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

Abstract

We consider viscoelastic wave equations of the Kirchhoff type 𝑢_𝑡𝑡−𝑀(‖∇𝑢‖²²)Δ𝑢+∫^𝑡0𝑔(𝑡−𝑠)Δ𝑢(𝑠)d𝑠+𝑢_𝑡=|𝑢|^𝑝−1𝑢 with Dirichlet boundary conditions, where ‖⋅‖_𝑝 denotes the norm in the Lebesgue space 𝐿^𝑝. 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 lim_𝑡→𝑇^∗−(‖𝑢(𝑡)‖²²+∫^𝑡0‖𝑢(𝑠)‖²²d𝑠)=∞ for some 0<𝑇^∗<+∞.

1. Introduction

In this paper we consider the following problem:𝑢_𝑡𝑡−𝑀‖∇𝑢‖²²Δ𝑢𝑢+(𝑥,^𝑡00𝑢𝑔)((=𝑥𝑡,−𝑢𝑡₀)𝑠=)Δ0𝑢,(𝑠()𝑥d,𝑠𝑡)+∈||𝑢𝜕_𝑡||Ω×^𝑚−1𝑢_𝑡[0=,∞|𝑢|),^𝑝−1𝑢,(𝑥,𝑡)∈Ω×(0,∞),(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥),𝑥∈Ω,(1.1) where Ω is a bounded domain in ℝ^𝑛 (𝑛≥1) with a smooth boundary 𝜕Ω,𝑝>1,𝑀(𝑠) is a nonnegative 𝐶¹ function like 𝑀(𝑠)=𝑎+𝑏𝑠^𝛾 for 𝑠≥0,𝑎≥0,𝑏≥0,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: 𝑢_𝑡𝑡−𝑀‖∇𝑢‖²²Δ𝑢+𝑔𝑢_𝑡=𝑓(𝑢),(𝑥,𝑡)∈Ω×(0,∞),(1.2) has been considered by Matsuyama and Ikehata in [1] for 𝑔(𝑢_𝑡)=𝛿|𝑢_𝑡|^𝑝−1𝑢_𝑡 and 𝑓(𝑢)=𝜇|𝑢|^𝑞−1𝑢. The authors proved existence of the global solutions by using Faedo-Galerkin method and the decay of energy based on the method of Nakao [2–4]. Later, Ono [5] investigated (1.2) for 𝑀(𝑠)=𝑏𝑠^𝛾 and 𝑓(𝑢)=|𝑢|^𝑝−2𝑢. When 𝑔(𝑢_𝑡)=−Δ𝑢_𝑡,𝑢_𝑡 or |𝑢_𝑡|^𝛽𝑢_𝑡, the author showed that the solutions blow up in finite time with 𝐸(0)≤0. For 𝑀(𝑠)=𝑎+𝑏𝑠^𝛾 and 𝑔(𝑢_𝑡)=𝑢_𝑡, 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(𝑢_𝑡)=𝑢_𝑡 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 𝑀≡1 and in the presence of the viscoelastic term (i.e., 𝑔≠0), the equation 𝑢_𝑡𝑡−Δ𝑢+^𝑡0𝑔(𝑡−𝑠)Δ𝑢(𝑠)d𝑠+||𝑢_𝑡||^𝑚−1𝑢_𝑡=|𝑢|^𝑝−1𝑢,(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 and^∞0𝑔(𝑠)d𝑠≤𝑝−1𝑝−1+1/(𝑝+1),(1.4)

while the solution continues to exist globally for any initial data in the appropriate space if m≥𝑝. 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: 𝑢_𝑡𝑡−Δ𝑢+^𝑡0𝑔(𝑡−𝑠)Δ𝑢(𝑠)d𝑠+𝑎(𝑥)𝑢_𝑡+|𝑢|^𝛾𝑢=0,(𝑥,𝑡)∈Ω×(0,∞),(1.5) where 𝑎∶Ω→𝑅⁺ is a function which may be null on a part of Ω. Under the condition that 𝑎(𝑥)≥𝑎₀>0 on 𝜔⊂Ω, with 𝜔 satisfying some geometric restrictions and −𝜉₁𝑔(𝑡)≤𝑔′(𝑡)≤−𝜉₂𝑔(𝑡), 𝑡≥0 to guarantee that ‖𝑔‖_𝐿¹_((0,∞)) is small enough, they proved an exponential decay rate.

When 𝑔≠0 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: 𝑢_𝑡𝑡−𝑀‖∇𝑢‖²²Δ𝑢+^𝑡0𝑔(𝑡−𝑠)Δ𝑢(𝑠)d𝑠−Δ𝑢_𝑡=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 [15–33] 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)𝑔∈𝐶¹([0,∞)) is a nonnegative and non-increasing function satisfying1−^∞0𝑔(𝑠)d𝑠=𝑙>0.(2.1)(A2) The function 𝑒^𝑡/2𝑔(𝑡) is of positive type in the following sense (see [10]): ^𝑡0𝑣(𝑠)^𝑠0𝑒^{(𝑠−𝑧)/2}𝑔(𝑠−𝑧)𝑣(𝑧)d𝑧d𝑠≥0,∀𝑣∈𝐶¹([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+12^∞0𝑔(𝜏)d𝜏𝑠≥𝑚₁𝑠,∀𝑠≥0,(2.3)where 𝑀(𝑠)=∫^𝑠0𝑀(𝜏)d𝜏.

Remark 2.2. It is clear that when 𝑀(𝑠)=𝑎+𝑏𝑠^𝛾 for 𝑠≥0,𝑎≥0,𝑏≥0, 𝑎+𝑏>0, 𝛾>0 and 𝑝>1+2𝛾, condition (A3) can be replaced by ^∞0𝑔(𝜏)d𝜏<⎧⎪⎨⎪⎩𝑝−1𝑝+(𝑝1−𝑎1,−2𝛾)𝑏if𝑎>0,𝑏≥0,𝐶^𝛾𝑝(𝑝+1)(𝛾+1)‖‖𝑢⁰‖‖^22𝛾,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 𝐶_𝑝 is the constant from the Poincaré inequality ‖𝑢(𝑡)‖²²≤𝐶_𝑝‖∇𝑢(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<𝑝≤𝑛/(n−2) when n≥3,1<𝑝<∞ when 𝑛=1,2. For 𝑢₀∈𝐻¹⁰(Ω)∩𝐻²(Ω), 𝑢₁∈𝐻¹⁰(Ω), and 𝑀(‖∇𝑢₀‖²²)>0, problem (1.1) has a unique local solution 𝑢∈𝐶[0,𝑇);𝐻¹⁰(Ω)∩𝐻²(Ω),𝑢_𝑡∈𝐶[0,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:𝐸(𝑡)∶==1𝐸(𝑢(𝑡))2‖‖𝑢^𝑡‖‖²²+12𝑀‖∇𝑢‖²²−12^𝑡0𝑔(𝑠)d𝑠‖∇𝑢‖²²+12(𝑔∘∇𝑢)(𝑡)−𝑝+11‖𝑢‖^𝑝+1𝑝+1,(2.6)𝐼(𝑢)=𝑀‖∇𝑢‖²²‖∇𝑢‖²²−‖𝑢‖^𝑝+1𝑝+1,(2.7) where(𝑔∘𝑤)(𝑡)=^𝑡0𝑔(𝑡−𝑠)‖𝑤(𝑡,⋅)−𝑤(𝑠,⋅)‖²²d𝑠.(2.8)

As in [7, 10], we can get𝑑d𝑡𝐸(𝑡)=−‖‖𝑢^𝑡‖‖²²−12𝑔(𝑡)‖∇𝑢‖²²+12𝑔^∘∇𝑢(𝑡)≤0,∀𝑡≥0.(2.9) Then we have𝐸(𝑡)=𝐸(0)−^𝑡0‖‖𝑢_𝑠‖‖²²d𝑠+12^𝑡0𝑔^∘∇𝑢(𝑠)d𝑠−12^𝑡0𝑔(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<𝑝≤𝑛/(𝑛−2) when 𝑛≥3,1<𝑝<∞ when 𝑛=1,2. Let 𝑢 be a solution of problem (1.1) with initial data 𝑢₀∈𝐻¹⁰(Ω)∩𝐻²(Ω),𝑢₁∈𝐻¹⁰(Ω) and 𝑀(‖∇𝑢₀‖²²)>0, and further assume that 𝐼𝐸𝑢(0)>0,(2.11)₀<0,(2.12)_Ω‖‖𝑢𝑢₀𝑢₁𝑑𝑥>0,(2.13)₀‖‖²²>(𝑝+1)𝐶^𝑝𝑚₁𝐸(0).(2.14) Then the solution of problem (1.1) blows up in finite time 0<𝑇^∗<+∞, which means that lim_{𝑡→𝑇∗−}‖𝑢(𝑡)‖²²+^𝑡0‖𝑢(𝑠)‖²²𝑑𝑠=∞,(2.15) where 𝐶_𝑝 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^(𝐻𝑡)>(0)^𝑡0>𝑔0(,𝑡−𝐻𝑠)__Ω∇𝑢(𝑠)∇𝑢(𝑡)d𝑥d𝑠,(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 𝑢₀∈𝐻¹⁰(Ω)∩𝐻²(Ω) and 𝑢₁∈𝐻¹⁰(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 12d𝑑𝑡²²_Ω||𝑢(𝑥,𝑡)||²d𝑥=_Ω||𝑢^𝑡||²+𝑢𝑢_𝑡𝑡=‖‖𝑢d𝑥_𝑡‖‖²²−𝑀‖∇𝑢‖²²‖∇𝑢‖²²+‖𝑢‖^𝑝+1𝑝+1>+−^𝑡0_Ω𝑔(𝑡−𝑠)_Ω∇𝑢(𝑠)∇𝑢(𝑡)d𝑥d𝑠−_Ω𝑢𝑢_𝑡d𝑥𝑢𝑢_𝑡d𝑥+^𝑡0𝑔(𝑡−𝑠)_Ω∇𝑢(𝑠)∇𝑢(𝑡)d𝑥d𝑠,(3.4) where the last inequality is derived by (3.3). Then we get 𝑑²d𝑡²_Ω||𝑢(𝑥,𝑡)||²d𝑥+𝑑d𝑡_Ω||𝑢(𝑥,𝑡)||²d𝑥>^𝑡0𝑔(𝑡−𝑠)_Ω∇𝑢(𝑠)∇𝑢(𝑡)d𝑥d𝑠.(3.5) Therefore, by using Lemma 3.1, we finish our proof.

Lemma 3.3. If 𝑢₀∈𝐻¹⁰(Ω)∩𝐻²(Ω) and 𝑢₁∈𝐻¹⁰(Ω) satisfy the assumptions in Theorem 2.4, then the solution 𝑢(𝑡) of problem (1.1) satisfies ‖𝑢‖₂₂𝐼(𝑢(𝑡))<0,(3.6)>(𝑝+1)𝐶^𝑝𝑚₁𝐸(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)𝐶^𝑝𝑚₁𝐸(0).(3.9) And by the continuity of ∫_Ω𝑢²d𝑥 on 𝑡, we note that _Ω𝑢²𝑡₀d𝑥≥(𝑝+1)𝐶^𝑝𝑚₁𝐸(0).(3.10) On the other hand, by (2.6) and (2.9), we get 𝑀‖‖∇𝑢𝑡₀‖‖²²−^𝑡0₀𝑔(𝑠)d𝑠‖‖∇𝑢𝑡₀‖‖²²+(𝑔∘∇𝑢)𝑡₀−2𝑝+1‖‖𝑢𝑡⁰‖‖^𝑝+1𝑝+1≤2𝐸(0).(3.11) Combining (3.11) with (3.8) yields 𝑝+12𝑀‖‖∇𝑢𝑡₀‖‖²²−𝑝+12^𝑡0₀𝑔(𝑠)d𝑠‖‖∇𝑢𝑡₀‖‖²²+𝑝+1−𝑀‖‖∇𝑢𝑡₀2(𝑔∘∇𝑢)𝑡⁰‖‖²²‖‖∇𝑢𝑡⁰‖‖²²≤(𝑝+1)𝐸(0).(3.12) By (A3), we get ‖‖∇𝑢𝑡₀‖‖²²<𝑝+1𝑚₁𝐸(0).(3.13) By Poincaré’s inequality, we have ‖‖𝑢𝑡₀‖‖²²<(𝑝+1)C^𝑝𝑚₁𝐸(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 _Ω𝑢²d𝑥≥_Ω𝑢²⁰d𝑥>(𝑝+1)𝐶^𝑝𝑚₁𝐸(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‖𝑢(𝑠)‖²²d𝑠+𝑇₀−𝑡‖‖𝑢₀‖‖²²+𝛽𝑡₂+𝑡²,𝑡∈0,𝑇₀,(3.17) where 𝑇₀,𝑡₂, and 𝛽 are positive constants, which will be chosen later.
A straightforward calculation gives 𝐺^(𝑡)==22_ΩΩ𝑢𝑢_𝑡d𝑥+‖𝑢(𝑡)‖²²−‖‖𝑢₀‖‖²²+2𝛽𝑡₂+𝑡𝑢𝑢_𝑡d𝑥+2^𝑡0𝑢(𝑠),𝑢_𝑠(𝑠)d𝑠+2𝛽𝑡₂+𝑡,(3.18) consequently, 𝐺^(𝑡=)=2‖‖2𝑢_Ω𝑡||𝑢_𝑡||²d𝑥+2_Ω𝑢𝑢_𝑡𝑡d𝑥+2_Ω𝑢𝑢_𝑡d𝑥+2𝛽−2_Ω‖‖²²−2𝑀‖∇𝑢‖²²‖∇𝑢‖²²+2^𝑡0𝑔(𝑡−𝑠)_Ω∇𝑢(𝑠)∇𝑢(𝑡)d𝑥d𝑠=2‖‖𝑢_𝑡𝑢𝑢_𝑡d𝑥+2_Ω𝑢𝑢_𝑡d𝑥+2‖𝑢‖^𝑝+1𝑝+1+2𝛽‖‖²²−2𝑀‖∇𝑢‖²²‖∇𝑢‖²²+2‖𝑢‖^𝑝+1𝑝+1+2^𝑡0𝑔(𝑡−𝑠)d𝑠‖∇𝑢‖²²+2^𝑡0𝑔(𝑡−𝑠)_Ω∇𝑢(𝑡)(∇𝑢(𝑠)−∇𝑢(𝑡))d𝑥d𝑠+2𝛽.(3.19) By using Young’s inequality, we obtain ^𝑡0𝑔(𝑡−𝑠)_Ω||∇𝑢(𝑡)||||∇𝑢(𝑠)−∇𝑢(𝑡)||d𝑥d𝑠≤^𝑡0𝑔(𝑠)d𝑠‖∇𝑢(𝑡)‖²²+14(𝑔∘∇𝑢)(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^(𝑡)≥(𝑝+3)‖‖𝑢_𝑡‖‖²²+(𝑝+1)𝑀‖∇𝑢‖²²−2𝑀‖∇𝑢‖²²‖∇𝑢‖²²−(𝑝+1)^𝑡0𝑔(𝑠)d𝑠‖∇𝑢‖²²−2(𝑝+1)𝐸(𝑡)+𝑝+12(𝑔∘∇𝑢)(𝑡)+2𝛽.(3.21) By (A3), we deduce 𝐺^(𝑡)>(𝑝+3)‖‖𝑢_𝑡‖‖²²+2𝑚₁‖∇𝑢‖²²−2(𝑝+1)𝐸(𝑡)+𝑝+12(𝑔∘∇𝑢)(𝑡)+2𝛽.(3.22) Noting that (2.10), we obtain that −𝐸(𝑡)≥−𝐸(0)+^𝑡0‖‖𝑢_𝑠‖‖²²d𝑠.(3.23) Combining (3.22)-(3.23) yields 𝐺^(𝑡)>+(𝑝+𝑝3+)‖‖1𝑢_𝑡‖‖²²+2𝑚₁‖∇𝑢‖²²−2(𝑝+1)𝐸(0)+2(𝑝+21)(𝑔_𝑡0∘∇𝑢)(𝑡)‖‖𝑢_𝑠‖‖²²d𝑠+2𝛽.(3.24) By Poincaré’s inequality, Lemma 3.2, and (2.14), we see that 2𝑚₁‖∇𝑢‖²²−2(𝑝+1)𝐸(0)+𝑝+1>2𝑚₁2(𝑔∘∇𝑢)(𝑡)𝐶_𝑝‖‖𝑢₀‖‖²²−2(𝑝+1)𝐸(0)>0,(3.25) by (3.24)-(3.25), we get 𝐺^(𝑡)>(𝑝+3)‖‖𝑢_𝑡‖‖²²+2𝑚¹𝐶_𝑝‖‖𝑢₀‖‖²²−2(𝑝+1)𝐸(0)+2(𝑝+1)^𝑡0‖‖𝑢_𝑠‖‖²²d𝑠+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<2𝑚¹𝐶_𝑝‖‖𝑢₀‖‖²²−2(𝑝+1)𝐸(0),(3.27) consequently, 𝐺^(𝑡)>(𝑝+3)‖‖𝑢_𝑡‖‖²²+2(𝑝+1)^𝑡0‖‖𝑢_𝑠‖‖²²d𝑠+(𝑝+3)𝛽.(3.28) As far as 𝛽 is fixed, we select 𝑡₂ large enough satisfying 𝑝−12_Ω𝑢₀𝑢₁d𝑥+𝛽𝑡₂>‖‖𝑢⁰‖‖²².(3.29) From (3.17), (3.18), and (3.29), we now choose 𝑇₀ such that 𝑇₀>(‖𝑢₀‖²²+𝛽𝑡²²)/(((𝑝−1)/2)(∫_Ω𝑢₀𝑢₁d𝑥+𝛽𝑡₂)−‖𝑢₀‖²²)>0, which ensures that 𝑇₀>4𝑝−1𝐺𝐺^((00)).(3.30) Letting 𝐴∶=‖𝑢(𝑡)‖²²𝐵+∶^𝑡0=‖𝑢1(𝑠)‖²²d𝑠+𝛽𝑡₂+𝑡²,2𝐺^𝐶∶=‖‖𝑢_𝑡(𝑡),(𝑡)‖‖²²+^𝑡0‖‖𝑢_𝑠(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+34𝐺^(𝑡)²≥(𝑝+3)𝐴𝐶−𝐵².(3.33) Furthermore, we have 𝐴𝑟²−2𝐵𝑟+𝐶=_Ω𝑟𝑢(𝑡)−𝑢_𝑡(𝑡)²d𝑥+^𝑡0‖‖𝑟𝑢(𝑠)−𝑢_𝑠(𝑠)‖‖²²d𝑠+𝛽𝑟𝑡₂+𝑡−1²≥0,(3.34) for every 𝑟∈ℝ, which implies that 𝐵²−𝐴𝐶≤0. Thus, we obtain 𝐺^(𝑡)𝐺(𝑡)−𝑝+34𝐺^(𝑡)²≥0.(3.35) As (𝑝+3)/4>1, letting 𝜃=(𝑝−1)/4, we have 𝐺^(𝑡)𝐺(𝑡)−(1+𝜃)𝐺^(𝑡)²≥0.(3.36) According to concavity technique, there exists a real number 𝑇^∗ such that 𝑇^∗≤𝐺(0)/𝜃𝐺′(0)<𝑇₀ and we have lim_{𝑡→𝑇∗−}𝐺(𝑡)=∞,(3.37) that is, lim_{𝑡→𝑇∗−}‖𝑢(𝑡)‖²²+^𝑡0‖𝑢(𝑠)‖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+𝑏𝑠^𝛾 for 𝑠≥0, 𝑎≥0,𝑏≥0,𝑎+𝑏>0, 𝛾>0 and 𝑝>1+2𝛾, by straightforward calculation, we obtain 𝑝+12𝑀(𝑠)−𝑀(𝑠)+𝑝+12^∞0=𝑝+1𝑔(𝜏)d𝜏𝑠2𝑎𝑠+𝛾+𝑏1𝑠^𝛾+1−𝑎𝑠−𝑏𝑠^𝛾+1−(𝑝+1)𝑠2^∞0=𝑝−1𝑔(𝜏)d𝜏2𝑎𝑠+(𝑝−2(1𝛾−+21𝛾))𝑏𝑠^𝛾+1−(𝑝+1)𝑠2^∞0𝑔(𝜏)d𝜏.(4.1) If 𝑎>0 and 𝑏≥0, it follows from (2.4) that ∫^∞0𝑔(𝜏)d𝜏<((𝑝−1)/(𝑝+1))𝑎. Thus, we have 𝑝+12𝑀(𝑠)−𝑀(𝑠)+𝑝+12^∞0>𝑝−1𝑔(𝜏)d𝜏𝑠2𝑎𝑠+(𝑝−2(1𝛾−+21𝛾))𝑏𝑠^𝛾+1−(𝑝+1)𝑠2𝑝𝑝+−11𝑎−((𝑝−1)/(𝑝+1))𝑎−∫^∞0𝑔(𝜏)d𝜏=(𝑝−1−2𝛾)𝑏22(𝛾+1)𝑠^𝛾+1+𝑝+14𝑝𝑝+−11𝑎−^∞0𝑔(𝜏)d𝜏𝑠≥𝑝+14𝑝𝑝−+11𝑎−^∞0𝑔(𝜏)d𝜏𝑠,(4.2)where we have used a obvious conclusion: 𝑚<𝑛⇒𝑚<𝑛−(𝑛−𝑚)/2. Therefore, we can choose 𝑚₁=((𝑝+1)/4)[((𝑝−1)/(𝑝+1))𝑎−∫^∞0𝑔(𝜏)d𝜏] in condition (A3).

If 𝑎=0 and 𝑏>0, then𝑝+12𝑀(𝑠)−𝑀(𝑠)+𝑝+12^∞0=(𝑝−1−2𝛾)𝑏𝑔(𝜏)d𝜏s2(𝛾+1)𝑠^𝛾+1−(𝑝+1)𝑠2^∞0𝑔(𝜏)d𝜏>(𝑝−1−2𝛾)𝑏2(𝛾+1)𝑠^𝛾+1−(𝑝+1)𝑠×⎡⎢⎢⎢⎢⎣(𝑝−1−2𝛾)𝑏2𝐶^𝛾𝑝(𝑝+1)(𝛾+1)‖‖𝑢⁰‖‖^22𝛾−(𝑝−1−2𝛾)𝑏/𝐶^𝛾𝑝(𝑝+1)(𝛾+1)‖‖𝑢₀‖‖^22𝛾−∫^∞0𝑔(𝜏)d𝜏=(𝑝−1−2𝛾)𝑏22(𝛾+1)𝑠𝑠^𝛾−1𝐶^𝛾𝑝‖‖𝑢₀‖‖^22𝛾+(𝑝+41)𝑠𝐶^𝛾𝑝(𝑝(𝑝−+11−)(2𝛾𝛾+)𝑏1)‖‖𝑢⁰‖‖^22𝛾−^∞0𝑔(𝜏)d𝜏.(4.3)

Taking 𝑠=‖∇𝑢(𝑡)‖²², applying Lemma 3.2 and Poincaré’s inequality, we can get 𝑝+12𝑀‖∇𝑢(𝑡)‖²²−𝑀‖∇𝑢(𝑡)‖²²+𝑝+12^∞0𝑔(𝜏)d𝜏‖∇𝑢(𝑡)‖²²>(𝑝−1−2𝛾)𝑏2(𝛾+1)‖∇𝑢(𝑡)‖²²‖∇𝑢(𝑡)‖^22𝛾−1𝐶^𝛾𝑝‖‖𝑢₀‖‖^22𝛾+𝑝+14‖∇𝑢(𝑡)‖²²𝐶^𝛾𝑝(𝑝(𝑝−+11−)(2𝛾𝛾+)𝑏1)‖‖𝑢⁰≥(𝑝−1−2𝛾)𝑏‖‖^22𝛾−^∞0𝑔(𝜏)d𝜏2(𝛾+1)‖∇𝑢(𝑡)‖²²‖∇𝑢(𝑡)‖^22𝛾−1𝐶^𝛾𝑝‖𝑢(𝑡)‖^22𝛾+𝑝+14‖∇𝑢(𝑡)‖²²𝐶^𝛾𝑝(𝑝(𝑝−+11−)(2𝛾𝛾+)𝑏1)‖‖𝑢⁰≥(𝑝−1−2𝛾)𝑏‖‖^22𝛾−^∞0𝑔(𝜏)d𝜏2(𝛾+1)‖∇𝑢(𝑡)‖²²‖∇𝑢(𝑡)‖^22𝛾+𝑝+1−‖∇𝑢(𝑡)‖^22𝛾4‖∇𝑢(𝑡)‖²²𝐶^𝛾𝑝(𝑝(𝑝−+11−)(2𝛾𝛾+)𝑏1)‖‖𝑢⁰=𝑝+1‖‖^22𝛾−^∞0𝑔(𝜏)d𝜏4𝐶^𝛾𝑝(𝑝(𝑝−+11−)(2𝛾𝛾+)𝑏1)‖‖𝑢⁰‖‖^22𝛾−^∞0𝑔(𝜏)d𝜏‖∇𝑢(𝑡)‖²².(4.4)So, we can choose 𝑚₁=((𝑝+1)/4)[(𝑝−1−2𝛾)𝑏/(𝐶^𝛾𝑝(𝑝+1)(𝛾+1)‖𝑢₀‖^22𝛾)−∫^∞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+𝑠^𝛾, 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+𝑠^𝛾, conditions (2.11)–(2.14) become𝐸(0)=12‖‖𝑢¹‖‖²²+12‖‖∇𝑢⁰‖‖²²+12(𝛾+1)‖‖∇𝑢⁰‖‖^22(𝛾+1)−1𝑝+1‖‖𝑢⁰𝐼𝑢‖‖^𝑝+1𝑝+1>0,₀=‖‖∇𝑢₀‖‖²²+‖‖∇𝑢₀‖‖^22(𝛾+1)−‖‖𝑢₀‖‖^𝑝+1𝑝+1<0,_Ω‖‖𝑢𝑢₀𝑢₁d𝑥>0,₀‖‖²²>(𝑝+1)𝐶^𝑝𝑚₁𝐸(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 have𝐸(0)=12𝛼²‖‖𝑣₁‖‖²²+12𝜆²‖‖∇𝑣₀‖‖²²+12(𝛾+1)𝜆^2(𝛾+1)‖‖∇𝑣₀‖‖^22(𝛾+1)−1𝑝+1𝜆^𝑝+1𝐼𝑢₀‖‖𝑣₀‖‖^𝑝+1𝑝+1,‖‖𝑢₀=𝜆²‖‖∇𝑣₀‖‖²²+𝜆^2(𝛾+1)‖‖∇𝑣₀‖‖^22(𝛾+1)−𝜆^𝑝+1‖‖𝑣₀‖‖^𝑝+1𝑝+1,‖‖²²=𝜆²‖‖𝑣₀‖‖²².(4.6)

We suppose that 𝜆^2(𝛾+1)<𝜆^𝑝+1 (i.e., 𝑝>1+2𝛾) for 𝜆>1, so we can choose sufficiently large 𝜆 such that 12𝜆²‖‖∇𝑣₀‖‖²²+12(𝛾+1)𝜆^2(𝛾+1)‖‖∇𝑣₀‖‖^22(𝛾+1)−1𝑝+1𝜆^𝑝+1𝐼𝑢‖‖𝑣₀‖‖^𝑝+1𝑝+1<0,₀=𝐼𝜆𝑣₀‖‖=𝑢𝜆²‖‖∇𝑣₀‖‖²²+𝜆^2(𝛾+1)‖‖∇𝑣₀‖‖^22(𝛾+1)−𝜆^𝑝+1‖‖𝑣₀‖‖^𝑝+1𝑝+1<0,₀‖‖²²=𝜆²‖‖𝑣₀‖‖²²>(𝑝+1)𝐶^𝑝𝑚₁𝐸(0).(4.7)And when 𝜆 is fixed, we may choose 𝛼 such that 𝐸(0)=𝑑.

Similarly, in the case 𝑀(𝑠)=𝑠^𝛾, 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).