# Asymptotic Behavior for a Nondissipative and Nonlinear System of the Kirchhoff Viscoelastic Type

Authors: Nasser-Eddine Tatar
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

A wave equation of the Kirchhoff type with several nonlinearities is stabilized by a viscoelastic damping. We consider the case of nonconstant (and unbounded) coefficients. This is a nondissipative case, and as a consequence the nonlinear terms cannot be estimated in the usual manner by the initial energy. We suggest a way to get around this difficulty. It is proved that if the solution enters a certain region, which we determine, then it will be attracted exponentially by the equilibrium.


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## Body

## 1. Introduction

We will consider the following wave equation with a viscoelastic damping term:(1.1)utt+∑i=1mbi(t)|u|piu=(1+∑j=1kaj(t)‖∇u‖22qj)Δu-∫0th(t-s)Δu(s)ds,inΩ×R+,u=0,on⁡Γ×R+,u(x,0)=u0(x),ut(x,0)=u1(x),inΩ,
where Ω is a bounded domain in Rn with smooth boundary Γ=∂Ω and pi,qj>0,i=1,…,m,j=1,…,k. The functions u0(x) and u1(x) are given initial data, and the (nonnegative) functions aj(t),bi(t), and h(t) are at least absolutely continuous and will be specified later on. This problem arises in viscoelasticity where it has been shown by experiments that when subject to sudden changes, the viscoelastic response not only does depend on the current state of stress but also on all past states of stress. This gives rise to the integral term called the memory term. One may find a rich literature in this regard (with or without the Kirchhoff terms) treating mainly the stabilization of such systems for different classes of functions h. We refer the reader to [1–25] and the references therein. For problems of the Kirchhoff type, one can consult [26–35] and in particular [36–46] where the equations are supplemented by a nonlinear source. Several questions, such as well-posedness and asymptotic behavior, have been discussed in these references, to cite but a few.As is clear from the equation in (1.1), we consider here several nonlinearities and the relaxation function is not necessarily decreasing or even nonincreasing. These issues are important but do not constitute the main contribution in the present paper. In case that  aj(t) and bi(t) are not nonincreasing, then we are in a nondissipative situation. This is the case also when the relaxation function oscillates (in case aj(t),bi(t) are nonincreasing). Our argument here is simple and flexible. It relies on a Gronwall-type inequality involving several nonlinearities. We prove that there exists a sufficiently large T>0 and a constant U after which (the modified energy of) global solutions are bounded below by U or decay to zero exponentially. We were not able to find conditions directly on the initial data because the Gronwall inequality is applicable only after some large values of time.For simplicity we shall consider the simpler casep1=p,pi=0,b1=b,bi=0,i=2,…,m and q1=q,qj=0,a1=a,aj=0,j=2,…,k.The local existence and uniqueness may be found in [36, 37].Theorem 1.1.
Assume that(u0,u1)∈H01(Ω)×L2(Ω) and h(t) is a nonnegative summable kernel. If 0<p<2/(n-2) when n≥3 and p>0 when n=1,2, then there exists a unique solution u to problem (1.1) such that
(1.2)u∈C([0,T];H01(Ω))∩C1([0,T];L2(Ω))
for T small enough.The plan of the paper is as follows. In the next section we prepare some materials needed to prove our result. Section3 is devoted to the statement and proof of our theorem.


## 2. Preliminaries

In this section we define the different functionals we will work with. We prove an equivalence result between two functionals. Further, some useful lemmas are presented. We define the (classical) energy by(2.1)E(t)=12(‖ut‖22+‖∇u‖22)+a(t)2(q+1)‖∇u‖22(q+1)+b(t)p+2‖u‖p+2p+2,t≥0,
where ∥·∥p denotes the norm in Lp(Ω). Then by  (1.1)  it is easy to see that for t≥0(2.2)E′(t)=∫Ω∇ut⋅∫0th(t-s)∇u(s)dsdx+a′(t)2(q+1)‖∇u‖22(q+1)+b′(t)p+2‖u‖p+2p+2.
The first term in the right-hand side of (2.2) may be written as the derivative of some expression; namely,
(2.3)∫Ω∇ut⋅∫0th(t-s)∇u(s)dsdx=12∫Ω(h′□∇u)dx-12h(t)‖∇u‖22-12ddt{∫Ω(h□∇u)dx-(∫0th(s)ds)‖∇u‖22},
where
(2.4)(h□v)(t):=∫0th(t-s)|v(t)-v(s)|2ds.
Therefore, if we modify E(t) to
(2.5)E(t):=12{‖ut‖22+(1-∫0th(s)ds)‖∇u‖22+∫Ω(h□∇u)dx}+a(t)2(q+1)‖∇u‖22(q+1)+b(t)p+2‖u‖p+2p+2,
we obtain for t≥0(2.6)E′(t)=12∫Ω((h′□∇u)-h(t)|∇u|2)dx+a′(t)2(q+1)‖∇u‖22(q+1)+b′(t)p+2‖u‖p+2p+2.
Assuming that
(2.7)1-∫0+∞h(s)ds=:1-κ>0
makes ℰ(t) a nonnegative functional. The following functionals
(2.8)Φ1(t):=∫Ωutudx,Φ2(t):=-∫Ωut∫0th(t-s)(u(t)-u(s))dsdx
are standard and will be used here. The next ones have been introduced by the present author in [24]
(2.9)Φ3(t):=∫0tHγ(t-s)‖∇u(s)‖22ds,Φ4(t):=∫0tΨγ(t-s)‖∇u(s)‖22ds,
where
(2.10)Hγ(t):=γ(t)-1∫t∞h(s)γ(s)ds,Ψγ(t):=γ(t)-1∫t∞ξ(s)γ(s)ds,t≥0,
and γ(t) and ξ(t) are two nonnegative functions which will be precised later (see (H2), (H3)). The functional
(2.11)L(t):=E(t)+∑i=14λiΦi(t)
for some λi>0,i=1,2,3,4, to be determined is equivalent to ℰ(t)+Φ3(t)+Φ4(t).Proposition 2.1.
There existρi>0,i=1,2 such that
(2.12)ρ1[E(t)+Φ3(t)+Φ4(t)]≤L(t)≤ρ2[E(t)+Φ3(t)+Φ4(t)]
for all t≥0 and small λi,i=1,2.Proof.
By the inequalities(2.13)Φ1(t)=∫Ωutudx≤12‖ut‖22+Cp2‖∇u‖22,Φ2(t)≤12‖ut‖22+Cpκ2∫Ω(h□∇u)dx,
where Cp is the Poincaré constant, we have
(2.14)L(t)≤12(1+λ1+λ2)‖ut‖22+12(1-∫0th(s)ds+λ1Cp)‖∇u‖22+b(t)p+2‖u‖p+2p+2+12(1+λ2Cpκ)∫Ω(h□∇u)dx+a(t)2(q+1)‖∇u‖22(q+1)+λ3Φ3(t)+λ4Φ4(t),t≥0.
On the other hand,
(2.15)2L(t)≥(1-λ1-λ2)‖ut‖22+(1-λ2Cpκ)∫Ω(h□∇u)dx+2b(t)p+2‖u‖p+2p+2+a(t)q+1‖∇u‖22(q+1)+[1-κ-λ1Cp]‖∇u‖22+2λ3Φ3(t)+2λ4Φ4(t).
Therefore, ρ1[ℰ(t)+Φ3(t)+Φ4(t)]≤L(t)≤ρ2[ℰ(t)+Φ3(t)+Φ4(t)] for some constant ρi>0,i=1,2 and small λi,i=1,2 such that λ1<min{1,(1-κ)/Cp} and λ2<min{1/Cpκ,1-λ1}.
The identity to follow is easy to justify and is helpful to prove our result.Lemma 2.2.
One has forh∈C(0,∞) and v∈C((0,∞);L2(Ω))(2.16)∫Ωv(t)∫0th(t-s)v(s)dsdx=12(∫0th(s)ds)‖v(t)‖22+12∫0th(t-s)‖v(s)‖22ds-12∫Ω(h□v)dx,t≥0.The next lemma is crucial in estimating (partially) our nonlinear terms. It can be found in [47].LetI⊂R, and let g1,g2:I→R∖{0}. We write g1∝g2 if g2/g1 is nondecreasing in I.Lemma 2.3.
Leta(t) be a positive continuous function in J:=[α,β),kj(t),j=1,…,n are nonnegative continuous functions, gj(u),j=1,…,n are nondecreasing continuous functions in R+, with gj(u)>0 for u>0, and u(t) is a nonnegative continuous functions in J. If g1∝g2∝⋯∝gn in (0,∞), then the inequality
(2.17)u(t)≤a(t)+∑j=1n∫αtkj(s)gj(u(s))ds,t∈J,
implies that
(2.18)u(t)≤cn(t),α≤t<β0,
where c0(t):=sup0≤s≤ta(s),
(2.19)cj(t):=Gj-1[Gj(cj-1(t))+∫αtkj(s)ds],j=1,…,n,Gj(u):=∫ujudxgj(x),u>0(uj>0,j=1,...,n),
and β0 is chosen so that the functions cj(t),j=1,…,n are defined for α≤t<β0.Lemma 2.4.
Assume that2≤q<+∞ if n=1,2 or 2≤q<2n/(n-2) if n≥3. Then there exists a positive constant Ce=Ce(Ω,q) such that
(2.20)‖u‖q≤Ce‖∇u‖2
for u∈H01(Ω).


## 3. Asymptotic Behavior

In this section we state and prove our result. To this end we need some notation. For every measurable set𝒜⊂R+, we define the probability measure h^ by
(3.1)h^(A):=1κ∫A‍h(s)ds.
The nondecreasingness set and the non-decreasingness rate of h are defined by
(3.2)Qh:={s∈R+:h(s)>0,h′(s)≥0},(3.3)Rh:=h^(Qh),
respectively.The following assumptions on the kernelh(t) will be adopted.(H1)
h(t)≥0 for all t≥0 and 0<κ=∫0+∞h(s)ds<1.(H2)
h is absolutely continuous and of bounded variation on (0,∞) and h′(t)≤ξ(t) for some nonnegative summable function ξ(t) (=max{0,h′(t)} where h′(t) exists) and almost all t>0.(H3)
There exists a nondecreasing functionγ(t)>0 such that γ′(t)/γ(t)=η(t) is a nonincreasing function: ∫0+∞h(s)γ(s)ds<+∞ and ∫0+∞ξ(s)γ(s)ds<+∞.Note that a wide class of functions satisfies the assumption (H3). In particular, exponentially and polynomially (or power type) decaying functions are in this class.Lett*>0 be a number such that ∫0t*h(s)ds=h*>0. We denote by ℬt the set ℬt:=ℬ∩[0,t].Lemma 3.1.
One has fort≥t* and δi>0,i=1,…,5(3.4)Φ2′(t)≤(1-h*)[δ1+32∫Qth(t-s)ds]‖∇u‖22+(δ3-h*)‖ut‖22+[(1-h*)κ4δ1+(1+1δ2)κ]∫Ω∫Ath(t-s)|∇u(t)-∇u(s)|2dsdx+12(1-h*)∫Qth(t-s)‖∇u(s)‖22ds-Cp4δ3BV[h]∫Ω(h′□∇u)dx+(1+δ2)(∫Qth(t-s)ds)∫Ω∫Qth(t-s)|∇u(t)-∇u(s)|2dsdx+2p+1δ4Ceb2(t)(1-κ)p+1Ep+1(t)+Cpκ4δ4∫Ω(h□∇u)dx+δ5a(t)‖∇u‖22(q+1)+2q-1κa(t)δ5(1-κ)qEq+1(t)+Cp4δ3(∫Qtξ(t-s)ds)∫Ω∫Qtξ(t-s)|∇u(t)-∇u(s)|2dsdx,
where BV[h] is the total variation of h.Proof.
This lemma is proved by a direct differentiation ofΦ2(t) along solutions of (1.1) and estimation of the different terms in the obtained expression of the derivative. Indeed, we have
(3.5)Φ2′(t)=-∫Ωutt∫0th(t-s)(u(t)-u(s))dsdx-∫Ωut[∫0th′(t-s)(u(t)-u(s))ds+ut∫0th(s)ds]dx
or
(3.6)Φ2′(t)=-∫Ω[(1-∫0th(s)ds)Δu-b(t)|u|pu+a(t)‖∇u‖22qΔu+∫0th(t-s)(Δu(t)-Δu(s))ds]∫0th(t-s)(u(t)-u(s))dsdx-(∫0th(s)ds)‖ut‖22-∫Ωut∫0th′(t-s)(u(t)-u(s))dsdx,t≥0.
Therefore,
(3.7)Φ2′(t)=(1-∫0th(s)ds+a(t)‖∇u‖22q)×∫Ω∇u⋅∫0th(t-s)(∇u(t)-∇u(s))dsdx-(∫0th(s)ds)‖ut‖22+b(t)∫Ω|u|pu∫0th(t-s)(u(t)-u(s))dsdx-∫Ωut∫0th′(t-s)(u(t)-u(s))dsdx+∫Ω|∫0th(t-s)(∇u(t)-∇u(s))ds|2dx,t≥0.
For all measurable sets 𝒜 and 𝒬 such that 𝒜=R+∖𝒬, it is clear that
(3.8)∫Ω∇u⋅∫0th(t-s)(∇u(t)-∇u(s))dsdx=∫Ω∇u⋅∫A∩[0,t]h(t-s)(∇u(t)-∇u(s))dsdx+∫Ω∇u⋅∫Q∩[0,t]h(t-s)(∇u(t)-∇u(s))dsdx=∫Ω∇u⋅∫A∩[0,t]h(t-s)(∇u(t)-∇u(s))dsdx+(∫Q∩[0,t]h(t-s)ds)‖∇u‖22-∫Ω∇u⋅∫Q∩[0,t]h(t-s)∇u(s)dsdx,t≥0.
For δ1>0, the first term in the right-hand side of (3.8) satisfies
(3.9)∫Ω∇u⋅∫Ath(t-s)(∇u(t)-∇u(s))dsdx≤δ1‖∇u‖22+κ4δ1∫Ω∫Ath(t-s)|∇u(t)-∇u(s)|2dsdx,t≥0,
and the third one fulfills
(3.10)∫Ω∇u⋅∫Qth(t-s)∇u(s)dsdx≤12(∫Qth(t-s)ds)‖∇u‖22+12∫Qth(t-s)‖∇u(s)‖22ds,t≥0.
Back to (3.8) we may write
(3.11)∫Ω∇u⋅∫0th(t-s)(∇u(t)-∇u(s))dsdx≤δ1‖∇u‖22+κ4δ1∫Ω∫Ath(t-s)|∇u(t)-∇u(s)|2dsdx+32‖∇u‖22∫Qth(t-s)ds+12∫Qth(t-s)‖∇u(s)‖22ds,t≥0.
The last term in the right-hand side of (3.7) will be estimated as follows:
(3.12)∫Ω|∫0th(t-s)(∇u(t)-∇u(s))ds|2dx≤(1+1δ2)κ∫Ω∫Ath(t-s)|∇u(t)-∇u(s)|2dsdx+(1+δ2)(∫Qth(t-s)ds)∫Ω∫Qth(t-s)|∇u(t)-∇u(s)|2dsdx,δ2>0.
For the fourth term in (3.7), it holds that
(3.13)-∫Ωut∫0th′(t-s)(u(t)-u(s))dsdx≤δ3‖ut‖22-Cp4δ3BV[h]∫Ω(h′□∇u)dx+Cp4δ3(∫Qtξ(t-s)ds)∫Ω∫Qtξ(t-s)|∇u(t)-∇u(s)|2dsdx,δ3>0,t≥0.
Moreover, from Lemma 2.4, for p>0 if n=1,2 and 0<p<2/(n-2) if n≥3, we find
(3.14)b(t)∫Ω|u|pu∫0th(t-s)(u(t)-u(s))dsdx≤δ4b2(t)‖u‖2(p+1)2(p+1)+Cp4δ4(∫0th(s)ds)∫Ω(h□∇u)dx≤δ4Ceb2(t)‖∇u‖22(p+1)+Cpκ4δ4∫Ω(h□∇u)dx≤2p+1δ4Ceb2(t)(1-κ)p+1Ep+1(t)+Cpκ4δ4∫Ω(h□∇u)dx,δ4>0,t≥0.
The definition of ℰ(t) in (2.5) allows us to write
(3.15)a(t)‖∇u‖22q∫Ω∇u⋅∫0th(t-s)(∇u(t)-∇u(s))dsdx≤a(t)‖∇u‖22q{δ5‖∇u‖22+κ4δ5∫Ω(h□∇u)dx}≤δ5a(t)‖∇u‖22(q+1)+2q-1κa(t)δ5(1-κ)qEq+1(t),δ5>0,t≥0.
Gathering all the relations (3.11)–(3.15) together with (3.7), we obtain for t≥t*(3.16)Φ2′(t)≤(1-h*)[δ1+32∫Qth(t-s)ds]‖∇u‖22+(δ3-h*)‖ut‖22+[(1-h*)κ4δ1+(1+1δ2)κ]∫Ω∫Ath(t-s)|∇u(t)-∇u(s)|2dsdx+12(1-h*)∫Qth(t-s)‖∇u(s)‖22ds-Cp4δ3BV[h]∫Ω(h′□∇u)dx+(1+δ2)(∫Qth(t-s)ds)∫Ω∫Qth(t-s)|∇u(t)-∇u(s)|2dsdx+2p+1δ4Ceb2(t)(1-κ)p+1Ep+1(t)+Cpκ4δ4∫Ω(h□∇u)dx+δ5a(t)‖∇u‖22(q+1)+2q-1κa(t)δ5(1-κ)qEq+1(t)+Cp4δ3(∫Qtξ(t-s)ds)∫Ω∫Qtξ(t-s)|∇u(t)-∇u(s)|2dsdx.In the following theorem we will assume thatp<q just to fix ideas. The result is also valid for p>q. It suffices to interchange p↔q and A(t)↔B(t) in the proof following it. The case p=q is easier.We will make use of the following hypotheses for some positive constantsA,B,U, and V to be determined.(A)
a(t) is a continuously differentiable function such that a′(t)<Aa(t),t≥0.(B)
b(t) is a continuously differentiable function such that b′(t)<Bb(t),t≥0.(C)
p>0 if n=1,2 and 0<p<2/(n-2) if n≥3.(D)
[∫0∞‍a(s)e-qsds]1/q[∫0∞‍b2(s)e-psds]1/p<U.(E)
[∫0∞‍a(s)e-q∫0s‍η(τ)dτds]1/q[∫0∞‍b2(s)e-p∫0s‍η(τ)dτds]1/p<V.Theorem 3.2.
Assume that the hypotheses (H1)–(H3), (A)–(C) hold andℛh<1/4. If limt→∞η(t)=η¯≠0, then, for global solutions and small ∫𝒬ξ(s)ds, there exist T1>0 and U>0 such that L(t)>U,t≥T1 or
(3.17)E(t)≤M1e-ν1t,t≥0
for some positive constants M1 and ν1 as long as (D) holds. If  η¯=0, then there exist T2>0 and V>0 such that L(t)>V,t≥T2 or
(3.18)E(t)≤M2e-ν2∫0t‍η(s)ds,t≥0
for some positive constants M2 and ν2 as long as (E) holds.Proof.
A differentiation ofΦ1(t) with respect to t along trajectories of (1.1) gives
(3.19)Φ1′(t):=‖ut‖22-‖∇u‖22+∫Ω∇u⋅∫0th(t-s)∇u(s)dsdx-a(t)‖∇u‖22(q+1)-b(t)‖u‖p+2p+2,
and Lemma 2.2 implies
(3.20)Φ1′(t)≤‖ut‖22-(1-κ2)‖∇u‖22+12∫0th(t-s)‖∇u(s)‖22ds-12∫Ω(h□∇u)dx-a(t)‖∇u‖22(q+1)-b(t)‖u‖p+2p+2,t≥0.
Next, a differentiation of Φ3(t) and Φ4(t) yields
(3.21)Φ3′(t)=Hγ(0)‖∇u‖22+∫0tHγ′(t-s)‖∇u(s)‖22ds=Hγ(0)‖∇u‖22-∫0tγ′(t-s)γ(t-s)Hγ(t-s)‖∇u(s)‖22ds-∫0th(t-s)‖∇u(s)‖22ds≤Hγ(0)‖∇u‖22-η(t)Φ3(t)-∫0th(t-s)‖∇u(s)‖22ds,t≥0,Φ4′(t)=Ψγ(0)‖∇u‖22+∫0tΨγ′(t-s)‖∇u(s)‖22ds=Ψγ(0)‖∇u‖22-∫0tγ′(t-s)γ(t-s)Ψγ(t-s)‖∇u(s)‖22ds-∫0tξ(t-s)‖∇u(s)‖22ds≤Ψγ(0)‖∇u‖22-η(t)Φ4(t)-∫0tξ(t-s)‖∇u(s)‖22ds,t≥0.
Taking into account Lemma3.1 and the relations (2.6), (3.20)-(3.21), we see that
(3.22)L′(t)≤(12-Cp4δ3λ2BV[h])∫Ω(h′□∇u)dx+[λ1+(δ3-h*)λ2]‖ut‖22+{λ2(1-h*)[δ1+32∫Qth(t-s)ds]+λ3Hγ(0)+λ4Ψγ(0)-λ1(1-κ2)}×‖∇u‖22+(λ12+λ2(1-h*)2-λ3)∫0th(t-s)‖∇u(s)‖22ds+[a′(t)2(q+1)+δ5λ2a(t)-λ1a(t)]‖∇u‖22(q+1)+[(1+δ2)λ2∫Qth(t-s)ds-λ12]×∫Ω(h□∇u)dx+λ2κ[1+1-h*4δ1+1δ2]∫Ω∫Ath(t-s)|∇u(t)-∇u(s)|2dsdx-λ3η(t)Φ3(t)-λ4η(t)Φ4(t)+[b′(t)p+2-λ1b(t)]‖u‖p+2p+2+λ2Cpκ4δ4∫Ω(h□∇u)dx+2q-1κa(t)δ5(1-κ)qλ2Eq+1(t)+2p+1δ4Ceb2(t)(1-κ)p+1λ2Ep+1(t)-λ4∫0tξ(t-s)‖∇u(s)‖22ds+λ2Cp4δ3(∫Qtξ(t-s)ds)∫Qtξ(t-s)‖∇u(t)-∇u(s)‖22ds,t≥t*.
Next, as in [17], we introduce the sets
(3.23)An:={s∈R+:nh′(s)+h(s)≤0},n∈N,
and observe that
(3.24)⋃n‍An=R+∖{Qh∪Nh},
where 𝒩h is the null set where h′ is not defined and 𝒬h is as in (3.2). Furthermore, if we denote 𝒬n:=R+∖𝒜n, then limn→∞h^(𝒬n)=h^(𝒬h) because 𝒬n+1⊂𝒬n for all n and ⋂n𝒬n=𝒬h∪𝒩h. Moreover, we designate by A~nt the sets
(3.25)A~nt:={s∈R+:0≤s≤t,nh′(t-s)+h(t-s)≤0},n∈N.
In (3.22), we take 𝒜t:=A~nt and 𝒬t:=𝒬~nt. Choosing λ1=(h*-ε)λ2, it is clear that
(3.26)(1+δ2)λ2κh^(Qn)-λ12≤0
for small ε and δ2, large n and t*, if h^(𝒬)<1/4. We deduce that
(3.27)(1+δ2)λ2∫Q~nth(t-s)ds-λ12<0.
Furthermore, if h^(𝒬)<1/4, then
(3.28)3(1-h*)2∫Q~nth(t-s)ds<δh*(1-κ2)
with
(3.29)δ=3(1-h*)κ4(2-κ)h*+β
and a small β>0. Pick
(3.30)λ3=12[λ1+λ2(1-h*)]
and Hγ(0) such that
(3.31)λ3Hγ(0)<λ2(1-δ)h*(2-κ)2.
Note that this is possible if t* is so large that h*>7κ/(8-κ) even though
(3.32)Hγ(0)=γ(0)-1∫0∞h(s)γ(s)ds≥∫0∞‍h(s)ds=κ.
Taking the relations (3.22)–(3.30) into account and selecting λ2<δ3/CpBV[h] so that
(3.33)12-Cpλ24δ3BV[h]≥14,
and small enough so that
(3.34)λ2κ(1+1-h*4δ1+1δ2+Cp4δ4)<14n,
we find for δ3=ε/2, large δ4, small Ψγ(0), and t≥t*(3.35)L′(t)≤-C1{‖ut‖22+‖∇u‖22+∫Ω(h□∇u)dx}+2p+1δ4Ceb2(t)(1-κ)p+1λ2Ep+1(t)+[a′(t)2(q+1)+δ5λ2a(t)-λ1a(t)]‖∇u‖22(q+1)+2q-1κa(t)δ5(1-κ)qλ2Eq+1(t)-λ3η(t)Φ3(t)-λ4η(t)Φ4(t)+12[1+λ2Cp2δ3∫Qtξ(t-s)ds]∫Ω∫Qnth′(t-s)|∇u(t)-∇u(s)|2dsdx+[b′(t)p+2-λ1b(t)]‖u‖p+2p+2-λ4∫0tξ(t-s)‖∇u(s)‖22ds
for some positive constant C1. Take λ4>1+(λ2Cp/2δ3)∫𝒬ξ(s)ds,δ5,∫𝒬ξ(s)ds small, and
(3.36)a′(t)2(q+1)<(λ1-α)a(t)
(i.e., A=2(q+1)(λ1-α) for some 0<α<λ1) and
(3.37)b′(t)p+2<(λ1-β)b(t)
(i.e., B=(p+2)(λ1-β) for some 0<β<λ1) to derive that
(3.38)L′(t)≤-C2E(t)+2p+1δ4Ceb2(t)(1-κ)p+1λ2Ep+1(t)+2q-1κa(t)δ5(1-κ)qλ2Eq+1(t)-λ3η(t)Φ3(t)-λ4η(t)Φ4(t)
for some positive constant C2.
Iflimt→∞η(t)≠0, then there exist a t^≥t* and C3>0 such that η(t)≥C3 for t≥t^. Thus, in virtue of Proposition 2.1, for C3>0, we have
(3.39)L′(t)≤-C3L(t)+B(t)Lp+1(t)+A(t)Lq+1(t),
where
(3.40)B(t):=2p+1δ4Ceλ2(1-κ)p+1ρ1p+1b2(t),A(t):=2q-1κλ2δ5(1-κ)qρ1q+1a(t).
If there exists a T≥t^ such that L(T)<[p∫0∞‍B~(s)ds]-1/p, where B~(t):=B(t)e-pC3(t-T), then from (3.39)
(3.41)[eC3(t-T)L(t)]′≤eC3(t-T)B(t)Lp+1(t)+eC3(t-T)A(t)Lq+1(t),
and it follows that for t≥T(3.42)L~(t)≤L(T)+∫Tt‍e-pC3(s-T)B(s)L~p+1(s)ds+∫Tt‍e-qC3(s-T)A(s)L~q+1(s)ds
with L~(t):=eC3(t-T)L(t). Now we apply Lemma 2.3 to get
(3.43)L~(t)≤[N-q-q∫0t‍A~(s)ds]-1/q,t≥T,
where A~(t):=e-qC3(t-T)A(t) and N:=[L(T)-p-p∫0∞‍B~(s)ds]-1/p. If, in addition, q∫0∞‍A~(s)ds<L(T)-q, then L~(t) is uniformly bounded by a positive constant C4. Thus
(3.44)L(t)≤C4e-C3t,t≥T,
and by continuity (3.44) holds for all t≥0.
Iflimt→∞η(t)=0, then for any C>0 there exists a t¯(C)≥t* such that η(t)≤C for t≥t¯(C). Therefore,
(3.45)L′(t)≤-C5η(t)L(t)+B(t)Lp+1(t)+A(t)Lq+1(t),t≥t¯=t¯(C2)
for some C5>0. The previous argument carries out with eC3(t-T) replaced by eC5∫Tt‍η(d)ds.
In case thatq<p, we reverse the roles of p and q in the argument above. The case p=q is clear.Remark 3.3.
The case where the derivative of the kernel does not approach zero on𝒜 (as is the case, for instance, when h′≤-Ch on 𝒜) is interesting. Indeed, the right-hand side in condition (3.34) will be replaced by C/4 with a possibly large constant C.Remark 3.4.
The argument clearly works for all kinds of kernels previously treated where derivatives cannot be positive or even take the value zero. In these cases there will be no need for the smallness conditions on the kernels. This work shows that derivatives may be positive (i.e., kernels may be increasing) on some “small” subintervals and open the door for (optimal) estimations and improvements of these sets.Remark 3.5.
The assumptionsa′(t)<2(q+1)(λ1-α)a(t) and b′(t)<(p+2)(λ1-β)b(t) may be relaxed to a′(t)<2(q+1)λ1a(t) and b′(t)<(p+2)λ1b(t), respectively. In this case α=α(t) and β=β(t) would depend on t.Remark 3.6.
The assertion in Theorem3.2 is an “alternative” statement. As a next step it would be nice to discuss the (sufficient conditions of) occurrence of each case in addition to the global existence.



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*Source: 936140-2012-08-16.xml*