# Global Nonexistence for a Nonlinear Viscoelastic Equation with Nonlinear Damping and Velocity-Dependent Material Density

Authors: Jian Dang; Qingying Hu; Hongwei Zhang
Journal: Journal of Function Spaces
 (2019)

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## Abstract

We consider the initial boundary value problem of a nonlinear viscoelastic equation of Kirchhoff-type with nonlinear damping and velocity-dependent material density. We establish a nonexistence result of global solutions with positive initial energy and negative initial energy, respectively.


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## Body

## 1. Introduction

In this paper we study the global nonexistence for solutions of the nonlinear viscoelastic equation problem with nonlinear damping and velocity-dependent material density as follows:(1)utρutt-M∇u2Δu+αΔ2u-βΔutt-γΔut+∫0tgt-sΔusds+hut=fu,x∈Ω,t>0,(2)ux,t=∂u∂ν=0,x∈∂Ω,t>0,(3)ux,0=u0x,utx,0=u1x,x∈Ω,where Ω is a bounded domain in Rn(n≥1) with smooth boundary ∂Ω so that the divergence theorem can be applied and g(t) is a positive function that represents the kernel of the memory term. Here ρ>0,α>0,β>0, and γ>0 are positive constants. In this paper, we take M(s)=1+bsr,h(s)=|s|m-2s, and f(u)=|s|p-2s, where m≥2,p>2,r>0, and b>0 are positive constants.This model describes a small amplitude vibration of an elastic string (see Kirchhoff [1]) and derives from the description of the vibrations of thin rods whose density depends on the velocity ut(see, e.g., [2]). The motivation of our work is the results regarding viscoelastic equations of Kirchhoff type by Zhang et al. [3]. They study the global existence and general decay of the energy for solutions to a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density(4)utρutt-M∇u2Δu+αΔ2u-βΔutt-γΔut+∫0tgt-sΔusds+axutm-2ut+up-2u=0.They also give a detailed review of literatures. However, they do not take the global nonexistence of the solution to (4) with nonlinear source into consideration. For more information about this, readers can refer to [3–18] and references therein. It is well known that the nonlinear source term |u|p-2u causes global nonexistence of solutions when either the condition γ=0 or h(ut)=0 holds in (1)(see [5–15] and references cited therein). Liu and Wang [19] consider the initial boundary value problem of (5)utρutt-Δut-div⁡ax∇uα-2∇u-div⁡∇utβ-2∇ut+Qut=fuand they establish a blow up result for certain solutions with nonpositive initial energy as well as positive initial energy. Said-Houari [20] studies the following system of nonlinear viscoelastic wave equations:(6)utjutt-Δu-Δutt+∫0tgt-sΔusds+utmut=f1u,v,vtjvtt-Δv-Δvtt+∫0tht-sΔvsds+vtrvt=f2u,v.They prove that the energy of the system will grow up as an exponential function as time goes to infinity, provided that the initial data are large enough. They do not get the blow-up result. Tahamtani and Pwyravi [6] study the following system:(7)utρutt+Δ2u-Δutt+∫0tg1t-sΔ2usds-Δut+utmut=f1u,v,vtρvtt+Δ2v-Δvtt+∫0tg2t-sΔ2vsds-Δvt+vtrvt=f2u,v,and they prove uniform decay of solution energy under some restrictions on the initial data and the relaxation functions. Moreover, they establish a growth result for certain solutions with positive initial energy, but they do not get the blow-up result. Therefore, a problem of whether the solution of problem (1)-(3) still blows up in finite time for appropriately bounded initial energy when introducing both presences of nonlinear weak damping term h(ut)=utm-2ut and linear strong damping term Δut(i.e., γ>0) arises. For the following special case, Gazzola and Squassina [21] consider the initial boundary problem:(8)utt-ωΔut-Δu+μut=up-2uand obtain the polynomial decay and finite time blow-up result under certain initial values. Later, Gerbi and Houari [22] obtain the exponential decay based on a small perturbation of energy. Chen and Liu [23] extend the linear damping term to the nonlinear damping term |ut|m-2ut and obtain the energy decay rate and the exponential growth. Similarly, [24] is also under the dynamic boundary conditions. From the physics points of view, the strong damping term Δut and the nonlinear dissipative damping term h(ut) play a dissipative or inhibitive part in the energy accumulation in the configurations. But the nonlinear source term f(u) leads to energy gathering in the configurations. If the energy accumulation arising from the nonlinear source term and other nonlinear factors cannot be dissipated synchronously, the remaining energy accumulation may cause the configurations to break or burn out in finite time; i.e., the solutions of problem (1)-(3) blow up in finite time. However introducing both presences of nonlinear weak damping term h(ut) and linear strong damping term Δut makes the problem interesting but difficult. Indeed, a strong impact of dissipative terms could make the existence of global solutions easier since they play the role of stabilizing terms and their smoothing effect makes the blow-up more difficult [11, 17, 25]. The most frequently used technique "concavity argument" in the proof of blow-up is no longer effective, and the technique in papers mentioned above cannot be used directly here either. At present only a few results are known for the interaction between the weak damping term h(ut) and the strong damping term Δut.The main purpose of this paper is to investigate the nonexistence result of global solutions for problem (1)-(3) with both terms Δut and h(ut). More precisely, we shall show global nonexistence results for problem (1)-(3), and we should overcome the difficulties brought by the treatment of the nonlinear terms |ut|ρutt,M(∇u2),h(ut),f(u) and interaction among the damping term -γΔut,h(ut), memory term ∫0tgt-sΔusds, and the source term f(u). The outline of this article is as follows. In Section 2, we introduce some notations, assumptions, and preliminaries. In Section 3, we show the main results of this article. For simplicity, we assume α=β=γ=1.


## 2. Preliminaries

In this section, we give some assumptions and known results in order to state the main results of this article. Throughout this article, the following notations are used for precise statements:Lp(Ω)(1<p<∞) denotes the usual space of all Lp-functions on Ω with norm uLp(Ω)=up and inner product (u,v)=∫Ωuvdx. For simplicity, we denote uL2(Ω)=u and we take α=β=γ=1. The constant C used throughout this paper is positive generic constants, which may vary in different situation. We also denote that(9)k=∫0tgsds,g∗∇ut=∫0tgt-s∇ut-∇us2ds.First, we present the following assumptions:(A1)M(s)=1+bsr, b>0,r>0, for all s≥0.(A2)g:R+→R+ is a C1 function satisfying(10)g0=0,-ξ1gs≤g′s≤-ξ2gs,a-∫0∞gsds=l>0,and here ξ1 and ξ2 are positive constants.(A3)0<ρ<n/n-2 if n≥3, and ρ>0 if n=1,2.(A4)p>max⁡2r+1,m,ρ+2 and 1<m<p≤2(n-1)/n-2 if n≥3, and 1<m<p≤∞ if n=1,2.Next, we present the following local existence theorem that can be found in [3](see also [26]).Theorem 1 (see [3]).
Supposing that (A1)-(A4) hold and thatu0∈H4∩H01, and u1∈H02, then problem (1)-(3) admits a unique solution(11)u∈L∞0,T;H4∩H02;ut∈L∞0,T;H02;utt∈L20,T;H01.Now, we define the energy of the problem (1)-(3) by(12)Et=Eu=1ρ+2utρ+2ρ+2+12∇ut2+Ju,where(13)Ju=12Δu2+121-k∇u2+12g∗∇ut+b2r+1∇u2r+1-1pupp.Similar to the proof of Lemma 4.1 in [12] and the proof of Lemma 3.1 in [20], multiplying the first equation of (1) by ut, integrating over Ω, and using integrating by parts, we have the following results.Lemma 2.
E ( t ) is a nonincreasing function on [0,∞) and(14)E′t=-∇ut2-utmm+12g′∗∇ut-12gt∇u2≤0.
From (A2) and Poincare inequality, we get(15)Et≥12Δu2+l∇u2+br+1∇u2r+1+g∗∇ut-Bplp/2p∇up>Gλ,for t≥0, where B=c∗/l, c∗ is the Poincare constant, and(16)Gλ=12λ2-Bppλp,λ=Δu2+l∇u2+br+1∇u2r+1+g∗∇ut1/2.It is easy to verify that G(λ) has a maximum at λ1=B-p/p-2 and the maximum value is E1=1/2-1/pλ12. Before we prove the main result, we need the following lemma, which is similar to the proof of Lemma 5.1 in [12], the proof of Lemma 3.2 in [15], and the proof of Lemma 3.2 in [20].Lemma 3.
Assume that (A1)-(A4) hold,u0∈H4∩H01,u1∈H02, and let u be a solution of problem (1)-(3) with initial data satisfying E(0)<E1 and Δu02+l∇u02+b/r+1∇u02(r+1)≥λ12. Then there exists a constant λ2>λ1, such that(17)Δu2+l∇u2+br+1∇u2r+1+g∗∇ut>λ22,∀t∈0,T.


## 3. Main Results

In this section, we prove blow-up result for the solution of problem (1)-(3) with positive but appropriately bounded and negative initial energy, respectively. In order to state our main result, we make an extra assumption on g:(18)∫0∞gsds<min⁡2p-22p-3,2lpE1-E02p-3λ12.Theorem 4.
Letr<min⁡{1-l/4l,p-2/p}. Assuming that (A1)-(A4) and (18) hold, u0∈H4∩H01, and u1∈H02, then any solution u of problem (1)-(3) with initial data satisfying E(0)<E1 and Δu02+l∇u02+b/r+2∇u02(r+1)≥λ12 will blow up in finite time.Proof.
We set(19)Ht=E2-Et,fort≥0,where E2=E(0)+E1/2. From (14) and (19), we get(20)H′t=-E′t=∇ut2+utmm-12g′∗∇ut+12gt∇u2>0and then H(t) is an increasing function and(21)Ht≥H0=E2-E0>0.On the other hand, by Lemma 3, we have(22)Ht<E2-1ρ+2utρ+2ρ+2-12Δu2+1-k∇u2+br+1∇u2r+1+g∗∇ut+1pupp≤E1-12λ12+1pupp=-1pλ12+1pupp≤1pupp.Hence, combining (21) and (22) with the embedding H01↪Lp, we have(23)0<H0≤Ht≤1pupp≤Bpp∇up.We set(24)Gt=u,ut+12∇u2and then define(25)Lt=Hρ01-αt+ϵGt,where ρ0,α,ϵ>0 are small parameters to be chosen later. By the definition of the solution, we have(26)G′t=utρ+2ρ+2-Δu2-∇u2-b∇u2r+1+∫0tgt-s∇ut,∇usds-∫Ωutm-2utudx+upp=utρ+2ρ+2-Δu2-∇u2-b∇u2r+1+∫0tgt-s∇ut2ds+∫0tgt-s∇ut,∇us-∇utds-∫Ωutm-2utudx+upp.Using Schwarz’s inequality and Young’s inequality, (26) takes on the form(27)G′t≥utρ+2ρ+2-Δu2-∇u2-b∇u2r+1+∫0tgt-s∇ut2ds-∫Ωutm-2utudx-∫0tgt-s∇ut∇us-∇utds+upp≥utρ+2ρ+2-Δu2-1-k∇u2-b∇u2r+1-∫Ωutm-2utudx-∫0tgt-s∇ut∇us-∇utds+upp≥utρ+2ρ+2-Δu2-1-k∇u2-b∇u2r+1-∫Ωutm-2utudx-14k∇u2-g∗∇ut+upp.Adding the term p(H(t)-E2+E(t)) and using the definition of E(t), then (27) becomes(28)G′t≥utρ+2ρ+2-Δu2-1-k∇u2-b∇u2r+1-∫Ωutm-2utudx-14k∇u2-g∗∇ut+p2utρ+2ρ+2+Δu2+a-k∇u2+br+1∇u2r+1+g∗∇ut+pHt-pE2=p+22utρ+2ρ+2+p-22Δu2+p2-11-k-14k∇u2+pb2r+1-b∇u2r+1+p2-1g∗∇ut-∫Ωutm-2utudx+pHt-pE2.We denote a1=p+2/2,a2=p-2/2,a3=1/l[p-2/2-2p-3/4∫0∞g(s)ds]<1/l[(p/2-1)(1-k)-1/4k], and a4=p-2(r+1)/2=(pb/2(r+1)-b)r+1/b. By p>2 and (18), we observe that a2>0,a3>0, and a4>0. Due to the restriction on r, we have a3=min{a2,a3,a4}>0, and then (28) becomes(29)G′t≥a1utρ+2ρ+2+a3Δu2+l∇ut2+br+1∇u2r+1+g∗∇ut-∫Ωutm-2utudx+pHt-pE2.Using the fact that Δu2+l∇u2+b/r+1∇u2(r+1)+(g∗∇u)(t)>λ22 by Lemma 3, we get(30)a3Δu2+l∇ut2+br+1∇u2r+1+g∗∇ut-pE2=a3λ22-λ12λ22Δu2+l∇u2+br+1∇u2r+1+g∗∇ut+a3λ12Δu2+l∇u2+b/r+2∇u2r+1+g∗∇utλ22-pE2≥a3λ22-λ12λ22Δu2+l∇u2+br+1∇u2r+1+g∗∇ut+a3λ12-pE2≥c1Δu2+l∇u2+br+1∇u2r+1+g∗∇ut+c2,where c1=a3λ22-λ12/λ22 and c2=a3λ12-pE2. Moreover, by the assumption of p, Lemma 3, 0<l<1, the definition of E2, and (18), we see that c1>0 and(31)c2=a3λ12-pE2=1lp-22-2p-34∫0∞gsdsλ12-pE2=p-22l-2p-34l∫0∞gsdsλ12-pE2≥p-22λ12-pE2-2p-34l∫0∞gsdsλ12=pE1-E02-2p-34l∫0∞gsdsλ12>0.Thus, (29) yields (32)G′t≥a1utρ+2ρ+2+c1Δu2+l∇u2+br+1∇u2r+1+g∗∇ut-∫Ωutm-2utudx+pHt.By Hölder inequality and (23) we have(33)∫Ωutm-2utudt≤∫Ωutm-1udx≤umutmm-1≤Cupp/mup1-p/mutmm-1≤Cupp/mH1/p-1/mtutmm-1.Considering (23), Young’s inequality, and the fact that utmm≤H′(t), we get(34)∫Ωutm-2utudt≤Cϵ1mupp+ϵ1-m/m-1H′tH-α1t,where α1=1/m-1/p>0,ϵ1>0. Now, we make α and ρ0 satisfy(35)0<α<min⁡α1,12-1p,p-mpm-1,max⁡ρρ+2,1-α1,1r+1,1ρ+2<ρ01-α<1,and then we have(36)1-ρ01-α-α1<0,1<1ρ01-α<r+1,ρ01-α>1ρ+2.Furthermore, from (34) and (23), we have(37)∫Ωutm-2utudt≤Cϵ1mH-α10upp+ϵ1-m/m-1H1-ρ01-α-α10Hρ01-α-1tH′t.By differentiating (25), from (32) and (37), we get(38)L′t≥ρ01-α-ϵCϵ1-m/m-1H1-ρ01-α-α10Hρ01-α-1tH′t+ϵa1utρ+2ρ+2+pHt+ϵc1Δu2+l∇u2+br+1∇u2r+1+g∗∇ut-ϵCϵ1mH-α10upp.Letting a=1/2min⁡{a1,c1,p/2}, decomposing ϵpH(t) in (38) by ϵpH(t)=2aϵH(t)+p-2aϵH(t), and noting(39)Ht≥1pupp-1ρ+2utρ+2ρ+2-12Δu2+1-k∇u2+br+1∇u2r+1+g∗∇ut, from (23) and (38), we find that(40)L′t≥ρ01-α-ϵCϵ1-m/m-1H1-ρ01-α-α10Hρ01-α-1tH′t+ϵ2ap-Cϵ1mH-α10upp+ϵa1-aρ+2utρ+2ρ+2+ϵc1-aΔu2+l∇u2+br+1∇u2r+1+g∗∇ut+p-2aϵHt.Choosing ϵ1>0 small enough so that ϵ1m<a/pCHα1(0) and 0<ϵ<ρ0(1-α)/2CH-(1-ρ0(1-α)-α1)(0)ϵ1m/m-1, from (40), we have(41)L′t≥Cϵupp+utρ+2ρ+2+Δu2+∇u2+∇u2r+1+g∗∇ut+Ht.Therefore, L(t) is a nondecreasing function for t≥0. Letting ϵ in (25) small enough, we get L(0)>0. Consequently, we obtain L(t)≥L(0)>0 for t≥0.
We claim the inequality(42)L′t≥CLt1/ρ01-α.For the proof of (42), we consider two alternatives:
(i) If there exists at>0 so that G(t)<0, then(43)Lt1/ρ01-α=Hρ01-αt+ϵGt1/ρ01-α≤Ht.Thus (42) follows (43).
(ii) If there exists at>0 so that G(t)≥0, since 1<1/ρ0(1-α)<1+r by (36), then we deduce from (25), Young inequality, Hölder inequality, and the embedding Lp↪L2 that(44)Lt1/ρ01-α≤CHt+upτ/ρ01-α+uts/ρ01-α+∇u2/ρ01-αfor 1/τ+1/s=1,τ>0,s>0. We take s=(ρ+2)ρ0(1-α), by the restrictions on α in (35) and (36), then(45)s>1,τρ01-α=ρ+2ρ+2ρ01-α-1<ρ+2<p,2ρ01-α<2r+1, and(46)uts/ρ01-α=utρ+2<Cutρ+2ρ+2,upτ/ρ01-α=up2/2ρ01-α-1.Thus from (44), (46), and (23), we have(47)Lt1/ρ01-α≤CHt+utρ+2ρ+2+up2/2ρ01-α-1-pupp+∇u2/ρ01-α-2r+1∇u2r+1≤CHt+utρ+2ρ+2+pH01/p2/2ρ01-α-1-pupp+pBpH01/p2/ρ01-α-2r+1∇u2r+1≤CHt+utρ+2ρ+2+upp+∇u2r+1.This inequality together with (41) implies (42).
Then, by integrating both sides of (42) over [0,t], it follows that there exists a T0>0 such that(48)limt→T0-⁡Lt=limt→T0-⁡Hρ01-αt+ϵGt=∞.This limit combining with (47), (43), and (23) gives (49)limt→T0-upp+Δu2+∇u2+∇u2r+1+utρ+2ρ+2=∞.This theorem is proved.Theorem 5.
Letr<min⁡{1-l/4l,p-2/p}. Assuming that (A1)-(A3), E(0)<0, and ∫0∞g(s)ds<2(p-2)/2p-3 hold and u0,u1∈H2∩H01, then the local solution of problem (1)-(3) blows up in finite time.Proof.
SettingH(t)=-E(t) instead of H(t) in (19) and then applying the same arguments as that in Theorem 4, we get the desired result.



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