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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 530861, 15 pages
http://dx.doi.org/10.1155/2012/530861
Research Article

Global Nonexistence of Solutions for Viscoelastic Wave Equations of Kirchhoff Type with High Energy

College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received 12 February 2012; Accepted 26 March 2012

Academic Editor: Amol Sasane

Copyright © 2012 Gang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider viscoelastic wave equations of the Kirchhoff type 𝑢𝑡𝑡𝑀(𝑢22)Δ𝑢+𝑡0𝑔(𝑡𝑠)Δ𝑢(𝑠)d𝑠+𝑢𝑡=|𝑢|𝑝1𝑢 with Dirichlet boundary conditions, where 𝑝 denotes the norm in the Lebesgue space 𝐿𝑝. Under some suitable assumptions on 𝑔 and the initial data, we establish a global nonexistence result for certain solutions with arbitrarily high energy, in the sense that lim𝑡𝑇(𝑢(𝑡)22+𝑡0𝑢(𝑠)22d𝑠)= for some 0<𝑇<+.

1. Introduction

In this paper we consider the following problem:𝑢𝑡𝑡𝑀𝑢22Δ𝑢+𝑡0||𝑢𝑔(𝑡𝑠)Δ𝑢(𝑠)d𝑠+𝑡||𝑚1𝑢𝑡=|𝑢|𝑝1[𝑢,(𝑥,𝑡)Ω×(0,),𝑢(𝑥,𝑡)=0,(𝑥,𝑡)𝜕Ω×0,),𝑢(𝑥,0)=𝑢0(𝑥),𝑢𝑡(𝑥,0)=𝑢1(𝑥),𝑥Ω,(1.1) where Ω is a bounded domain in 𝑛 (𝑛1) with a smooth boundary 𝜕Ω,𝑝>1,𝑀(𝑠) is a nonnegative 𝐶1 function like 𝑀(𝑠)=𝑎+𝑏𝑠𝛾 for 𝑠0,𝑎0,𝑏0,𝑎+𝑏>0,𝛾>0 and 𝑔(𝑡) represents the kernel of memory term.

Problem (1.1) without the viscoelastic term (i.e., 𝑔=0) has been extensively studied and many results concerning global existence, decay, and blowup have been established. For example, the following equation: 𝑢𝑡𝑡𝑀𝑢22𝑢Δ𝑢+𝑔𝑡=𝑓(𝑢),(𝑥,𝑡)Ω×(0,),(1.2) has been considered by Matsuyama and Ikehata in [1] for 𝑔(𝑢𝑡)=𝛿|𝑢𝑡|𝑝1𝑢𝑡 and 𝑓(𝑢)=𝜇|𝑢|𝑞1𝑢. The authors proved existence of the global solutions by using Faedo-Galerkin method and the decay of energy based on the method of Nakao [24]. Later, Ono [5] investigated (1.2) for 𝑀(𝑠)=𝑏𝑠𝛾 and 𝑓(𝑢)=|𝑢|𝑝2𝑢. When 𝑔(𝑢𝑡)=Δ𝑢𝑡,𝑢𝑡 or |𝑢𝑡|𝛽𝑢𝑡, the author showed that the solutions blow up in finite time with 𝐸(0)0. For 𝑀(𝑠)=𝑎+𝑏𝑠𝛾 and 𝑔(𝑢𝑡)=𝑢𝑡, this model was considered by the same author in [6]. By applying the potential well method he obtained the blow-up properties with positive initial energy 𝐸(0). Recently, Zeng et al. [7] studied (1.2) for the case 𝑔(𝑢𝑡)=𝑢𝑡 with the same initial and boundary conditions as that of problem (1.1). By using the concavity argument, they proved that the solutions to (1.2) blow up in finite time with arbitrarily high energy.

In the case of 𝑀1 and in the presence of the viscoelastic term (i.e., 𝑔0), the equation 𝑢𝑡𝑡Δ𝑢+𝑡0||𝑢𝑔(𝑡𝑠)Δ𝑢(𝑠)d𝑠+𝑡||𝑚1𝑢𝑡=|𝑢|𝑝1𝑢,(𝑥,𝑡)Ω×(0,),(1.3) was studied by Messaoudi in [8], where the author proved that any weak solution with negative initial energy blows up in finite time if 𝑝>𝑚 and0𝑔(𝑠)d𝑠𝑝1𝑝1+1/(𝑝+1),(1.4)

while the solution continues to exist globally for any initial data in the appropriate space if m𝑝. This blow-up result was improved by the same author in [9] for positive initial energy under suitable conditions on 𝑔,𝑚, and 𝑝. More recently, Wang [10] investigated (1.3) and established a blow-up result with arbitrary positive initial energy. In the related work, Cavalcanti et al. [11] studied the following equation: 𝑢𝑡𝑡Δ𝑢+𝑡0𝑔(𝑡𝑠)Δ𝑢(𝑠)d𝑠+𝑎(𝑥)𝑢𝑡+|𝑢|𝛾𝑢=0,(𝑥,𝑡)Ω×(0,),(1.5) where 𝑎Ω𝑅+ is a function which may be null on a part of Ω. Under the condition that 𝑎(𝑥)𝑎0>0 on 𝜔Ω, with 𝜔 satisfying some geometric restrictions and 𝜉1𝑔(𝑡)𝑔(𝑡)𝜉2𝑔(𝑡), 𝑡0 to guarantee that 𝑔𝐿1((0,)) is small enough, they proved an exponential decay rate.

When 𝑔0 and 𝑀 is not a constant function, problems related to (1.1) have been treated by several authors. Wu and Tsai [12] considered the global existence, asymptotic behavior, and blow-up properties for the following equation: 𝑢𝑡𝑡𝑀𝑢22Δ𝑢+𝑡0𝑔(𝑡𝑠)Δ𝑢(𝑠)d𝑠Δ𝑢𝑡=𝑓(𝑢),(𝑥,𝑡)Ω×(0,),(1.6) with the same initial and boundary conditions as that of problem (1.1). They obtained the blow-up properties of local solution with small positive initial energy by using the direct method of [13]. Global existence and decay properties of the solutions were also obtained there. In [14], Wu then extended the decay result of [12] under a weaker condition on 𝑔.

For other papers related to existence, uniform decay and blowup of solutions of nonlinear wave equations, see [1533] and references therein.

Motivated by the above research, we consider problem (1.1) for 𝑚=1 in this paper and establish a global nonexistence result for certain solutions with arbitrarily high energy by using concavity technique. In this way, we can extend the result of [7] to nonzero term 𝑔 and the result of [10] to nonconstant 𝑀(𝑠). Throughout the rest of this paper, we always assume that 𝑚=1.

The structure of this paper is as follows. In Section 2, we present some assumptions, notations and the main result. In Section 3, we give the proof of the main result. Some further remarks are stated in Section 4.

2. Preliminaries and Main Result

In this section, we will give some assumptions, notations and state the main result. We first give the following assumptions:(A1)𝑔𝐶1([0,)) is a nonnegative and non-increasing function satisfying10𝑔(𝑠)d𝑠=𝑙>0.(2.1)(A2) The function 𝑒𝑡/2𝑔(𝑡) is of positive type in the following sense (see [10]): 𝑡0𝑣(𝑠)𝑠0𝑒(𝑠𝑧)/2𝑔(𝑠𝑧)𝑣(𝑧)d𝑧d𝑠0,𝑣𝐶1([0,)),𝑡>0.(2.2)

Remark 2.1. Assumption (A2) is needed to prove Lemma 3.1 below.

In order to prove our result, we make the following assumption on 𝑀 and 𝑔:(A3) there exists a positive constant 𝑚1 such that 𝑝+12𝑀(𝑠)𝑀(𝑠)+𝑝+120𝑔(𝜏)d𝜏𝑠𝑚1𝑠,𝑠0,(2.3)where 𝑀(𝑠)=𝑠0𝑀(𝜏)d𝜏.

Remark 2.2. It is clear that when 𝑀(𝑠)=𝑎+𝑏𝑠𝛾 for 𝑠0,𝑎0,𝑏0, 𝑎+𝑏>0, 𝛾>0 and 𝑝>1+2𝛾, condition (A3) can be replaced by 0𝑔(𝜏)d𝜏<𝑝1𝑝+1𝑎,if𝑎>0,𝑏0,(𝑝12𝛾)𝑏𝐶𝛾𝑝𝑢(𝑝+1)(𝛾+1)022𝛾,if𝑎=0,𝑏>0,(2.4) which is the same as the one in [10, Theorem  1.1] for the case 𝑎=1 and 𝑏=0, where 𝐶𝑝 is the constant from the Poincaré inequality 𝑢(𝑡)22𝐶𝑝𝑢(𝑡)22. Then, the possible choice of the positive constant 𝑚1 in (A3) can be easily obtained (see Section 4.1 for details).
It is necessary to state the local existence theorem for problem (1.1), whose proof follows the arguments in [12, 34].

Theorem 2.3. Assume that (A1) holds, and 1<𝑝𝑛/(n2) when n3,1<𝑝< when 𝑛=1,2. For 𝑢0𝐻10(Ω)𝐻2(Ω), 𝑢1𝐻10(Ω), and 𝑀(𝑢022)>0, problem (1.1) has a unique local solution [𝑢𝐶0,𝑇);𝐻10(Ω)𝐻2(Ω),𝑢𝑡[𝐶0,𝑇);𝐿2(Ω)𝐿2[0,𝑇);𝐻10(Ω),(2.5)for the maximum existence time 𝑇>0.

The energy functional 𝐸(𝑡) and an auxiliary functional 𝐼(𝑢) of the solution 𝑢(𝑡) of problem (1.1) are defined as follows:=1𝐸(𝑡)=𝐸(𝑢(𝑡))2𝑢𝑡22+12𝑀𝑢2212𝑡0𝑔(𝑠)d𝑠𝑢22+12(1𝑔𝑢)(𝑡)𝑝+1𝑢𝑝+1𝑝+1,(2.6)𝐼(𝑢)=𝑀𝑢22𝑢22𝑢𝑝+1𝑝+1,(2.7) where(𝑔𝑤)(𝑡)=𝑡0𝑔(𝑡𝑠)𝑤(𝑡,)𝑤(𝑠,)22d𝑠.(2.8)

As in [7, 10], we can get𝑑𝑢d𝑡𝐸(𝑡)=𝑡2212𝑔(𝑡)𝑢22+12𝑔𝑢(𝑡)0,𝑡0.(2.9) Then we have𝐸(𝑡)=𝐸(0)𝑡0𝑢𝑠221d𝑠+2𝑡0𝑔(1𝑢𝑠)d𝑠2𝑡0𝑔(𝑠)𝑢(𝑠)22d𝑠.(2.10) Now we are in a position to state the main result.

Theorem 2.4. Assume that (A1) holds and 1<𝑝𝑛/(𝑛2) when 𝑛3,1<𝑝< when 𝑛=1,2. Let 𝑢 be a solution of problem (1.1) with initial data 𝑢0𝐻10(Ω)𝐻2(Ω),𝑢1𝐻10(Ω) and 𝑀(𝑢022)>0, and further assume that 𝐼𝑢𝐸(0)>0,(2.11)0<0,(2.12)Ω𝑢0𝑢1𝑢𝑑𝑥>0,(2.13)022>(𝑝+1)𝐶𝑝𝑚1𝐸(0).(2.14) Then the solution of problem (1.1) blows up in finite time 0<𝑇<+, which means that lim𝑡𝑇(𝑢𝑡)22+𝑡0(𝑢𝑠)22𝑑𝑠=,(2.15) where 𝐶𝑝 is a constant from the Poincaré inequality and 𝑚1 comes from condition (𝐴3).

Remark 2.5. We note that the set of the initial data which satisfy conditions (2.11)–(2.14) is not empty (see Section 4.2 for details).

3. Proof of the Main Result

In this section we prove our main result, Theorem 2.4, whose proof follows the ideas already used in [7, 10] and relies on the following lemmas.

Lemma 3.1 (see [10, Lemma  2.1]). Assume that 𝑔(𝑡) satisfies assumptions (A1)-(A2), and 𝐻(𝑡) is a function which is twice continuously differentiable satisfying 𝐻(𝑡)+𝐻(𝑡)>𝑡0𝑔(𝑡𝑠)Ω𝑢(𝑠)𝑢(𝑡)d𝑥d𝑠,𝐻(0)>0,𝐻(0)>0,(3.1) for every 𝑡[0,𝑇), where 𝑢(𝑡) is the corresponding solution of problem (1.1) with 𝑢0 and 𝑢1. Then the function 𝐻(𝑡) is strictly increasing on [0,𝑇).

Lemma 3.2. Suppose that 𝑢0𝐻10(Ω)𝐻2(Ω) and 𝑢1𝐻10(Ω) satisfy Ω𝑢0𝑢1d𝑥>0.(3.2) If the solution 𝑢(𝑡) of problem (1.1) exists on [0,𝑇) and satisfies 𝐼(𝑢(𝑡))<0,(3.3) then 𝑢(𝑡)22 is strictly increasing on [0,𝑇).

Proof. Since 𝑢(𝑡) is the solution of problem (1.1), by a simple computation, we have 12𝑑2d𝑡2Ω||||𝑢(𝑥,𝑡)2d𝑥=Ω||𝑢𝑡||2+𝑢𝑢𝑡𝑡=𝑢d𝑥𝑡22𝑀𝑢22𝑢22+𝑢𝑝+1𝑝+1+𝑡0𝑔(𝑡𝑠)Ω𝑢(𝑠)𝑢(𝑡)d𝑥d𝑠Ω𝑢𝑢𝑡d𝑥>Ω𝑢𝑢𝑡d𝑥+𝑡0𝑔(𝑡𝑠)Ω𝑢(𝑠)𝑢(𝑡)d𝑥d𝑠,(3.4) where the last inequality is derived by (3.3). Then we get 𝑑2d𝑡2Ω||||𝑢(𝑥,𝑡)2𝑑d𝑥+d𝑡Ω||||𝑢(𝑥,𝑡)2d𝑥>𝑡0𝑔(𝑡𝑠)Ω𝑢(𝑠)𝑢(𝑡)d𝑥d𝑠.(3.5) Therefore, by using Lemma 3.1, we finish our proof.

Lemma 3.3. If 𝑢0𝐻10(Ω)𝐻2(Ω) and 𝑢1𝐻10(Ω) satisfy the assumptions in Theorem 2.4, then the solution 𝑢(𝑡) of problem (1.1) satisfies 𝐼(𝑢(𝑡))<0,(3.6)𝑢22>(𝑝+1)𝐶𝑝𝑚1𝐸(0),(3.7) for all 𝑡[0,𝑇).

Proof. We will prove the above lemma by contradiction. First we assume that (3.6) is not true over [0,𝑇), it means that there exists a time 𝑡0 such that 𝑡0=min{𝑡(0,𝑇)𝐼(𝑢(𝑡))=0}.(3.8) Since 𝐼(𝑢(𝑡))<0 on [0,𝑡0), by Lemma 3.2, we see that Ω𝑢2d𝑥 is strictly increasing over [0,𝑡0), which implies Ω𝑢2d𝑥>Ω𝑢20d𝑥>(𝑝+1)𝐶𝑝𝑚1𝐸(0).(3.9) And by the continuity of Ω𝑢2d𝑥 on 𝑡, we note that Ω𝑢2𝑡0d𝑥(𝑝+1)𝐶𝑝𝑚1𝐸(0).(3.10) On the other hand, by (2.6) and (2.9), we get 𝑀𝑡𝑢022𝑡00𝑡𝑔(𝑠)d𝑠𝑢022𝑡+(𝑔𝑢)02𝑢𝑡𝑝+10𝑝+1𝑝+12𝐸(0).(3.11) Combining (3.11) with (3.8) yields 𝑝+12𝑀𝑡𝑢022𝑝+12𝑡00𝑡𝑔(𝑠)d𝑠𝑢022+𝑝+12(𝑡𝑔𝑢)0𝑡𝑀𝑢022𝑡𝑢022(𝑝+1)𝐸(0).(3.12) By (A3), we get 𝑡𝑢022<𝑝+1𝑚1𝐸(0).(3.13) By Poincaré’s inequality, we have 𝑢𝑡022<(𝑝+1)C𝑝𝑚1𝐸(0).(3.14) Obviously, there is a contradiction between (3.10) and (3.14), thus we prove that 𝐼(𝑢(𝑡))<0,(3.15) for every 𝑡(0,𝑇). By Lemma 3.2, it follows that Ω𝑢2d𝑥 is strictly increasing on [0,𝑇), which implies that Ω𝑢2d𝑥Ω𝑢20d𝑥>(𝑝+1)𝐶𝑝𝑚1𝐸(0),(3.16) for every 𝑡[0,𝑇). This completes the proof of Lemma 3.3.

Proof of Theorem 2.4. We prove our main result by adopting concavity method. We assume by contradiction that the 𝑇 is sufficiently large. Then we consider the auxiliary function 𝐺(𝑡)=𝑢(𝑡)22+𝑡0(𝑢𝑠)22𝑇d𝑠+0𝑢𝑡022𝑡+𝛽2+𝑡2,𝑡0,𝑇0,(3.17) where 𝑇0,𝑡2, and 𝛽 are positive constants, which will be chosen later.
A straightforward calculation gives 𝐺(𝑡)=2Ω𝑢𝑢𝑡d𝑥+𝑢(𝑡)22𝑢022𝑡+2𝛽2+𝑡=2Ω𝑢𝑢𝑡d𝑥+2𝑡0𝑢(𝑠),𝑢𝑠𝑡(𝑠)d𝑠+2𝛽2,+𝑡(3.18) consequently, 𝐺(𝑡)=2Ω||𝑢𝑡||2d𝑥+2Ω𝑢𝑢𝑡𝑡d𝑥+2Ω𝑢𝑢𝑡𝑢d𝑥+2𝛽=2𝑡222𝑀𝑢22𝑢22+2𝑡0𝑔(𝑡𝑠)Ω𝑢(𝑠)𝑢(𝑡)d𝑥d𝑠2Ω𝑢𝑢𝑡d𝑥+2Ω𝑢𝑢𝑡d𝑥+2𝑢𝑝+1𝑝+1𝑢+2𝛽=2𝑡222𝑀𝑢22𝑢22+2𝑢𝑝+1𝑝+1+2𝑡0𝑔(𝑡𝑠)d𝑠𝑢22+2𝑡0𝑔(𝑡𝑠)Ω𝑢(𝑡)(𝑢(𝑠)𝑢(𝑡))d𝑥d𝑠+2𝛽.(3.19) By using Young’s inequality, we obtain 𝑡0𝑔(𝑡𝑠)Ω||||||||𝑢(𝑡)𝑢(𝑠)𝑢(𝑡)d𝑥d𝑠𝑡0𝑔(𝑠)d𝑠𝑢(𝑡)22+14(𝑔𝑢)(𝑡).(3.20) Substituting (2.6) and (3.20) for the third and the fifth terms of the right hand side of (3.19), respectively, we have 𝐺(𝑢𝑡)(𝑝+3)𝑡22+(𝑝+1)𝑀𝑢222𝑀𝑢22𝑢22(𝑝+1)𝑡0𝑔(𝑠)d𝑠𝑢2212(𝑝+1)𝐸(𝑡)+𝑝+2(𝑔𝑢)(𝑡)+2𝛽.(3.21) By (A3), we deduce 𝐺𝑢(𝑡)>(𝑝+3)𝑡22+2𝑚1𝑢2212(𝑝+1)𝐸(𝑡)+𝑝+2(𝑔𝑢)(𝑡)+2𝛽.(3.22) Noting that (2.10), we obtain that 𝐸(𝑡)𝐸(0)+𝑡0𝑢𝑠22d𝑠.(3.23) Combining (3.22)-(3.23) yields 𝐺𝑢(𝑡)>(𝑝+3)𝑡22+2𝑚1𝑢22+12(𝑝+1)𝐸(0)𝑝+2(𝑔𝑢)(𝑡)+2(𝑝+1)𝑡0𝑢𝑠22d𝑠+2𝛽.(3.24) By Poincaré’s inequality, Lemma 3.2, and (2.14), we see that 2𝑚1𝑢2212(𝑝+1)𝐸(0)+𝑝+2>(𝑔𝑢)(𝑡)2𝑚1𝐶𝑝𝑢0222(𝑝+1)𝐸(0)>0,(3.25) by (3.24)-(3.25), we get 𝐺(𝑢𝑡)>(𝑝+3)𝑡22+2𝑚1𝐶𝑝𝑢0222(𝑝+1)𝐸(0)+2(𝑝+1)𝑡0𝑢𝑠22d𝑠+2𝛽,(3.26) which means that 𝐺(𝑡)>0 for every 𝑡[0,𝑇0]. Thus, by 𝐺(0)>0 and 𝐺(0)>0, we get 𝐺(𝑡) and 𝐺(𝑡) are strictly increasing on [0,𝑇0].
We first choose 𝛽 small enough satisfying (𝑝+1)𝛽<2𝑚1𝐶𝑝𝑢0222(𝑝+1)𝐸(0),(3.27) consequently, 𝐺(𝑢𝑡)>(𝑝+3)𝑡22+2(𝑝+1)𝑡0𝑢𝑠22d𝑠+(𝑝+3)𝛽.(3.28) As far as 𝛽 is fixed, we select 𝑡2 large enough satisfying 𝑝12Ω𝑢0𝑢1d𝑥+𝛽𝑡2>𝑢022.(3.29) From (3.17), (3.18), and (3.29), we now choose 𝑇0 such that 𝑇0>(𝑢022+𝛽𝑡22)/(((𝑝1)/2)(Ω𝑢0𝑢1d𝑥+𝛽𝑡2)𝑢022)>0, which ensures that 𝑇0>4𝑝1𝐺(0)𝐺.(0)(3.30) Letting 𝐴=𝑢(𝑡)22+𝑡0(𝑢𝑠)22𝑡d𝑠+𝛽2+𝑡2,1𝐵=2𝐺(𝑢𝑡),𝐶=𝑡(𝑡)22+𝑡0𝑢𝑠(𝑠)22d𝑠+𝛽.(3.31) Since we have assumed that the solution 𝑢(𝑡) to problem (1.1) exists for every 𝑡[0,𝑇), where 𝑇 is sufficiently large, we have 𝐺𝐺(𝑡)𝐴,(𝑡)(𝑝+3)𝐶,(3.32) for every 𝑡[0,𝑇0]. Then it follows that 𝐺(𝑡)𝐺(𝑡)𝑝+34𝐺(𝑡)2(𝑝+3)𝐴𝐶𝐵2.(3.33) Furthermore, we have 𝐴𝑟22𝐵𝑟+𝐶=Ω𝑟𝑢(𝑡)𝑢𝑡(𝑡)2d𝑥+𝑡0𝑟𝑢(𝑠)𝑢𝑠(𝑠)22𝑟𝑡d𝑠+𝛽2+𝑡120,(3.34) for every 𝑟, which implies that 𝐵2𝐴𝐶0. Thus, we obtain 𝐺(𝑡)𝐺(𝑡)𝑝+34𝐺(𝑡)20.(3.35) As (𝑝+3)/4>1, letting 𝜃=(𝑝1)/4, we have 𝐺𝐺(𝑡)𝐺(𝑡)(1+𝜃)(𝑡)20.(3.36) According to concavity technique, there exists a real number 𝑇 such that 𝑇𝐺(0)/𝜃𝐺(0)<𝑇0 and we have lim𝑡𝑇𝐺(𝑡)=,(3.37) that is, lim𝑡𝑇(𝑢𝑡)22+𝑡0(𝑢𝑠)22d𝑠=,(3.38) which contradicts the assumption that the 𝑇 is sufficiently large.
This completes the proof of Theorem 2.4.

4. Some Further Remarks

4.1. The Possible Choice of the Positive Constant 𝑚1 in (A3)

When 𝑀(𝑠)=𝑎+𝑏𝑠𝛾 for 𝑠0, 𝑎0,𝑏0,𝑎+𝑏>0, 𝛾>0 and 𝑝>1+2𝛾, by straightforward calculation, we obtain 𝑝+12𝑀(𝑠)𝑀(𝑠)+𝑝+120𝑠=𝑔(𝜏)d𝜏𝑝+12𝑏𝑎𝑠+𝑠𝛾+1𝛾+1𝑎𝑠𝑏𝑠𝛾+1(𝑝+1)𝑠20=𝑔(𝜏)d𝜏𝑝12𝑎𝑠+(𝑝12𝛾)𝑏𝑠2(𝛾+1)𝛾+1(𝑝+1)𝑠20𝑔(𝜏)d𝜏.(4.1) If 𝑎>0 and 𝑏0, it follows from (2.4) that 0𝑔(𝜏)d𝜏<((𝑝1)/(𝑝+1))𝑎. Thus, we have 𝑝+12𝑀(𝑠)𝑀(𝑠)+𝑝+120𝑠>𝑔(𝜏)d𝜏𝑝12(𝑎𝑠+𝑝12𝛾)𝑏2𝑠(𝛾+1)𝛾+1(𝑝+1)𝑠2𝑝1𝑝+1𝑎((𝑝1)/(𝑝+1))𝑎0𝑔(𝜏)d𝜏2=(𝑝12𝛾)𝑏𝑠2(𝛾+1)𝛾+1+𝑝+14𝑝1𝑝+1𝑎0𝑔(𝜏)d𝜏𝑠𝑝+14𝑝1𝑝+1𝑎0𝑔(𝜏)d𝜏𝑠,(4.2)where we have used a obvious conclusion: 𝑚<𝑛𝑚<𝑛(𝑛𝑚)/2. Therefore, we can choose 𝑚1=((𝑝+1)/4)[((𝑝1)/(𝑝+1))𝑎0𝑔(𝜏)d𝜏] in condition (A3).

If 𝑎=0 and 𝑏>0, then𝑝+12𝑀(𝑠)𝑀(𝑠)+𝑝+120s=𝑔(𝜏)d𝜏(𝑝12𝛾)𝑏𝑠2(𝛾+1)𝛾+1(𝑝+1)𝑠20𝑔(𝜏)d𝜏>(𝑝12𝛾)𝑏𝑠2(𝛾+1)𝛾+1(𝑝+1)𝑠2×(𝑝12𝛾)𝑏𝐶𝛾𝑝𝑢(𝑝+1)(𝛾+1)022𝛾(𝐶𝑝12𝛾)𝑏/𝛾𝑝(𝑢𝑝+1)(𝛾+1)022𝛾0𝑔(𝜏)d𝜏2=(𝑝12𝛾)𝑏2𝑠𝑠(𝛾+1)𝛾1𝐶𝛾𝑝𝑢022𝛾+(𝑝+1)𝑠4(𝑝12𝛾)𝑏𝐶𝛾𝑝𝑢(𝑝+1)(𝛾+1)022𝛾0.𝑔(𝜏)d𝜏(4.3)

Taking 𝑠=𝑢(𝑡)22, applying Lemma 3.2 and Poincaré’s inequality, we can get 𝑝+12𝑀𝑢(𝑡)22𝑀𝑢(𝑡)22+𝑝+120𝑔(𝜏)d𝜏𝑢(𝑡)22>(𝑝12𝛾)𝑏2(𝛾+1)𝑢(𝑡)22𝑢(𝑡)22𝛾1𝐶𝛾𝑝𝑢022𝛾+𝑝+14𝑢(𝑡)22(𝑝12𝛾)𝑏𝐶𝛾𝑝𝑢(𝑝+1)(𝛾+1)022𝛾0𝑔(𝜏)d𝜏(𝑝12𝛾)𝑏(2(𝛾+1)𝑢𝑡)22(𝑢𝑡)22𝛾1𝐶𝛾𝑝(𝑢𝑡)22𝛾+𝑝+14𝑢(𝑡)22(𝑝12𝛾)𝑏𝐶𝛾𝑝𝑢(𝑝+1)(𝛾+1)022𝛾0𝑔(𝜏)d𝜏(𝑝12𝛾)𝑏2(𝛾+1)𝑢(𝑡)22𝑢(𝑡)22𝛾𝑢(𝑡)22𝛾+𝑝+14𝑢(𝑡)22(𝑝12𝛾)𝑏𝐶𝛾𝑝𝑢(𝑝+1)(𝛾+1)022𝛾0=𝑔(𝜏)d𝜏𝑝+14(𝑝12𝛾)𝑏𝐶𝛾𝑝𝑢(𝑝+1)(𝛾+1)022𝛾0𝑔(𝜏)d𝜏𝑢(𝑡)22.(4.4)So, we can choose 𝑚1=((𝑝+1)/4)[(𝑝12𝛾)𝑏/(𝐶𝛾𝑝(𝑝+1)(𝛾+1)𝑢022𝛾)0𝑔(𝜏)d𝜏] in condition (A3).

4.2. The Set of the Initial Data Satisfying Conditions (2.11)–(2.14) Is Not Empty

For any real value of the initial energy 𝐸(0)=𝑑>0, there exists such initial data which leads to blow up in finite time.

For instance, in the case 𝑀(𝑠)=1+𝑠𝛾, then for any (𝑢0,𝑢1)(𝐻10(Ω)𝐻2(Ω))×𝐻10(Ω) with Ω𝑢0𝑢1d𝑥>0, we may take some 𝜆,𝛼>0, such that (𝑢0,𝑢1) satisfies the above conditions (2.11)–(2.14).

Indeed, for 𝑀(𝑠)=1+𝑠𝛾, conditions (2.11)–(2.14) become1𝐸(0)=2𝑢122+12𝑢022+12(𝛾+1)𝑢022(𝛾+1)1𝑢𝑝+10𝑝+1𝑝+1𝐼𝑢>0,0=𝑢022+𝑢022(𝛾+1)𝑢0𝑝+1𝑝+1<0,Ω𝑢0𝑢1𝑢d𝑥>0,022>(𝑝+1)𝐶𝑝𝑚1𝐸(0).(4.5)

Now taking (𝑣0,𝑣1)(𝐻10(Ω)𝐻2(Ω))×𝐻10(Ω) such that Ω𝑣0𝑣1d𝑥>0, and letting (𝑢0,𝑢1)=(𝜆𝑣0,𝛼𝑣1) for any scaling parameter 𝜆>0 and 𝛼>0, then we have1𝐸(0)=2𝛼2𝑣122+12𝜆2𝑣022+12𝜆(𝛾+1)2(𝛾+1)𝑣022(𝛾+1)1𝜆𝑝+1𝑝+1𝑣0𝑝+1𝑝+1,𝐼𝑢0=𝜆2𝑣022+𝜆2(𝛾+1)𝑣022(𝛾+1)𝜆𝑝+1𝑣0𝑝+1𝑝+1,𝑢022=𝜆2𝑣022.(4.6)

We suppose that 𝜆2(𝛾+1)<𝜆𝑝+1 (i.e., 𝑝>1+2𝛾) for 𝜆>1, so we can choose sufficiently large 𝜆 such that 12𝜆2𝑣022+12𝜆(𝛾+1)2(𝛾+1)𝑣022(𝛾+1)1𝜆𝑝+1𝑝+1𝑣0𝑝+1𝑝+1𝐼𝑢<0,0=𝐼𝜆𝑣0=𝜆2𝑣022+𝜆2(𝛾+1)𝑣022(𝛾+1)𝜆𝑝+1𝑣0𝑝+1𝑝+1𝑢<0,022=𝜆2𝑣022>(𝑝+1)𝐶𝑝𝑚1𝐸(0).(4.7)And when 𝜆 is fixed, we may choose 𝛼 such that 𝐸(0)=𝑑.

Similarly, in the case 𝑀(𝑠)=𝑠𝛾, we can also take initial data (𝑢0,𝑢1) satisfying the above conditions (2.11)–(2.14) (see [7, Remark  1.4]).

Thus the set of the initial data which satisfy conditions (2.11)–(2.14) is not empty.

Acknowledgments

This work was partly supported by the Tianyuan Fund of Mathematics (Grant no. 11026211) and the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant no. 09KJB110005).

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