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

Authors: Gang Li; Linghui Hong; Wenjun Liu
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We consider viscoelastic wave equations of the Kirchhoff typeutt-M(∥∇u∥22)Δu+∫0tg(t-s)Δu(s)ds+ut=|u|p-1u 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 limt→T*-(∥u(t)∥22+∫0t∥u(s)∥22ds)=∞ for some 0<T*<+∞.


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## Body

## 1. Introduction

In this paper we consider the following problem:(1.1)utt-M(‖∇u‖22)Δu+∫0tg(t-s)Δu(s)ds+|ut|m-1ut=|u|p-1u,(x,t)∈Ω×(0,∞),u(x,t)=0,(x,t)∈∂Ω×[0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,
where Ω is a bounded domain in ℝn (n≥1) with a smooth boundary ∂Ω,p>1,M(s) is a nonnegative C1 function like M(s)=a+bsγ 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:
(1.2)utt-M(‖∇u‖22)Δu+g(ut)=f(u),(x,t)∈Ω×(0,∞),
has been considered by Matsuyama and Ikehata in [1] for g(ut)=δ|ut|p-1ut 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)=bsγ and f(u)=|u|p-2u. 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 ofM≡1 and in the presence of the viscoelastic term (i.e., g≠0), the equation
(1.3)utt-Δu+∫0tg(t-s)Δu(s)ds+|ut|m-1ut=|u|p-1u,(x,t)∈Ω×(0,∞),
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(1.4)∫0∞g(s)ds≤p-1p-1+1/(p+1),while the solution continues to exist globally for any initial data in the appropriate space ifm≥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:
(1.5)utt-Δu+∫0tg(t-s)Δu(s)ds+a(x)ut+|u|γu=0,(x,t)∈Ω×(0,∞),
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.Wheng≠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:
(1.6)utt-M(‖∇u‖22)Δu+∫0tg(t-s)Δu(s)ds-Δut=f(u),(x,t)∈Ω×(0,∞),
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 Section2, 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 satisfying(2.1)1-∫0∞g(s)ds=l>0.(A2)
The functionet/2g(t) is of positive type in the following sense (see [10]):
(2.2)∫0tv(s)∫0se(s-z)/2g(s-z)v(z)dzds≥0,∀v∈C1([0,∞)),∀t>0.Remark 2.1.
Assumption (A2) is needed to prove Lemma3.1 below.In order to prove our result, we make the following assumption onM and g:(A3)
there exists a positive constantm1 such that
(2.3)p+12M¯(s)-[M(s)+p+12∫0∞g(τ)dτ]s≥m1s,∀s≥0,where M¯(s)=∫0sM(τ)dτ.Remark 2.2.
It is clear that whenM(s)=a+bsγ for s≥0,a≥0,b≥0, a+b>0, γ>0 and p>1+2γ, condition (A3) can be replaced by
(2.4)∫0∞g(τ)dτ<{p-1p+1a,ifa>0,b≥0,(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ,ifa=0,b>0,
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)∥22≤Cp∥∇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∈H01(Ω)∩H2(Ω), u1∈H01(Ω), and M(∥∇u0∥22)>0, problem (1.1) has a unique local solution
(2.5)u∈C([0,T);H01(Ω)∩H2(Ω)),ut∈C([0,T);L2(Ω))∩L2([0,T);H01(Ω)),for the maximum existence time T>0.The energy functionalE(t) and an auxiliary functional I(u) of the solution u(t) of problem (1.1) are defined as follows:(2.6)E(t)∶=E(u(t))=12‖ut‖22+12M¯(‖∇u‖22)-12∫0tg(s)ds‖∇u‖22+12(g∘∇u)(t)-1p+1‖u‖p+1p+1,(2.7)I(u)=M(‖∇u‖22)‖∇u‖22-‖u‖p+1p+1,
where(2.8)(g∘w)(t)=∫0tg(t-s)‖w(t,⋅)-w(s,⋅)‖22ds.As in [7, 10], we can get(2.9)ddtE(t)=-‖ut‖22-12g(t)‖∇u‖22+12(g′∘∇u)(t)≤0,∀t≥0.
Then we have(2.10)E(t)=E(0)-∫0t12‖us‖22ds+12∫0t12(g′∘∇u)(s)ds-12∫0t12g(s)‖∇u(s)‖22ds.
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∈H01(Ω)∩H2(Ω),u1∈H01(Ω) and  M(∥∇u0∥22)>0, and further assume that
(2.11)E(0)>0,(2.12)I(u0)<0,(2.13)∫Ωu0u1dx>0,(2.14)‖u0‖22>(p+1)Cpm1E(0).
Then the solution of problem (1.1) blows up in finite time  0<T*<+∞, which means that
(2.15)lim⁡t→T*-(‖u(t)‖22+∫0t‖u(s)‖22ds)=∞,
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, Theorem2.4, 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 thatg(t) satisfies assumptions (A1)-(A2), and H(t) is a function which is twice continuously differentiable satisfying
(3.1)H′′(t)+H′(t)>∫0tg(t-s)∫Ω∇u(s)∇u(t)dxds,H(0)>0,H′(0)>0,
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 thatu0∈H01(Ω)∩H2(Ω) and u1∈H01(Ω) satisfy
(3.2)∫Ωu0u1dx>0.
If the solution u(t) of problem (1.1) exists on [0,T) and satisfies
(3.3)I(u(t))<0,
then ∥u(t)∥22 is strictly increasing on [0,T).Proof.
Sinceu(t) is the solution of problem (1.1), by a simple computation, we have
(3.4)12d2dt2∫Ω|u(x,t)|2dx=∫Ω(|ut|2+uutt)dx=‖ut‖22-M(‖∇u‖22)‖∇u‖22+‖u‖p+1p+1+∫0tg(t-s)∫Ω∇u(s)∇u(t)dxds-∫Ωuutdx>-∫Ωuutdx+∫0tg(t-s)∫Ω∇u(s)∇u(t)dxds,
where the last inequality is derived by (3.3). Then we get
(3.5)d2dt2∫Ω|u(x,t)|2dx+ddt∫Ω|u(x,t)|2dx>∫0tg(t-s)∫Ω∇u(s)∇u(t)dxds.
Therefore, by using Lemma 3.1, we finish our proof.Lemma 3.3.
Ifu0∈H01(Ω)∩H2(Ω) and u1∈H01(Ω) satisfy the assumptions in Theorem 2.4, then the solution u(t) of problem (1.1) satisfies
(3.6)I(u(t))<0,(3.7)‖u‖22>(p+1)Cpm1E(0),
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
(3.8)t0=min⁡{t∈(0,T):I(u(t))=0}.
Since I(u(t))<0 on [0,t0), by Lemma 3.2, we see that ∫Ωu2dx is strictly increasing over [0,t0), which implies
(3.9)∫Ωu2dx>∫Ωu02dx>(p+1)Cpm1E(0).
And by the continuity of ∫Ωu2dx on t, we note that
(3.10)∫Ωu2(t0)dx≥(p+1)Cpm1E(0).
On the other hand, by (2.6) and (2.9), we get
(3.11)M¯(‖∇u(t0)‖22)-∫0t0g(s)ds‖∇u(t0)‖22+(g∘∇u)(t0)-2p+1‖u(t0)‖p+1p+1≤2E(0).
Combining (3.11) with (3.8) yields
(3.12)p+12M¯(‖∇u(t0)‖22)-p+12∫0t0g(s)ds‖∇u(t0)‖22+p+12(g∘∇u)(t0)-M(‖∇u(t0)‖22)‖∇u(t0)‖22≤(p+1)E(0).
By (A3), we get
(3.13)‖∇u(t0)‖22<p+1m1E(0).
By Poincaré’s inequality, we have
(3.14)‖u(t0)‖22<(p+1)Cpm1E(0).
Obviously, there is a contradiction between (3.10) and (3.14), thus we prove that
(3.15)I(u(t))<0,
for every t∈(0,T). By Lemma 3.2, it follows that ∫Ωu2dx is strictly increasing on [0,T), which implies that
(3.16)∫Ω(p+1)Cpm1u2dx≥∫Ω(p+1)Cpm1u02dx>(p+1)Cpm1E(0),
for every t∈[0,T). This completes the proof of Lemma 3.3.Proof of Theorem2.4.
We prove our main result by adopting concavity method. We assume by contradiction that theT is sufficiently large. Then we consider the auxiliary function
(3.17)G(t)=‖u(t)‖22+∫0t‖u(s)‖22ds+(T0-t)‖u0‖22+β(t2+t)2,t∈[0,T0],
where T0,t2, and β are positive constants, which will be chosen later.
A straightforward calculation gives(3.18)G′(t)=2∫Ωuutdx+‖u(t)‖22-‖u0‖22+2β(t2+t)=2∫Ωuutdx+2∫0t(u(s),us(s))ds+2β(t2+t),
consequently,
(3.19)G′′(t)=2∫Ω∇|ut|2dx+2∫Ωuuttdx+2∫Ωuutdx+2β=2‖ut‖22-2M(‖∇u‖22)‖∇u‖22+2∫0tg(t-s)∫Ω∇u(s)∇u(t)dxds-2∫Ωuutdx+2∫Ωuutdx+2‖u‖p+1p+1+2β=2‖ut‖22-2M(‖∇u‖22)‖∇u‖22+2‖u‖p+1p+1+2∫0tg(t-s)ds‖∇u‖22+2∫0tg(t-s)∫Ω∇u(t)(∇u(s)-∇u(t))dxds+2β.
By using Young’s inequality, we obtain
(3.20)∫0tg(t-s)∫Ω|∇u(t)||∇u(s)-∇u(t)|dxds≤∫0tg(s)ds‖∇u(t)‖22+14(g∘∇u)(t).
Substituting (2.6) and (3.20) for the third and the fifth terms of the right hand side of (3.19), respectively, we have
(3.21)G′′(t)≥(p+3)‖ut‖22+(p+1)M¯(‖∇u‖22)-2M(‖∇u‖22)‖∇u‖22-(p+1)∫0tg(s)ds‖∇u‖22-2(p+1)E(t)+(p+12)(g∘∇u)(t)+2β.
By (A3), we deduce
(3.22)G′′(t)>(p+3)‖ut‖22+2m1‖∇u‖22-2(p+1)E(t)+(p+12)(g∘∇u)(t)+2β.
Noting that (2.10), we obtain that
(3.23)-E(t)≥-E(0)+∫0t‖us‖22ds.
Combining (3.22)-(3.23) yields
(3.24)G′′(t)>(p+3)‖ut‖22+2m1‖∇u‖22-2(p+1)E(0)+(p+12)(g∘∇u)(t)+2(p+1)∇∫0tG(0)G′(0)‖us‖22ds+2β.
By Poincaré’s inequality, Lemma 3.2, and (2.14), we see that
(3.25)2m1‖∇u‖22-2(p+1)E(0)+(p+12)(g∘∇u)(t)>2m1Cp‖u0‖22-2(p+1)E(0)>0,
by (3.24)-(3.25), we get
(3.26)G′′(t)>(p+3)‖ut‖22+2m1Cp‖u0‖22-2(p+1)E(0)+2(p+1)∫0t‖us‖22ds+2β,
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 chooseβ small enough satisfying
(3.27)(p+1)β<2m1Cp‖u0‖22-2(p+1)E(0),
consequently,
(3.28)G′′(t)>(p+3)‖ut‖22+2(p+1)∫0t‖us‖22ds+(p+3)β.
As far as β is fixed, we select t2 large enough satisfying
(3.29)p-12(∫Ωu0u1dx+βt2)>‖u0‖22.
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
(3.30)T0>4p-1G(0)G′(0).
Letting
(3.31)A∶=‖u(t)‖22+∫0t‖u(s)‖22ds+β(t2+t)2,B∶=12G′(t),C∶=‖ut(t)‖22+∫0t‖us(s)‖22ds+β.
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
(3.32)G(t)≥A,G′′(t)≥(p+3)C,
for every t∈[0,T0]. Then it follows that
(3.33)G′′(t)G(t)-p+34(G′(t))2≥(p+3)(AC-B2).
Furthermore, we have
(3.34)Ar2-2Br+C=∫Ω(ru(t)-ut(t))2dx+∫0t‖ru(s)-us(s)‖22ds+β[r(t2+t)-1]2≥0,
for every r∈ℝ, which implies that B2-AC≤0. Thus, we obtain
(3.35)G′′(t)G(t)-p+34(G′(t))2≥0.
As (p+3)/4>1, letting θ=(p-1)/4, we have
(3.36)G′′(t)G(t)-(1+θ)(G′(t))2≥0.
According to concavity technique, there exists a real number T* such that T*≤G(0)/θG′(0)<T0 and we have
(3.37)lim⁡t→T*-⁡G(t)=∞,
that is,
(3.38)lim⁡t→T*-⁡(‖u(t)‖22+∫0t‖u(s)‖22ds)=∞,
which contradicts the assumption that the T is sufficiently large.
This completes the proof of Theorem2.4.


## 4. Some Further Remarks

### 4.1. The Possible Choice of the Positive Constantm1 in (A3)

WhenM(s)=a+bsγ for s≥0, a≥0,b≥0,a+b>0, γ>0 and p>1+2γ, by straightforward calculation, we obtain
(4.1)p+12M¯(s)-[M(s)+p+12∫0∞g(τ)dτ]s=p+12(as+bγ+1sγ+1)-as-bsγ+1-(p+1)s2∫0∞g(τ)dτ=p-12as+(p-1-2γ)b2(γ+1)sγ+1-(p+1)s2∫0∞g(τ)dτ.
If a>0 and b≥0, it follows from (2.4) that ∫0∞g(τ)dτ<((p-1)/(p+1))a. Thus, we have
(4.2)p+12M¯(s)-[M(s)+p+12∫0∞g(τ)dτ]s>p-12as+(p-1-2γ)b2(γ+1)sγ+1-(p+1)s2[p-1p+1a-((p-1)/(p+1))a-∫0∞g(τ)dτ2]=(p-1-2γ)b2(γ+1)sγ+1+p+14[p-1p+1a-∫0∞g(τ)dτ]s≥p+14[p-1p+1a-∫0∞g(τ)dτ]s,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-∫0∞g(τ)dτ] in condition (A3).Ifa=0 and b>0, then(4.3)p+12M¯(s)-[M(s)+p+12∫0∞g(τ)dτ]s=(p-1-2γ)b2(γ+1)sγ+1-(p+1)s2∫0∞g(τ)dτ>(p-1-2γ)b2(γ+1)sγ+1-(p+1)s2×[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ2(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-((p-1-2γ)b/(Cpγ(p+1)(γ+1)))‖u0‖22γ-∫0∞g(τ)dτ2]=(p-1-2γ)b2(γ+1)s(sγ-1Cpγ‖u0‖22γ)+(p+1)s4[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ].Takings=∥∇u(t)∥22, applying Lemma 3.2 and Poincaré’s inequality, we can get
(4.4)p+12M¯(‖∇u(t)‖22)-[M(‖∇u(t)‖22)+p+12∫0∞g(τ)dτ]‖∇u(t)‖22>(p-1-2γ)b2(γ+1)‖∇u(t)‖22(‖∇u(t)‖22γ-1Cpγ‖u0‖22γ)+p+14‖∇u(t)‖22[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ]≥(p-1-2γ)b2(γ+1)‖∇u(t)‖22(‖∇u(t)‖22γ-1Cpγ‖u(t)‖22γ)+p+14‖∇u(t)‖22[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ]≥(p-1-2γ)b2(γ+1)‖∇u(t)‖22(‖∇u(t)‖22γ-‖∇u(t)‖22γ)+p+14‖∇u(t)‖22[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ]=p+14[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ]‖∇u(t)‖22.So, we can choose m1=((p+1)/4)[(p-1-2γ)b/(Cpγ(p+1)(γ+1)∥u0∥22γ)-∫0∞g(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 energyE(0)=d>0, there exists such initial data which leads to blow up in finite time.For instance, in the caseM(s)=1+sγ, then for any (u0,u1)∈(H01(Ω)∩H2(Ω))×H01(Ω) with ∫Ωu0u1dx>0, we may take some λ,α>0, such that (u0,u1) satisfies the above conditions (2.11)–(2.14).Indeed, forM(s)=1+sγ, conditions (2.11)–(2.14) become(4.5)E(0)=12‖u1‖22+12‖∇u0‖22+12(γ+1)‖∇u0‖22(γ+1)-1p+1‖u0‖p+1p+1>0,I(u0)=‖∇u0‖22+‖∇u0‖22(γ+1)-‖u0‖p+1p+1<0,∫Ωu0u1dx>0,‖u0‖22>(p+1)Cpm1E(0).Now taking(v0,v1)∈(H01(Ω)∩H2(Ω))×H01(Ω) such that ∫Ωv0v1dx>0, and letting (u0,u1)=(λv0,αv1) for any scaling parameter λ>0 and α>0, then we have(4.6)E(0)=12α2‖v1‖22+12λ2‖∇v0‖22+12(γ+1)λ2(γ+1)‖∇v0‖22(γ+1)-1p+1λp+1‖v0‖p+1p+1,I(u0)=λ2‖∇v0‖22+λ2(γ+1)‖∇v0‖22(γ+1)-λp+1‖v0‖p+1p+1,‖u0‖22=λ2‖v0‖22.We suppose thatλ2(γ+1)<λp+1 (i.e., p>1+2γ) for λ>1, so we can choose sufficiently large λ such that
(4.7)12λ2‖∇v0‖22+12(γ+1)λ2(γ+1)‖∇v0‖22(γ+1)-1p+1λp+1‖v0‖p+1p+1<0,I(u0)=I(λv0)=λ2‖∇v0‖22+λ2(γ+1)‖∇v0‖22(γ+1)-λp+1‖v0‖p+1p+1<0,‖u0‖22=λ2‖v0‖22>(p+1)Cpm1E(0).And when λ is fixed, we may choose α such that E(0)=d.Similarly, in the caseM(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.


## 4.1. The Possible Choice of the Positive Constantm1 in (A3)

WhenM(s)=a+bsγ for s≥0, a≥0,b≥0,a+b>0, γ>0 and p>1+2γ, by straightforward calculation, we obtain
(4.1)p+12M¯(s)-[M(s)+p+12∫0∞g(τ)dτ]s=p+12(as+bγ+1sγ+1)-as-bsγ+1-(p+1)s2∫0∞g(τ)dτ=p-12as+(p-1-2γ)b2(γ+1)sγ+1-(p+1)s2∫0∞g(τ)dτ.
If a>0 and b≥0, it follows from (2.4) that ∫0∞g(τ)dτ<((p-1)/(p+1))a. Thus, we have
(4.2)p+12M¯(s)-[M(s)+p+12∫0∞g(τ)dτ]s>p-12as+(p-1-2γ)b2(γ+1)sγ+1-(p+1)s2[p-1p+1a-((p-1)/(p+1))a-∫0∞g(τ)dτ2]=(p-1-2γ)b2(γ+1)sγ+1+p+14[p-1p+1a-∫0∞g(τ)dτ]s≥p+14[p-1p+1a-∫0∞g(τ)dτ]s,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-∫0∞g(τ)dτ] in condition (A3).Ifa=0 and b>0, then(4.3)p+12M¯(s)-[M(s)+p+12∫0∞g(τ)dτ]s=(p-1-2γ)b2(γ+1)sγ+1-(p+1)s2∫0∞g(τ)dτ>(p-1-2γ)b2(γ+1)sγ+1-(p+1)s2×[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ2(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-((p-1-2γ)b/(Cpγ(p+1)(γ+1)))‖u0‖22γ-∫0∞g(τ)dτ2]=(p-1-2γ)b2(γ+1)s(sγ-1Cpγ‖u0‖22γ)+(p+1)s4[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ].Takings=∥∇u(t)∥22, applying Lemma 3.2 and Poincaré’s inequality, we can get
(4.4)p+12M¯(‖∇u(t)‖22)-[M(‖∇u(t)‖22)+p+12∫0∞g(τ)dτ]‖∇u(t)‖22>(p-1-2γ)b2(γ+1)‖∇u(t)‖22(‖∇u(t)‖22γ-1Cpγ‖u0‖22γ)+p+14‖∇u(t)‖22[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ]≥(p-1-2γ)b2(γ+1)‖∇u(t)‖22(‖∇u(t)‖22γ-1Cpγ‖u(t)‖22γ)+p+14‖∇u(t)‖22[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ]≥(p-1-2γ)b2(γ+1)‖∇u(t)‖22(‖∇u(t)‖22γ-‖∇u(t)‖22γ)+p+14‖∇u(t)‖22[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ]=p+14[(p-1-2γ)bCpγ(p+1)(γ+1)‖u0‖22γ-∫0∞g(τ)dτ]‖∇u(t)‖22.So, we can choose m1=((p+1)/4)[(p-1-2γ)b/(Cpγ(p+1)(γ+1)∥u0∥22γ)-∫0∞g(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 energyE(0)=d>0, there exists such initial data which leads to blow up in finite time.For instance, in the caseM(s)=1+sγ, then for any (u0,u1)∈(H01(Ω)∩H2(Ω))×H01(Ω) with ∫Ωu0u1dx>0, we may take some λ,α>0, such that (u0,u1) satisfies the above conditions (2.11)–(2.14).Indeed, forM(s)=1+sγ, conditions (2.11)–(2.14) become(4.5)E(0)=12‖u1‖22+12‖∇u0‖22+12(γ+1)‖∇u0‖22(γ+1)-1p+1‖u0‖p+1p+1>0,I(u0)=‖∇u0‖22+‖∇u0‖22(γ+1)-‖u0‖p+1p+1<0,∫Ωu0u1dx>0,‖u0‖22>(p+1)Cpm1E(0).Now taking(v0,v1)∈(H01(Ω)∩H2(Ω))×H01(Ω) such that ∫Ωv0v1dx>0, and letting (u0,u1)=(λv0,αv1) for any scaling parameter λ>0 and α>0, then we have(4.6)E(0)=12α2‖v1‖22+12λ2‖∇v0‖22+12(γ+1)λ2(γ+1)‖∇v0‖22(γ+1)-1p+1λp+1‖v0‖p+1p+1,I(u0)=λ2‖∇v0‖22+λ2(γ+1)‖∇v0‖22(γ+1)-λp+1‖v0‖p+1p+1,‖u0‖22=λ2‖v0‖22.We suppose thatλ2(γ+1)<λp+1 (i.e., p>1+2γ) for λ>1, so we can choose sufficiently large λ such that
(4.7)12λ2‖∇v0‖22+12(γ+1)λ2(γ+1)‖∇v0‖22(γ+1)-1p+1λp+1‖v0‖p+1p+1<0,I(u0)=I(λv0)=λ2‖∇v0‖22+λ2(γ+1)‖∇v0‖22(γ+1)-λp+1‖v0‖p+1p+1<0,‖u0‖22=λ2‖v0‖22>(p+1)Cpm1E(0).And when λ is fixed, we may choose α such that E(0)=d.Similarly, in the caseM(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.



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*Source: 530861-2012-04-26.xml*