# General Decay and Blow-Up of Solutions for a System of Viscoelastic Equations of Kirchhoff Type with Strong Damping

Authors: Wenjun Liu; Gang Li; Linghui Hong
Journal: Journal of Function Spaces
 (2014)

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## Abstract

The general decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping is considered. We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy by exploiting the convexity technique, the other is for certain solutions with arbitrarily positive initial energy based on the method of Li and Tsai. Then, we give a decay result of global solutions by the perturbed energy method under a weaker assumption on the relaxation functions.


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## Body

## 1. Introduction

In this work, we investigate the following system of viscoelastic equations of Kirchhoff type:(1)utt-M(∥∇u∥22)Δu+∫0t‍g1(t-s)Δu(s)ds-Δut=f1(u,v),(x,t)∈Ω×(0,∞),vtt-M(∥∇v∥22)Δv+∫0t‍g2(t-s)Δv(s)ds-Δvt=f2(u,v),(x,t)∈Ω×(0,∞),u(x,t)=0,v(x,t)=0,(x,t)∈∂Ω×[0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,v(x,0)=v0(x),vt(x,0)=v1(x),x∈Ω,
whereΩ⊂ℝn,n≥1, is a bounded domain with smooth boundary∂Ω,Mis a positive locally Lipschitz function, andgi(·):ℝ+→ℝ+(i=1,2),(f1,f2):ℝ2→ℝ2are given functions to be specified later.To motivate our work, let us recall some previous results regarding viscoelastic equations of Kirchhoff type. The following problem:(2)utt-M(∥∇u∥22)Δu+∫0t‍g(t-s)Δu(s)ds+h(ut)=f(u),(x,t)∈Ω×(0,∞),u=0,(x,t)∈∂Ω×[0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω
is a model to describe the motion of deformable solids as hereditary effect is incorporated. It was first studied by Torrejón and Yong [1] who proved the existence of a weakly asymptotic stable solution for large analytical datum. Later, Muñoz Rivera [2] showed the existence of global solutions for small datum and the total energy decays to zero exponentially under some restrictions. Then, Wu and Tsai [3] treated problem (2) forh(ut)=-Δut and proved the global existence, decay, and blow-up with suitable conditions on initial data. They obtained the blow-up properties of local solution with small positive initial energy by the direct method of [4]. To obtain the decay result, they assumed that the nonnegative kernelg′(t)≤-rg(t),∀t≥0for somer>0.This energy decay result was recently improved by Wu in [5] under a weaker condition on g (i.e.,g′(t)≤0for t≥0). For a single wave equation of Kirchhoff type without the viscoelastic term, we refer the reader to Matsuyama and Ikehata [6] and Ono [7–10].Many results concerning local existence, global existence, decay, and blow-up of solutions for a system of wave equations of Kirchhoff type without viscoelastic terms (i.e.,gi=0,i=1,2) have also been extensively studied. For example, Park and Bae [11] considered the system of wave equations with nonlinear dampings forf1(u,v)=μ|u|q-1uandf2(u,v)=μ|v|q-1v,q≥1, and showed the global existence and asymptotic behavior of solutions under some restrictions on the initial energy. Later, Benaissa and Messaoudi [12] discussed blow-up properties for negative initial energy. Recently, Wu and Tsai [13] studied the system (1) forgi=0(i=1,2). Under some suitable assumptions onfi(i=1,2), they proved local existence of solutions by applying the Banach fixed point theorem and the blow-up of solutions by using the method of Li and Tsai in [4], where three different cases on the sign of the initial energyE(0)are considered.In the case ofM≡1and in the presence of viscoelastic term (i.e.,g≠0), Cavalcanti et al. [14] studied the equation that was subject to a locally distributed dissipation
(3)utt-Δu+∫0t‍g(t-s)Δu(s)ds+a(x)ut+|u|γu=0,(x,t)∈Ω×(0,∞),
with the same initial and boundary conditions as that of (2), and proved an exponential decay rate. This work extended the result of Zuazua [15], in which he considered (3) with g=0and the localized linear damping. By using the piecewise multipliers method, Cavalcanti and Oquendo [16] investigated the equation
(4)utt-k0Δu+∫0t‍div[a(x)g(t-s)∇u(s)]ds+b(x)h(ut)+f(u)=0,(x,t)∈Ω×(0,∞),
with the same initial and boundary conditions as that of (2). Under the similar conditions on the relaxation functiongas above, and a(x)+b(x)≥δ>0 for all x∈Ω, they improved the results of [14] by establishing exponential stability for exponential decay functiongand linear functionh, and polynomial stability for polynomial decay functiongand nonlinear functionh, respectively.Concerning blow-up results, Messaoudi [17] considered the equation
(5)utt-Δu+∫0t‍g(t-s)Δu(s)ds+aut|ut|m-2=b|u|r-2u,(x,t)∈Ω×(0,∞).
He proved that any weak solution with negative initial energy blows up in finite time ifr>mand
(6)∫0∞‍g(s)ds≤r-2r-2+1/r,
while exists globally for any initial data in the appropriate space if m≥r.This result was improved by the same author in [18] for positive initial energy under suitable conditions on g,m, andr. Recently, Liu [19] studied the equation
(7)utt-Δu+∫0t‍g(t-s)Δu(s)ds-ωΔut+μut=|u|r-2u,(x,t)∈Ω×(0,∞),
with the same initial and boundary condition as that of (2). By virtue of convexity technique and supposing that
(8)∫0∞‍g(s)ds≤r-2r-2+1/[(1-δ^)2r+2δ(1-δ^)],
whereδ^=max{0,δ}, he proved that the solution with nonpositive initial energy as well as positive initial energy blows up in finite time.We should mention that the following system:(9)utt-Δu+∫0t‍g1(t-τ)Δu(τ)dτ+|ut|m-1ut=f1(u,v),(x,t)∈Ω×(0,T),vtt-Δv+∫0t‍g2(t-τ)Δv(τ)dτ+|vt|γ-1vt=f2(u,v),(x,t)∈Ω×(0,T),u(x,t)=0,v(x,t)=0,(x,t)∈∂Ω×[0,T),u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,v(x,0)=v0(x),vt(x,0)=v1(x),x∈Ω
was considered by Han and Wang in [20], where Ω is a bounded domain with smooth boundary ∂Ω in ℝn,n=1,2,3. Under suitable assumptions on the functions gi,fi(i=1,2), the initial data and the parameters in the above problem established local existence, global existence, and blow-up property (the initial energyE(0)<0). This latter blow-up result has been improved by Messaoudi and Said-Houari [21] into certain solutions with positive initial energy. Recently, Liang and Gao in [22] investigated the following problem:
(10)utt-Δu+∫0t‍g1(t-τ)Δu(τ)dτ-Δut=f1(u,v),(x,t)∈Ω×(0,T),vtt-Δv+∫0t‍g2(t-τ)Δv(τ)dτ-Δvt=f2(u,v),(x,t)∈Ω×(0,T),
with the same initial and boundary conditions as that of (9). Under suitable assumptions on the functionsgi,fi(i=1,2)and certain initial data in the stable set, they proved that the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, they proved that there are solutions with positive initial energy that blow up in finite time. It is also worth mentioning the work [23] in which we studied system (1). Under suitable assumptions on the functionsgi,fi(i=1,2)and certain initial conditions, we showed that the solutions are global in time and the energy decays exponentially. For other papers related to existence, uniform decay, and blow-up of solutions of nonlinear wave equations, we refer the reader to [14, 24–29] for existence and uniform decay, to [17, 30–34] for blow-up, and to [35–40] for the coupled system. To the best of our knowledge, the general decay and blow-up of solutions for systems of viscoelastic equations of Kirchhoff type with strong damping have not been well studied.Motivated by the above mentioned research, we consider in the present work the coupled system (1) with nonzerogi(i=1,2)and nonconstantM(s). We note that in such a coupled system case we should overcome the additional difficulties brought by the treatment of the nonlinear coupled terms. We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy, the other is for certain solutions with arbitrarily positive initial energy. Then, we give a decay result of global solutions under a weaker assumption on the relaxation functionsgi(t)(i=1,2).This paper is organized as follows. In the next section we present some assumptions, notations and known results and state the main results: Theorems4, 5, 6, and 7. The two blow-up results, Theorems 5 and 6, are proved in Sections 3 and 4, respectively. Section 5 is devoted to the proof of the decay result—Theorem 7.


## 2. Preliminaries and Main Result

In this section we present some assumptions, notations, and known results and state the main results. First, we make the following assumptions.(A1)
M(s)is a positive locally Lipschitz function fors≥0with the Lipschitz constantLsatisfying
(11)M(s)≥m0>0.(A2)
gi(t):[0,∞)→(0,∞),(i=1,2)are strictly decreasingC1functions such that
(12)m0-∫0∞‍gi(s)ds=li>0.(A3)
There exist two positive differentiable functionsξ1(t) and ξ2(t)such that
(13)gi′(t)≤-ξi(t)gi(t),(i=1,2),fort≥0,andξi(t)satisfies
(14)ξi′(t)≤0,∀t>0,∫0+∞‍ξi(t)dt=∞,(i=1,2).(A4)
We make the following extra assumption onM:
(15)2(p+2)M¯(s)-2M(s)s≥2(p+1)m0s,∀s≥0,whereM¯(s)=∫0s‍M(τ)dτ.Remark 1.
It is clear thatM(s)=m0+m1sγ(which occurs physically in the study of vibrations of damped flexible space structures in a bounded domain inℝn) satisfies (A4) fors≥0,m0>0,m1≥0,γ≥0 as long as p+1>γ.Indeed, by straightforward calculations, we obtain
(16)2(p+2)M¯(s)-2M(s)s=2(p+2)(m0s+m1γ+1sγ+1)-2m0s-2m1sγ+1=2(p+1)m0s+2(p+1-γ)γ+1m1sγ+1≥2(p+1)m0s.Next, we introduce some notations. Consider(17)(gi⋄∇w)(t)=∫0t‍gi(t-s)∥∇w(t)-∇w(s)∥22ds,l=min{l1,l2},g0(t)=max{g1(t),g2(t)},∀t≥0,∥·∥q=∥·∥Lq(Ω),1≤q≤∞,
the Hilbert spaceL2(Ω)endowed with the inner product
(18)(u,v)=∫Ω‍u(x)v(x)dx,
and the functionsf1(u,v)andf2(u,v)(see also [21]):
(19)f1(u,v)=[a|u+v|2(p+1)(u+v)+b|u|pu|v|p+2],f2(u,v)=[a|u+v|2(p+1)(u+v)+b|u|p+2|v|pv],
wherea,b>0are constants andpsatisfies
(20)-1<p,ifn=1,2,-1<p≤(3-n)(n-2),ifn≥3.One can easily verify that(21)uf1(u,v)+vf2(u,v)=2(p+2)F(u,v),∀(u,v)∈ℝ2,
where
(22)F(u,v)=12(p+2)[a|u+v|2(p+2)+2b|uv|p+2].Then, we give two lemmas which will be used throughout this work.Lemma 2 (Sobolev-Poincaré inequality [41]).
If2≤ρ≤2n/(n-2), then
(23)∥u∥ρ≤C∥∇u∥2foru∈H01(Ω)
holds with some constantC.Lemma 3 ([21, Lemma 3.2]).
Assume that (20) holds. Then there existsη1>0 such that for any(u,v)∈(H01(Ω)∩H2(Ω))×(H01(Ω)∩H2(Ω)), one has
(24)a∥u+v∥2(p+2)2(p+2)+2b∥uv∥p+2p+2≤η1(l1∥∇u∥22+l2∥∇v∥22)p+2.We now state a local existence theorem for system (1), whose proof follows the arguments in [3, 13].Theorem 4.
Suppose that (20), (A1), and (A2)hold, and thatu0,v0∈H01(Ω)∩H2(Ω) and u1,v1∈L2(Ω).Then problem (1) has a unique local solution
(25)u,v∈C([0,T);H01(Ω)∩H2(Ω)),ut,vt∈C([0,T);L2(Ω))∩L2([0,T);H01(Ω)),
for someT>0.Moreover, at least one of the following statements is valid:
(26)(1)T=∞,(2)limt→T-(∥ut∥22+∥vt∥22+∥Δu∥22+∥Δv∥22)=∞.The energy associated with system (1) is given by
(27)E(t)=12(∥ut∥22+∥vt∥22)-12(∫0t‍g1(s)ds)∥∇u∥22-12(∫0t‍g2(s)ds)∥∇v∥22-∫Ω‍F(u,v)dx+12[(g1⋄∇u)(t)+(g2⋄∇v)(t)]+12M¯(∥∇u∥22)+12M¯(∥∇v∥22),fort≥0.
As in [5], we can get
(28)E′(t)=-(∥∇ut∥22+∥∇vt∥22)-12[g1(t)∥∇u∥22+g2(t)∥∇v∥22]+12[(g1′⋄∇u)(t)+(g2′⋄∇v)(t)]≤0,∀t≥0.
Then we have
(29)E(t)=E(0)-∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ-12∫0t‍[g1(τ)∥∇u(τ)∥22+g2(τ)∥∇v(τ)∥22]dτ+12∫0t‍[(g1′⋄∇u)(τ)+(g2′⋄∇v)(τ)]dτ.We introduce(30)B1=η11/2(p+2),α*=B1-(p+2)/(p+1),E1=(12-12(p+2))α*2,
where η1 is the optimal constant in (24).Our first result is concerned with the blow-up for certain local solutions with nonpositive initial energy as well as positive initial energy.Theorem 5.
Assume that(A1)-(A2),(A4), and (20) hold. Let(u,v)be the unique local solution to system (1). For any fixedδ<1, assuming that we can choose(u0,v0),(u1,v1)satisfy
(31)E(0)<δE1,(l1∥∇u0∥22+l2∥∇v0∥22)1/2>α*.
Suppose further that
(32)∫0∞‍gi(s)ds≤2(p+1)m02(p+1)+1/{2[(1-δ^)2(p+2)+δ^(1-δ^)]},(i=1,2),
where δ^=max{0,δ}, then T<∞.Our second result shows that certain local solutions with arbitrarily positive initial energy can also blow up.Theorem 6.
Suppose that (A1)-(A2), (A4), (20), and
(33)∫0∞‍gi(s)ds≤2(p+1)m02(p+1)+1/2(p+2),(i=1,2),
hold. Assume further thatu0,v0∈H01(Ω)∩H2(Ω) and u1,v1∈L2(Ω) and satisfy
(34)0<E(0)<min{(∫Ω‍u0u1dx+∫Ω‍v0v1dx)22[∥u0∥22+∥v0∥22+T1(∥∇u0∥22+∥∇v0∥22)],p+3(p+2)(p+3-p+1)∫Ω‍(u0u1+v0v1)dx-p+32(p+2)(∥u0∥22+∥v0∥22+∥∇u0∥22+∥∇v0∥22)(∫Ω‍u0u1dx+∫Ω‍v0v1dx)22[∥u0∥22+∥v0∥22+T1(∥∇u0∥22+∥∇v0∥22)]},
then the solution of problem (1) blows up at a finite timeT*in the sense of (119) below. Moreover, the upper bounds forT*can be estimated by
(35)T*≤2(3p+5)/2(p+1)(p+1)c2a1{1-[1+ch(0)]-1/(p+1)},
whereh(0)=[∥u0∥22+∥v0∥22+T1(∥∇u0∥22+∥∇v0∥22)]-(p+1)/2anda1is given in (116) below.Finally, we state the general decay result. For convenience, we choose especiallyM(s)=m0+m1sγ,m0>0,m1≥0, and γ≥0. Then the energy functional E(t), defined by (27), becomes
(36)E(t)∶=E(u(t),v(t))=12(∥ut∥22+∥vt∥22)+12[(m0-∫0t‍g1(s)ds)∥∇u∥22+(m0-∫0t‍g2(s)ds)∥∇v∥22]+12[m1γ+1(∥∇u∥22(γ+1)+∥∇v∥22(γ+1))(g1⋄∇u)(t)+(g2⋄∇v)(t)+m1γ+1(∥∇u∥22(γ+1)+∥∇v∥22(γ+1))]-∫Ω‍F(u,v)dx.Theorem 7.
Suppose that (20),(A1)-(A2), and (A3) hold, and that (u0,u1)∈(H01(Ω)∩H2(Ω))×L2(Ω) and (v0,v1)∈(H01(Ω)∩H2(Ω))×L2(Ω) and satisfy E(0)<E1 and
(37)(l1∥∇u0∥22+l2∥∇v0∥22)1/2<α*.
Then for eacht0>0, there exist two positive constantsKandksuch that the energy of (1) satisfies
(38)E(t)≤Ke-k∫t0t‍ξ(s)ds,t≥t0,
whereξ(t)∶=min{ξ1(t),ξ2(t)}.To achieve general decay result we will use a Lyapunov type technique for some perturbation energy following the method introduced in [42]. This result improves the one in Li et al. [23] in which only the exponential decay rates are considered.


## 3. Blow-Up of Solutions with Initial Data in the Unstable Set

In this section, we prove a finite time blow-up result for initial data in the unstable set. We need the following lemmas.Lemma 8.
Suppose that (20),(A1),(A2), and(A4)hold. Let(u,v)be the solution of system (1). Assume further thatE(0)<E1and
(39)(l1∥∇u0∥22+l2∥∇v0∥22)1/2>α*.
Then there exists a constantα3>α*such that
(40)(l1∥∇u(t)∥22+l2∥∇v(t)∥22)1/2≥α3,fort∈[0,T).Proof.
We first note that, by (27), (24) and the definition ofB1, we have
(41)E(t)≥12(m0-∫0t‍g1(s)ds)∥∇u(t)∥22+12(m0-∫0t‍g2(s)ds)∥∇v(t)∥22-12(p+2)(a∥u+v∥2(p+2)2(p+2)+2b∥uv∥p+2p+2)≥12[l1∥∇u(t)∥22+l2∥∇v(t)∥22]-12(p+2)(a∥u+v∥2(p+2)2(p+2)+2b∥uv∥p+2p+2)≥12[l1∥∇u(t)∥22+l2∥∇v(t)∥22]-B12(p+2)2(p+2)(l1∥∇u(t)∥22+l2∥∇v(t)∥22)p+2=12α2-B12(p+2)2(p+2)α2(p+2)∶=G(α),
where we have usedM¯(s)=∫0s‍M(τ)dτ≥m0s≥0and
(42)α=(l1∥∇u(t)∥22+l2∥∇v(t)∥22)1/2.
It is easy to verify that G(α) is increasing in (0,α*), decreasing in (α*,∞), and that G(α)→-∞, as α→∞, and
(43)G(α)max=G(α*)=12α*2-B12(p+2)2(p+2)α*2(p+2)=E1,
whereα*is given in (30). SinceE(0)<E1, there exists α3>α* such that G(α3)=E(0). Set α0=(l1∥∇u0∥22+l2∥∇v0∥22)1/2, then from (41) we can get G(α0)≤E(0)=G(α3), which implies that α0≥α3.
Now we establish (40) by contradiction. First we assume that (40) is not true over[0,T), then there existst0∈(0,T)such that
(44)(l1∥∇u(t0)∥22+l2∥∇v(t0)∥22)1/2<α3.
By the continuity ofl1∥∇u(t)∥22+l2∥∇v(t)∥22we can chooset0such that
(45)(l1∥∇u(t0)∥22+l2∥∇v(t0)∥22)1/2>α*.
Again, the use of (41) leads to
(46)E(t0)≥G[(l1∥∇u(t0)∥22+l2∥∇u(t0)∥22)1/2]>G(α3)=E(0).
This is impossible sinceE(t)≤E(0)for all t∈[0,T).
Hence (40) is established.Lemma 9 (see [43, 44]).
LetL(t) be a positive, twice differentiable function, which satisfies, for t>0, the inequality
(47)L(t)L′′(t)-(1+σ)L′(t)2≥0
for some σ>0. If L(0)>0 and L′(0)>0, then there exists a time T*≤L(0)/[σL′(0)] such that limt→T*-L(t)=∞.Proof of Theorem5.
Assume by contradiction that the solution(u,v)is global. Then we considerL(t):[0,T]→ℝ+ defined by
(48)L(t)=∥u(t)∥22+∥v(t)∥22+∫0t‍∥∇u(τ)∥22dτ+∫0t‍∥∇v(τ)∥22dτ+(T-t)(∥∇u0∥22+∥∇v0∥22)+β(t+T0)2,
where T,  β, and T0 are positive constants to be chosen later. ThenL(t)>0 for all t∈[0,T]. Furthermore
(49)L′(t)=2∫Ω‍u(t)ut(t)dx+2∫Ω‍v(t)vt(t)dx+∥∇u(t)∥22+∥∇v(t)∥22-(∥∇u0∥22+∥∇v0∥22)+2β(t+T0)=2∫Ω‍u(t)ut(t)dx+2∫Ω‍v(t)vt(t)dx+2∫0t‍(∇u(τ),∇ut(τ))dτ+2∫0t‍(∇v(τ),∇vt(τ))dτ+2β(t+T0),
and, consequently,
(50)L′′(t)=2∫Ω‍u(t)utt(t)dx+2∫Ω‍v(t)vtt(t)dx+2∥ut(t)∥22+2∥vt(t)∥22+2(∇u(t),∇ut(t))+2(∇v(t),∇vt(t))+2β
for almost every t∈[0,T]. Testing the first equation of system (1) with u and the second equation of system (1) with v, integrating the results over Ω, using integration by parts, and summing up, we have
(51)(utt(t),u(t))+(vtt(t),v(t))+(∇u(t),∇ut(t))+(∇v(t),∇vt(t))=-M(∥∇u(t)∥22)∥∇u(t)∥22-M(∥∇v(t)∥22)∥∇v(t)∥22-∫Ω‍∫0t‍g1(t-s)Δu(s)u(t)dsdx-∫Ω‍∫0t‍g2(t-s)Δv(s)v(t)dsdx+2(p+2)∫Ω‍F(u,v)dx,
which implies
(52)L′′(t)=2∥ut(t)∥22+2∥vt(t)∥22-2M(∥∇u(t)∥22)∥∇u(t)∥22-2M(∥∇v(t)∥22)∥∇v(t)∥22+2β-2∫Ω‍∫0t‍g1(t-s)Δu(s)u(t)dsdx-2∫Ω‍∫0t‍g2(t-s)Δv(s)v(t)dsdx+4(p+2)∫Ω‍F(u,v)dx.
Therefore, we have
(53)L(t)L′′(t)-p+32L′(t)2=2L(t)(∥ut(t)∥22+∥vt(t)∥22-M(∥∇u(t)∥22)∥∇u(t)∥22-M(∥∇v(t)∥22)∥∇v(t)∥22+β)-2L(t)(∫Ω‍∫0t‍g1(t-s)Δu(s)u(t)dsdx+∫Ω‍∫0t‍g2(t-s)Δv(s)v(t)dsdx-2(p+2)∫Ω‍F(u,v)dx∫0t‍)-2(p+3)×[∫0t‍∫Ω‍u(t)ut(t)dx+∫Ω‍v(t)vt(t)dx+β(t+T0)+∫0t‍(∇u(τ),∇ut(τ))dτ+∫0t‍(∇v(τ),∇vt(τ))dτ]2=2L(t)(∥ut(t)∥22+∥vt(t)∥22-M(∥∇u(t)∥22)∥∇u(t)∥22-M(∥∇v(t)∥22)∥∇v(t)∥22+β)-2L(t)(∫Ω‍∫0t‍g1(t-s)Δu(s)u(t)dsdx+∫Ω‍∫0t‍g2(t-s)Δv(s)v(t)dsdx-2(p+2)∫Ω‍F(u,v)dx∫0t‍)+2(p+3){H(t)-[L(t)-(T-t)(∥∇u0∥22+∥∇v0∥22)]×Ψ(t)},
where Ψ(t),H(t):[0,T]→ℝ+ are the functions defined by
(54)Ψ(t)=∥ut(t)∥22+∥vt(t)∥22+∫0t‍∥∇ut(τ)∥22dτ+∫0t‍∥∇vt(τ)∥22dτ+β,H(t)=(∥u(t)∥22+∥v(t)∥22+∫0t‍∥∇u(τ)∥22dτ+∫0t‍∥∇v(τ)∥22dτ+β(t+T0)2)Ψ(t)-[∫0t‍∫Ω‍[u(t)ut(t)+v(t)vt(t)]dx+∫0t‍[(∇u(τ),∇ut(τ))+(∇v(τ),∇vt(τ))]dτ+β(t+T0)∫0t‍]2.
Using the Cauchy-Schwarz inequality, we obtain
(55)(∫Ω‍u(t)ut(t)dx)2≤∥u(t)∥22∥ut(t)∥22,(the  similar  inequality  forv(t)holdstrue)(∫0t‍(∇u(τ),∇ut(τ))dτ)2≤∫0t‍∥∇u(τ)∥22dτ∫0t‍∥∇ut(τ)∥22dτ(thesimilar  inequality  forv(t)),∫Ω‍u(t)ut(t)dx∫Ω‍v(t)vt(t)dx≤∥u(t)∥2∥vt(t)∥2∥ut(t)∥2∥v(t)∥2≤12∥u(t)∥22∥vt(t)∥22+12∥ut(t)∥22∥v(t)∥22.
Similarly, we have(56)∫Ω‍u(t)ut(t)dx∫0t‍(∇u(τ),∇ut(τ))dτ≤12∥u(t)∥22∫0t‍∥∇ut(τ)∥22dτ+12∥ut(t)∥22∫0t‍∥∇u(τ)∥22dτ,(the similar inequality forv(t)holds true)∫Ω‍u(t)ut(t)dx∫0t‍(∇v(τ),∇vt(τ))dτ≤12∥u(t)∥22∫0t‍∥∇vt(τ)∥22dτ+12∥ut(t)∥22∫0t‍∥∇v(τ)∥22dτ,∫Ω‍v(t)vt(t)dx∫0t‍(∇u(τ),∇ut(τ))dτ≤12∥v(t)∥22∫0t‍∥∇ut(τ)∥22dτ+12∥vt(t)∥22∫0t‍∥∇u(τ)∥22dτ,∫0t‍(∇u(τ),∇ut(τ))dτ∫0t‍(∇v(τ),∇vt(τ))dτ≤12∫0t‍∥∇u(τ)∥22dτ∫0t‍∥∇vt(τ)∥22dτ+12∫0t‍∥∇ut(τ)∥22dτ∫0t‍∥∇v(τ)∥22dτ.
By Ho¨lder’s inequality and Young’s inequality, we obtain
(57)β(t+T0)∫Ω‍u(t)ut(t)dx≤β(t+T0)∥u(t)∥2∥ut(t)∥2≤12β∥u(t)∥22+12β(t+T0)2∥ut(t)∥22.(the  similar  inequality  forv(t)holds  true)β(t+T0)∫0t‍(∇u(τ),∇ut(τ))dτ≤β(t+T0)∫0t‍∥∇u(τ)∥2∥∇ut(τ)∥2dτ≤12β∫0t‍∥∇u(τ)∥22dτ+12β(t+T0)2∫0t‍∥∇ut(τ)∥22dτ.(thesimilarinequalityforv(t)holdstrue).
The previous inequalities imply thatH(t)≥0for everyt∈[0,T].Using (53), we get
(58)L(t)L′′(t)-p+32L′(t)2≥L(t)Φ(t)
for almost every t∈[0,T], where Φ(t):[0,T]→ℝ+ is the map defined by
(59)Φ(t)=-2(p+2)(∥ut(t)∥22+∥vt(t)∥22)-2M(∥∇u(t)∥22)∥∇u(t)∥22-2M(∥∇v(t)∥22)∥∇v(t)∥22-2(∫Ω‍∫0t‍g1(t-s)Δu(s)u(t)dsdx+∫Ω‍∫0t‍g2(t-s)Δv(s)v(t)dsdx)-2(p+2)β-2(p+3)(∫0t‍∥∇ut(τ)∥22dτ+∫0t‍∥∇vt(τ)∥22dτ)+4(p+2)∫Ω‍F(u,v)dx.
For the fourth term on the right hand side of (59), we have
(60)-∫Ω‍∫0t‍g1(t-s)Δu(s)u(t)dsdx=∫0t‍g1(t-s)∫Ω‍∇u(s)·∇u(t)dxds=∫0t‍g1(t-s)∫Ω‍∇u(t)·(∇u(s)-∇u(t))dxds+(∫0t‍g1(s)ds)∥∇u(t)∥22.
Similarly,
(61)-∫Ω‍∫0t‍g2(t-s)Δv(s)v(t)dsdx=∫0t‍g2(t-s)∫Ω‍∇v(t)·(∇v(s)-∇v(t))dxds+(∫0t‍g2(s)ds)∥∇v(t)∥22.
Combining (59), (60) with (61), we get
(62)Φ(t)=-2(p+2)(∥ut(t)∥22+∥vt(t)∥22)-2M(∥∇u(t)∥22)∥∇u(t)∥22-2M(∥∇v(t)∥22)∥∇v(t)∥22+2∫0t‍g1(s)ds∥∇u(t)∥22+2∫0t‍g2(s)ds∥∇v(t)∥22+4(p+2)∫Ω‍F(u,v)dx-2(p+2)β+2∫0t‍g1(t-s)∫Ω‍∇u(t)(∇u(s)-∇u(t))dxds-2(p+3)∫0t‍∥∇ut(τ)∥22dτ+2∫0t‍g2(t-s)∫Ω‍∇v(t)(∇v(s)-∇v(t))dxds-2(p+3)∫0t‍∥∇vt(τ)∥22dτ.
Since, we have
(63)2∫0t‍gi(t-s)∫Ω‍∇u(t)·(∇u(s)-∇u(t))dxds≥-ɛ∫0t‍gi(t-s)∥∇u(s)-∇u(t)∥22ds-1ɛ(∫0t‍gi(s)ds)∥∇u(t)∥22=-ɛ(gi⋄∇u)(t)-1ɛ(∫0t‍gi(s)ds)∥∇u(t)∥22,i=1,2,
for any ɛ>0, inserting (63) into (62) and utilizing (27), we have
(64)Φ(t)≥-2(p+2)(∥ut(t)∥22+∥vt(t)∥22)-ɛ[(g1⋄∇u)(t)+(g2⋄∇v)(t)]+4(p+2)∫Ω‍F(u,v)dx+(2-1ɛ)×[∫0t‍g1(s)ds∥∇u(t)∥22+∫0t‍g2(s)ds∥∇v(t)∥22]-2(p+3)(∫0t‍∥∇ut(τ)∥22dτ+∫0t‍∥∇vt(τ)∥22dτ)-2M(∥∇u(t)∥22)∥∇u(t)∥22-2M(∥∇v(t)∥22)∥∇v(t)∥22-2(p+2)β-4(p+2)E(t)+4(p+2)E(t)=-4(p+2)E(t)+2(p+2)M¯(∥∇u(t)∥22)-2M(∥∇u(t)∥22)∥∇u(t)∥22+2(p+2)M¯(∥∇v(t)∥22)-2M(∥∇v(t)∥22)∥∇v(t)∥22-2(p+1+12ɛ)∫0t‍g1(s)ds∥∇u(t)∥22-2(p+2)β-2(p+1+12ɛ)∫0t‍g2(s)ds∥∇v(t)∥22-2(p+3)∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ+[2(p+2)-ɛ][(g1⋄∇u)(t)+(g2⋄∇v)(t)].
Using (29) and (A4), we have
(65)Φ(t)≥-4(p+2)E(0)+2(p+1)×(m0-∫0t‍g1(s)ds)∥∇u(t)∥22-1ɛ(∫0t‍g1(s)ds)∥∇u(t)∥22+2(p+1)(m0-∫0t‍g2(s)ds)∥∇v(t)∥22-1ɛ(∫0t‍g2(s)ds)∥∇v(t)∥22-2(p+2)β+2(p+1)∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ+[2(p+2)-ɛ][(g1⋄∇u)(t)+(g2⋄∇v)(t)]=-4(p+2)E(0)+[2(p+1)m0-(2(p+1)+1ɛ)∫0t‍g1(s)ds]×∥∇u(t)∥22-2(p+2)β+[2(p+1)m0-(2(p+1)+1ɛ)∫0t‍g2(s)ds]×∥∇v(t)∥22+[2(p+2)-ɛ][(g1⋄∇u)(t)+(g2⋄∇v)(t)]+2(p+1)∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ.
Ifδ<0,  that is, E(0)<0, we choose ɛ=2(p+2) in (65) and β small enough such that β≤-2E(0). Then by (32), we have
(66)Φ(t)≥[(2(p+1)+12(p+2))∫0t‍2(p+1)m0-(2(p+1)+12(p+2))∫0t‍g1(s)ds]×∥∇u(t)∥22+[(2(p+1)+12(p+2))∫0t‍2(p+1)m0-(2(p+1)+12(p+2))∫0t‍g2(s)ds]×∥∇v(t)∥22+2(p+1)∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ+2(p+2)(-2E(0)-β)≥0.
If0≤δ<1, that is, 0≤E(0)<δE1<E1, we choose ɛ=2(p+2)(1-δ)+2δ and β=2(δE1-E(0)) in (65). Then, we get
(67)Φ(t)≥-4(p+2)δE1+[∫0t‍2(p+1)m0-(2(p+1)+12(p+2)(1-δ)+2δ)×∫0t‍g1(s)ds(2(p+1)+12(p+2)(1-δ)+2δ)]×∥∇u(t)∥22+[∫0t‍2(p+1)m0-(2(p+1)+12(p+2)(1-δ)+2δ)×∫0t‍g2(s)ds(2(p+1)+12(p+2)(1-δ)+2δ)]×∥∇v(t)∥22+2(p+1)∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ+2(p+1)δ[(g1⋄∇u)(t)+(g2⋄∇v)(t)].
By (32), we have (notice that δ^=δ due to δ≥0)
(68)∫0t‍gi(s)ds≤∫0∞‍gi(s)ds≤2(p+1)m02(p+1)+1/{2[(1-δ)2(p+2)+δ(1-δ)]}=2(p+1)m0-2δ(p+1)m02(1-δ)(p+1)+1/{2(1-δ)(p+2)+2δ},(i=1,2);
that is,
(69)2δ(p+1)[m0-∫0t‍gi(s)ds]≤2(p+1)m0-(2(p+1)+12(p+2)(1-δ)+2δ)∫0t‍gi(s)ds,
for i=1,2. Therefore, from the above two inequalities and (A2), we can get
(70)Φ(t)≥-4(p+2)δE1+2δ(p+1)(l1∥∇u(t)∥22+l2∥∇v(t)∥22).
Since
(71)(l1∥∇u0∥22+l2∥∇v0∥22)1/2>α*,
by Lemma 8, there exists a constant α3>α*such that
(72)(l1∥∇u(t)∥22+l2∥∇v(t)∥22)1/2≥α3,
which implies
(73)p+12(p+2)(l1∥∇u(t)∥22+l2∥∇v(t)∥22)≥p+12(p+2)α32>p+12(p+2)α*2>E1.
It follows from (67) and (73) that
(74)Φ(t)≥4(p+2)[δE1-δE1]≥0.
Therefore, by (58), (66), and (74), we obtain
(75)L(t)L′′(t)-p+32L′(t)2≥0
for almost everyt∈[0,T].By (49), we then chooseT0sufficiently large such that
(76)(p+1)(∫Ω‍u0u1dx+∫Ω‍v0v1dx+βT0)>∥∇u0∥22+∥∇v0∥22>0,
consequently,
(77)L′(0)=2∫Ω‍u0u1dx+2∫Ω‍v0v1dx+2βT0>0.
Then by (76) and (77), we choose T large enough so that
(78)T>(∥u0∥22+∥v0∥22+βT02)×((p+1)(∫Ω‍u0u1dx+∫Ω‍v0v1dx+βT0)-(∥∇u0∥22+∥∇v0∥22)∫Ω‍)-1>0,
which ensures thatT>(2/(p+1))(L(0)/L′(0)). As (p+3)/2>1, letting σ=(p+1)/2, we can select T* such that T*≤L(0)/σL′(0)≤T. By using Lemma 9, we get limt→T*-L(t)=∞. This implies that
(79)limt→T*-(∥∇u(t)∥22+∥∇v(t)∥22)=∞,
which is a contradiction. Thus, T<∞.


## 4. Blow-Up of Solutions with Arbitrarily Positive Initial Energy

In this section, we prove the second blow-up result (Theorem6) for solutions with arbitrarily positive initial energy. In order to attain our aim, we need the following three lemmas.Lemma 10 (see [4]).
Letδ*>0 and B(t)∈C2(0,∞) be a nonnegative function satisfying
(80)B′′(t)-4(δ*+1)B′(t)+4(δ*+1)B(t)≥0.
If
(81)B′(0)>r2B(0)+K0,
then
(82)B′(t)>K0
for t>0, where K0 is a constant and r2=2(δ*+1)-2(δ*+1)δ* is the smallest root of the equation
(83)r2-4(δ*+1)r+4(δ*+1)=0.Lemma 11 (see [4]).
Ifh(t) is a nonincreasing function on [0,∞) and satisfies the differential inequality
(84)h′(t)2≥a*+b*h(t)2+1/δ*fort≥0,
where a*>0,δ*>0, and b*>0, then there exists a finite time T* such that limt→T*-h(t)=0 and the upper bound of T* is estimated by
(85)T*≤2(3δ*+1)/2δ*δ*ca*{1-[1+ch(0)]-1/2δ*},
wherec=(a*/b*)2+1/δ*.For the next lemma, we define(86)K(t)∶=K(u(t),v(t))=∥u(t)∥22+∥v(t)∥22+∫0t‍∥∇u(τ)∥22dτ+∫0t‍∥∇v(τ)∥22dτ,t≥0.Lemma 12.
Assume that the conditions of Theorem6 hold and let(u,v)be a solution of (1), then
(87)K′(t)>∥∇u0∥22+∥∇v0∥22,t>0.Proof.
By (86), we have
(88)K′(t)=2∫Ω‍uutdx+2∫Ω‍vvtdx+∥∇u∥22+∥∇v∥22,(89)K′′(t)=2∥ut∥22+2∥vt∥22+2∫Ω‍uuttdx+2∫Ω‍vvttdx+2∫Ω‍∇u·∇utdx+2∫Ω‍∇v·∇vtdx.
Testing the first equation of system (1) with u and testing the second equation of system (1) with v and plugging the results into the expression of K′′(t) we obtain
(90)K′′(t)=2(∥ut∥22+∥vt∥22)-2M(∥∇u∥22)∥∇u∥22-2M(∥∇v∥22)∥∇v∥22+4(p+2)∫Ω‍F(u,v)dx+2∫0t‍∫Ω‍g1(t-s)∇u(s)·∇u(t)dxds+2∫0t‍∫Ω‍g2(t-s)∇v(s)·∇v(t)dxds.
By (27), (29), and (90), we get
(91)K′′(t)-2(p+3)(∥ut∥22+∥vt∥22)≥-4(p+2)E(0)+4(p+2)∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ+2(p+2)[M¯(∥∇u∥22)+M¯(∥∇v∥22)]-[2M(∥∇u∥22)+2(p+2)∫0t‍g1(s)ds]∥∇u∥22-[2M(∥∇v∥22)+2(p+2)∫0t‍g2(s)ds]∥∇v∥22+2(p+2)[(g1⋄∇u)(t)+(g2⋄∇v)(t)]-2(p+2)∫0t‍[(g1′⋄∇u)(τ)+(g2′⋄∇v)(τ)]dτ+2∫0t‍∫Ω‍g1(t-s)∇u(s)·∇u(t)dxds+2∫0t‍∫Ω‍g2(t-s)∇v(s)·∇v(t)dxds.
By using Ho¨lder’s inequality and Young’s inequality, we have
(92)2∫0t‍∫Ω‍g1(t-s)∇u(s)·∇u(t)dxds=2∫0t‍g1(t-s)∫Ω‍∇u(t)(∇u(s)-∇u(t))dxds+2(∫0t‍g1(s)ds)∥∇u(t)∥22≥-2(p+2)(g1⋄∇u)(t)+(2-12(p+2))(∫0t‍g1(s)ds)∥∇u(t)∥22.
Similarly,
(93)2∫0t‍∫Ω‍g2(t-s)∇v(s)·∇v(t)dxds≥-2(p+2)(g2⋄∇v)(t)+(2-12(p+2))(∫0t‍g2(s)ds)∥∇v(t)∥22.
Then taking (92) and (93) into account, we obtain
(94)K′′(t)-2(p+3)(∥ut∥22+∥vt∥22)≥-4(p+2)E(0)+4(p+2)×∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ+{+(2(p+1)+12(p+2))∫0t‍g1(s)ds]2(p+2)M¯(∥∇u∥22)-[∫0t‍2M(∥∇u∥22)+(2(p+1)+12(p+2))∫0t‍g1(s)ds]×∥∇u∥22+(2(p+1)+12(p+2))∫0t‍g1(s)ds]}+{+(2(p+1)+12(p+2))∫0t‍g2(s)ds]2(p+2)M¯(∥∇v∥22)-[∫0t‍2M(∥∇v∥22)+(2(p+1)+12(p+2))∫0t‍g2(s)ds]×∥∇v∥22+(2(p+1)+12(p+2))∫0t‍g2(s)ds]}.
Thus, by (A4) and (33), we get
(95)K′′(t)-2(p+3)(∥ut∥22+∥vt∥22)≥-4(p+2)E(0)+4(p+2)×∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ.
We note that(96)∥∇u(t)∥22-∥∇u0∥22=2∫0t‍∫Ω‍∇u·∇utdxdτ,∥∇v(t)∥22-∥∇v0∥22=2∫0t‍∫Ω‍∇v·∇vtdxdτ.
By Ho¨lder’s inequality and Young’s inequality, we obtain from (96)
(97)∥∇u(t)∥22+∥∇v(t)∥22≤∥∇u0∥22+∥∇v0∥22+∫0t‍(∥∇u(τ)∥22+∥∇v(τ)∥22)dτ+∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ.
By Ho¨lder’s inequality and Young’s inequality again, it follows from (86), (88), and (97) that
(98)K′(t)≤K(t)+∥∇u0∥22+∥∇v0∥22+∥ut∥22+∥vt∥22+∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ.
In view of (95) and (98), we have
(99)K′′(t)-2(p+3)K′(t)+2(p+3)K(t)+L4≥0,
where
(100)L4∶=4(p+2)E(0)+2(p+3)(∥∇u0∥22+∥∇v0∥22).
Let
(101)B(t)=K(t)+L42(p+3),t>0.
Then B(t) satisfies (80) for δ*=(p+1)/2. By (81), we see that if
(102)K′(0)>(p+3-(p+3)(p+1))[K(0)+L42(p+3)]+∥∇u0∥22+∥∇v0∥22,
that is, if
(103)E(0)<p+3(p+2)(p+3-p+1)∫Ω‍(u0u1+v0v1)dx-p+32(p+2)(∥u0∥22+∥v0∥22+∥∇u0∥22+∥∇v0∥22),
which is satisfied by the second hypothesis to Theorem 6, then we get from Lemma 10 that K′(t)>∥∇u0∥22+∥∇v0∥22,t>0.Thus, the proof of Lemma 12 is completed.In what follows, we find an estimate for the life span ofK(t)and prove Theorem 6.Proof of Theorem6.
Let(104)h(t)=[K(t)+(T1-t)(∥∇u0∥22+∥∇v0∥22)]-δ*,fort∈[0,T1],
where δ*=(p+1)/2 and T1>0 is a certain constant which will be specified later. Then we have
(105)h′(t)=-δ*h(t)1+1/δ*(K′(t)-∥∇u0∥22-∥∇v0∥22),(106)h′′(t)=-δ*h(t)1+2/δ*V(t),
where
(107)V(t)=K′′(t)[K(t)+(T1-t)(∥∇u0∥22+∥∇v0∥22)]-(1+δ*)(K′(t)-∥∇u0∥22-∥∇v0∥22)2.
For simplicity, we denote
(108)P=∥u(t)∥22+∥v(t)∥22,Q=∫0t‍(∥∇u(τ)∥22+∥∇v(τ)∥22)dτ,R=∥ut(t)∥22+∥vt(t)∥22,S=∫0t‍(∥∇ut(τ)∥22+∥∇vt(τ)∥22)dτ.
It follows from (88), (96), Ho¨lder’s inequality, and Young’s inequality that
(109)K′(t)≤2(PR+QS)+∥∇u0∥22+∥∇v0∥22.
By (95), we get
(110)K′′(t)≥(-4-8δ*)E(0)+4(1+δ*)(R+S).
Applying (107)–(110), we obtain
(111)V(t)≥[(-4-8δ*)E(0)+4(1+δ*)(R+S)]×[K(t)+(T1-t)(∥∇u0∥22+∥∇v0∥22)]-4(1+δ*)(PR+QS)2.
From (104) and (98) we deduce that
(112)V(t)≥(-4-8δ*)E(0)h(t)-1/δ*+4(1+δ*)(R+S)(T1-t)(∥∇u0∥22+∥∇v0∥22)+4(1+δ*)[(R+S)(P+Q)-(PR+QS)2].
By Cauchy-Schwarz inequality, the last term in the above inequality is nonnegative. Hence, we have
(113)V(t)≥(-4-8δ*)E(0)h(t)-1/δ*,t≥0.
Therefore by (106) and (113), we get
(114)h′′(t)≤δ*(4+8δ*)E(0)h(t)1+1/δ*,t≥0.
Note that by Lemma 12, h′(t)<0fort>0. Multiplying (114) byh′(t) and integrating from 0 to t, we obtain
(115)h′(t)2≥a1+b1h(t)2+1/δ*,fort≥0,
where
(116)a1=δ*2h(0)2+2/δ*×[-8E(0)h(0)-1/δ*(K′(0)-∥∇u0∥22-∥∇v0∥22)2-8E(0)h(0)-1/δ*(K′(0)-∥∇u0∥22-∥∇v0∥22)2],(117)b1=8δ*2E(0).
We observe that a1>0 if
(118)E(0)<(K′(0)-∥∇u0∥22-∥∇v0∥22)28[K(0)+T1(∥∇u0∥22+∥∇v0∥22)]=(∫Ω‍u0u1dx+∫Ω‍v0v1dx)22[∥u0∥22+∥v0∥22+T1(∥∇u0∥22+∥∇v0∥22)].
Then by Lemma 11, there exists a finite time T* such that limt→T*-h(t)=0. Moreover, the upper bounds of T* are estimated by
(119)T*≤2(3δ*+1)/2δ*δ*ca1{1-[1+ch(0)]-1/2δ*}.
Therefore
(120)limt→T*-K(t)=limt→T*-(∫∥u(t)∥22+∥v(t)∥22+∫0t‍∥∇u(τ)∥22dτ+∫0t‍∥∇v(τ)∥22dτ∫)=∞.
This completes the proof.Remark 13.
The choice ofT1 in (104) is possible provided that T1≥T*.


## 5. General Decay of Solutions

In this section, we prove the general decay of solutions of system (1). The method of proof is similar to that of [42, Theorem 3.5]. We first state a lemma which is similar to the one first proved by Vitillaro in [33] to study a class of a scalar wave equation.Lemma 14 ([23, Lemma 3.2]).
Suppose that (20), (A1), and(A2)hold. Let(u,v)be the solution of system (1). Assume further that E(0)<E1 and
(121)(l1∥∇u0∥22+l2∥∇v0∥22)1/2<α*.
Then
(122)(l1∥∇u(t)∥22+l2∥∇v(t)∥22+(g1⋄∇u)(t)+(g2⋄∇v)(t))1/2<α*
for allt∈[0,T).The auxiliary functionalsI(t),J(t)of problem (1) are defined as(123)I(t)∶=I(u(t),v(t))=(m0-∫0t‍g1(s)ds)∥∇u(t)∥22+(m0-∫0t‍g2(s)ds)∥∇v(t)∥22+(g1⋄∇u)(t)+(g2⋄∇v)(t)-2(p+2)∫Ω‍F(u,v)dx,J(t)∶=J(u(t),v(t))=12[(m0-∫0t‍g1(s)ds)∥∇u∥22+(m0-∫0t‍g2(s)ds)∥∇v∥22]+12[m1γ+1(g1⋄∇u)(t)+(g2⋄∇v)(t)+m1γ+1(∥∇u∥22(γ+1)+∥∇v∥22(γ+1))]-∫Ω‍F(u,v)dx.Lemma 15.
Suppose that (20), (A1), and (A2)hold. Let (u,v) be the solution of system (1). Assume further that E(0)<E1and
(124)(l1∥∇u0∥22+l2∥∇v0∥22)1/2<α*.
Then
(125)l(∥∇u(t)∥22+∥∇v(t)∥22)≤2(p+2)p+1E(t),(126)(g1⋄∇u)(t)+(g2⋄∇v)(t)≤2(p+2)p+1E(t)
for all t∈[0,T).Proof.
SinceE(0)<E1and
(127)(l1∥∇u0∥22+l2∥∇v0∥22)1/2<α*,
it follows from Lemma 14 and (30) that
(128)l1∥∇u(t)∥22+l2∥∇v(t)∥22≤l1∥∇u(t)∥22+l2∥∇v(t)∥22+(g1⋄∇u)(t)+(g2⋄∇v)(t)<α*2=η1-1/(p+1),
which implies that
(129)I(t)≥(m0-∫0∞‍g1(s)ds)∥∇u∥22+(m0-∫0∞‍g2(s)ds)∥∇v∥22+(g1⋄∇u)(t)+(g2⋄∇v)(t)-2(p+2)∫Ω‍F(u,v)dx≥l1∥∇u(t)∥22+l2∥∇v(t)∥22-2(p+2)∫Ω‍F(u,v)dx=l1∥∇u(t)∥22+l2∥∇v(t)∥22-(a∥u+v∥2(p+2)2(p+2)+2b∥uv∥p+2p+2)≥l1∥∇u(t)∥22+l2∥∇v(t)∥22-η1(l1∥∇u(t)∥22+l2∥∇v(t)∥22)p+2=(l1∥∇u(t)∥22+l2∥∇v(t)∥22)×[1-η1(l1∥∇u(t)∥22+l2∥∇v(t)∥22)p+1]≥0
for t∈[0,T), where we have used (24). Further, by (123), we have
(130)J(t)≥12[(m0-∫0t‍g1(s)ds)∥∇u∥22+(m0-∫0t‍g2(s)ds)∥∇v∥22+(g1⋄∇u)(t)+(g2⋄∇v)(t)∫0t‍]-∫Ω‍F(u,v)dx-12(p+2)I(t)+12(p+2)I(t)≥(12-12(p+2))×[(m0-∫0t‍g1(s)ds)∥∇u∥22+(m0-∫0t‍g2(s)ds)∥∇v∥22]+(12-12(p+2))[(g1⋄∇u)(t)+(g2⋄∇v)(t)]+12(p+2)I(t)≥p+12(p+2)×[∫0t‍l1∥∇u(t)∥22+l2∥∇v(t)∥22+(g1⋄∇u)(t)+(g2⋄∇v)(t)∫0t‍]+12(p+2)I(t)≥0.
From (130) and (36), we deduce that
(131)l(∥∇u(t)∥22+∥∇v(t)∥22)≤l1∥∇u(t)∥22+l2∥∇v(t)∥22≤2(p+2)p+1J(t)≤2(p+2)p+1E(t),(g1⋄∇u)(t)+(g2⋄∇v)(t)≤2(p+2)p+1J(t)≤2(p+2)p+1E(t).We note that the following functional was introduced in [42]:
(132)G1(t)∶=E(t)+ε1ϕ(t)+ε2ψ(t),
where ε1 and ε2 are some positive constants and
(133)ϕ(t)=ξ1(t)∫Ω‍u(t)ut(t)dx+ξ2(t)∫Ω‍v(t)vt(t)dx,(134)ψ(t)=-ξ1(t)∫Ω‍ut(t)∫0t‍g1(t-s)(u(t)-u(s))dsdx-ξ2(t)∫Ω‍vt(t)∫0t‍g2(t-s)(v(t)-v(s))dsdx.
Here, we use the same functional (132) but chooseξi(t)≡1(i=1,2) in (133) and (134).Lemma 16.
There exist two positive constantsβ1andβ2such that
(135)β1G1(t)≤E(t)≤β2G1(t),∀t≥0.Proof.
From Ho¨lder’s inequality, Young’s inequality, Lemma 2, (36), and (125), we deduce
(136)|ϕ(t)|≤∥u∥2∥ut∥2+∥v∥2∥vt∥2≤12(∥u∥22+∥ut∥22+∥v∥22+∥vt∥22)≤12C2(∥∇u∥22+∥∇v∥22)+12(∥ut∥22+∥vt∥22)≤C2p+2l(p+1)E(t)+E(t)=(1+C2p+2l(p+1))E(t).
Applying Ho¨lder’s inequality, Young’s inequality, and Lemma 2 again, it follows that
(137)∫Ω‍ut(t)∫0t‍g1(t-s)(u(t)-u(s))dsdx≤∥ut∥2×[∫Ω‍(∫0t‍g1(t-s)(u(t)-u(s))ds)2dx]1/2≤12∥ut∥22+12∫Ω‍(∫0t‍g1(t-s)(u(t)-u(s))ds)2dx≤12∥ut∥22+12(m0-l)∫Ω‍∫0t‍g1(t-s)(u(t)-u(s))2dsdx≤12∥ut∥22+12C2(m0-l)(g1⋄∇u)(t).
Similarly, applying Ho¨lder’s inequality, Young’s inequality, and Lemma 2 again, we have
(138)∫Ω‍vt(t)∫0t‍g2(t-s)(v(t)-v(s))dsdx≤12∥vt(t)∥22+12C2(m0-l)(g2⋄∇v)(t).
It follows from (137), (138), (36), and (126) that
(139)|ψ(t)|≤12(∥ut∥22+∥vt∥22)+12C2(m0-l)[(g1⋄∇u)(t)+(g2⋄∇v)(t)]≤E(t)+p+2p+1C2(m0-l)E(t)=(1+p+2p+1C2(m0-l))E(t).
If we take ϵ1 and ϵ2 to be sufficiently small, then (135) follows from (132), (136), and (139).Lemma 17.
Under the conditions of Theorem7, the functional ϕ(t), defined by (133) (withξi(t)≡1(i=1,2)), satisfies
(140)ϕ′(t)≤∥ut∥22+∥vt∥22-[l-ϵ(m0-l+12)]∥∇u∥22-[l-ϵ(m0-l+12)]∥∇v∥22+14ϵ[(g1⋄∇u)(t)+(g2⋄∇v)(t)]+12ϵ(∥∇ut∥22+∥∇vt∥22)+2(p+2)∫Ω‍F(u,v)dx-m1(∥∇u∥22(γ+1)+∥∇v∥22(γ+1)),∀ϵ>0.Proof.
By using the equations in (1) and setting ξi(t)≡1(i=1,2) in (133), we easily see that
(141)ϕ′(t)=∥ut∥22+∥vt∥22+∫Ω‍uuttdx+∫Ω‍vvttdx=∥ut∥22+∥vt∥22-M(∥∇u∥22)∥∇u∥22-M(∥∇v∥22)∥∇v∥22+∫Ω‍∇u(t)·∫0t‍g1(t-s)∇u(s)dsdx+∫Ω‍∇v(t)·∫0t‍g2(t-s)∇v(s)dsdx-∫Ω‍∇ut·∇udx-∫Ω‍∇vt·∇vdx+2(p+2)×∫Ω‍F(u,v)dx.
By Cauchy-Schwarz inequality and Young’s inequality, we have
(142)∫Ω‍∇u(t)·∫0t‍g1(t-s)∇u(s)dsdx≤(∫0t‍g1(t-s)ds)∥∇u(t)∥22+∫Ω‍∫0t‍g1(t-s)|∇u(t)|×|∇u(s)-∇u(t)|dsdx≤(1+ϵ)(∫0t‍g1(s)ds)×∥∇u(t)∥22+14ϵ(g1⋄∇u)(t)≤(1+ϵ)(m0-l)×∥∇u(t)∥22+14ϵ(g1⋄∇u)(t),∀ϵ>0.
In the same way, we can get
(143)∫Ω‍∇v(t)·∫0t‍g2(t-s)∇v(s)dsdx≤(1+ϵ)(m0-l)∥∇v(t)∥22+14ϵ(g2⋄∇v)(t),(144)∫Ω‍∇ut·∇udx≤ϵ2∥∇u∥22+12ϵ∥∇ut∥22,(145)∫Ω‍∇vt·∇vdx≤ϵ2∥∇v∥22+12ϵ∥∇vt∥22,∀ϵ>0.
Using (141)–(145), we obtain
(146)ϕ′(t)≤∥ut∥22+∥vt∥22-m0∥∇u∥22-m0∥∇v∥22-m1(∥∇u∥22(γ+1)+∥∇v∥22(γ+1))+(1+ϵ)(m0-l)(∥∇u∥22+∥∇v(t)∥22)+14ϵ((g1⋄∇u)(t)+(g2⋄∇v)(t))+ϵ2(∥∇u∥22+∥∇v∥22)+12ϵ(∥∇ut∥22+∥∇vt∥22)+2(p+2)∫Ω‍F(u,v)dx=∥ut∥22+∥vt∥22-[l-ϵ(m0-l+12)]∥∇u∥22-[l-ϵ(m0-l+12)]×∥∇v∥22+14ϵ(g1⋄∇u)(t)+14ϵ(g2⋄∇v)(t)+12ϵ(∥∇ut∥22+∥∇vt∥22)-m1(∥∇u∥22(γ+1)+∥∇v∥22(γ+1))+2(p+2)∫Ω‍F(u,v)dx,∀ϵ>0.
So, Lemma 17 is established.Lemma 18.
Under the conditions of Theorem7, the functional ψ(t), defined by (134) (withξi(t)≡1(i=1,2)), satisfies
(147)ψ′(t)≤[η(m0-l)(3m0-2l)+3CηC7]×(∥∇u∥22+∥∇v∥22)+η(∥∇ut∥22+∥∇vt∥22)+ηm1(m0-l)(∥∇u∥22(γ+1)+∥∇v∥22(γ+1))+[(2η+2+C24η)(m0-l)+C64η]×[(g1⋄∇u)(t)+(g2⋄∇v)(t)]-g0(0)4ηC2[(g1′⋄∇u)(t)+(g2′⋄∇v)(t)]+(η-∫0t‍g1(s)ds)∥ut∥22+(η-∫0t‍g2(s)ds)∥vt∥22,∀η>0,
where
(148)g0(0)=max{g1(0),g2(0)},C6∶=m0+m1(2(p+2)l(p+1)E(0))γ,C7∶=C2(2p+3)[2(p+2)l(p+1)E(0)]2(p+1).Proof.
By using the equations in (1) and setξi(t)≡1(i=1,2)in (134), we observe that
(149)ψ′(t)=-∫Ω‍utt(t)∫0t‍g1(t-s)(u(t)-u(s))dsdx-∫Ω‍vtt(t)∫0t‍g2(t-s)(v(t)-v(s))dsdx-∫Ω‍ut(t)∫0t‍g1′(t-s)(u(t)-u(s))dsdx-∫Ω‍vt(t)∫0t‍g2′(t-s)(v(t)-v(s))dsdx-∫Ω‍ut(t)∫0t‍g1(t-s)ut(t)dsdx-∫Ω‍vt(t)∫0t‍g2(t-s)vt(t)dsdx=∫Ω‍M(∥∇u∥22)∇u(t)·∫0t‍g1(t-s)(∇u(t)-∇u(s))dsdx+∫Ω‍M(∥∇v∥22)∇v(t)·∫0t‍g2(t-s)(∇v(t)-∇v(s))dsdx-∫Ω‍(∫0t‍g1(t-s)∇u(s)ds)×[∫0t‍g1(t-s)(∇u(t)-∇u(s))ds]dx-∫Ω‍(∫0t‍g2(t-s)∇v(s)ds)×[∫0t‍g2(t-s)(∇v(t)-∇v(s))ds]dx+[∫Ω‍∇ut(t)∫0t‍g1(t-s)(∇u(t)-∇u(s))dsdx+∫Ω‍∇vt(t)∫0t‍g2(t-s)(∇v(t)-∇v(s))dsdx]-[∫Ω‍f1(u,v)∫0t‍g1(t-s)(u(t)-u(s))dsdx+∫Ω‍f2(u,v)∫0t‍g2(t-s)(v(t)-v(s))dsdx]-[∫Ω‍ut(t)∫0t‍g1′(t-s)(u(t)-u(s))dsdx+∫Ω‍vt(t)∫0t‍g2′(t-s)(v(t)-v(s))dsdx]-(∫0t‍g1(s)ds)∥ut(t)∥22-(∫0t‍g2(s)ds)∥vt(t)∥22∶=I1+⋯+I7-(∫0t‍g1(s)ds)∥ut(t)∥22-(∫0t‍g2(s)ds)∥vt(t)∥22.
In what follows we will estimate I1,…,I7 in (149). By Young’s inequality, (A2), and (125), we get
(150)|I1|≤ηM(∥∇u∥22)∥∇u∥22∫0t‍g1(s)ds+14ηM(∥∇u∥22)(g1⋄∇u)(t)=η(m0∥∇u∥22+m1∥∇u∥22(γ+1))∫0t‍g1(s)ds+14η(m0+m1∥∇u∥22γ)(g1⋄∇u)(t)≤η(m0-l)(m0∥∇u∥22+m1∥∇u∥22(γ+1))+14η[m0+m1(2(p+2)l(p+1)E(0))γ](g1⋄∇u)(t)≤ηm0(m0-l)∥∇u∥22+ηm1(m0-l)∥∇u∥22(γ+1)+C64η(g1⋄∇u)(t),∀η>0.
Similarly, we have
(151)|I2|≤ηm0(m0-l)∥∇v∥22+ηm1(m0-l)∥∇v∥22(γ+1)+C64η(g2⋄∇v)(t),∀η>0.
For I3 in (149), applying (A2), Ho¨lder’s inequality, and Young’s inequality, we deduce
(152)|I3|≤η∫Ω‍(∫0t‍g1(t-s)∇u(s)ds)2dx+14η∫Ω‍(∫0t‍g1(t-s)(∇u(t)-∇u(s))ds)2dx≤η∫Ω‍[∫0t‍g1(t-s)(|∇u(s)-∇u(t)|+|∇u(t)|)ds]2dx+14η(m0-l)(g1⋄∇u)(t)≤2η(∫0t‍g1(s)ds)2∥∇u∥22+(2η+14η)(m0-l)(g1⋄∇u)(t)≤2η(m0-l)2∥∇u∥22+(2η+14η)(m0-l)(g1⋄∇u)(t),∀η>0.
Similarly,
(153)|I4|≤2η(m0-l)2∥∇v∥22+(2η+14η)(m0-l)(g2⋄∇v)(t),∀η>0.
In the same way, we have
(154)|I5|≤η(∥∇ut∥22+∥∇vt∥22)+14η(m0-l)[(g1⋄∇u)(t)+(g2⋄∇v)(t)],∀η>0.
By Young’s inequality, Lemma 2, and (125) we see that
(155)|∫Ω‍f1(u,v)∫0t‍g1(t-s)(u(t)-u(s))ds dx|≤η∫Ω‍(a|u+v|2p+3+b|u|p+1|v|p+2)2dx+C2(m0-l)4η(g1⋄∇u)(t)≤Cη∫Ω‍(|u|2(2p+3)+|v|2(2p+3)+|u|2(p+1)|v|2(p+2))dx+C2(m0-l)4η(g1⋄∇u)(t)≤Cη[∥u∥2(2p+3)2(2p+3)+∥v∥2(2p+3)2(2p+3)+∥u∥2(2p+3)2(p+1)∥v∥2(2p+3)2(p+2)]+C2(m0-l)4η(g1⋄∇u)(t)≤CηC2(2p+3)×[∥∇u∥22(2p+3)+∥∇v∥22(2p+3)+∥∇u∥22(p+1)∥∇v∥22(p+2)]+C2(m0-l)4η(g1⋄∇u)(t)≤CηC2(2p+3)(2(p+2)l(p+1)E(0))2p+2×[∥∇u∥22+∥∇v∥22+∥∇u∥2∥∇v∥2]+C2(m0-l)4η(g1⋄∇u)(t)≤32CηC7[∥∇u∥22+∥∇v∥22]+C2(m0-l)4η(g1⋄∇u)(t),∀η>0.
Hence, we infer that
(156)|I6|≤3CηC7(∥∇u∥22+∥∇v∥22)+C2(m0-l)4η[(g1⋄∇u)(t)+(g2⋄∇v)(t)],∀η>0.
By Young’s inequality and Lemma 2 again, we obtain
(157)|I7|≤η∥ut∥22+14η∫Ω‍(∫0t‍g1′(t-s)(u(t)-u(s))ds)2dx+η∥vt∥22+14η∫Ω‍(∫0t‍g2′(t-s)(v(t)-v(s))ds)2dx≤η(∥ut∥22+∥vt∥22)+C24η(∫0t‍g1′(s)ds)(g1′⋄∇u)(t)+C24η(∫0t‍g2′(s)ds)(g2′⋄∇v)(t)≤η(∥ut∥22+∥vt∥22)-g1(0)C24η(g1′⋄∇u)(t)-g2(0)C24η(g2′⋄∇v)(t),∀η>0.
Combining (149)–(157), we complete the proof of Lemma 18.Proof of Theorem7.
Sincegiare positive, we have, for anyt0>0,(158)∫0t‍gi(s)ds≥∫0t0‍gi(s)ds(i=1,2),t≥t0,denote(159)g3=min{∫0t0‍g1(s)ds,∫0t0‍g2(s)ds}.
By using (28), (132), (140), and (147), a series of computations, fort≥t0, we have
(160)G1′(t)=E′(t)+ε1ϕ′(t)+ε2ψ′(t)≤-{[(2η+2+C24η)(m0-l)+C64η]ε1[l-ϵ(m0-l+12)]-ε2η[(m0-l)(3m0-2l)+3CC7][(2η+2+C24η)(m0-l)+C64η]}×[∥∇u∥22+∥∇v∥22]-[m1ε1-m1ε2η(m0-l)]×(∥∇u∥22(γ+1)+∥∇v∥22(γ+1))-(1-ε12ϵ-ε2η)(∥∇ut∥22+∥∇vt∥22)+{ε14ϵ+ε2[(2η+2+C24η)(m0-l)+C64η]}×[(g1⋄∇u)(t)+(g2⋄∇v)(t)]-(ε2g3-ε2η-ε1)(∥ut∥22+∥vt∥22)+2ε1(p+2)∫Ω‍F(u,v)dx+(12-g0(0)4ηε2C2)×[(g1′⋄∇u)(t)+(g2′⋄∇v)(t)].
We chooseε1,ε2, andηsmall enough, such that
(161)ε2η(m0-l)<ε1<ε2(g3-η)
andϵ=ε1; then we can check that
(162)C8=ε2g3-ε2η-ε1>0,C10=m1ε1-m1ε2η(m0-l)>0,C11=12-g0(0)4ηε2C2>0,C12=1-ε12ϵ-ε2η>0.
We chooseη,ε1, andε2so small that (162) remains valid and, further(163)C9=ε1[l-ϵ(m0-l+12)]-ε2η[(m0-l)(3m0-2l)+3CC7]>0.
So, we arrive at
(164)G1′(t)≤-C8(∥ut∥22+∥vt∥22)-C9(∥∇u∥22+∥∇v∥22)+2ε1(p+2)∫Ω‍F(u,v)dx-C10(∥∇u∥22(γ+1)+∥∇v∥22(γ+1))+C11[(g1′⋄∇u)(t)+(g2′⋄∇v)(t)]+{ε14ϵ+ε2[(2η+2+C24η)(m0-l)+C64η]}×[(g1⋄∇u)(t)+(g2⋄∇v)(t)],
which yields (if needed, one can choose ε2 sufficiently small)
(165)G1′(t)≤-c*E(t)+C[(g1⋄∇u)(t)+(g2⋄∇v)(t)],∀t≥t0,
where c* is some positive constant. It follows from (165), (A3), and (28) that
(166)ξ(t)G1′(t)≤-c*ξ(t)E(t)+Cξ(t)[(g1⋄∇u)(t)+(g2⋄∇v)(t)]≤-c*ξ(t)E(t)+Cξ1(t)(g1⋄∇u)(t)+Cξ2(t)(g2⋄∇v)(t)≤-c*ξ(t)E(t)-C[(g1′⋄∇u)(t)+(g2′⋄∇v)(t)]≤-c*ξ(t)E(t)-CE′(t),∀t≥t0.
That is
(167)L*′(t)≤-c*ξ(t)E(t)≤-kξ(t)L*(t),∀t≥t0,
whereL*(t)=ξ(t)G1(t)+CE(t)is equivalent toE(t)due to (135) andkis a positive constant. A simple integration of (167) leads to
(168)L*(t)≤L*(t0)e-k∫t0t‍ξ(s)ds,∀t≥t0.
This completes the proof.



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*Source: 284809-2014-02-04.xml*