# Blow-Up and Global Existence Analysis for the Viscoelastic Wave Equation with a Frictional and a Kelvin-Voigt Damping

Authors: Fosheng Wang; Chengqiang Wang
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We are concerned in this paper with the initial boundary value problem for a quasilinear viscoelastic wave equation which is subject to a nonlinear action, to a nonlinear frictional damping, and to a Kelvin-Voigt damping, simultaneously. By utilizing a carefully chosen Lyapunov functional, we establish first by the celebrated convexity argument a finite time blow-up criterion for the initial boundary value problem in question; we prove second by an a priori estimate argument that some solutions to the problem exists globally if the nonlinearity is “weaker,” in a certain sense, than the frictional damping, and if the viscoelastic damping is sufficiently strong.


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## Body

## 1. Introduction

In this paper, we are concerned with the initial boundary value problem (IBVP) for a quasilinear viscoelastic wave equation whose energy is inhibited by a nonlinear frictional damping and by a Kelvin-Voigt damping. More precisely, we consider IBVP(1)∂tuρ∂t2u-Δu+∫0tgt-τΔu·,τdτ-Δ∂tu+∂tum-2∂tu=up-2uinΩ×0,+∞,u=0onΓ×0,+∞,u·,0=u0,∂tu·,0=u1inΩ,where u=u(x,t) is the unknown, Ω, having the boundary Γ∈C2, is a nonempty bounded open subset of RN with N a positive integer, ρ>0, m>2, p>2 are constants, and g:[0,+∞)→[0,+∞), the so-called relaxation function or viscoelastic kernel, is a decreasing function. The restrictions on ρ, p, m and g would be added when necessary. Eq. (1)1 was used to describe the vibration of beams or plates, whose wave propagation speed depends on the velocity of vibration, which is subject to nonlinear frictional damping force and viscoelastic damping force; see [1–3]. In particular, the term |∂tu|ρ∂t2u with ρ>0, by Boltzmann’s superposition principle, indicates that the wave speed depends on the velocity of the wave; ∫0tg(t-τ)Δu(·,τ)dτ, the so-called viscoelastic damping, is incorporated in IBVP (1) to characterize the hereditary properties.In recent years, the wave equation has been frequently chosen as the model to study the competition between different damping terms and the source term|u|p-2u. In [4], Messaoudi considered the general decay problem for IBVP(2)∂t2u-Δu+∫0tgt-τΔu·,τdτ=0inΩ×0,+∞,u=0onΓ×0,+∞,u·,0=u0,∂tu·,0=u1inΩ,and he proved, under certain additional restrictions on the relaxation function g, that the energy of IBVP (2) decreases to zero. The results and methods in [4] were later extended by Messaoudi and Al-Khulaifi [5] to IBVP(3)∂tuρ∂t2u-Δu-Δ∂t2u+∫0tgt-τΔu·,τdτ=0inΩ×0,+∞,u=0onΓ×0,+∞,u·,0=u0,∂tu·,0=u1inΩ;more precisely, they established some general decay results for IBVP (3). It was shown in both [4, 5] that the energy decay rate is closely related to the peculiarity of the relaxation function g. Different from (2)1, the wave speed of (3)1 depends on the velocity of the wave. The study of [5] was highly inspired by [6], in which Cavalcanti, Domingos Cavalcanti, and Ferreira studied the IBVP(4)∂tuρ∂t2u-Δu-Δ∂t2u+∫0tgt-τΔu·,τdτ-γΔ∂tu=0inΩ×0,+∞,u=0onΓ×0,+∞,u·,0=u0,∂tu·,0=u1inΩ,and they proved that the solutions of IBVP (4) exist globally if γ⩾0 and their energy decreases to zero if γ>0. Amongst the afore-mentioned references, there are vast literatures concerning energy decay for viscoelastic wave equations; see [7–14] and the references cited therein. The key ingredient in proving energy decay estimates is an appropriate modified energy functional associated with the initial boundary value problem under consideration. The usual idea to obtain decay for the afore-chosen energy functional is to exploit various perturbed energy functionals for the above energy functional. This idea is important in our later development.In the afore-mentioned references, the source term|u|p-2u, in one form or another, is “beaten” by the damping terms ∫0tg(t-τ)Δu(·,τ)dτ, -Δ∂tu and/or |∂tu|m-2∂tu. In some cases, these damping terms were also proved to be “beaten” by the above pure power source term; see [15–25]. As with the procedure to obtain energy decay, the main step in proving finite time blow-up results is to choose suitable Lyapunov functional L(t) which is closely related to the so-called modified energy functional frequently used in obtaining energy decay, and to employ an convexity argument to obtain L′t⩾C0Lt1/1-α0 with C0>0 and α0∈(0,1). This idea would be useful in proving our finite time blow-up result.More recently, Song [26] established a global nonexistence of solutions with positive initial data for IBVP(5)∂tuρ∂t2u-Δu+∫0tgt-τΔu·,τdτ+∂tum-2∂tu=up-2uinΩ×0,+∞,u=0onΓ×0,+∞,u·,0=u0,∂tu·,0=u1inΩ.Inspired by [26], Hao and Wei [27] studied IBVP(6)∂tuρ∂t2u-Δu+∫0tgt-τΔu·,τdτ-Δ∂tu=up-2uinΩ×0,+∞,u=0onΓ×0,+∞,u·,0=u0,∂tu·,0=u1inΩ,and they proved a global existence and a blow-up criterion for solutions to IBVP (6). Both Refs. [26, 27] assume the same restrictions on ρ, g, m and p. The results obtained by [26, 27] mean that the damping terms |∂tu|m-2∂tu and -Δ∂tu are beaten, in the sense that some solutions blow up in finite time, independently by the source term |u|p-2u in the “arena” |∂tu|ρ∂t2u-Δu+∫0tg(t-τ)Δu(·,τ)dτ=0. We are tempted to compare the damping term |∂tu|m-2∂tu-Δ∂tu and the source term |u|p-2u. The later one was shown in a certain condition to be “stronger” than the former one in the arena of viscoelastic plate equation ∂t2u+Δ2u-∫0tg(t-τ)Δ2u(·,τ)dτ-Δ∂t2u=0. More, precisely, Li et al. [28] provided a finite time blow-up result for IBVP(7)∂t2u+Δ2u-∫0tgt-τΔ2u·,τdτ-Δ∂tu-Δ∂t2u-∂tum-1∂tu=up-1uinΩ×0,+∞,u=0,∂νu=0onΓ×0,+∞,u·,0=u0,∂tu·,0=u1inΩ.Inspired by [26–28], we seek to prove that some solutions with positive initial data to IBVP (1) blow up in finite time under certain conditions (see Theorem 11); all solutions to IBVP (1) exist globally under other conditions (see Theorem 12). Different from [26], we provide a global existence result under some conditions, in addition to a finite time blow-up result. Compared with the finite time blow-up results in [26, 27], our finite time blow-up result shows that the source beats the combination of the frictional damping and the Kelvin-Voigt damping.Notations 1.
Throughout the paper,C is a generic positive constant and it can take a different value at every new occurrence. Lp(Ω) (resp.H01(Ω)) is the classical Lebsegue (resp., Sobolev) space.The rest of our paper is planned as follows. In Section2, we collect the main results and some preliminaries necessary in proving the results. In Section 3, we prove in a certain detail all the main results. In Section 4, we present an open question which confuses us recently.


## 2. Preliminaries and the Main Results

In this section, we collect the main assumptions on the relaxation functiong and the parameters in IBVP (1), some results which can simplify our later development and the main results of this paper. Due to their pervasion in the textbooks on PDEs or in the references cited below, all the proofs of the following lemmas are omitted here. Let us recall first Young’s inequality with a δ>0.Lemma 2.
For every pair(a,b)∈[0,+∞)2, and every θ∈(0,1), we have (8)a1-θbθ⩽δa+θ1-θδ1/θ-1b,∀δ∈0,+∞.We list here all the assumptions which are necessary for our claimed results.Assumption 3.
The relaxation functiong:[0,+∞)→[0,+∞) is absolutely continuous and decreasing and satisfies (9)l=1-∫0+∞gτdτ>0.Assumption 4.
m > 2; p>2 and ρ>0 if N=1,2; 2<p⩽2(n-1)/n-2 and 0<ρ⩽2/n-2 if N⩾3.Assumption 5.
p > max ⁡ ( m , ρ + 2 ) and ∫0+∞g(τ)dτ<p/2-1/p/2-1+1/2p.Assumption 6.
2 < p ⩽ ρ + 2 and ρ+2⩽2(n-1)/n-2.Assumption 7.
E ( 0 ) < 1 - 1 / ( p - 2 ) p 1 - l / l 1 / 2 - 1 / p ( B 1 ) - 2 p / p - 2 and ∇u0L2(Ω,RN)⩾(B1)-p/p-2 where E(t) is given by (14):(10)B1=Bl,(11)B=inf⁡C∈0,+∞∣φLpΩ⩽C∇φL2Ω;RNwheneverφ∈H01Ω.Remark 8.
By the well-known Poincaré’s embedding,B is a number in (0,+∞).With the above assumptions prepared, we are now in a position to state the main results of this paper. What comes first is the local existence theory.Theorem 9.
Let Assumptions3 and 4 be fulfilled. For every pair (u0,u1)∈H01(Ω)×L2(Ω), there exists a (unique) Tmax∈(0,+∞] such that IBVP (1) admits a unique solution, (12)u∈C0,Tmax;H01Ω∩C10,Tmax;L2Ω,and that [0,Tmax) is the maximal time interval of existence of u.Remark 10.
T m a x is called the lifespan of u. In the situation Tmax=+∞, the solution u is said to exist globally; in the situation Tmax<+∞, the solution u is said to blow up in finite time.Our second main result is the following finite time blow-up result.Theorem 11.
Let Assumptions3, 4, and 5 be fulfilled. Then every solution u to IBVP (1) blows up in finite time whenever it satisfies Assumption 7: If (u0,u1) satisfies Assumption 7, then we have Tmax<+∞. And in this situation, we have furthermore (13)limt→Tmax⁡∂tu·,tLρ+2Ωρ+2+∇u·,tL2Ω;RN2+u·,tLpΩp=+∞.The third main result is the following global existence result.Theorem 12.
Let Assumptions3, 4, and 6 be fulfilled. Then for every pair (u0,u1)∈H01(Ω)×L2(Ω), IBVP (1) admits a unique global solution u∈C([0,+∞);H01(Ω))∩C1([0,+∞);L2(Ω)).In the proofs of the above theorems, the following modified energy functionals are very important:(14)Et=1ρ+2∂tu·,tLρ+2Ωρ+2+121-∫0tgτdτ∇u·,tL2Ω;RN2+12g⋄∇ut-1pu·,tLpΩp,∀t∈0,+∞;here and hereafter, we write for every ψ∈Lloc2[0,+∞);L2(Ω;Rn)(15)g⋄ψt=∫0tgt-τψ·,t-ψ·,τL2Ω;Rn2dτ,∀t∈0,+∞.The following lemma would be used several times later.Lemma 13.
LetE(t) be defined by (14). Then along the solution u to IBVP (1), we have(16)E′t=-∂tu·,tLmΩm-∇∂tu·,tL2Ω;RN2+12g′⋄∇ut-12gt∇u·,tL2Ω;RN2,∀t∈0,+∞.Remark 14.
It should be noted thatE(t) is decreasing whenever g fulfills Assumption 3.


## 3. Proofs of the Main Results

In this section, we present the proofs of our main results.Proof of Theorem9.
Since one can prove the local existence by mimicking the steps in [6] and by using the classical Faedo-Galerkin method, the details are omitted here. By combining the local existence result and a standard continuation method, we can show that there exists a maximal time Tmax of existence of the solution u for every pair (u0,u1)∈H01(Ω)×L2(Ω).Before proving Theorem11, we introduce the following lemma.Lemma 15 (see [26]).
Let Assumptions3, 4, and 5 be fulfilled. There exists a β∈(B1)-p/p-2,+∞ such that (17)1-∫0tgτdτ∇u·,tL2Ω;RN2⩾β2,u·,tLpΩ⩾B1β,∀t∈0,+∞whenever u is a solution to IBVP (1) verifying E(0)<E1 and ∇u0L2(Ω,Rn)⩾(B1)-p/p-2 where E1 is given by(18)E1=12-1pB1-2p/p-2.Proof of Theorem11.
Let(u0,u1)∈H01(Ω)×L2(Ω). We write u for the solution to IBVP (1) with its initial datum (u0,u1) and write T∗ for the maximal time of existence of u. Our goal is to show by a contradiction argument that T∗<+∞.
AssumeT∗=+∞. Let us introduce the following functional for u:(19)Lt=E2-Et1-α+ε2∇u·,tL2Ω;RN2+ερ+1∫Ωux,t∂tux,tρ∂tux,tdx,∀t∈0,+∞,where E(t) is given by (14) and E2∈(E(0),E1), ε>0 and α∈(0,1/m-1/p) are yet to be determined later. Differentiate both hand sides of (19), and conduct some routine but tedious calculations, to obtain(20)L′t=-1-αE2-Et-αE′t+ε∫Ω∇ux,t·∇∂tux,tdx+ερ+1∂tu·,tLρ+2Ωρ+2+εu·,t,∂tu·,tρ∂t2u·,tH01Ω,H-1Ω=1-αE2-Et-α∂tu·,tLmΩm+∇∂tu·,tL2Ω;RN2-12g′⋄∇ut+12gt∇u·,tL2Ω;RN2+ε∫Ω∇ux,t·∇∂tux,tdx+ερ+1∂tu·,tLρ+2Ωρ+2-ε∇u·,tL2Ω;RN2+ε∫0tgt-τ∫Ω∇ux,t·∇ux,τdxdτ-ε∫Ωux,t∂tux,tm-2∂tux,tdx-ε∫Ω∇ux,t·∇∂tux,tdx+εu·,tLpΩp⩾1-αE2-Et-α∂tu·,tLmΩm+ερ+1∂tu·,tLρ+2Ωρ+2-ε∇u·,tL2Ω;RN2+ε∫0tgt-τ∫Ω∇ux,t·∇ux,τdxdτ-ε∫Ωux,t∂tux,tm-2∂tux,tdx+εu·,tLpΩp,∀t∈0,+∞.We next treat ∫Ωu(x,t)|∂tu(x,t)|m-2∂tu(x,t)dx and ∫0tg(t-τ)∫Ω∇u(x,t)·∇u(x,τ)dxdτ term-by-term. By utilizing the algebraic identity (21)ab=a2-aa-b=a2-2·a12p1-δ1·a-bp1-δ12=a2+a12p1-δ1-a-bp1-δ122-a22p1-δ1-p1-δ1a-b22=a12p1-δ1-a-bp1-δ122+1-12p1-δ1a2-p1-δ12a-b2,for every a,b∈R, we conclude that(22)∫0tgt-τ∫Ω∇ux,t·∇ux,τdxdτ=∫0tgt-τ∫Ω12p1-δ1∇ux,t-p1-δ12∇ux,t-∇ux,τ2+1-12p1-δ1∇ux,t2-p1-δ12∇ux,t-∇ux,τ2dxdτ⩾1-12p1-δ1∫0tgτdτ∇u·,tL2Ω;RN2-p1-δ12g⋄∇ut,∀t∈0,+∞,where δ1∈(0,1) is yet to be selected later. By the definition (14) of E(t), we have(23)-p2g⋄∇ut=-pE2+pE2-Et+pρ+2∂tu·,tLρ+2Ωρ+2+p21-∫0tgτdτ∇u·,tL2Ω;RN2-u·,tLpΩp,∀t∈0,+∞.This, together with (22), implies(24)
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+
∞
.Set (25)δ1∗=p-1-1/lp.Thanks to (26)1-l=∫0+∞gτdτ<p/2-1p/2-1+1/2p, we have, for every δ1∈(0,δ1∗), (27)p1-δ1/2-11-∫0tgτdτ-1/2p1-δ1∫0tgτdτ1-∫0tgτdτ1-∫0tgτdτ∇u·,tL2Ω;RN2⩾lp1-δ1/2-1-1-l/2p1-δ1lβ2=lp1-δ1/2-1-1-l/2p1-δ1lB1-2p/p-2-p1-δ1E2+p1-δ1E2,∀t∈0,+∞,where “⩾” follows from Lemma 15. We fix δ1∗∗∈(0,δ1∗) and E2∗∈(E(0),E1) such that (28)lp1-δ1∗∗/2-1-1-l/2p1-δ1∗∗lB1-2p/p-2-p1-δ1∗∗E2∗>0,and therefore(29)p1-δ1/2-11-∫0tgτdτ-1/2p1-δ1∫0tgτdτ1-∫0tgτdτ1-∫0tgτdτ∇u·,tL2Ω;RN2>p1-δ1E2.The existence of δ1∗∗ and E2∗ can be justified by the following observation (30)limδ1,E2∈0,δ1∗×E0,E2,δ1,E2→0,E0⁡lp1-δ1/2-1-1-l/2p1-δ1lB1-2p/p-2-p1-δ1E2=p1-1p-2p1-ll12-1pB1-2p/p-2-E0>0. By Lemma 2, for every δ2>0, (31)∫Ωux,t∂tux,tm-2∂tux,tdx⩽u·,tLmΩ∂tux,tm-2∂tux,tLm/m-1Ω=u·,tLmΩ∂tu·,tLmΩm-1⩽Cu·,tLpΩ∂tu·,tLmΩm-1=Cu·,tLpΩp1/p-1/mu·,tLpΩp/m∂tu·,tLmΩm-1⩽Cδ2E2-E01/p-1/mu·,tLpΩp+CCδ2E2-Et1/p-1/m∂tu·,tLmΩm,∀t∈0,+∞.By Remark 14 and by the assumption that p>m, we have (32)E2-Et1/p-1/m=E2-Et-αE2-Etα-1/m-1/p⩽E2-Et-αE2-Etα-1/m-1/p⩽E2-Et-αE2-E0α-1/m-1/p,∀t∈0,+∞.This, together with (31), implies(33)∫Ωux,t∂tux,tm-2∂tux,tdx⩽Cδ2E2-E01/p-1/mu·,tLpΩp+CCδ2E2-E0α-1/m-1/pE2-Et-α∂tu·,tLmΩm,∀t∈0,+∞.Combine (20), (24), (29), and (33), to obtain for δ1=δ1∗∗, E2=E2∗ and every δ2>0 and every ε>0,(34)L′t⩾1-α-εCCδ2E2-E0α-1/m-1/pE2-Et-α∂tu·,tLmΩm+ε1ρ+1+p1-δ1ρ+2∂tu·,tLρ+2Ωρ+2+pε1-δ1E2-Et+εδ1-Cδ2E2-E01/p-1/mu·,tLpΩp,∀t∈0,+∞.Let δ1=δ1∗∗, E2=E2∗ and choose sufficiently small δ2>0 and ε>0 such that(35)L′t⩾C∂tu·,tLρ+2Ωρ+2+E2-Et+u·,tLpΩp,∀t∈0,+∞,where C>0 depends on δ1∗∗, E2∗, δ2, ε, E(0), p, ρ and α. Since the function [0,+∞)∋ζ↦ζ1/1-α∈[0,+∞) is convex, by Jensen’s inequality, we have (36)Lt1/1-α⩽E2-Et1-α+ε2∇u·,tL2Ω;RN2+ερ+1∫Ωux,t∂tux,tρ∂tux,tdx1/1-α⩽3α/1-αE2-Et+ε21/1-α∇u·,tL2Ω;RN21/1-α+ερ+11/1-α∫Ωux,t∂tux,tρ∂tux,tdx1/1-α⩽CαE2-Et+∇u·,tL2Ω;RN21/1-α+∫Ωux,t∂tux,tρ∂tux,tdx1/1-α,∀t∈0,+∞.Since T∗=+∞, ∂tu(·,t)Lρ+2(Ω)ρ+2+∇u(·,t)L2(Ω;RN)2+u(·,t)Lp(Ω)p is bounded by some constant C. Since E2-E(t) is increasing, we obtain (37)∇u·,tL2Ω;RN21/1-α⩽C1/1-α⩽C1/1-αE2-E0E2-Et,∀t∈0,+∞.On the other hand, we have the following sequence of inequalities:(38)∫Ωux,t∂tux,tρ∂tux,tdx1/1-α⩽u·,tLρ+2Ω∂tux,tρ∂tux,tLρ+2/ρ+1Ω1/1-α=u·,tLρ+2Ω1/1-α∂tu·,tLρ+2Ωρ+1/1-α⩽Cu·,tLpΩ1/1-α∂tu·,tLρ+2Ωρ+1/1-α⩽Cu·,tLpΩ1/1-α·1-αρ+2/1-αρ-2α+∂tu·,tLρ+2Ωρ+1/1-α·1-αρ+2/ρ+1=Cu·,tLpΩρ+2/1-αρ-2α+∂tu·,tLρ+2Ωρ+2,∀t∈0,+∞,where C>0 depends on the measure measΩ of Ω; “⩽” in the first line follows from Hölder’s inequality; “⩽” in the third line follows from the continuous embedding Lρ+2(Ω)↪Lp(Ω); “⩽” follows from Hölder’s inequality; “⩽” in the fourth line follows from Young’s inequality (see Lemma 2) and from (39)1-αρ-2α1-αρ+2+ρ+11-αρ+2=1. We fix α such that (40)0<α<min⁡1ρ+2-1p,1m-1p, and hence (41)ρ+21-αρ-2α=ρ+21-αρ-2α-p+p=pρ+21-αρ-2αα-1ρ+2-1p+p<p. By Remark 14, this implies(42)u·,tLpΩρ+2/1-αρ-2α=u·,tLpΩpu·,tLpΩρ+2/1-αρ-2α-p=u·,tLpΩpu·,tLpΩpρ+2/1-αρ-2αα-1/ρ+2-1/p⩽Cu·,tLpΩpE2-Etρ+2/1-αρ-2αα-1/ρ+2-1/p⩽Cu·,tLpΩpE2-E0ρ+2/1-αρ-2αα-1/ρ+2-1/p,∀t∈0,+∞,where “⩽” in the third line follows from (43)E2-Et=E2-1ρ+2∂tu·,tLρ+2Ωρ+2-121-∫0tgτdτ∇u·,tL2Ω;RN2-12g⋄∇ut+1pu·,tLpΩp⩽E1-121-∫0tgτdτ∇u·,tL2Ω;RN2+1pu·,tLpΩp⩽E1-12β2+1pu·,tLpΩp⩽12-1pB1-2p/p-2-12B1-2p/p-2+1pu·,tLpΩp=-1pB1-2p/p-2+1pu·,tLpΩp⩽1pu·,tLpΩp,∀t∈0,+∞.Combine (38) and (42), to yield (44)∫Ωux,t∂tux,tρ∂tux,tdx1/1-α⩽C∂tu·,tLρ+2Ωρ+2+u·,tLpΩpE2-E0ρ+2/1-αρ-2αα-1/ρ+2-1/p,∀t∈0,+∞.This, together with (37), implies directly the following differential inequality: (45)L′t⩾CLt1/1-α,∀t∈0,+∞,which implies in turn(46)Lt⩾L0α/α-1-Cαt1-α-1-α/α,∀t∈0,+∞,where C>0 depends on p, ρ and α. The proof is complete.Proof of Theorem12.
Let(u0,u1)∈H01(Ω)×L2(Ω), and let u∈C([0,Tmax);H01(Ω))∩C1([0,Tmax);L2(Ω)), where Tmax is the maximal time of existence, be the solution to IBVP (1) associated with the initial datum (u0,u1)∈H01(Ω)×L2(Ω). Our objective is to prove Tmax=+∞. We achieve this goal by contradiction. Assume to the contrary Tmax<+∞, and we associate the following functional to u(47)Ft=1ρ+2∂tu·,tLρ+2Ωρ+2+121-∫0tgτdτ∇u·,tL2Ω;RN2+12g⋄∇ut+1pu·,tLpΩp=Et+2pu·,tLpΩp,∀t∈0,Tmax,where E(t) is given as in (14). By using (16), we can deduce from Assumption 3 the following inequality(48)F′t=E′t-2∫Ωux,tp-2ux,t∂tux,tdx=-∂tu·,tLmΩm-∇∂tu·,tL2Ω;RN2+12g′⋄∇ut-12gt∇u·,tL2Ω;RN2+2∫Ωux,tp-2ux,t∂tux,tdx⩽2∫Ωux,tp-2ux,t∂tux,tdx-∇∂tu·,tL2Ω;RN2,∀t∈0,Tmax.By Young’s inequality (see Lemma 2), we have(49)∫Ωux,tp-2ux,t∂tux,tdx⩽∫Ωux,tp-2ux,t∂tux,tdx⩽∫Ωux,tp-1∂tux,tdx⩽∫Ωδ3∂tux,tp+Cδ3ux,tpdx=δ3∂tu·,tLpΩp+Cδ3u·,tLpΩp=δ3ρ+2-pρ∂tu·,tLpΩ2+δ3p-2ρ∂tu·,tLpΩρ+2+Cδ3u·,tLpΩp⩽δ3B2∇∂tu·,tLpΩ;RN2+Cδ3∂tu·,tLρ+2Ωρ+2+Cδ3u·,tLpΩp,∀t∈0,Tmax,where δ3 is a positive number which would be specified explicitly later and B is the best Poincaré’s embedding constant (see (11) for the definition). Plug (49) into (48) and write the result compactly, to yield (50)F′t⩽-1-2δ3B2∇∂tu·,tL2Ω;RN2+2Cδ3∂tu·,tLρ+2Ωρ+2+2Cδ3u·,tLpΩp,∀t∈0,Tmax.By choosing δ3=1/2B2, we obtain the following differential inequality (51)F′t⩽CFt,∀t∈0,Tmax,which implies immediately (52)Ft⩽eCtF0⩽eCTmaxF0,∀t∈0,Tmax,where C>0 is a constant depending on B, p and ρ. By the definition (47), we have (53)∂tu·,tLρ+2Ωρ+2+∇u·,tL2Ω;RN2⩽max⁡ρ+2,2lFt⩽max⁡ρ+2,2leCTmaxF0,∀t∈0,Tmax.Now choose t0∈(0,Tmax) such that(54)Tmax-t0<Tmax,where Tmax is the maximal time of existence of solutions whose initial data’s norms are bounded by max⁡(ρ+2,2/l)eCTmaxF(0). By Theorem 9, IBVP (1) supplemented by the initial condition (55)u~·,0,∂tu~·,0=u·,t0,∂tu·,t0admits a unique solution (56)u~∈C0,Tmax;H01Ω∩C10,Tmax;L2Ωwith Tmax the maximal time of existence of the solution. Write (57)u^x,t=ux,tifx,t∈Ω×0,t0,u~x,tifx,t∈Ω×t0,t0+Tmax.By the construction, u^ is the unique solution to IBVP (1) on 0,t0+Tmax associated with the initial datum (u0,u1)∈H01(Ω)×L2(Ω) which is given at the very beginning of the proof. By the definition of Tmax, Tmax⩾t0+Tmax. This contradicts (54).
Therefore,Tmax=+∞. The proof is complete.


## 4. Conclusion

In this paper, we studied a class of viscoelastic wave equations which incorporate a pure power source term, a nonlinear frictional damping term, and a strong linear damping term of Kelvin-Voigt type. We proved first a finite time blow-up result by a convexity argument and obtained second a global existence result by a priori estimates for the problem in question. The modified energy functional (14) played an important role in finishing the proofs of the finite time blow-up result and the global existence result.There remains a problem concerning IBVP (1), that is, the problem of finding or estimating the lifespan for every solution u to IBVP (1). When Assumptions 3, 4, and 6 are fulfilled, all the solutions exist globally, and hence their lifespans are +∞. When Assumptions 3, 4, and 5 are fulfilled, the problem of finding lifespans becomes very complicated; in this case, for those initial data satisfying the hypothesis of Theorem 12, we can deduce from (46) the upper bound estimate for the lifespans:(58)Tu=1-αCαL0α/α-1,where C>0 depends on p, ρ, α and the initial data (u0,u1). In view of this, we are tempted to understand better the following questions: (i)
How to obtain an estimate better than (58) upon lifespans?(ii)
Let Assumptions3, 4, and 5 be fulfilled. How to obtain an estimate upon lifespans for solutions whose initial data do not satisfy the hypothesis of Theorem 12?



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