑢_𝑡−2𝛼𝐝𝐢𝐯𝜎^→(∇𝑢),𝑢_𝑡+4𝛼𝑢_𝑡,𝐴𝑢+𝐝𝐢𝐯𝜎^→(∇𝑢),𝑢_𝑡()𝑑−2𝛼+1𝐺(𝑢)𝑑𝑡=2𝛼∇𝑢_𝑡,∇𝑢_𝑡+4𝛼𝑢_𝑡+,𝐴𝑢𝐝𝐢𝐯𝜎^→(∇𝑢),𝑢_𝑡−()𝑑4𝛼+2𝐺(𝑢).𝑑𝑡 (37) Using the positive semidefiniteness of 𝐴, Lemma 3, and (14), we have 𝐻^′(𝑡)=𝐻(0)+2𝛼^𝑡₀∇𝑢_𝜂,∇𝑢_𝜂𝑑𝜂+4𝛼^𝑡₀𝑢_𝜂+,𝐴𝑢𝑑𝜂_𝑡0𝐝𝐢𝐯𝜎^→(∇𝑢),𝑢_𝜂−()𝑑4𝛼+2𝐺(𝑢=()𝐺𝑢𝑑𝜂)𝑑𝜂4𝛼+2₀−(2𝛼+1)𝑢₀,𝐴𝑢₀+𝑢₁,𝑢₁+∇𝑢₁,∇𝑢₁−(2𝛼+1)𝛽+2𝛼^𝑡₀∇𝑢_𝜂,∇𝑢_𝜂()+𝑑𝜂+2𝛼𝑢,𝐴𝑢𝐝𝐢𝐯𝜎^→()−()𝐺(𝑢≥()𝐺𝑢∇𝑢,𝑢4𝛼+2)4𝛼+2₀−(2𝛼+1)𝑢₀,𝐴𝑢₀−()𝛽(𝐺𝑢2𝛼+1=22𝛼+1)₀−12𝐴𝑢₀,𝑢₀−𝛽2. (38) Thus, from what has been discussed above, we have 𝐹𝐹^′′−(𝛼+1)𝐹^′2≥4(2𝛼+1)𝐹𝐺𝑢₀−12𝐴𝑢₀,𝑢₀−𝛽2. (39) Therefore, for any 𝛽>0 such that 𝛽≤2𝐺𝑢₀−𝐴𝑢₀,𝑢₀=2¹₀𝐝𝐢𝐯𝜎^→𝜌∇𝑢₀,𝑢₀−∇𝑑𝜌₃𝑢₀²_𝐿2(Ω)+∇𝑢₀²_𝐿2(Ω)−𝜇∇²𝑢₀²_𝐿2(Ω), (40)𝐹𝐹^′′−(𝛼+1)(𝐹^′)²≥0 and (𝐹^−𝛼(𝑡))^′′≤0. We see that 𝐹(𝑡)>0, for all 𝑡∈[0,𝑇₀) and 𝐹^′(0)=2(𝑢₀,𝑢₁)+2𝛽𝜏>0, if 𝜏 is sufficiently large. Since a concave function must always lie below any tangent line, so we have 𝐹^−𝛼(𝑡)≤𝐹^−𝛼(0)+𝐹^−𝛼(0)^′𝑡 (41) or 𝐹(𝑡)≥𝐹^1+(1/𝛼)(0)𝐹(0)−𝛼𝑡𝐹^′(0)^−1/𝛼; (42) we may choose 𝑇₀ such that 𝑇₀≥(𝐹(0)/𝛼(𝐹^′(0)))≡𝑇_𝛽𝜏. Thus, we see that the interval of existence of 𝑢 must be contained in [0,𝐹(0)/𝛼(𝐹^′(0))) and that the finite time blow-up of solution of (9) is proved. Let 𝛽₀=2𝐺𝑢₀−𝐴𝑢₀,𝑢₀=2¹₀𝐝𝐢𝐯𝜎^→𝜌∇𝑢₀,𝑢₀−∇𝑑𝜌₃𝑢₀²_𝐿2(Ω)+∇𝑢₀²_𝐿2(Ω)−𝜇∇²𝑢₀²_𝐿2(Ω),𝑇_𝛽𝜏=𝐹(0)𝛼𝐹^′(0)=𝑢₀,𝑢₀+𝑇₀∇𝑢₀,∇𝑢₀+𝛽𝜏²2𝛼𝑢₀,𝑢₁.+2𝛼𝛽𝜏 (43) Since 𝑇₀≥𝑇_𝛽𝜏, we have (𝐹^−𝛼(𝑡))^′′≤0. Even if we take 𝑇₀=𝑇_𝛽𝜏, we have 𝑇_𝛽𝜏=𝑢₀,𝑢₀+𝛽𝜏²2𝛼𝑢₀,𝑢₁+2𝛼𝛽𝜏−∇𝑢₀,∇𝑢₀⁻¹; (44) thus we must choose 𝜏 so large such that 2𝛼𝛽𝜏>(∇𝑢₀,∇𝑢₀)−2𝛼(𝑢₀,𝑢₁). As a function, 𝑇_𝛽𝜏 has a minimum at 𝛽=𝜏=𝜏∇𝑢₀,∇𝑢₀−2𝛼𝑢₀,𝑢₁+∇𝑢₀,∇𝑢₀−2𝛼𝑢₀,𝑢₁²+4𝛼²𝛽𝑢₀,𝑢₀×2𝛼𝛽⁻¹, (45) and this minimum is 𝑇_{𝛽𝜏(𝛽)}=4𝛼²𝛽𝑢₀,𝑢₀+∇𝑢₀,∇𝑢₀−2𝛼𝑢₀,𝑢₁+∇𝑢₀,∇𝑢₀−2𝛼𝑢₀,𝑢₁²+4𝛼²𝛽𝑢₀,𝑢₀²×4𝛼²𝛽∇𝑢₀,∇𝑢₀−2𝛼𝑢₀,𝑢₁²+4𝛼²𝛽𝑢₀,𝑢₀^1/2−1. (46) Since 𝛽 is restricted to [0,𝛽₀], we see that 𝑇_{𝛽𝜏(𝛽)} attains its minimum at 𝛽=𝛽₀. Thus 𝑇 cannot exceed 𝑇_{𝛽𝜏(𝛽0)}.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.