# Boundary Stabilization of a Nonlinear Viscoelastic Equation with Interior Time-Varying Delay and Nonlinear Dissipative Boundary Feedback

Authors: Zaiyun Zhang; Jianhua Huang; Zhenhai Liu; Mingbao Sun
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We investigate a nonlinear viscoelastic equation with
interior time-varying delay and nonlinear dissipative boundary feedback.
Under suitable assumptions on the relaxation function and time-varying delay effect together with nonlinear dissipative boundary feedback, we
prove the global existence of weak solutions and asymptotic behavior of
the energy by using the Faedo-Galerkin method and the perturbed energy
method, respectively. This result improves earlier ones in the literature,
such as Kirane and Said-Houari (2011)
and Ammari et al. (2010). Moreover, we give an positive answer to the open
problem given by Kirane and Said-Houari (2011).


---

## Body

## 1. Introduction

In this paper, we consider the global existence and asymptotic behavior of a nonlinear viscoelastic equation with interior time-varying delay and nonlinear dissipative boundary feedback as follows:(1)utt-Δu+∫0t‍h(t-s)Δu(s)ds+aut(x,t-τ(t))=0,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkx∈Ω,t>0,u(x,t)=0,on⁡Γ0×(0,∞),∂u∂ν+g(ut(x,t))=0,on⁡Γ1×[0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,ut(x,t-τ(t))=f0(x,t),x∈Ω,-τ(0)≤t≤0,
where Ω is a bounded domain of Rn(n≥1) with a smooth boundary ∂Ω of C2,a is a positive real constant, τ(t)>0 represents the time-varying delay effect and the initial data u0,u1,f0 are given functions belonging to suitable spaces, h(t) is a positive function that represents the kernel of the memory term, g(ut) is nonlinear dissipative boundary feedback, and f0,h,g satisfy suitable assumptions (see in Section 2).This model appears in viscoelasticity (see [1, 2]). In the case of velocity-dependent material density (i.e., ρ=0) as well as presence of μ2=0 and in the absence of the memory effect (i.e., g=0), (1) reduces to the wave equation. There is large literature on the global existence and uniform stabilization of wave equations. We refer the readers to [3–5]. It is worth mentioning that Zhang and Miao [3] considered the nonlinear wave equation with dissipative term and boundary damping
(2)utt-Δu+a(x)ut+f(u)=0,inΩ×[0,∞),u=0,on⁡Γ1×[0,∞),∂u∂ν+g(ut)=0,on⁡Γ0×[0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),inΩ,
and they proved the existence and uniform decay of strong and weak solutions by using the Glerkin method and the multiplier technique, respectively. Later on, Zhang et al. [4] improved earlier ones in [3]. More precisely, they investigated the global existence and uniform stabilization of generalized dissipative Klein-Gordon equation with boundary damping
(3)utt-Δu+b(x)ut+f(u)+h(∇u)=0,inΩ×[0,∞),u=0,on⁡Γ1×[0,∞),∂u∂ν+g(ut)=0,on⁡Γ0×[0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),inΩ,
and they proved the existence and uniform decay of strong and weak solutions by using the nonlinear semigroup method, the perturbed energy method, and the multiplier technique. Quite recently, Cavalcanti et al. [6] considered the following model:
(4)utt-ΔMu+a(x)g(ut)=0,on⁡M×(0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),forx∈M,
where M is a smooth oriented embedded compact surface without boundary in R3 and ΔM  is the Laplace-Beltrami operator on manifold M; furthermore, they obtained explicit and optimal decay rates of the energy. Later on, Cavalcanti et al. [7] extended the result forn-dimensional compact Riemannian manifolds (M,g) with boundary in two ways: (i) by reducing arbitrarily the region where the dissipative effect lies (this gives us a totally sharp result with respect to the boundary measure and interior measure where the damping is effective) and (ii) by controlling the existence of subsets on the manifold that can be left without any dissipative mechanism, namely, a precise part of radially symmetric subsets. An analogous result holds for compact Riemannian manifolds without boundary.In the caseρ=0 and in the absence of delay (i.e., μ2=0), there is large literature on the existence and decay of nonlinear viscoelastic equation during the past decades. In [8], Cavalcanti et al. considered the exponential decay for the solution of viscoelastic wave equation with localized damping
(5)|ut|ρutt-Δu+∫0t‍g(t-s)Δu(s)ds+a(x)ut+u|u|r=0,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkx∈Ω,t>0.
Under the condition that a(x)≥a0>0 on ω⊂Ω, with ω satisfying some geometry restrictions and
(6)-ξ1g(t)≤g′(t)≤-ξ2g(t),t≥0,they proved an exponential decay result for the energy. Berrimi and Messaoudi [9] improved Cavalcanti’s result by introducing a differential functional which allowed to weaken the conditions on both a(x) and g. In [10], Cavalcanti and Oquendo studied
(7)|ut|ρutt-k0Δu+∫0t‍div⁡[a(x)g(t-s)∇u(s)]ds+b(x)h(ut)+f(u)=0,x∈Ω,t>0.
Under some geometric restrictions on ω and assuming that
(8)a(x)≥a0>0,∀x∈ω,-ξ1g(t)≤g′(t)≤-ξ2g(t),t≥0,a(x)+b(x)≥ρ>0,∀x∈Ω,
they established an exponential stability for the relaxation function g decaying exponentially and h linear and polynomial stability for g decaying polynomially and h nonlinear. It is worth mentioning that Zhang et al. [11] studied the following initial boundary value problem:
(9)utt+Au+∫0t‍g(t-s)Auds=0inΩ×(0,∞),u=0on⁡Γ×(0,∞),u(0)=u0,ut(0)=u1.
Furthermore, they showed that the solutions of (9) decay uniformly in time, with rates depending on the rate of decay of the kernel g. More precisely, the solution decays exponentially to zero provided that g decays exponentially to zero. When g decays polynomially, we show that the corresponding solution also decays polynomially to zero with the same rate of decay. For other related works, we refer the readers to [12–21] and the references therein.On the other hand, concerning the study of the following nonlinear viscoelastic equation with memory, there are a substantial number of contributions:(10)|ut|ρutt-Δu+∫0t‍g(t-s)Δu(s)ds+F(u,ut,utt)=0.
Recently, Han and Wang [22] investigated the following problem:
(11)|ut|ρutt-Δu-Δutt+∫0t‍g(t-s)Δu(s)ds=b|u|p-2u.
By introducing a new functional and using potential well method, the authors established the global existence and uniform decay if the initial data are in a suitable stable set. Cavalcanti et al. [23] studied a related problem with strong damping as follows:
(12)|ut|ρutt-Δu-Δutt+∫0t‍g(t-s)Δu(s)ds-γΔut=0.
By assuming 0<ρ≤2/(n-2), if n≥3 or ρ>0 and if n=1,2 and g(t) decays exponentially, they established that the global existence resulted for γ≥0 and the exponential decay of the energy for γ>0. This result has been extended to a situation γ=0 by Messaoudi and Tatar [24] and exponential decay and polynomial decay results have been shown in the absence as well as presence of a source term. Later on, inspired by the ideas of [25–27], Han and Wang [22] investigated the general decay of solutions of energy for the nonlinear viscoelastic equation
(13)|ut|ρutt-Δu-Δutt+∫0t‍g(t-s)Δu(s)ds+ut|ut|k=0.In recent years, the control of partial differential equation with time delay effects has become an active area of research; see, for instance, [28, 29] and the references therein. The presence of delay may be a source of instability. For instance, it was proved in [30–34] that an arbitrarily small delay may destabilize a system which is uniformly asymptotically stable in the absence of delay unless additional conditions or control terms have been used. In [32], Nicaise and Pignotti examined (1) with ρ=0,g≡0,μ1>0,μ2>0, and τ(t)=τ being a constant delay in the case of mixed homogeneous Dirichlet-Neumann boundary conditions, under a geometric condition on the Neumann part of the boundary. More precisely, they investigated the following system with linear frictional damping term and internal constant delay:
(14)utt(x,t)-Δu(x,t)+μ1ut(x,t)+μ2ut(x,t-τ)=0,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkx∈Ω,t>0u(x,t)=0,x∈Γ0,t>0∂u∂ν(x,t)=0,x∈Γ1,t>0
or with boundary constant delay
(15)utt(x,t)-Δu(x,t)=0,x∈Ω,t>0,u(x,t)=0,x∈Γ0,t>0,∂u∂ν(x,t)+μ1ut(x,t)+μ2ut(x,t-τ)=0,kkkkkkkkkkkkkkkkkkkkx∈Γ1,t>0.
In the presence of delay (μ2>0), Nicaise and Pignotti [32] examined systems (14) and (15) and proved under the assumptions μ2<μ1 that the energy is exponentially stable. Otherwise, they constructed a sequence of delays for which the corresponding solution is instable. The main approach used there is an observability inequality together with a Carleman estimate. See also [35] for treatment to these problems in more general abstract form and [36] for analogous results in the case of boundary time-varying delay. We also recall the result by Nicaise et al. [36], where the researchers proved the same result as in [32] for the one space dimension by applying the spectral analysis approach. Recently, Kirane and Said-Houari [37] considered (1) with ρ=0,μ1>0,μ2>0, and τ(t)≡τ being a constant delay in the case of the initial and Dirichlet boundary wave equation with a linear damping and a delay term as follows:
(16)utt-Δu+∫0t‍g(t-s)Δu(s)ds+μ1ut(x,t)+μ2ut(x,t-τ)=0,x∈Ω,t>0,u(x,t)=0,x∈∂Ω,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,ut(x,t-τ)=f0(x,t-τ),x∈Ω,t∈(0,τ).
Under an assumption between the weight of the delay term in the feedback and the weight of the term without delay, using the Faedo-Galerkin method combined with some energy estimate, they proved the global existence of (16). Also, they proved exponential decay of (16) via suitable Lyapunov functionals.Recently, the stability of PDEs with time-varying delays was studied in [38–44]. In [40], Nicaise and Pignotti investigated the stabilization problem by interior damping of the wave equation with internal time-varying delay feedback and established exponential stability estimates by introducing suitable Lyapunov functionals, under the condition |μ2|<1-dμ1 in which the positivity of the coefficient μ1 is not necessary. In [41], Nicaise et al. showed the exponential stability of the heat and wave equations with time-varying boundary delay in 1-D, under the condition 0≤μ2<1-dμ1, where d is a constant such that τ′(t)≤d<1.The rest of the paper is organized as follows. In Section2, we show some assumptions and state our main result. In Section 3, we present the proof of our main result. That is, we will prove the global existence by using Faedo-Galerkin method and establish the general decay result (including exponential decay and polynomial decay) by using the perturbed energy method. Finally, in Section 4, we give further remarks on this context.


## 2. Some Assumptions and Main Results

In this section, before proceeding to our analysis, we present some assumptions and state the main result. We use the standard Hilbert spaceL2(Ω) and the Sobolev space H01(Ω) with their usual scalar products and norms. Throughout this paper, Ci is used to denote a generic positive constant from line to line.For the relaxation functionh, we assume that(G1)
h(t):(0,∞)→(0,∞) is a nonincreasing differentiable function such that
(17)1-∫0∞‍h(s)ds=l>0;(G2)
there exists a nonincreasing differentiable functionζ(t) such that
(18)h′(t)≤-ζ(t)hp(t),1≤p<32,t≥0.We assume that ρ satisfies
(19)0<ρ≤2n-2,ifn≥3;ρ>0,ifn=1,2.
For the time-varying delay, we assume that there exist positive constant τ0,τ¯ such that
(20)0<τ0≤τ(t)≤τ¯,∀t>0.
Furthermore, we assume that the delay satisfies
(21)τ′(t)≤d<1,∀t>0,
that
(22)τ(t)∈W2,∞([0,T]),∀T>0,
and that μ1,μ2 satisfy
(23)|μ2|<1-dμ1.Remark 1.
We show an example of functions satisfying (G2) as follows:(24)h(s)=e-σs,p=1,h(s)=ϑ(1+s)-1/(p-1),p>1,
for σ,ϑ>0 to be chosen properly; see [2].Remark 2.
Conditionp<3/2 is imposed so that ∫0∞‍h2-p(s)ds<∞.Now, we are in a position to state our main results.Theorem 3.
Let (20)–(23) be satisfied and h satisfy (G2). Then, given (u0,u1)∈H01(Ω)×L2(Ω),f0∈L2(Ω×(0,1)), and T>0, there exists a unique weak solution u(x,t) such that
(25)u∈C(0,T;H01(Ω))∩C1(0,T;L2(Ω)),ut∈L2(0,T;H01(Ω))∩L2((0,T)×Ω).
Moreover, if (20)–(23) hold and h satisfies (G1) and (G2), then there exist two positive constants K,k such that for any solution of the problem (1) of the energy satisfies
(26)E(t)≤Ke-kt,p=1,t≥t0,(27)E(t)≤K(1+t)-1/(p-1),p>1,t≥t0.


## 3. Proof of the Main Result

In this section, we will divide our proof into two steps. In Step1, we prove the global existence of weak solutions by using Faedo-Galerkin method benefited from the ideas of [2, 3, 37]. In Step 2, we establish the general decay of energy by introducing the new energy functional and using the perturbed energy method inspired by the contributions; see, for instance, [2–4, 11, 39].Step 1 (global existence of weak solutions).
Let{ω}j∞ be an orthogonal basis of H01(Ω) with ωj being the eigenfunction of the following problem:
(28)-Δωj=λjωj,x∈Ω,ωj=0,x∈∂Ω.
Denote Wn=Span⁡{ω1,ω2,…,ωn} for subspace generated by the first n vectors of the basis of {ω}j∞. Then, we construct approximation of the solution (u,z) as follows:
(29)un(x,t)=∑j=1n‍gjn(t)ωj,zn(x,t,ρ)=∑j=1m‍hjn(t)ϕj(x,ρ)
and we choose two sequences u0n and u1n in Wn and a sequence z0n in Vn such that u0n→u0 strongly in H01(Ω),u1n→u1 strongly in L2(Ω), and z0n→f0 strongly in L2(Ω×(0,1)). Define the sequence ϕj(x,ρ) as follows: ϕj(x,0)=ϕj(x). Then, from [37, pp 1069], we may extend ϕj(x,0) by ϕj(x,ρ) over L2(Ω×(0,1)) and denote Vn=Span⁡{ϕ1,ϕ2,…,ϕn}.To facilitate further our analysis, we introduce as in [32, 36, 39] the new variable
(30)z(x,θ,t)=ut(x,t-τ(t)θ),x∈Ω,θ∈(0,1),t>0.
Then, we get
(31)τ(t)z(x,θ,t)+(1-τ′(t)θ)zθ(x,θ,t)=0,kkkkkkkkkkkx∈Ω,θ∈(0,1),t>0.
Therefore, the problem (1) can be rewritten as follows:
(32)|ut|ρutt-Δu+∫0t‍h(t-s)Δu(s)ds+μ1ut(x,t)+μ2z(x,1,t)=0,x∈Ω,t>0,τ(t)z(x,θ,t)+(1-τ′(t)θ)zθ(x,θ,t)=0,kkkkkkkkkkkkx∈Ω,θ∈(0,1),t>0u(x,t)=0,x∈∂Ω,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,kkkkkkkkkz(x,θ,0)=f0(x,-τ(0)),x∈Ω,θ∈(0,1),-τ(0)≤t≤0.
Hence, (un(t),zn(t)) are solutions to the following Cauchy problem as follows:
(33)∫Ω‍|utn|ρuttnωjdx+∫Ω‍∇un∇ωjdx-∫0t‍h(t-s)∇u(s)∇ωjdxds+∫Ω‍[μ1utn(x,t)+μ2zn(x,1,t)]ωjdx=0,zn(x,0,t)=utn(x,t),(un(0),utn(0))=(u0n,u1n),(34)∫Ω‍[τ(t)znt(x,θ,t)+(1-τ′(t)θ)znθ(x,θ,t)]ϕjdx=0,zn,0=z0n.
By standard method of ODE, we know that there exists only one local solution of the Cauchy problem (33) and (34) on some interval [0,tn),0<tn<T, for arbitrary T>0; then, this solution can be extended to the whole interval [0,T] by a priori estimates below.To facilitate further our analysis, we need some notations and technical Lemmas4 and 6. Let us first introduce some notations
(35)(ϕ⋆ψ)(t)=∫0t‍ϕ(t-s)ψ(s)ds;(ϕ⋄ψ)(t)=∫0t‍ϕ(t-s)|ψ(t)-ψ(s)|ds,(ϕ∘ψ)(t)=∫0t‍ϕ(t-s)∫Ω‍|ψ(t)-ψ(s)|2dxds,
with these notations; we have the following lemma given in [2, 11].Lemma 4.
Forϕ∈C1(R) and ψ∈H1(0,T), one has
(36)(ϕ⋆ψ)(t)·ψ(t)=-12ϕ(t)∥ψ(t)∥2+12(ϕ′♢ψ)(t)-12ddt[∫0t‍(ϕ♢ψ)(t)kkkkkkkkk-(∫0t‍ϕ(s)ds)|ϕ|2dx].Remark 5.
In fact, the proof of this lemma follows by differentiating the termg♢ϕ. More details are presented in [2, 11, 37].Lemma 6.
Assuming thatv∈L∞(0,T;H1(Ω)),h is a continuous function such that
(37)∫0∞‍h1-α(s)ds<∞,0≤α≤1.
Then, we have
(38)(h∘∇v)(t)≤2[∫0t‍∥∇v(s)∥22ds+t∥∇v(t)∥22](p-1)/p×((h∘∇v)(t))1/p.Proof.
It suffices to observe that, forq>1,0≤α≤1,
(39)(h∘∇v)(t)=∫0t‍h(t-s)∥∇v(t)-∇v(s)∥22ds=∫0t‍h(1-α)/q(t-s)∥∇v(t)-∇v(s)∥22/qh(q-1+α)/qkkkk×(t-s)∥∇v(t)-∇v(s)∥22(q-1)/qds.
By applying Hölder inequality, we obtain
(40)(h∘∇v)(t)≤(∫0t‍h(1-α)/q(t-s)∥∇v(t)-∇v(s)∥22ds)1/q×(∫0t‍h(q-1+α)/(q-1)(t-s)∥∇v(t)-∇v(s)∥22ds)(q-1)/q.
Taking q=(p-1+α)/(p-1), we get
(41)(h∘∇v)(t)≤(∫0t‍h(1-α)(p-1)/(p-1+α)(t-s)kkkkk×∥∇v(t)-∇v(s)∥22ds∫0t‍)(p-1)/(p-1+α)×(∫0t‍hpα/(p-1+α)(t-s)∥∇v(t)-∇v(s)∥22ds)α/(p-1+α).
Finally, taking α=1 in the above equality, Lemma 6 is completed.


### 3.1. A Priori Estimate

Takingωj=utn in (33) and integrating over (0,t), using integration by parts and Lemma 4, we obtain
(42)12[(1-∫0t‍h(s)ds)∥∇un∥22+2ρ+2∥utn∥ρ+2ρ+2+(h∘∇un)(t)]+μ1∫0t‍∥utn∥22ds+μ2∫0t‍∫Ω‍zn(x,1,s)utn(x,s)dxds+12∫0t‍h(s)∥∇un(s)∥22ds-12∫0t‍(h′∘∇un)(s)ds=12(∥∇u0∥22+2ρ+2∥u1∥ρ+2ρ+2).
Taking ϕj=zn(ξ/τ(t)) in (34) and integrating over (0,t), we get
(43)ξ2∫Ω‍∫01‍zn2(x,θ,t)dθdx+ξ∫0t‍∫Ω‍∫01‍1-τ′(t)θτ(t)znθzn(x,θ,s)dθdxds=ξ2∥z0n∥L2(Ω×(0,1))2.
Now, integrating by parts, we obtain
(44)∫0t‍∫Ω‍∫01‍1-τ′(t)θτ(t)znθzn(x,θ,s)dθdxds=12∫0t‍∫Ω‍∫01‍(∂∂θzn2(x,θ,s)1-τ′(t)θτ(t))dθdxds=-12∫0t‍∫Ω‍1-τ′(t)θτ(t)zn2(x,θ,s)dsdx+12∫0t‍∫Ω‍[(1-τ′(t)θ)zn2(x,1,s)-zn2(x,0,s)τ(t)]dxds.
It follows from (43) and (44) that
(45)ξ2∫Ω‍∫01‍zn2(x,θ,t)dθdx+ξ∫0t‍∫Ω‍∫01‍τ′(t)θ-1τ(t)zn2(x,θ,s)dxds+12∫0t‍∫Ω‍[(1-τ′(t)θ)zn2(x,1,s)-zn2(x,0,s)τ(t)]dxds=ξ2∥z0n∥L2(Ω×(0,1))2.
Summing up (42) and (45), we conclude that
(46)En(t)+μ1∫0t‍∥utn∥22ds+μ2∫0t‍∫Ω‍zn(x,1,s)utn(x,s)dxds+12∫0t‍h(s)∥∇un(s)∥22ds-12∫0t‍(h′∘∇un)(s)ds+ξ2∫0t‍∫Ω‍τ′(t)θ-1τ(t)zn2(x,θ,s)dxds+12∫0t‍∫Ω‍[(1-τ′(t)θ)zn2(x,1,s)-zn2(x,0,s)τ(t)]dxds=12∥∇u0∥22+1ρ+2∥utn∥ρ+2ρ+2+ξ2∥z0∥L2(Ω×(0,1))2=En(0),
where
(47)En(t)=12[(1-∫0t‍h(s)ds)∥∇un∥22+2ρ+2∥utn∥ρ+2ρ+2kk+(h∘∇un)(t)∫0t‍]+ξ2∥zn∥L2(Ω×(0,1))2.
Using Young’s inequality and noticing (20) and (21), we arrive at
(48)(μ1-μ2ξ2)∫0t‍∥utn∥22ds+∫0t‍∫Ω‍[ξ1-τ′(t)2τ(t)-μ22ξ]zn2(x,1,s)+12∫0t‍h(s)∥∇un(s)∥22ds-12∫0t‍(h′∘∇un)(s)ds+ξ2∫0t‍∫Ω‍τ′(t)θ-1τ(t)zn2(x,θ,s)dxds=En(0).
Choosing some value of τ(t)>0 and θ and noticing (20) and (21), we have (τ′(t)θ-1)/τ(t)>0. Moreover, choosing some value of τ(t)>0 and ξ, we obtain
(49)μ1-μ2ξ2>0,ξ1-τ′(t)2τ(t)-μ22ξ>0.
That is,
(50)μ2τ(t)1-τ′(t)<ξ<2μ1μ2.
In fact, by (20) and (21), we get μ2τ(t)0/(1-d)<ξ<2μ1/μ2. From (48) and (50), (G1), and (G1) and Lemma 6, we conclude that we can find a positive C independent of n, such that
(51)En(t)≤C.
Hence, using the fact that 1-∫0t‍h(s)ds≥l, the estimate (51), and equality (47), we deduce
(52)unis uniformly bounded inL∞(0,T;H01(Ω)),utnis uniformly bounded inL∞(0,T;L2(Ω)),znis uniformly bounded inL∞(0,T;L2(Ω×(0,1))).
By (52), we infer that there exist two subsequences un,zn (still denoted by un,zn) and two functions u and z, such that
(53)un⇀uweakly star inL∞(0,T;H01(Ω)),utn⇀utweakly star inL∞(0,T;L2(Ω)),zn⇀zweakly star inL∞(0,T;L2(Ω×(0,1))).
From (52), we have un is bounded in L2(0,T;H01(Ω)) and utn is bounded in L2(0,T;L2(Ω)). Consequently, un is bounded in L2(0,T;L2(Ω)). More details are present in [37, pp 1072].Since the Sobolev embeddingH1(0,T;H1(Ω))↪L2(0,T;L2(Ω)) is compact, using Aubin-Lions theorem (see [45]), we can extract a subsequence of un (still denoted by un), such that
(54)un⟶ustrongly inL2(0,T;L2(Ω)),utn⟶utstrongly inL2(0,T;L2(Ω)),
which implies utn→ut almost everywhere in Ω×(0,T).Hence,(55)|utn|ρutn⟶|ut|ρutalmost everywhere inΩ×(0,T).
On the other hand, by the Sobolev embedding theorem and estimate (51), this yields
(56)∥|utn|ρutn∥L2(0,T;L2(Ω))=∫0T‍∫Ω‍|utn|2(ρ+1)dxdt≤CS2(ρ+1)∫0T∥∇utn∥22(ρ+1)dt≤CS2(ρ+1)Cρ+1T,
where CS is the Sobolev embedding constant. Thus, using (55), (56), and Lions Lemma [46], we get
(57)|utn|ρutn⇀|ut|ρutweakly inL2(0,T;L2(Ω)).
Let D(0,T) be the space of C∞ functions with compact support in (0,T). Multiplying the first equation in (33) by Θ(t)∈D(0,T) and integrating over (0,T), we conclude that
(58)-1ρ+1∫0T‍(|utn|ρutn,ωj)Θt(t)dt+∫0T‍(∇utn,∇ωj)Θ(t)dt-∫0T‍∫0t‍h(t-s)(∇utn,∇ωj)Θt(t)dsdt+∫0T‍(μ1utn+μ2zn,ωj)Θ(t)dt=0.
Noticing that {ωj}j∞ is a basis of H01(Ω), via convergence (53) and (57), we can pass to the limit in (58) and obtain
(59)|ut|ρutt-Δu+∫0t‍h(t-s)Δu(s)ds+μ1ut(x,t)+μ2z(x,1,t)=0.
Similarly, we get
(60)τ(t)zt(x,θ,t)+(1-τ′(t)θ)zθ(x,θ,t)=0.
From (53) and given the label of lemma in [46], we obtain
(61)un(0)⇀u(0)weakly inH01(Ω);utn(0)⇀ut(0)weakly inL2(Ω).
Therefore, we have u(0)=u0,ut(0)=u1. Consequently, the global existence of weak solution is established.Step 2 (general decay of the energy).
First, we introduce the new energy functionalE(t) and the perturbed energy Eε(t); then we apply the perturbed energy method to establish general decay of the energy. More precisely, the method used is based on the construction of suitable Lyapunov functionals E(t) and Eε(t) satisfying
(62)ddtEε(t)≤-C1Eε(t)+C2E(t)-rt
for some positive constants C1,C2,R. More details are present in [3, pp 1017] or [2, 4, 16].Now, we introduce the new energy functional as follows:(63)E(t)=E(u,z,t)=12[2ρ+2∥ut∥ρ+2ρ+2+(1-∫0t‍h(s)ds)∥∇u∥22+(h∘∇u)(t)]+ξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx,
where ξ,λ are suitable positive constants.Next, we will fixξ such that
(64)2μ1-|μ2|1-d-ξ>0,ξ-|μ2|1-d>0,λ<1τ¯|log⁡⁡-|μ2|ξ1-d|.Remark 7.
In fact, the existence of such a constantξ is guaranteed by the assumption (23).Therefore, we have the following lemma.Lemma 8.
Let (20)–(23) be satisfied and h satisfy (G1). Then, for the solution of problem (1), the energy functional defined by (63) is nonincreasing and satisfies
(65)E′(t)≤12(h∘∇u)(t)-12h(t)∫Ω‍|∇u|dx-C1∫Ω‍[ut2(x,t)+ut2(x,t-τ(t))]dx-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx≤0,
for some positive constant C1.Proof of Lemma8.
Differentiating (63) and noticing the first equation in (1) together with
(66)(h∘∇u)(t)=∫Ω‍∫0t‍h(t-s)|∇u(t)-∇u(s)|2dsdx,
we obtain
(67)E′(t)=∫Ω‍|ut|ρ+1utt-12h(t)∫Ω‍|∇u|2dx+(1-∫0t‍h(s)ds)∫Ω‍∇u·∇utdx+∫0t‍h(t-s)ds∫Ω‍[∇u(t)-∇u(s)]dsdx+12∫0t‍h′(t-s)∫Ω‍|∇u(t)-∇u(s)|2dsdx+ξ2∫Ω‍ut2(x,t)dx-ξ2∫Ω‍e-λτ(t)ut2(x,t-τ(t))(1-τ′(t))dx-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx=∫Ω‍ut[Δu-∫0t‍h(t-s)Δu(s)ds-μ1ut(x,t)kkkkkkk-μ2ut(x,t-τ(t))∫0t‍]dx-12h(t)∫Ω‍|∇u|2dx+∫Ω‍∇u·∇utdx-∫0t‍h(s)ds∫Ω‍∇u·∇utdx+∫0t‍h(t-s)ds∫Ω‍[∇u(t)-∇u(s)]dsdx+12(h′∘∇u)(t)+ξ2∫Ω‍ut2(x,t)dx-ξ2∫Ω‍e-λτ(t)ut2(x,t-τ(t))(1-τ′(t))dx-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx=-μ1∫Ω‍ut2(x,t)dx-μ2∫Ω‍ut(x,t)ut(x,t-τ(t))dx-12h(t)∫Ω‍|∇u|2dx+12(h′∘∇u)(t)+ξ2∫Ω‍ut2(x,t)dx-ξ2∫Ω‍e-λτ(t)ut2(x,t-τ(t))(1-τ′(t))dx-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx.
Applying Young’s inequality, we obtain
(68)-μ2∫Ω‍ut(x,t)ut(x,t-τ(t))dx≤|μ2|21-d∫Ω‍ut2(x,t)dx+|μ2|1-d2∫Ω‍ut2(x,t-τ(t))dx.
Integrating by parts, using the assumption (20), (21) and (67), (68), we arrive at
(69)E′(t)≤-μ1∫Ω‍ut2(x,t)dx-μ2∫Ω‍ut(x,t)ut(x,t-τ(t))dx-12h(t)∫Ω‍|∇u|2dx+12(h′∘∇u)(t)+ξ2∫Ω‍ut2(x,t)dx-ξ2(1-d)e-λτ¯×∫Ω‍ut2(x,t-τ(t))dxhhhhhh(0<τ0≤τ(t)≤τ¯,τ′(t)≤d<1)-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx≤12(h′∘∇u)(t)-12h(t)∫Ω‍|∇u|2dx-(μ1-|μ2|21-d-ξ2)∫Ω‍ut2(x,t)dx-(ξ2(1-d)e-λτ¯-|μ2|1-d2)×∫Ω‍ut2(x,t-τ(t))dx-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx.
Combining (64) and (69) and the assumptions (G1) and (G2), (65) is established.Next, we introduce the following functionals:(70)Φ(t)=1ρ+1∫Ω‍|ut|ρ+1udx,(71)Ψ(t)=-1ρ+1∫Ω‍|ut|ρ+1∫0t‍h(t-s)[u(t)-u(s)]dsdx.
Set
(72)L(t)=NE(t)+εΦ(t)+Ψ(t),
where N and ε are suitable positive constants to be determined later.Remark 9.
Indeed, we easily see that, forε small enough while N large enough, there exist two positive constants α0,α1, such that
(73)α0E(t)≤L(t)≤α1E(t),∀t≥0.
Concerning the estimates of Φ(t),Ψ(t), we have the following lemmas.Lemma 10.
Under the assumption (G1), the functionalΦ(t) satisfies the estimate
(74)Φ′(t)≤-l2∫Ω‍|∇u|2dx+C2∫Ω‍[ut2(x,t)+ut2(x,t-τ(t))]dx+C3(h∘∇u).Proof of Lemma10.
Differentiating (70) and integrating by parts, we get
(75)Φ′(t)=∫Ω‍|ut|ρ+1uttudx+1ρ+1∫Ω‍|ut|ρ+2dx=∫Ω‍u[Δu-∫0t‍h(t-s)Δu(s)ds-μ1ut(x,t)kkkkkkkk-μ2ut(x,t-τ(t))∫0t‍]dx+1ρ+1∫Ω‍|ut|ρ+2dx=-∫Ω‍|∇u|2dx+∫Ω‍∇u·∫0t‍h(t-s)∇u(s)dsdx-μ1∫Ω‍uut(x,t)dx-μ2∫Ω‍uut(x,t-τ(t))dx1ρ+1∫Ω‍|ut|ρ+2dx=-l∫Ω‍|∇u|2dx+∫Ω‍∇u·∫0t‍h(t-s)[∇u(s)-∇u(t)]dsdx-μ1∫Ω‍uut(x,t)dx-μ2∫Ω‍uut(x,t-τ(t))dx+1ρ+1∫Ω‍|ut|ρ+2dx.
Using Young’s inequality and (G1), we obtain (see [2])
(76)∫Ω‍∇u·∫0t‍h(t-s)[∇u(s)-∇u(t)]dsdx≤δ∫Ω‍|∇u|2dx+14δ∫Ω‍[∫0t‍h(t-s)|∇u(s)-∇u(t)|ds]2dx≤δ∫Ω‍|∇u|2dx+1-l4(h∘∇u)(t),∀δ>0.
Also, applying Young’s and Poincaré’s inequality yields
(77)-μ1∫Ω‍uut(x,t)dx≤δ∫Ω‍|∇u|2dx+C(δ)∫Ω‍ut2(x,t)dx,-μ2∫Ω‍uut(x,t-τ(t))dx≤δ∫Ω‍|∇u|2dx+C(δ)∫Ω‍ut2(x,t-τ(t))dx.
Noticing (75)–(77) and choosing δ small enough, we obtain estimate (74).Lemma 11.
Under the assumption (G1), the functionalΨ(t) satisfies the estimate
(78)Ψ′(t)≤-(∫0t‍h(s)ds-2δ)∫Ωut2dx+δ∫Ω‍|∇u|2dx+C4δ(h∘∇u)(t)-C5δ(h′∘∇u)(t)+δ∫Ω‍ut2(x,t-τ(t))dx.Proof of Lemma11.
Differentiating (71), integrating by parts, and noticing the first equation in (1), we have
(79)Ψ′(t)=-∫Ω‍|ut|ρutt∫0t‍h(t-s)[u(t)-u(s)]dsdx-1ρ+1∫Ω‍|ut|ρut∫0t‍h′(t-s)[u(t)-u(s)]dsdx-(∫0t‍h(s)ds)∫Ω‍1ρ+1|ut|ρ+2dx=∫Ω‍[-Δu+∫0t‍h(t-s)Δu(s)dskkkkk+μ1ut(x,t)+μ2ut(x,t-τ(t))∫0t‍]×∫0t‍h(t-s)[u(t)-u(s)]dsdx-1ρ+1∫Ω‍|ut|ρut∫0t‍h′(t-s)[u(t)-u(s)]dsdx-(∫0t‍h(s)ds)∫Ω‍1ρ+1|ut|ρ+2dx=∫Ω‍∇u·∫0t‍h(t-s)[∇u(t)-∇u(s)]dsdx+∫Ω‍∫0t‍h(t-s)Δu(s)dskkkkk×∫0t‍h(t-s)[u(t)-u(s)]dsdx+∫Ω‍[∫0t‍h(t-s)[u(t)-u(s)]ds]kkkkk×[μ1ut(x,t)+μ2ut(x,t-τ(t))]dx-1ρ+1∫Ω‍|ut|ρut∫0t‍h′(t-s)[u(t)-u(s)]dsdx-(∫0t‍h(s)ds)∫Ω‍1ρ+1|ut|ρ+2dx.
Observe that
(80)∫Ω‍∫0t‍h(t-s)Δu(s)ds∫0t‍h(t-s)[u(t)-u(s)]dsdx=-∫Ω‍[∫0t‍h(t-s)∇u(s)dskkkkkkkkk×∫0t‍h(t-s)[∇u(t)-∇u(s)]ds]dx=-∫Ω‍[∫0t‍h(t-s)[∇u(s)-∇u(t)+∇u(s)]dskkkkkkkkk×∫0t‍h(t-s)[∇u(t)-∇u(s)]ds]dx=-∫Ω‍[∫0t‍h(t-s)[∇u(s)-∇u(t)]ds]2dx-(∫0t‍h(t)ds)×∫Ω‍∇u·∫0t‍h(t-s)[∇u(t)-∇u(s)]dsdx.
It follows from (79) and (80) that
(81)Ψ′(t)=(1-∫0t‍h(s)ds)×∫Ω‍∇u·∫0t‍h(t-s)[∇u(t)-∇u(s)]dsdx+∫Ω‍[∫0t‍h(t-s)[∇u(s)-∇u(t)]ds]2dx+∫Ω‍[∫0t‍h(t-s)[u(t)-u(s)]ds]kkkkk×[μ1ut(x,t)+μ2ut(x,t-τ(t))]dx-1ρ+1∫Ω‍|ut|ρut∫0t‍h′(t-s)[u(t)-u(s)]dsdx-(∫0t‍h(s)ds)∫Ω‍1ρ+1|ut|ρ+2dx.
Using Young’s and Poincaré’s inequality, we get (see [2])
(82)(1-∫0t‍h(s)ds)∫Ω‍∇u·∫0t‍h(t-s)[∇u(t)-∇u(s)]dsdx≤δ∫Ω‍|∇u|2dx+Cδ(h∘∇u)(t),-1ρ+1∫Ω‍|ut|ρut∫0t‍h′(t-s)[u(t)-u(s)]dsdx≤δ∫Ω‍ut2dx-Cδ(h′∘∇u)(t).
From (81) and (82), we derive Lemma 11.Now, we are ready to finalize our proof of general decay of the energy. Sinceh is positive, we have
(83)∫0t‍h(s)ds≥∫0t0‍h(s)ds=g0,∀t≥t0.
It follows from (65), (72), (74), and (78) that
(84)L′(t)=NE(t)+εΦ′(t)+Ψ′(t)≤N2(h′∘∇u)(t)-N2h(t)∫Ω‍|∇u|2dx-NC1∫Ω‍[ut2(x,t)+ut2(x,t-τ(t))]dx-λξN2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx+εC2∫Ω‍[ut2(x,t)+ut2(x,t-τ(t))]dx-εl2∫Ω‍|∇u|2dxP+εC3(h∘∇u)(t)-(∫0t‍h(s)ds-2δ)∫Ω‍ut2dx+δ∫Ω‍|∇u|2dx+C4δ(h∘∇u)(t)-C5δ(h′∘∇u)(t)+δ∫Ω‍ut2(x,t-τ(t))dx=-[(NC1+g0)-2δ-εC2]∫Ω‍ut2(x,t)dx+(εC3+C4δ)(h∘∇u)(t)+(N2-C5δ)(h′∘∇u)(t)-(εl2-δ)∫Ω‍|∇u|2dx-(NC1-δ-εC2)∫Ω‍ut2(x,t-τ(t))-λξN2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx.
If we choose some constants in the inequality (84), such that
(85)a1=(NC1+g0)-2δ-εC2>0,a3=NC1-δ-εC2>0,a2=εl2-δ>0,a4=N2-C5δ>0,a5=εC3+C4δ>0,a6=λξN2,
then we conclude that
(86)L′(t)≤-a1∫Ω‍ut2(x,t)dx-a2∫Ω‍|∇u|2dx+a4(h′∘∇u)(t)+a5(h∘∇u)(t)-a6∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx.
Hence, we have two cases to consider the general decay results as follows.Case 1 (p=1).
Choosing some values ofa1,a2,a3,a4,a5,a6 and noticing the definition of E(t) (see (63)), we conclude that there exists a constant β1>0, such that
(87)L′(t)≤-β1E(t),∀t≥0.
Therefore, by Remark 9 and (87), we get
(88)L′(t)≤-β1α1E(t),∀t≥0.
Integrating (88) over (0,t), we obtain
(89)L(t)≤L(0)e-(β1/α1)t,∀t≥0.
Observing Remark 9 (i.e., α0E(t)≤L(t)≤α1E(t)) and (89), we derive
(90)α0E(t)≤L(t)≤L(0)e-(β1/α1)t,∀t≥0.
That is,
(91)E(t)≤L(0)α0e-(β1/α1)t=˙Kekt,p=1,∀t≥0.
Assuming K=L(0)/α0,k=β1/α1, we obtain the exponential decay of the energy. So, (26) is established.Case 2 (1<p<3/2).
Due to (G2), we easily see that(92)∫0∞‍h1-r(s)ds<∞,0≤r≤2-p.
From the sketch of proof of Lemma 6, we observe that
(93)(h∘∇u)(t)≤C[∫0∞‍h1-α(s)dsE(0)](p-1+α)/(p-1)×[(hp∘∇u)(t)]α/(p-1+α).
Thus, forσ>1, using (63) and (93), we get
(94)Eσ(t)≤C[Eσ-1(0)(∥ut∥ρ+2ρ+2+∥∇u∥22+∥ut∥22)kkkk+(h∘∇u)σ(t)∥ut∥ρ+2ρ+2]≤CEσ-1(0)[∥ut∥ρ+2ρ+2+∥∇u∥22+∥ut∥22]+C[(∫0∞‍h1-α(s)ds)E(0)](p-1+α)/(p-1)×[(hp∘∇u)(t)]σα/(p-1+α).Choosingα=1/2,σ=2p-1 (i.e., σα/(p-1+α)=1) (94) reduces to
(95)Eσ(t)≤C[∥ut∥ρ+2ρ+2+∥∇u∥22+∥ut∥22+(hp∘∇u)(t)].
Combining (86) and (87) with Remark 9, we obtain
(96)L′(t)≤-β1Cα1σLσ(t),∀t≥0.
A simple integration of (96) over (0,t) yields
(97)L′(t)≤C6(1+t)-1/(σ-1),∀t≥0.
As a consequence of (97), we obtain
(98)∫0∞‍L(t)dt+sup⁡t≥0tF(t)<∞.
So, by using Lemma 6, we have
(99)(h∘∇u)(t)≤C[∫0t‍∥u(s)∥H1(s)ds+t∥u∥H1(Ω)](p-1)/p×(hp∘∇u)1/p(t)≤C[∫0t‍F(s)ds+tF(t)](p-1)/p(hp∘∇u)1/p(t)≤C(hp∘∇u)1/p(t)
which implies that
(100)(hp∘∇u)(t)≥C(h∘∇u)p(t).
Consequently, from (86) and (100), we have
(101)L′(t)≤-C7[∥ut∥ρ+2ρ+2+∥∇u∥22+∥ut∥22+(h∘∇u)p(t)],kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk∀t≥0.
On the other hand, similarly to (95), we
(102)Ep(t)≤C8[∥ut∥ρ+2ρ+2+∥∇u∥22+∥ut∥22+(h∘∇u)p(t)],kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk∀t≥0.
Then, it follows from Remark 9, (101), and (102) that
(103)L′(t)≤-C9Lp(t),∀t≥0.
A simple integration of (103) over (0,t) gives
(104)L′(t)≤K(1+t)-1/(p-1),∀t≥0.
By (104) and Remark 9, we obtain the polynomial decay of the energy. That is,
(105)E(t)≤K(1+t)-1/(p-1),∀t≥0.
Thus, our main result is completed.Remark 12.
Our novel contribution is to show that our work improves earlier result in [37] in which only the exponential decay was investigated. More precisely, Kirane and Said-Houari [37] considered the exponential decay of problem (1) with a constant delay (i.e., τ(t)=τ) and velocity-independent material density (i.e., ρ=0).Remark 13.
By using the fact that energyE is bounded on [0,t0], we can easily show that estimates (26) and (27) hold for t≥0. (See, for instance, [2].)


## 3.1. A Priori Estimate

Takingωj=utn in (33) and integrating over (0,t), using integration by parts and Lemma 4, we obtain
(42)12[(1-∫0t‍h(s)ds)∥∇un∥22+2ρ+2∥utn∥ρ+2ρ+2+(h∘∇un)(t)]+μ1∫0t‍∥utn∥22ds+μ2∫0t‍∫Ω‍zn(x,1,s)utn(x,s)dxds+12∫0t‍h(s)∥∇un(s)∥22ds-12∫0t‍(h′∘∇un)(s)ds=12(∥∇u0∥22+2ρ+2∥u1∥ρ+2ρ+2).
Taking ϕj=zn(ξ/τ(t)) in (34) and integrating over (0,t), we get
(43)ξ2∫Ω‍∫01‍zn2(x,θ,t)dθdx+ξ∫0t‍∫Ω‍∫01‍1-τ′(t)θτ(t)znθzn(x,θ,s)dθdxds=ξ2∥z0n∥L2(Ω×(0,1))2.
Now, integrating by parts, we obtain
(44)∫0t‍∫Ω‍∫01‍1-τ′(t)θτ(t)znθzn(x,θ,s)dθdxds=12∫0t‍∫Ω‍∫01‍(∂∂θzn2(x,θ,s)1-τ′(t)θτ(t))dθdxds=-12∫0t‍∫Ω‍1-τ′(t)θτ(t)zn2(x,θ,s)dsdx+12∫0t‍∫Ω‍[(1-τ′(t)θ)zn2(x,1,s)-zn2(x,0,s)τ(t)]dxds.
It follows from (43) and (44) that
(45)ξ2∫Ω‍∫01‍zn2(x,θ,t)dθdx+ξ∫0t‍∫Ω‍∫01‍τ′(t)θ-1τ(t)zn2(x,θ,s)dxds+12∫0t‍∫Ω‍[(1-τ′(t)θ)zn2(x,1,s)-zn2(x,0,s)τ(t)]dxds=ξ2∥z0n∥L2(Ω×(0,1))2.
Summing up (42) and (45), we conclude that
(46)En(t)+μ1∫0t‍∥utn∥22ds+μ2∫0t‍∫Ω‍zn(x,1,s)utn(x,s)dxds+12∫0t‍h(s)∥∇un(s)∥22ds-12∫0t‍(h′∘∇un)(s)ds+ξ2∫0t‍∫Ω‍τ′(t)θ-1τ(t)zn2(x,θ,s)dxds+12∫0t‍∫Ω‍[(1-τ′(t)θ)zn2(x,1,s)-zn2(x,0,s)τ(t)]dxds=12∥∇u0∥22+1ρ+2∥utn∥ρ+2ρ+2+ξ2∥z0∥L2(Ω×(0,1))2=En(0),
where
(47)En(t)=12[(1-∫0t‍h(s)ds)∥∇un∥22+2ρ+2∥utn∥ρ+2ρ+2kk+(h∘∇un)(t)∫0t‍]+ξ2∥zn∥L2(Ω×(0,1))2.
Using Young’s inequality and noticing (20) and (21), we arrive at
(48)(μ1-μ2ξ2)∫0t‍∥utn∥22ds+∫0t‍∫Ω‍[ξ1-τ′(t)2τ(t)-μ22ξ]zn2(x,1,s)+12∫0t‍h(s)∥∇un(s)∥22ds-12∫0t‍(h′∘∇un)(s)ds+ξ2∫0t‍∫Ω‍τ′(t)θ-1τ(t)zn2(x,θ,s)dxds=En(0).
Choosing some value of τ(t)>0 and θ and noticing (20) and (21), we have (τ′(t)θ-1)/τ(t)>0. Moreover, choosing some value of τ(t)>0 and ξ, we obtain
(49)μ1-μ2ξ2>0,ξ1-τ′(t)2τ(t)-μ22ξ>0.
That is,
(50)μ2τ(t)1-τ′(t)<ξ<2μ1μ2.
In fact, by (20) and (21), we get μ2τ(t)0/(1-d)<ξ<2μ1/μ2. From (48) and (50), (G1), and (G1) and Lemma 6, we conclude that we can find a positive C independent of n, such that
(51)En(t)≤C.
Hence, using the fact that 1-∫0t‍h(s)ds≥l, the estimate (51), and equality (47), we deduce
(52)unis uniformly bounded inL∞(0,T;H01(Ω)),utnis uniformly bounded inL∞(0,T;L2(Ω)),znis uniformly bounded inL∞(0,T;L2(Ω×(0,1))).
By (52), we infer that there exist two subsequences un,zn (still denoted by un,zn) and two functions u and z, such that
(53)un⇀uweakly star inL∞(0,T;H01(Ω)),utn⇀utweakly star inL∞(0,T;L2(Ω)),zn⇀zweakly star inL∞(0,T;L2(Ω×(0,1))).
From (52), we have un is bounded in L2(0,T;H01(Ω)) and utn is bounded in L2(0,T;L2(Ω)). Consequently, un is bounded in L2(0,T;L2(Ω)). More details are present in [37, pp 1072].Since the Sobolev embeddingH1(0,T;H1(Ω))↪L2(0,T;L2(Ω)) is compact, using Aubin-Lions theorem (see [45]), we can extract a subsequence of un (still denoted by un), such that
(54)un⟶ustrongly inL2(0,T;L2(Ω)),utn⟶utstrongly inL2(0,T;L2(Ω)),
which implies utn→ut almost everywhere in Ω×(0,T).Hence,(55)|utn|ρutn⟶|ut|ρutalmost everywhere inΩ×(0,T).
On the other hand, by the Sobolev embedding theorem and estimate (51), this yields
(56)∥|utn|ρutn∥L2(0,T;L2(Ω))=∫0T‍∫Ω‍|utn|2(ρ+1)dxdt≤CS2(ρ+1)∫0T∥∇utn∥22(ρ+1)dt≤CS2(ρ+1)Cρ+1T,
where CS is the Sobolev embedding constant. Thus, using (55), (56), and Lions Lemma [46], we get
(57)|utn|ρutn⇀|ut|ρutweakly inL2(0,T;L2(Ω)).
Let D(0,T) be the space of C∞ functions with compact support in (0,T). Multiplying the first equation in (33) by Θ(t)∈D(0,T) and integrating over (0,T), we conclude that
(58)-1ρ+1∫0T‍(|utn|ρutn,ωj)Θt(t)dt+∫0T‍(∇utn,∇ωj)Θ(t)dt-∫0T‍∫0t‍h(t-s)(∇utn,∇ωj)Θt(t)dsdt+∫0T‍(μ1utn+μ2zn,ωj)Θ(t)dt=0.
Noticing that {ωj}j∞ is a basis of H01(Ω), via convergence (53) and (57), we can pass to the limit in (58) and obtain
(59)|ut|ρutt-Δu+∫0t‍h(t-s)Δu(s)ds+μ1ut(x,t)+μ2z(x,1,t)=0.
Similarly, we get
(60)τ(t)zt(x,θ,t)+(1-τ′(t)θ)zθ(x,θ,t)=0.
From (53) and given the label of lemma in [46], we obtain
(61)un(0)⇀u(0)weakly inH01(Ω);utn(0)⇀ut(0)weakly inL2(Ω).
Therefore, we have u(0)=u0,ut(0)=u1. Consequently, the global existence of weak solution is established.Step 2 (general decay of the energy).
First, we introduce the new energy functionalE(t) and the perturbed energy Eε(t); then we apply the perturbed energy method to establish general decay of the energy. More precisely, the method used is based on the construction of suitable Lyapunov functionals E(t) and Eε(t) satisfying
(62)ddtEε(t)≤-C1Eε(t)+C2E(t)-rt
for some positive constants C1,C2,R. More details are present in [3, pp 1017] or [2, 4, 16].Now, we introduce the new energy functional as follows:(63)E(t)=E(u,z,t)=12[2ρ+2∥ut∥ρ+2ρ+2+(1-∫0t‍h(s)ds)∥∇u∥22+(h∘∇u)(t)]+ξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx,
where ξ,λ are suitable positive constants.Next, we will fixξ such that
(64)2μ1-|μ2|1-d-ξ>0,ξ-|μ2|1-d>0,λ<1τ¯|log⁡⁡-|μ2|ξ1-d|.Remark 7.
In fact, the existence of such a constantξ is guaranteed by the assumption (23).Therefore, we have the following lemma.Lemma 8.
Let (20)–(23) be satisfied and h satisfy (G1). Then, for the solution of problem (1), the energy functional defined by (63) is nonincreasing and satisfies
(65)E′(t)≤12(h∘∇u)(t)-12h(t)∫Ω‍|∇u|dx-C1∫Ω‍[ut2(x,t)+ut2(x,t-τ(t))]dx-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx≤0,
for some positive constant C1.Proof of Lemma8.
Differentiating (63) and noticing the first equation in (1) together with
(66)(h∘∇u)(t)=∫Ω‍∫0t‍h(t-s)|∇u(t)-∇u(s)|2dsdx,
we obtain
(67)E′(t)=∫Ω‍|ut|ρ+1utt-12h(t)∫Ω‍|∇u|2dx+(1-∫0t‍h(s)ds)∫Ω‍∇u·∇utdx+∫0t‍h(t-s)ds∫Ω‍[∇u(t)-∇u(s)]dsdx+12∫0t‍h′(t-s)∫Ω‍|∇u(t)-∇u(s)|2dsdx+ξ2∫Ω‍ut2(x,t)dx-ξ2∫Ω‍e-λτ(t)ut2(x,t-τ(t))(1-τ′(t))dx-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx=∫Ω‍ut[Δu-∫0t‍h(t-s)Δu(s)ds-μ1ut(x,t)kkkkkkk-μ2ut(x,t-τ(t))∫0t‍]dx-12h(t)∫Ω‍|∇u|2dx+∫Ω‍∇u·∇utdx-∫0t‍h(s)ds∫Ω‍∇u·∇utdx+∫0t‍h(t-s)ds∫Ω‍[∇u(t)-∇u(s)]dsdx+12(h′∘∇u)(t)+ξ2∫Ω‍ut2(x,t)dx-ξ2∫Ω‍e-λτ(t)ut2(x,t-τ(t))(1-τ′(t))dx-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx=-μ1∫Ω‍ut2(x,t)dx-μ2∫Ω‍ut(x,t)ut(x,t-τ(t))dx-12h(t)∫Ω‍|∇u|2dx+12(h′∘∇u)(t)+ξ2∫Ω‍ut2(x,t)dx-ξ2∫Ω‍e-λτ(t)ut2(x,t-τ(t))(1-τ′(t))dx-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx.
Applying Young’s inequality, we obtain
(68)-μ2∫Ω‍ut(x,t)ut(x,t-τ(t))dx≤|μ2|21-d∫Ω‍ut2(x,t)dx+|μ2|1-d2∫Ω‍ut2(x,t-τ(t))dx.
Integrating by parts, using the assumption (20), (21) and (67), (68), we arrive at
(69)E′(t)≤-μ1∫Ω‍ut2(x,t)dx-μ2∫Ω‍ut(x,t)ut(x,t-τ(t))dx-12h(t)∫Ω‍|∇u|2dx+12(h′∘∇u)(t)+ξ2∫Ω‍ut2(x,t)dx-ξ2(1-d)e-λτ¯×∫Ω‍ut2(x,t-τ(t))dxhhhhhh(0<τ0≤τ(t)≤τ¯,τ′(t)≤d<1)-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx≤12(h′∘∇u)(t)-12h(t)∫Ω‍|∇u|2dx-(μ1-|μ2|21-d-ξ2)∫Ω‍ut2(x,t)dx-(ξ2(1-d)e-λτ¯-|μ2|1-d2)×∫Ω‍ut2(x,t-τ(t))dx-λξ2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx.
Combining (64) and (69) and the assumptions (G1) and (G2), (65) is established.Next, we introduce the following functionals:(70)Φ(t)=1ρ+1∫Ω‍|ut|ρ+1udx,(71)Ψ(t)=-1ρ+1∫Ω‍|ut|ρ+1∫0t‍h(t-s)[u(t)-u(s)]dsdx.
Set
(72)L(t)=NE(t)+εΦ(t)+Ψ(t),
where N and ε are suitable positive constants to be determined later.Remark 9.
Indeed, we easily see that, forε small enough while N large enough, there exist two positive constants α0,α1, such that
(73)α0E(t)≤L(t)≤α1E(t),∀t≥0.
Concerning the estimates of Φ(t),Ψ(t), we have the following lemmas.Lemma 10.
Under the assumption (G1), the functionalΦ(t) satisfies the estimate
(74)Φ′(t)≤-l2∫Ω‍|∇u|2dx+C2∫Ω‍[ut2(x,t)+ut2(x,t-τ(t))]dx+C3(h∘∇u).Proof of Lemma10.
Differentiating (70) and integrating by parts, we get
(75)Φ′(t)=∫Ω‍|ut|ρ+1uttudx+1ρ+1∫Ω‍|ut|ρ+2dx=∫Ω‍u[Δu-∫0t‍h(t-s)Δu(s)ds-μ1ut(x,t)kkkkkkkk-μ2ut(x,t-τ(t))∫0t‍]dx+1ρ+1∫Ω‍|ut|ρ+2dx=-∫Ω‍|∇u|2dx+∫Ω‍∇u·∫0t‍h(t-s)∇u(s)dsdx-μ1∫Ω‍uut(x,t)dx-μ2∫Ω‍uut(x,t-τ(t))dx1ρ+1∫Ω‍|ut|ρ+2dx=-l∫Ω‍|∇u|2dx+∫Ω‍∇u·∫0t‍h(t-s)[∇u(s)-∇u(t)]dsdx-μ1∫Ω‍uut(x,t)dx-μ2∫Ω‍uut(x,t-τ(t))dx+1ρ+1∫Ω‍|ut|ρ+2dx.
Using Young’s inequality and (G1), we obtain (see [2])
(76)∫Ω‍∇u·∫0t‍h(t-s)[∇u(s)-∇u(t)]dsdx≤δ∫Ω‍|∇u|2dx+14δ∫Ω‍[∫0t‍h(t-s)|∇u(s)-∇u(t)|ds]2dx≤δ∫Ω‍|∇u|2dx+1-l4(h∘∇u)(t),∀δ>0.
Also, applying Young’s and Poincaré’s inequality yields
(77)-μ1∫Ω‍uut(x,t)dx≤δ∫Ω‍|∇u|2dx+C(δ)∫Ω‍ut2(x,t)dx,-μ2∫Ω‍uut(x,t-τ(t))dx≤δ∫Ω‍|∇u|2dx+C(δ)∫Ω‍ut2(x,t-τ(t))dx.
Noticing (75)–(77) and choosing δ small enough, we obtain estimate (74).Lemma 11.
Under the assumption (G1), the functionalΨ(t) satisfies the estimate
(78)Ψ′(t)≤-(∫0t‍h(s)ds-2δ)∫Ωut2dx+δ∫Ω‍|∇u|2dx+C4δ(h∘∇u)(t)-C5δ(h′∘∇u)(t)+δ∫Ω‍ut2(x,t-τ(t))dx.Proof of Lemma11.
Differentiating (71), integrating by parts, and noticing the first equation in (1), we have
(79)Ψ′(t)=-∫Ω‍|ut|ρutt∫0t‍h(t-s)[u(t)-u(s)]dsdx-1ρ+1∫Ω‍|ut|ρut∫0t‍h′(t-s)[u(t)-u(s)]dsdx-(∫0t‍h(s)ds)∫Ω‍1ρ+1|ut|ρ+2dx=∫Ω‍[-Δu+∫0t‍h(t-s)Δu(s)dskkkkk+μ1ut(x,t)+μ2ut(x,t-τ(t))∫0t‍]×∫0t‍h(t-s)[u(t)-u(s)]dsdx-1ρ+1∫Ω‍|ut|ρut∫0t‍h′(t-s)[u(t)-u(s)]dsdx-(∫0t‍h(s)ds)∫Ω‍1ρ+1|ut|ρ+2dx=∫Ω‍∇u·∫0t‍h(t-s)[∇u(t)-∇u(s)]dsdx+∫Ω‍∫0t‍h(t-s)Δu(s)dskkkkk×∫0t‍h(t-s)[u(t)-u(s)]dsdx+∫Ω‍[∫0t‍h(t-s)[u(t)-u(s)]ds]kkkkk×[μ1ut(x,t)+μ2ut(x,t-τ(t))]dx-1ρ+1∫Ω‍|ut|ρut∫0t‍h′(t-s)[u(t)-u(s)]dsdx-(∫0t‍h(s)ds)∫Ω‍1ρ+1|ut|ρ+2dx.
Observe that
(80)∫Ω‍∫0t‍h(t-s)Δu(s)ds∫0t‍h(t-s)[u(t)-u(s)]dsdx=-∫Ω‍[∫0t‍h(t-s)∇u(s)dskkkkkkkkk×∫0t‍h(t-s)[∇u(t)-∇u(s)]ds]dx=-∫Ω‍[∫0t‍h(t-s)[∇u(s)-∇u(t)+∇u(s)]dskkkkkkkkk×∫0t‍h(t-s)[∇u(t)-∇u(s)]ds]dx=-∫Ω‍[∫0t‍h(t-s)[∇u(s)-∇u(t)]ds]2dx-(∫0t‍h(t)ds)×∫Ω‍∇u·∫0t‍h(t-s)[∇u(t)-∇u(s)]dsdx.
It follows from (79) and (80) that
(81)Ψ′(t)=(1-∫0t‍h(s)ds)×∫Ω‍∇u·∫0t‍h(t-s)[∇u(t)-∇u(s)]dsdx+∫Ω‍[∫0t‍h(t-s)[∇u(s)-∇u(t)]ds]2dx+∫Ω‍[∫0t‍h(t-s)[u(t)-u(s)]ds]kkkkk×[μ1ut(x,t)+μ2ut(x,t-τ(t))]dx-1ρ+1∫Ω‍|ut|ρut∫0t‍h′(t-s)[u(t)-u(s)]dsdx-(∫0t‍h(s)ds)∫Ω‍1ρ+1|ut|ρ+2dx.
Using Young’s and Poincaré’s inequality, we get (see [2])
(82)(1-∫0t‍h(s)ds)∫Ω‍∇u·∫0t‍h(t-s)[∇u(t)-∇u(s)]dsdx≤δ∫Ω‍|∇u|2dx+Cδ(h∘∇u)(t),-1ρ+1∫Ω‍|ut|ρut∫0t‍h′(t-s)[u(t)-u(s)]dsdx≤δ∫Ω‍ut2dx-Cδ(h′∘∇u)(t).
From (81) and (82), we derive Lemma 11.Now, we are ready to finalize our proof of general decay of the energy. Sinceh is positive, we have
(83)∫0t‍h(s)ds≥∫0t0‍h(s)ds=g0,∀t≥t0.
It follows from (65), (72), (74), and (78) that
(84)L′(t)=NE(t)+εΦ′(t)+Ψ′(t)≤N2(h′∘∇u)(t)-N2h(t)∫Ω‍|∇u|2dx-NC1∫Ω‍[ut2(x,t)+ut2(x,t-τ(t))]dx-λξN2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx+εC2∫Ω‍[ut2(x,t)+ut2(x,t-τ(t))]dx-εl2∫Ω‍|∇u|2dxP+εC3(h∘∇u)(t)-(∫0t‍h(s)ds-2δ)∫Ω‍ut2dx+δ∫Ω‍|∇u|2dx+C4δ(h∘∇u)(t)-C5δ(h′∘∇u)(t)+δ∫Ω‍ut2(x,t-τ(t))dx=-[(NC1+g0)-2δ-εC2]∫Ω‍ut2(x,t)dx+(εC3+C4δ)(h∘∇u)(t)+(N2-C5δ)(h′∘∇u)(t)-(εl2-δ)∫Ω‍|∇u|2dx-(NC1-δ-εC2)∫Ω‍ut2(x,t-τ(t))-λξN2∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx.
If we choose some constants in the inequality (84), such that
(85)a1=(NC1+g0)-2δ-εC2>0,a3=NC1-δ-εC2>0,a2=εl2-δ>0,a4=N2-C5δ>0,a5=εC3+C4δ>0,a6=λξN2,
then we conclude that
(86)L′(t)≤-a1∫Ω‍ut2(x,t)dx-a2∫Ω‍|∇u|2dx+a4(h′∘∇u)(t)+a5(h∘∇u)(t)-a6∫t-τ(t)t‍∫Ω‍e-λ(t-s)ut2(x,s)dsdx.
Hence, we have two cases to consider the general decay results as follows.Case 1 (p=1).
Choosing some values ofa1,a2,a3,a4,a5,a6 and noticing the definition of E(t) (see (63)), we conclude that there exists a constant β1>0, such that
(87)L′(t)≤-β1E(t),∀t≥0.
Therefore, by Remark 9 and (87), we get
(88)L′(t)≤-β1α1E(t),∀t≥0.
Integrating (88) over (0,t), we obtain
(89)L(t)≤L(0)e-(β1/α1)t,∀t≥0.
Observing Remark 9 (i.e., α0E(t)≤L(t)≤α1E(t)) and (89), we derive
(90)α0E(t)≤L(t)≤L(0)e-(β1/α1)t,∀t≥0.
That is,
(91)E(t)≤L(0)α0e-(β1/α1)t=˙Kekt,p=1,∀t≥0.
Assuming K=L(0)/α0,k=β1/α1, we obtain the exponential decay of the energy. So, (26) is established.Case 2 (1<p<3/2).
Due to (G2), we easily see that(92)∫0∞‍h1-r(s)ds<∞,0≤r≤2-p.
From the sketch of proof of Lemma 6, we observe that
(93)(h∘∇u)(t)≤C[∫0∞‍h1-α(s)dsE(0)](p-1+α)/(p-1)×[(hp∘∇u)(t)]α/(p-1+α).
Thus, forσ>1, using (63) and (93), we get
(94)Eσ(t)≤C[Eσ-1(0)(∥ut∥ρ+2ρ+2+∥∇u∥22+∥ut∥22)kkkk+(h∘∇u)σ(t)∥ut∥ρ+2ρ+2]≤CEσ-1(0)[∥ut∥ρ+2ρ+2+∥∇u∥22+∥ut∥22]+C[(∫0∞‍h1-α(s)ds)E(0)](p-1+α)/(p-1)×[(hp∘∇u)(t)]σα/(p-1+α).Choosingα=1/2,σ=2p-1 (i.e., σα/(p-1+α)=1) (94) reduces to
(95)Eσ(t)≤C[∥ut∥ρ+2ρ+2+∥∇u∥22+∥ut∥22+(hp∘∇u)(t)].
Combining (86) and (87) with Remark 9, we obtain
(96)L′(t)≤-β1Cα1σLσ(t),∀t≥0.
A simple integration of (96) over (0,t) yields
(97)L′(t)≤C6(1+t)-1/(σ-1),∀t≥0.
As a consequence of (97), we obtain
(98)∫0∞‍L(t)dt+sup⁡t≥0tF(t)<∞.
So, by using Lemma 6, we have
(99)(h∘∇u)(t)≤C[∫0t‍∥u(s)∥H1(s)ds+t∥u∥H1(Ω)](p-1)/p×(hp∘∇u)1/p(t)≤C[∫0t‍F(s)ds+tF(t)](p-1)/p(hp∘∇u)1/p(t)≤C(hp∘∇u)1/p(t)
which implies that
(100)(hp∘∇u)(t)≥C(h∘∇u)p(t).
Consequently, from (86) and (100), we have
(101)L′(t)≤-C7[∥ut∥ρ+2ρ+2+∥∇u∥22+∥ut∥22+(h∘∇u)p(t)],kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk∀t≥0.
On the other hand, similarly to (95), we
(102)Ep(t)≤C8[∥ut∥ρ+2ρ+2+∥∇u∥22+∥ut∥22+(h∘∇u)p(t)],kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk∀t≥0.
Then, it follows from Remark 9, (101), and (102) that
(103)L′(t)≤-C9Lp(t),∀t≥0.
A simple integration of (103) over (0,t) gives
(104)L′(t)≤K(1+t)-1/(p-1),∀t≥0.
By (104) and Remark 9, we obtain the polynomial decay of the energy. That is,
(105)E(t)≤K(1+t)-1/(p-1),∀t≥0.
Thus, our main result is completed.Remark 12.
Our novel contribution is to show that our work improves earlier result in [37] in which only the exponential decay was investigated. More precisely, Kirane and Said-Houari [37] considered the exponential decay of problem (1) with a constant delay (i.e., τ(t)=τ) and velocity-independent material density (i.e., ρ=0).Remark 13.
By using the fact that energyE is bounded on [0,t0], we can easily show that estimates (26) and (27) hold for t≥0. (See, for instance, [2].)


## 4. Further Remarks

In this section, we address some interesting problems of nonlinear viscoelastic equation with time-varying delay effects and velocity-dependent material density. Here, we mention some of them.(1)
An interesting problem is to show the well-posedness and stabilization of the nonlinear viscoelastic equation with boundary feedback with respect to time-varying delay effects. What will happen if the controller with time-varying delay effects is in the equation instead of on the boundary? More precisely, in our forthcoming work, we will investigate the well-posedness and general decay properties of the solutions for the following nonlinear viscoelastic equation with velocity-dependent material density:(106)|ut|ρutt-Δu+∫0t‍h(t-s)Δu(s)ds=0,inΩ×[0,∞),u(x,t)=0,on⁡Γ0×(0,∞),∂u∂ν+μ1ut(x,t)+μ2ut(x,t-τ(t))=0,on⁡Γ1×[0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),inΩ,ut(x,t-τ(t))=f(x,t),on⁡Γ1×(-τ(0),0),where Ω is bounded domain of Rn and n≥1 with a smooth boundary Γ and let Γ0,Γ1 be a partition of Γ such that Γ0¯∩Γ1¯=∅,Γ0≠∅,Γ1≠∅,ν=(ν1,ν2⋯νn) denotes the unit outward normal to Γ.(2)
Another interesting problem is to give a positive answer of the open problem given by Kirane and Said-Houari [37]. That is, the linear damping term μ1ut in the first equation of (16) plays a decisive role in their proofs. Thus, the problem of whether the stability properties they have proved are preserved when μ1=0 is open. In order to overcome the above difficulty, our main idea is to contrast the effects of the time-varying delay by using the dissipative nonlinear boundary feedback. That is, in our future work, we investigate the following problem:
(107)|ut|ρutt-Δu+∫0t‍h(t-s)Δu(s)ds+μ2ut(x,t-τ(t))=0,inΩ×[0,∞),u(x,t)=0,on⁡Γ0×(0,∞),∂u∂ν+g(ut(x,t))=0,on⁡Γ1×[0,∞),u(x,0)=u0(x),ut(x,0)=u1(x),inΩ,ut(x,t-τ(t))=f(x,t),on⁡Γ1×(-τ(0),0),where μ2 is constant and g(ut) is the dissipative nonlinear boundary feedback.



---

*Source: 102594-2014-06-09.xml*