Blow-Up of Solutions for a Class of Nonlinear Pseudoparabolic Equations with a Memory Term

Abstract

We consider the nonlinear pseudoparabolic equation with a memory term 𝑢_𝑡−Δ𝑢−Δ𝑢_𝑡+∫^𝑡₀𝜆(𝑡−𝜏)Δ𝑢(𝜏)𝑑𝜏=div|∇𝑢|^𝑝−2𝑢+𝑢^1+𝛼, 𝑥∈Ω, 𝑡>0, with an initial condition and Dirichlet boundary condition. Under negative initial energy and suitable conditions on p, 𝛼, and the relaxation function 𝜆(𝑡), we prove a finite-time blow-up result by using the concavity method.

1. Introduction

In this paper, we consider the initial boundary value problem for a class of nonlinear pseudoparabolic equations with a memory term: 𝑢_𝑡−𝛽Δ𝑢−𝛾Δ𝑢_𝑡+^𝑡₀𝜆()()||𝑡−𝜏Δ𝑢𝑥,𝜏𝑑𝜏=𝛿div∇𝑢^𝑝−2(𝑢)𝑢()∇𝑢+𝑓,𝑥∈Ω,𝑡>0,𝑥,0=𝑢₀(𝑥)|,𝑥∈Ω,𝑢(𝑥,𝑡)_𝜕Ω=0,𝑥∈𝜕Ω,𝑡>0, (1) where Ω⊂R^𝑛 is a bounded domain with a smooth boundary 𝜕Ω, 𝜆∶R₊→R is a given continuous function, 𝛽, 𝛾, and 𝛿 are all real constant parameters, 𝑝>2, and div(|∇𝑢|^𝑝−2∇𝑢) is the so-called 𝑝-Laplace operator. This type of equations describes a variety of important physical processes, such as the analysis of heat conduction in materials with memory, electric signals in nonlinear telegraph line with nonlinear damping, viscous flow in materials with memory [1], vibration of nonlinear elastic rod with viscosity [2], nonlinear bidirectional shallow water waves [3], and the velocity evolution of ion-acoustic waves in a collisionless plasma when an ion viscosity is invoked [4].

Equation (1) includes many important mathematical physics models.

In the absence of the memory term, the viscous term, and 𝑝-Laplace operator term (𝛾=𝛿=0,𝜆(𝑠)=0),𝛽=1, the model reduces to semilinear parabolic equation: 𝑢_𝑡−Δ𝑢=𝑓(𝑢),𝑥∈Ω,𝑡>0. (2) On the existence, nonexistence, and the properties of solutions of (2), there have been many results [5–9].

In the absence of the memory term and 𝑝-Laplace operator term (𝛿=0,𝜆(𝑠)=0),𝛽=𝛾=1, the model reduces to semilinear pseudoparabolic equation: 𝑢_𝑡−Δ𝑢−Δ𝑢_𝑡=𝑓(𝑢),𝑥∈Ω,𝑡>0. (3) Kaikina et al. [10] discussed the periodic boundary value problem of (3) under some assumption forms of nonlinear function 𝑓. Cao et al. [11] investigated a class of periodic problems of pseudoparabolic type equations with nonlinear periodic sources. A rather complete classification of the exponent 𝑝 was given, in terms of the existence and nonexistence of nontrivial and nonnegative periodic solutions. Cao et al. [12] dealt with the Cauchy problem for semilinear pseudoparabolic equations. Existence and uniqueness of local solutions were proved, and the large-time behavior was investigated. Kaikina [13] and Xu and Su [14] discussed the initial boundary value problems of pseudoparabolic equation (3) under some classes of nonlinear function 𝑓(𝑢). They obtained some sufficient conditions of existence and uniqueness of local solutions and the large-time behavior of global solutions.

In the absence of the memory term and the viscous term (𝜆(𝑠)=0,𝛾=0),𝛽=0,𝛿=1, (1) becomes nonlinear parabolic equation with 𝑝-Laplace nonlinear term: 𝑢_𝑡=div|∇𝑢|^𝑝−2∇𝑢+𝑓(𝑢),𝑥∈Ω,𝑡>0. (4) Tsutsumi [15] studied the initial boundary value problem of (4) with 𝑓(𝑢)=𝑢^1+𝛼, where 𝑝<2+𝛼. He obtained the existence of global weak solutions by using the potential well method. Liu and Zhao [16] considered the same problem with critical initial conditions 𝐽(𝑢₀)=𝑑 or 𝐼(𝑢₀)=0 and proved the existence of global solution for this problem. Xu et al. [17] discussed (4) at the high energy level, where 𝑝<2+𝛼<∞ if 𝑛≤𝑝 and 𝑝<2+𝛼≤𝑛𝑝/(𝑛−𝑝) if 𝑛>𝑝. They proved the finite time blow-up of solutions by the comparison principle and variational methods. Messaoudi in [18] considers an initial boundary value problem related to (4) and proves, under suitable conditions on 𝑓, a blow-up result for solutions with vanishing or negative initial energy.

In the absence of the viscous term and 𝑝-Laplace operator (𝛾=0,𝛿=0),𝛽=0, Gripenberg [19] considered the nonlinear parabolic equation with Volterra integral term equation: 𝑢_𝑡=^𝑡₀𝑘(𝑡−𝑠)𝜎𝑢_𝑥(𝑥,𝑠)_𝑥(),()𝑑𝑠+𝑓𝑥,𝑡𝑥∈0,1,𝑡≥0. (5) He investigated the initial boundary value problem of (5) and established the global existence of a strong solution of the problem.

In the absence of the viscous term and 𝑝-Laplace operator (𝛾=0,𝛿=0), as 𝛽=1, the model reduces to the equation 𝑢_𝑡−Δ𝑢=^𝑡₀𝑏(𝑡−𝜏)Δ𝑢(𝜏)𝑑𝜏+𝑓(𝑢),𝑥∈Ω,𝑡>0. (6) Yin [20] obtained the global existence of a classical solution of (7) under the assumption of a one-sided growth condition. Messaoudi [21] investigated a semilinear parabolic equation with the viscoelastic memory term. He established the finite time blow-up result for the solution with negative or vanishing initial energy for nonlinear function 𝑓(𝑢)=|𝑢|^𝑝−2𝑢.

To the best of our knowledge, there are few works on the study of nonlinear pseudoparabolic equation with memory term of Volterra integral type. Shang and Guo [22–24] investigated the initial boundary value problem and initial value problem of the nonlinear pseudoparabolic equations with Volterra integral term: 𝑢_𝑡−𝑓(𝑢)_𝑥𝑥−𝑢_𝑥𝑥𝑡+^𝑡₀𝜆(𝑡−𝑠)𝜎𝑢(𝑥,𝑠),𝑢_𝑥(𝑥,𝑠)_𝑥𝑑𝑠=𝑓𝑥,𝑡,𝑢,𝑢_𝑥,𝑥∈(0,1),𝑡>0. (7) They proved the existence, uniqueness, and regularities of the global strong solution and gave some conditions of the nonexistence of global solution. In 2007, Ptashnyk [25] investigated the initial boundary value of degenerate quasilinear pseudoparabolic equations with memory term. He obtained some existence results of global solutions. Up to now, there are not any research works on the multidimensional nonlinear pseudoparabolic equations with memory term.

In the present work, we deal with the initial boundary problem of the nonlinear pseudoparabolic equation with the memory term of Volterra integral type, the damping term, and 𝑝-Laplace operator: 𝑢_𝑡−Δ𝑢−Δ𝑢_𝑡+^𝑡₀𝜆()(𝜏)||𝑡−𝜏Δ𝑢𝑑𝜏=div∇𝑢^𝑝−2∇𝑢+𝑢^1+𝛼𝑢(),𝑥∈Ω,𝑡>0,𝑥,0=𝑢₀(𝑥)|,𝑥∈Ω,𝑢(𝑥,𝑡)_𝜕Ω=0,𝑥∈𝜕Ω,𝑡>0, (8) where Ω⊂R^𝑛 is a bounded domain, 𝜆(𝑠)∶R₊→R is a given continuous function, 𝛽,𝛾,𝛿,>0, and div(|∇𝑢|^𝑝−2∇𝑢) is the so-called 𝑝-Laplace operator. By using the concavity method first introduced by Levine [5], under negative initial energy and suitable conditions on 𝑝,𝛼, and the relaxation function 𝜆(𝑡), we prove that there exists finite-time blow-up solution.

Without loss of generality, we choose 𝛽=𝛾=𝛿=1 in the following discussion.

2. Preliminaries and Main Results

In this section, we introduce some notations, basic definitions, and important lemmas which will be needed in this paper.

For functions 𝑢(𝑥,𝑡),𝑣(𝑥,𝑡) defined on Ω, we introduce (𝑢,𝑣)=_Ω𝑢𝑣𝑑𝑥,‖𝑢‖₂=_Ω|𝑢|²𝑑𝑥^1/2,‖𝑢‖_𝑝=_Ω|𝑢|^𝑝𝑑𝑥^1/𝑝,‖𝑢‖_𝐻^𝑚=_{|𝛼|≤𝑚}𝐷^𝛼𝑢²₂^1/2,‖𝑢‖_∞=esssup_𝑥∈Ω|𝑢(𝑥,𝜕)|Δ=²𝜕𝑥²₁𝜕+⋯+²𝜕𝑥²_𝑛𝜕,∇=𝜕𝑥₁𝜕,…,𝜕𝑥_𝑛. (9)

We now construct a space of functions as follows. Let 𝐻^𝑚(Ω) denote the Sobolev space with the norm ‖𝑢‖_𝐻^𝑚=(∑_{|𝛼|≤𝑚}‖𝐷^𝛼𝑢‖²₂)^1/2. 𝐶^∞₀(Ω) denotes the class of 𝐶^∞ functions with the compact support in Ω. 𝐻^𝑚₀(Ω) denotes the closure in 𝐻^𝑚(Ω) of 𝐶^∞₀(Ω). The Hilbert space 𝐻^𝑚₀(Ω) is a subspace of the Sobolev space 𝐻^𝑚(Ω).

The following are the basic hypotheses to establish the main results of this paper:(a)2<𝑝≤2+𝛼<∞;(b)𝜆 is a 𝐶¹ function satisfying 𝜆(𝜏)≥0,𝜆^′(𝜏)≤0,_∞0𝜆(𝜏)𝛼()𝑑𝜏<𝛼+2𝛼().𝛼+2+1 (10)(11)

To obtain the results of this paper, we will introduce the “modified” energy function: 𝐸(𝑡)=12(𝑡)+1𝜆∘∇𝑢)(21−^𝑡₀𝜆(𝜏)𝑑𝜏‖∇𝑢(𝑡)‖²₂+1𝑝‖∇𝑢(𝑡)‖^𝑝_𝑝−12+𝛼_Ω𝑢^2+𝛼𝑑𝑥, (12) where (𝜆∘𝑣)(𝑡)=^𝑡₀𝜆(𝑡−𝜏)‖𝑣(𝑡)−𝑣(𝜏)‖²₂𝑑𝜏. (13) The following lemma is similar to the lemma of [21] with slight modification.

Lemma 1. Assume that (10) hold. Let p satisfy (𝑎) and let 𝑢 be a solution of (8). Then 𝐸(𝑡) is nonincreasing function; that is 𝐸^′(𝑡)≤0. (14) Moreover, the following energy inequality holds: 𝐸(𝑡)+^𝑡₀𝑢_𝑡(𝜏)²_𝐻1𝑑𝜏<𝐸(0). (15)

Proof. By multiplying the equation in (8) by 𝑢_𝑡, integrating over Ω, we obtain 𝑑𝑑𝑡^𝑡₀𝑢_𝑡(𝜏)²₂𝑑𝜏+^𝑡₀∇𝑢_𝑡(𝜏)²₂1𝑑𝜏+2_Ω|∇𝑢|²+1𝑑𝑥𝑝‖∇𝑢‖^𝑝_𝑝−12+𝛼_Ω𝑢^2+𝛼−𝑑𝑥_𝑡0𝜆(𝑡−𝜏)_Ω∇𝑢_𝑡(𝑡)∇𝑢(𝜏)𝑑𝑥𝑑𝜏=0. (16) For the last term on the left side of (16), ^𝑡₀𝜆(𝑡−𝜏)_Ω∇𝑢_𝑡(𝑡)(𝜏)=∇𝑢𝑑𝑥𝑑𝜏_𝑡0𝜆(𝑡−𝜏)_Ω∇𝑢_𝑡(𝑡(𝜏)(𝑡+)[∇𝑢−∇𝑢)]𝑑𝑥𝑑𝜏_𝑡0𝜆(𝑡−𝜏)_Ω∇𝑢_𝑡(𝑡)(𝑡)1∇𝑢𝑑𝑥𝑑𝜏=−2^𝑡₀𝜆(𝑑𝑡−𝜏)𝑑𝑡_Ω|∇𝑢(𝜏)−∇𝑢(𝑡)|²+1𝑑𝑥𝑑𝜏2^𝑡₀𝜆(𝜏𝑑)𝑑𝑡_Ω|∇𝑢(𝑡)|²1𝑑𝑥𝑑𝜏=−2𝑑𝑑𝑡^𝑡₀𝜆(𝑡−𝜏)_Ω|∇𝑢(𝜏)−∇𝑢(𝑡)|²+1𝑑𝑥𝑑𝜏2𝑑𝑑𝑡^𝑡₀𝜆(𝜏)_Ω|∇𝑢(𝑡)|²+1𝑑𝑥𝑑𝜏2^𝑡₀𝜆^′(𝑡−𝜏)_Ω|∇𝑢(𝜏)−∇𝑢(𝑡)|²−1𝑑𝑥𝑑𝜏2𝜆(𝑡)_Ω|∇𝑢(𝑡)|²𝑑𝑥. (17) Inserting (17) into (16), we have 𝑑𝑑𝑡^𝑡₀𝑢_𝑡(𝜏)²₂𝑑𝜏+^𝑡₀∇𝑢_𝑡(𝜏)²₂1𝑑𝜏+2_Ω|∇𝑢|²+1𝑑𝑥𝑝‖∇𝑢‖^𝑝_𝑝−12+𝛼_Ω𝑢^2+𝛼+1𝑑𝑥2𝑑𝑑𝑡^𝑡₀𝜆(𝑡−𝜏)_Ω|∇𝑢(𝜏)−∇𝑢(𝑡)|²−1𝑑𝑥𝑑𝜏2𝑑𝑑𝑡^𝑡₀𝜆(𝜏)_Ω|∇𝑢(𝑡)|²=1𝑑𝑥𝑑𝜏2^𝑡₀𝜆^′(𝑡−𝜏)_Ω|∇𝑢(𝜏)−∇𝑢(𝑡)|²−1𝑑𝑥𝑑𝜏2𝜆(𝑡)_Ω|∇𝑢(𝑡)|²𝑑𝑥≤0, (18) for regular solution. The proof of Lemma 1 is completed. This result is valid for weak solutions by a simple density argument.

Now we consider the finite time blow-up of solutions with 𝐸(0)<0 for the problem (8).

Theorem 2. Let 𝑝 satisfy (a) and let the relaxation function 𝜆(𝑠) be a 𝐶¹ function satisfying (10) and (11). Assume that 𝑢₀∈𝐻¹₀(Ω) such that 𝐸(0)<0. Then the solutions 𝑢(𝑥,𝑡) of the problem (8) blow up in finite time; that is, the maximum existence time 𝑇_max of 𝑢(𝑥,𝑡) is finite and lim_{𝑡↦𝑇max}‖𝑢(𝑥,𝑡)‖²_𝐻1=+∞. (19)

Proof. The proof makes use of the so-called “concavity” arguments. For any 𝑇₀>0, let 𝑀(𝑡)=^𝑡₀‖𝑢(𝜏)‖²₂+𝑑𝜏_𝑡0‖∇𝑢(𝜏)‖²₂𝑑𝜏+𝑇₀−𝑡∇𝑢₀²₂+𝑢₀²₂, (20) for 𝑡∈[0,𝑇₀].
A direct computation yields 𝑀^′(𝑡)=‖𝑢(𝑡)‖²₂+‖∇𝑢(𝑡)‖²₂−∇𝑢₀²₂+𝑢₀²₂=2^𝑡₀𝑢,𝑢_𝑡𝑑𝜏+2^𝑡₀∇𝑢,∇𝑢_𝑡𝑀𝑑𝜏,_′′(𝑡)=2𝑢,𝑢_𝑡+2∇𝑢,∇𝑢_𝑡. (21) By multiplying (8) with 𝑢 and integrating over Ω, 𝑢,𝑢_𝑡+∇𝑢,∇𝑢_𝑡=−‖∇𝑢(𝑡)‖²₂−_Ω^𝑡₀𝜆()(𝜏)(𝑡)−‖‖𝑡−𝜏Δ𝑢𝑑𝜏𝑢𝑑𝑥∇𝑢(𝑡)^𝑝_𝑝+_Ω𝑢^2+𝛼𝑑𝑥. (22) This implies that 𝑀^′′(𝑡)=−2‖∇𝑢(𝑡)‖²₂−2_Ω^𝑡₀𝜆()(𝜏)(𝑡)‖‖𝑡−𝜏Δ𝑢𝑑𝜏𝑢𝑑𝑥−2∇𝑢(𝑡)^𝑝_𝑝+_Ω𝑢^2+𝛼𝑑𝑥, (23) and we have 𝑀^′′(𝑡)𝑀(𝑡)−𝛼+44𝑀^′(𝑡)²=2𝑀(𝑡)−‖∇𝑢(𝑡)‖²₂−_Ω^𝑡₀𝜆()(𝜏)(𝑡)−‖‖𝑡−𝜏Δ𝑢𝑑𝜏𝑢𝑑𝑥∇𝑢(𝑡)^𝑝_𝑝+_Ω𝑢^2+𝛼−𝑑𝑥𝛼+442^𝑡₀𝑢,𝑢_𝑡𝑑𝜏+2^𝑡₀∇𝑢,∇𝑢_𝑡𝑑𝜏²=2𝑀(𝑡)−‖∇𝑢(𝑡)‖²₂−_Ω^𝑡₀𝜆()(𝜏)(𝑡)−‖(𝑡𝑡−𝜏Δ𝑢𝑑𝜏𝑢𝑑𝑥∇𝑢)‖^𝑝_𝑝+_Ω𝑢^2+𝛼+()×𝐻(𝑡)−𝑀(𝑡)−𝑇𝑑𝑥𝛼+4₀−𝑡∇𝑢₀²₂+𝑢₀²₂×^𝑡₀𝑢_𝑡,𝑢_𝑡𝑑𝜏+^𝑡₀∇𝑢_𝑡,∇𝑢_𝑡𝑑𝜏, (24) where 𝐻(𝑡)=^𝑡₀(𝑢,𝑢)𝑑𝜏+^𝑡₀()×∇𝑢,∇𝑢𝑑𝜏_𝑡0𝑢_𝑡,𝑢_𝑡𝑑𝜏+^𝑡₀∇𝑢_𝑡,∇𝑢_𝑡−𝑑𝜏_𝑡0𝑢,𝑢_𝑡𝑑𝜏+^𝑡₀∇𝑢,∇𝑢_𝑡𝑑𝜏². (25) Using Schwartz’s inequality, we have ^𝑡₀𝑢,𝑢_𝑡𝑑𝜏²≤^𝑡₀‖𝑢‖²₂𝑑𝜏^𝑡₀𝑢_𝑡²₂𝑑𝜏,_𝑡0(∇𝑢,∇𝑢_𝑡)𝑑𝜏²≤^𝑡₀‖∇𝑢‖²₂𝑑𝜏^𝑡₀∇𝑢_𝑡²₂𝑑𝜏,_𝑡0𝑢,𝑢_𝑡𝑑𝜏^𝑡₀∇𝑢,∇𝑢_𝑡≤1𝑑𝜏2^𝑡₀‖𝑢‖²₂𝑑𝜏^𝑡₀∇𝑢_𝑡²₂+1𝑑𝜏2^𝑡₀𝑢_𝑡²₂𝑑𝜏^𝑡₀‖∇𝑢‖²₂𝑑𝜏. (26) By (26), we have 𝐻(𝑡)≥0,𝑡∈0,𝑇₀. (27) Thus, 𝑀^′′(𝑡)𝑀(𝑡)−𝛼+44𝑀^′(𝑡)²≥𝑀(𝑡)𝜂(𝑡), (28) where 𝜂(𝑡)=−(𝛼+4)^𝑡₀𝑢_𝑡,𝑢_𝑡𝑑𝜏+^𝑡₀∇𝑢_𝑡,∇𝑢_𝑡‖‖𝑑𝜏−2∇𝑢(𝑡)²₂−2_Ω^𝑡₀𝜆()(𝜏)(𝑡)‖‖𝑡−𝜏Δ𝑢𝑑𝜏𝑢𝑑𝑥−2∇𝑢(𝑡)^𝑝_𝑝+2_Ω𝑢^2+𝛼𝑑𝑥. (29) For the third term on the left of (29), we have −_Ω^𝑡₀𝜆()(𝜏)(𝑡)=𝑡−𝜏Δ𝑢𝑑𝜏𝑢𝑑𝑥_𝑡0𝜆(𝑡−𝜏)_Ω(𝜏)(𝑡)=∇𝑢∇𝑢𝑑𝑥𝑑𝜏_𝑡0𝜆(𝑡−𝜏)_Ω∇𝑢(𝑡)∇[𝑢(𝜏)−𝑢(𝑡)]𝑑𝑥𝑑𝜏+^𝑡₀𝜆(𝑡−𝜏)‖∇𝑢(𝑡)‖²₂𝑑𝜏. (30) By (29) and (30), we have 𝜂(𝑡)=−(𝛼+4)^𝑡₀𝑢_𝑡²_𝐻1𝑑𝜏−21−^𝑡₀𝜆(𝑡−𝜏)𝑑𝜏‖∇𝑢(𝑡)‖²₂+2^𝑡₀𝜆(𝑡−𝜏)_Ω(𝑡(𝜏)(𝑡‖‖∇𝑢)[∇𝑢−∇𝑢)]𝑑𝑥𝑑𝜏−2∇𝑢^𝑝_𝑝+2_Ω𝑢^2+𝛼()𝑑𝑥≥−𝛼+4^𝑡₀𝑢_𝑡²_𝐻1𝑑𝜏−21−^𝑡₀𝜆(𝑡−𝜏)𝑑𝜏‖∇𝑢(𝑡)‖²₂−2‖∇𝑢‖^𝑝_𝑝+2_Ω𝑢^2+𝛼𝑑𝑥−2𝛼+22^𝑡₀𝜆(𝑡−𝜏)‖∇𝑢(𝜏)−∇𝑢(𝑡)‖²₂+1𝑑𝜏2(𝛼+2)^𝑡₀𝜆(𝑡−𝜏)‖∇𝑢(𝑡)‖²₂()𝐸(𝑡)+()𝑑𝜏=−2𝛼+2𝛼+21−^𝑡₀𝜆()×‖(𝑡𝑡−𝜏𝑑𝜏∇𝑢)‖²₂+2(𝛼+2)𝑝‖∇𝑢‖^𝑝_𝑝−(𝛼+4)^𝑡₀𝑢_𝑡²_𝐻11𝑑𝜏−𝛼+2^𝑡₀𝜆(𝜏)𝑑𝜏‖∇𝑢(𝑡)‖²₂−21−^𝑡₀𝜆(𝑡−𝜏)𝑑𝜏‖∇𝑢(𝑡)‖²₂−2‖∇𝑢‖^𝑝_𝑝=−2(𝛼+2)𝐸(𝑡)+𝛼1−^𝑡₀𝜆(𝑡−𝜏)𝑑𝜏‖∇𝑢(𝑡)‖²₂−1𝛼+2^𝑡₀𝜆(𝜏)𝑑𝜏‖∇𝑢(𝑡)‖²₂−(𝛼+4)^𝑡₀𝑢_𝑡²_𝐻1𝑑𝜏+2𝛼−2𝑝+4𝑝‖∇𝑢‖^𝑝_𝑝. (31) Using Lemma 1, we have 𝐸(𝑡)+^𝑡₀𝑢_𝑡²_𝐻1𝑑𝜏<𝐸(0), (32) and then 𝜂(𝑡)≥−2(𝛼+2)𝐸(0)+𝛼^𝑡₀𝑢_𝑡²_𝐻1+𝛼𝑑𝜏1−^𝑡₀𝜆(𝜏)−1𝑑𝜏𝛼+2^𝑡₀𝜆(𝜏)𝑑𝜏‖∇𝑢(𝑡)‖²₂=−2(𝛼+2)𝐸(0)+𝛼^𝑡₀𝑢_𝑡²_𝐻1+𝛼()𝑑𝜏𝛼−𝛼+2+1𝛼+2^𝑡₀𝜆(𝜏)𝑑𝜏‖∇𝑢(𝑡)‖²₂. (33) Since ^∞₀𝜆(𝜏)𝛼()𝑑𝜏<𝛼+2𝛼(),𝛼+2+1 (34) this implies that 𝜂(𝑡)>𝛿,0≤𝑡<∞, (35) where 𝛿 is a positive constant.
From the discussion above, we see that 𝑀^′′(𝑡)𝑀(𝑡)−𝛼+44𝑀^′(𝑡)²≥𝑀(𝑡)𝛿. (36) From the definition of 𝑀(𝑡), there exists 𝜌>0, such that 𝑀(𝑡)≥𝜌,for𝑡∈[0,𝑇), (37) and we have 𝑀^′′(𝑡)𝑀(𝑡)−𝛼+44𝑀^′(𝑡)²≥𝜌𝛿. (38) Thus, 𝑀(𝑡)^{−𝛼/4′′}=−𝛼4𝑀(𝑡)^{−𝛼/4−2}𝑀^′′(𝑡)𝑀(𝑡)−𝛼+44𝑀^′(𝑡)²≤−𝛼4𝜌𝛿𝑀(𝑇)^{−(𝛼+8)/4}<0. (39) Hence, this proves that 𝑀(𝑡)^−𝛼/4 reaches 0 in finite time as 𝑡→𝑇⁻₁. Since 𝑇₁ is independent of 𝑇, we may assume that 𝑇₁<𝑇.
This means lim_{𝑡→𝑇−1}𝑀(𝑡)=+∞ (40) or lim_{𝑡→𝑇−1}^𝑡₀‖𝑢(𝜏)‖²₂𝑑𝜏+^𝑡₀‖∇𝑢(𝜏)‖²₂+𝑇𝑑𝜏₀−𝑡∇𝑢₀²₂+𝑢₀²₂=+∞. (41) This implies that lim_{𝑡→𝑇−1}^𝑡₀‖𝑢(𝜏)‖²_𝐻1𝑑𝜏=+∞. (42) Then, the desired assertion immediately follows.
Remark 1. In the absence of the viscous term (Δ𝑢_𝑡) or 𝑝-Laplace operator (div(|∇𝑢|^𝑝−2∇𝑢)) for the problem (8), the equation reduced to(1)𝑢_𝑡−Δ𝑢+∫^𝑡₀𝜆(𝑡−𝜏)Δ𝑢(𝜏)𝑑𝜏=𝑢^1+𝛼, (2)𝑢_𝑡−Δ𝑢−Δ𝑢_𝑡+∫^𝑡₀𝜆(𝑡−𝜏)Δ𝑢(𝜏)𝑑𝜏=𝑢^1+𝛼, (3)𝑢_𝑡−Δ𝑢+∫^𝑡₀𝜆(𝑡−𝜏)Δ𝑢(𝜏)𝑑𝜏=div(|∇𝑢|^𝑝−2∇𝑢)+𝑢^1+𝛼, and from the process of the proof, we can see that the results of Theorem 2 still hold.

Conflict of Interests

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

Acknowledgments

This work is supported by the NSF of China (40890150 and 40890153) and the Scientific Program (2008B080701042) of Guangdong Province. The authors thank the anonymous referees for their helpful comments and suggestions.