# Polynomial Decay Rate for a Coupled Lamé System with Viscoelastic Damping and Distributed Delay Terms

Authors: Nadjat Doudi; Salah Mahmoud Boulaaras; Ahmad Mohammed Alghamdi; Bahri Cherif
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we prove a general energy decay results of a coupled Lamé system with distributed time delay. By assuming a more general of relaxation functions and using some properties of convex functions, we establish the general energy decay results to the system by using an appropriate Lyapunov functional.


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## Body

## 1. Introduction

In this work, we shall be concerned with studying the general decay rate of the following Lamé system inΩ×ℝ+:
(1)utt−Δeu+∫0tg1t−sΔusds−k1Δut−∫τ1τ2μ1ρΔutx,t−ρdρ=f1u,v,vtt−Δev+∫0tg2t−sΔvsds−k2Δvt−∫τ1τ2μ2ρΔvtx,t−ρdρ=f2u,v.Equations (1) are associated with the following boundary and initial conditions
(2)ux,t=vx,t=0,on∂Ω×ℝ+,ux,0=u0x,vx,0=v0x,utx,0=u1x,vtx,0=v1x,x∈Ω,utx,−t,vtx,−t=f0x,t,g0x,t,inΩ×0,τ2,where Ω is a bounded domain in ℝnn=1,2,3, with smooth boundary ∂Ω. The elasticity differential operator Δe is given by
(3)Δeu=μΔu+μ+λ∇divu,and the Lamé constants μ and λ are satisfying the following conditions
(4)μ>0,μ+λ>0.The parametersk1,k2,τ1, and τ2 are positive constants, with τ1<τ2. The functions μ1,μ2:τ1,τ2→ℝ are bounded. The functions f1u,v and f2u,v which represent the source terms will be specified later.After several authors have studied the problems of coupled systems and hyperbolic systems, their stability is associated with velocities and is proven under conditions imposed on the subgroup [1]. The researchers also studied behavior of the energy in a limited field with nonlinear damping and external force and a varying delay of time to find solutions to the Lame system [1–9].Recently, problems that contain viscoelasticity have been addressed, and many results have been found regarding the global existence and stability of solutions (see [2, 9]), under conditions on the relaxation function, whether exponential or polynomial decay. In addition, in [10], Boulaaras obtained the stability result of the global solution to the Lamé system with the flexible viscous term by adding logarithmic nonlinearity, even though the kernel is not necessarily decreasing in contrast to what he studied [2].Introducing a distributed delay term makes our problem different from those considered so far in the literature.The importance of this term appears in many works, and this is due to the fact that many phenomena depends on their past. Also, it is influence on the asymptotic behavior of the solution for the different types of problems such that Timoshenko system [3, 11–13], transmission problem [14], wave equation [15], and thermoelastic system [16, 17].In the present work, we extend the general decay result obtained by Feng in [18] to the case of distributed term delay, namely, we will make sure that the result is achieved if the distributed delay term exists.This paper is organized as follows. In the second section, we give some preliminaries related to problem (1). In Section 3, we prove our main result.


## 2. Preliminaries

In this section, we provide some materials and necessary assumptions which we need in the prove of our results. We use the standard Lebesgue and Sobolev spaces with their scaler products and norms. For simplicity, we would write. instead of .2. Throughout this work, we used a generic positive constant c.For the relaxation functionsg1 and g2, we assume, for i=1,2,(A1)git:ℝ+→ℝ+ are nonincreasing C1 functions satisfying
(5)gi0>0andμ−∫0∞gisds=li>0.We assume further that fori=1,2:(A2) There exist twoC1 functions Gi:ℝ+→ℝ+, with Gi0=Gi′0=0, which are linear or are strictly increasing and strictly convex functions of class C2ℝ+ on 0,r,r≤gi0, such that
(6)gi′t≤−ξitGigit,∀t≥0,where ξit are C1 functions satisfying
(7)ξit>0,ξi′t≤0,∀t≥0.(A3) For the source termsf1 and f2, we take
(8)f1u,v=αu+vp−1u+v+βup−3/2uvp+1/2,∀u,v∈ℝn2,f2u,v=αu+vp−1u+v+βvp−3/2vup+1/2,∀u,v∈ℝn2,with α,β>0. Clearly,
(9)uf1u,v+vf2u,v=p+1Fu,v,∀u,v∈ℝn2,where
(10)Fu,v=1p+1αu+vp+1+2βuvp+1/2,∀u,v∈ℝn2,f1u,v=∂F∂u,f2u,v=∂F∂v.Further, we assume that there is C>0, such that
(11)∂fi∂uu,v+∂fi∂vu,v≤Cup−1+vp−1,i=1,2where1≤p≤6.(A4)(12)ifn=1,2;p≥3,ifn=3;p=3.So, we have the embedding(13)H01Ω°LqΩfor2≤q≤2nn−2ifn≥3orq≥2ifn=1,2,Lr°Lqforq<r.Letcs the same embedding constant, so we have
(14)νq≤cs∇ν2,νq≤csνrforν∈H01Ω.Remark 1.
There exist two constantsΛ1>0 and Λ2>0 such that
(15)∫Ωfiu,v2dx≤Λil1∇u2+l2∇v2p,i=1,2.As in many papers, we introduce the following new variables(16)zx,ρ,ρ,t=utx,t−ρρ,yx,ρ,ρ,t=vtx,t−ρρ,then, we obtain
(17)ρztx,ρ,ρ,t+zρx,ρ,ρ,t=0,zx,0,ρ,t=utx,t,(18)ρytx,ρ,ρ,t+yρx,ρ,ρ,t=0,yx,0,ρ,t=vtx,t.Consequently, the problem (1) is equivalent to
(19)utt−Δeu+∫0tg1t−sΔusds−k1Δut−∫τ1τ2μ1ρΔzx,1,ρ,tdρ=f1u,v,vtt−Δev+∫0tg2t−sΔvsds−k2Δvt−∫τ1τ2μ2ρΔyx,1,ρ,tdρ=f2u,v,ρztx,ρ,ρ,t+zρx,ρ,ρ,t=0,ρytx,ρ,ρ,t+yρx,ρ,ρ,t=0,with the initial data and boundary conditions
(20)ux,0,vx,0=u0x,v0x,inΩ,utx,0,vtx,0=u1x,v1x,inΩ,utx,−t,vtx,−t=f0x,t,g0x,t,inΩ×0,τ2,ux,t=vx,t=0,in∂Ω×0,∞,zx,ρ,ρ,0,yx,ρ,ρ,0=f0x,ρρ,g0x,ρρ,inΩ×0,1×0,τ2,where
(21)x,ρ,ρ,t∈Ω×0,1×τ1,τ2×0,∞.We recall the following notations(22)h∗φ=∫0tht−sφsdsdx,h∘φt=∫0tht−sφt−φs2ds.Thus, we have the following important property(23)∫Ωh∗φφtdx=−12htφt2+12h′∘φt−12ddth∘φt−∫0thsdsφt2.The energy modified associated to the problem (19) is defined by
(24)Et=12ut2+μ−∫0tg1sds∇u2+λ+μdivu2+12g1∘∇ut+η∫Ω∫01∫τ1τ2ρμ1ρ∇zx,ρ,ρ,t2dρdρdx+12vt2+μ−∫0tg2sds∇v2+λ+μdivv2+12g2∘∇vt+η∫Ω∫01∫τ1τ2ρμ2ρ∇yx,ρ,ρ,t2dρdρdx−∫ΩFu,vdx.First, we prove in the following theorem the result of energy identity.Lemma 2.
Assume that(25)∫τ1τ2μiρdρ<ki,i=1,2.Then, the energy modified defined by (24) satisfies, along the solution u,v,z,y of (19), the estimate
(26)ddtEt≤−12g1t∇u2+12g1′∘∇ut−12g2t∇v2+12g2′∘∇vt−k1−η+12∫τ1τ2μ1ρdρ∇ut2−η−12∫Ω∫τ1τ2μ1ρ∇zx,1,ρ,t2dρdx−k2−η+12∫τ1τ2μ2ρdρ∇vt2−η−12∫Ω∫τ1τ2μ2ρ∇yx,1,ρ,t2dρdx≤0,for
(27)1<η<min2k1∫τ1τ2μ1ρdρ,2k2∫τ1τ2μ2ρdρ−1.Proof.
First multiplying the equation0.141 by ut and integrating by parts over Ω, we obtain
(28)12ddtut2+μ∇u2+λ+μdivu2−∫Ω∇ut∫0tg1t−s∇usds+k1∇ut2+∫Ω∇ut∫τ1τ2∣μ1ρ∇zx,1,ρ,tdρdx=∫Ωf1u,v.udx,by using (23), we obtain
(29)12ddtut2+μ−∫0tg1sds∇u2+λ+μdivu2+g1∘∇ut=−12g1∇u2+12g1′∘∇ut+∫Ωutf1u,vdx−k1∇ut2+∫Ω∇ut∫τ1τ2μ1ρ∇zx,1,ρ,tdρdx.
Similarly, multiplying the equation (19) byvt and integrating over Ω, we obtain
(30)12ddtvt2+μ−∫0tg2sds∇v2+λ+μdivv2+g2∘∇vt=−12g2∇v2+12g2′∘∇vt+∫Ωvtf2u,vdx−k2∇vt2+∫Ω∇vt∫τ1τ2μ2ρ∇yx,1,ρ,tdρdx.
Multiplying the equation (19) by −ημ1ρΔzx,ρ,ρ,t and integrating by parts over Ω×0,1×τ1,τ2, we obtain
(31)η∫Ω∫01∫τ1τ2ρμ1ρ∇zx,ρ,ρ,t∇ztx,ρ,ρ,tdρdρdx=−η∫Ω∫01∫τ1τ2μ1ρ∇zx,ρ,ρ,t∇zρx,ρ,ρ,tdρdρdx,therefore
(32)ddtη2∫Ω∫01∫τ1τ2ρμ1ρ∇zx,ρ,ρ,t2dρdρdx=−η2∫Ω∫01∫τ1τ2μ1ρddρ∇zx,ρ,ρ,t2dρdρdx=−η2∫Ω∫τ1τ2μ1ρ∇zx,1,ρ,t2dρdx+η2∫Ω∫τ1τ2μ1ρ∇utx,t2dρdx.
Multiplying the fourth equation of (19) by −ημ2ρΔyx,ρ,ρ,t and integrating over Ω×0,1×τ1,τ2, we obtain
(33)ddtη2∫Ω∫01∫τ1τ2ρμ2ρ∇yx,ρ,ρ,t2dρdρdx=−η2∫Ω∫01∫τ1τ2μ2ρddρ∇yx,ρ,ρ,t2dρdρdx=−η2∫Ω∫τ1τ2μ2ρ∇yx,1,ρ,t2dρdx+η2∫Ω∫τ1τ2μ2ρ∇vtx,t2dρdx.
For the source term, we have(34)∫Ωutf1u,vdx+∫Ωvtf2u,vdx=∫Ωutαu+vp−1u+v+βup−3/2uvp+1/2+∫Ωvtαu+vp−1u+v+βvp−3/2vup+1/2=∫Ωαu+vp−1u+vut+vt+βup−3/2u.utvp+1/2+βvp−3/2v.vtup+1/2dx=ddt∫Ωαp+1u+vp+1+2βp+1uvp+1/2dx=ddt∫ΩFu,vdx.
By collecting the previous equations (29)–(34), we get
(35)ddtEt=−12g1∇u2+12g1′∘∇ut−k1∇ut+∫Ω∇ut∫τ1τ2μ1ρ∇zx,1,ρ,tdρdx−12g2∇v2+12g2′∘∇vt−k2∇vt+∫Ω∇vt∫τ1τ2μ2ρ∇yx,1,ρ,tdρdx−η2∫Ω∫τ1τ2μ1ρ∇zx,1,ρ,t2dρdx+η2∫Ω∫τ1τ2μ1ρ∇utx,t2dρdx−η2∫Ω∫τ1τ2μ2ρ∇yx,1,ρ,t2dρdx+η2∫Ω∫τ1τ2μ2ρ∇vtx,t2dρdx.
Using Young’s inequality, we obtain(36)∫Ω∇ut∫τ1τ2μ1ρ∇zx,1,ρ,tdρdx≤∫Ω∫τ1τ2∇utμ1ρμ1ρ∇zx,1,ρ,tdρdx≤12∫τ1τ2μ1ρdρ∇ut+12∫Ω∫τ1τ2μ1ρ∇zx,1,ρ,t2dρdx,similarly
(37)∫Ω∇vt∫τ1τ2μ2ρ∇yx,1,ρ,tdρdx≤12∫τ1τ2μ2ρdρ∇vt+12∫Ω∫τ1τ2μ2ρ∇yx,1,ρ,t2dρdx.
This completes the proof.


## 3. General Decay

In this section we will prove that the solution of problems (19)–(20) decay generally to trivial solution. Using the energy method and suitable Lyapunov functional.In the following, we will present our main stability result:Theorem 3 (Decay rates of energy).
Assume that (A1)–(A3) hold. Then, for every t0>0, there exist two positive constants α1 and α2 such that the energy defined by (24) satisfies the following decay
(38)Et≤α2G4−1α1∫g−1rtξsds,∀t≥g−1r,where
(39)G4t=∫tr1sG0sds,G0t=minG1′t,G2′t,and ξt=minξ1t,ξ2t,gt=maxg1t,g2t.This theorem will be proved later after providing some remarks.Remark 4.
(1)
In case∫0∞ξitdt=∞, Theorem 3 ensures limt→∞Et=0.(2)
From (A2), we infer thatlimt→∞git=0. Then, there exists some t1≥0 large enough such that
(a)
(40)git1=r⇒git≤r,∀t≥t1.
AsGi is positive continuous functions, andgi and ξi are positive nonincreasing continuous functions, then, for all 0≤t≤t1,(41)0<git1≤git≤gi0and0<ξit1≤ξit≤ξi0,which implies for some positive constants ai and bi,
(42)ai≤ξitGigit≤bi.
Consequently,(43)gi′t≤−ξitGigit≤−aigi0gi0≤−aigi0git,fort∈0,t1.(3)
We also mention Johnson’s inequality, which is very important for proving our result. IfG is a convex function on a,b, g:Ω→a,b], we have(44)G1k∫Ωgxhxdx≤1k∫ΩGgxhxdx,where h is a function that satisfies
(45)hx≥0and∫Ωhxdx=k>0.To prove the desired result, we create a Lyapunov functional equivalent toE. For this, we define some functions that allow us to construct this Lyapunov function.As in Baowei [18] and Mustafa ([19, 20]), we define
(46)Cζ,i=∫0∞gi2sζgis−gi′sandhit=ζgit−gi′t,i=1,2,for any 0<ζ<1.Lemma 5.
Letu,v,z,y be a solution of the problem (19). Then, the functional
(47)φt=∫Ωututtdx+∫Ωvtvttdx,satisfies the estimate
(48)φ′t≤−l12∇ut2−l22∇vt2+utt2+vtt2−λ+μdivut2−λ+μdivvt2+3Cζ,12l1h1∘∇ut+3k122l1∇ut2+3k222l2∇vt2+3Cζ,22l2h2∘∇vt+p+1∫ΩFut,vtdx+3k12l1∫Ω∫τ1τ2μ1ρ∇zx,1,ρ,t2dρdx+3k22l2∫Ω∫τ1τ2μ2ρ∇yx,1,ρ,t2dρdx.Proof.
Taking the derivative of (47), we obtain
(49)φ′t=∫Ωutt2dx+∫Ωututttdx+∫Ωvtt2dx+∫Ωvtvtttdx.
From problem (19) and using integration by parts, we get
(50)φ′t=utt2+vtt2+∫ΩutΔeu−∫0tg1t−sΔusds+k1Δut+∫τ1τ2μ1ϱΔzx,1,ϱ,tdϱ+f1u,vdx+∫ΩvtΔev−∫0tg2t−sΔvsds+k2Δvt+∫τ1τ2μ2ϱΔyx,1,ϱ,tdϱ+f2u,vdx=utt2+vtt2−k1∫Ω∇u∇utdx−k2∫Ω∇v∇vtdx−u∇ut2−λ+μdivut2+∫Ω∇ut∫0tg1t−s∇usds−∫Ω∇ut∫τ1τ2μ1ϱ∇zx,1,ϱ,tdϱ+∫Ωutf1u,vdx−μ∇vt2−λ+μdivvt2+∫Ω∇vt∫0tg2t−s∇vsds−∫Ω∇vt∫τ1τ2μ2ϱ∇yx,1,ϱ,tdϱ+∫Ωvtf2u,vdx=utt2+vtt2−k1∫Ω∇u∇utdx−k2∫Ω∇v∇vtdx−μ−∫0tg1sds∇ut2−λ+μdivut2+∫Ω∇ut∫0tg1t−s∇us−∇utdsdx−∫Ω∇ut∫τ1τ2μ1ϱ∇zx,1,ϱ,tdϱdx+∫Ωutf1u,vdx−μ−∫0tg2sds∇vt2−λ+μdivvt2+∫Ω∇vt∫0tg2t−s∇vs−∇vtdsdx−∫Ω∇ut∫τ1τ2μ2ϱ∇yx,1,ϱ,tdϱdx+∫Ωvtf2u,vdx.
By using Hölder and Young’s inequalities, we have(51)∫Ω∇ut∫0tg1t−s∇us−∇utdsdx≤l16∇ut2+32l1∫Ω∫0tg1t−s∇us−∇utds2dx≤l16∇ut2+32l1∫Ω∫0tg1t−sζg1t−s−g1′t−sζg1t−s−g1′t−s∇us−∇utds2dx≤l16∇ut2+32l1∫0tg1sζg1s−g1′sds∫Ω∫0tζg1t−s−g1′t−s∇us−∇ut2ds≤l16∇ut2+3Cζ,12l1h1∘∇ut.
Similarly, we obtain(52)∫Ω∇vt∫0tg2t−s∇vs−∇vtdsdx≤l26∇vt2+3Cζ,22l2h2∘∇vt,∫Ω∇ut∫τ1τ2μ1ρ∇zx,1,ρ,tdρdx≤l16∇ut2+32l1∫Ω∫τ1τ2∣μ1ρ∣∇zx,1,ρ,tdρ2dx≤l16∇ut2+32l1∫Ω∫τ1τ2∣μ1ρ∣∣μ1ρ∣∇zx,1,ρ,tdρ2≤l16∇ut2+32l1∫τ1τ2∣μ1ρ∣dρ∫Ω∫τ1τ2∣μ1ρ∣∇zx,1,ρ,t2dρ≤l16∇ut2+3k12l1∫Ω∫τ1τ2∣μ1ρ∣∇zx,1,ρ,t2dρ,∫Ω∇vt∫τ1τ2μ2ρ∇yx,1,ρ,tdρdx≤l26∇vt2+32l2∫τ1τ2∣μ2ρ∣dρ∫Ω∫τ1τ2∣μ2ρ∣∇yx,1,ρ,t2dρ≤l26∇vt2+3k22l2∫Ω∫τ1τ2∣μ2ρ∣∇yx,1,ρ,t2dρ.
The Young’s inequality gives(53)k1∫Ω∇ut∇uttdx≤l16∇ut2+3k122l1∇utt2,k2∫Ω∇vt∇vttdx≤l26∇vt2+3k222l2∇vtt2.
For the source term, we have(54)∫Ωutf1u,vdx+∫Ωvtf2u,vdx=p+1∫ΩFu,vdx.
Combining the equations (51)–(54), thus, our proof is completed.Lemma 6.
Let (u, v, z, y) be a solution of the problem (19). Then, the functional
(55)ψt=∫Ωutt∫0tg1sus−utdsdx+∫Ωvtt∫0tg2svs−vtdsdx=ψ1t+ψ2t,
satisfies for anyδ>0 the estimate
(56)ψ′t≤δ+δΛ3l1∇ut2+δdivu2+δΛ3l2∇vt2+δ−∫0tg1sdsutt2+cCζ,1+1δh1∘∇ut+δk1∫Ω∫τ1τ2∣μ1ρ∣∇zx,1,ρ,t2dρdx+δk1∇ut2+δ+δΛ4l2∇vt2+δdivv2+δΛ4l1∇ut2+δ−∫0tg2sdsvtt2+cCζ,2+1δh2∘∇vt+δk2∫Ω∫τ1τ2∣μ2ρ∣∇yx,1,ρ,t2dρdx+δk1∇vt2,where Λ3 and Λ4 are two positive constants.Proof.
First, we begin to estimateψ1′t(57)ψ1′t=∫Ωuttt∫0tg1t−sus−utdsdx+∫Ωutt∫0tg1′t−sus−utdsdx−∫0tg1sdsutt2=∫ΩΔeu−∫0tg1t−sΔusds+k1Δu+∫τ1τ2μ1ρΔzx,1,ρ,tdρ+f1u,v×∫0tg1t−sus−utdsdx+∫Ωutt∫0tg1′t−sus−utdsdx−∫0tg1sdsutt2=∫Ωutt∫0tg1′t−sus−utdsdx−∫0tg1sdsutt2−μ∫Ω∇u∫0tg1t−s∇us−∇utdsdx−λ+μ∫Ωdivut∫0tg1t−sdivus−divutdsdx+∫0tg1t−s∇usds∫0tg1t−s∇us−∇utdsdx+∫Ω∫τ1τ2μ1ρ∇zx,1,ρ,tdρ∫0tg1t−s∇us−∇utdsdx+∫Ωf1u,v∫0tg1t−sus−utdsdx−k1∫Ω∇ut∫0tg1t−s∇us−∇utdsdx.
Then, we have(58)ψ1′t=∫Ωutt∫0tg1′t−sus−utdsdx−∫0tg1sdsutt2−μ−∫0tg1sds∫Ω∇u∫0tg1t−s∇us−∇utdsdx−λ+μ∫Ωdivut∫0tg1t−sdivus−divutdsdx+∫Ω∫0tg1t−s∇us−∇utdsdx2−k1∫Ω∇ut∫0tg1t−s∇us−∇utdsdx+∫Ω∫τ1τ2μ1ρ∇zx,1,ρ,tdρ.∫0tg1t−s∇us−∇utdsdx+∫Ωf1u,v∫0tg1t−sus−utdsdx.
As in previous proof and by using Young’s inequality, we conclude that for anyδ>0,
(59)μ−∫0tg1sds∫Ω∇u∫0tg1t−s∇us−∇utdsdx≤δ∇u2+c.Cζ,1δh1∘∇ut.
Similarly and by using the factdivu2≤c∇u2, we have
(60)λ+μ∫Ωdivut∫0tg1t−sdivus−divutdsdx≤δdivu2+c.Cζ,1δh1∘∇ut.
The same argument for(61)∫Ωf1u,v∫0tg1t−sus−utdsdx≤δ∫Ωf1u,v2dx+14δ∫Ω∫0tg1t−sus−utds2dx≤δΛ1l1∇u2+l2∇v2p+c.Cζ,1δh1∘∇ut≤δΛ12p+1p−1E0p−1l1∇u2+l2∇v2+c.Cζ,1δh1∘∇ut≤δΛ3l1∇u2+l2∇v2+c.Cζ,1δh1∘∇ut,
whereΛ3=2p+1/p−1E0p−1.
From (46), we have
(62)∫Ωutt∫0tg1′t−sus−utdsdx=∫Ωutt∫0th1t−sus−utdsdx−∫Ωutt∫0tζg1t−sus−utdsdx≤δutt2+cδ∫Ω∫0th1t−sh1t−sus−utds2dx+cζ2δ∫Ω∫0tg1t−sus−utds2dx≤δutt2+cδ∫0th1sdsh1∘ut+cζ2Cζ,1δh1∘ut≤δutt2+cδh1∘∇ut+cζ2Cζ,1δh1∘∇ut≤δutt2+cCζ,1+1δh1∘∇ut,∫Ω∫τ1τ2μ1ρ∇zx,1,ρ,tdρ.∫0tg1t−s∇us−∇utdsdx≤δ∫Ω∫τ1τ2∣μ1ρ∣∇zx,1,ρ,tdρ2dx+14δ∫Ω∫0tg1t−s∇us−∇utds2dx≤δ∫τ1τ2∣μ1ρ∣dρ∫Ω∫τ1τ2∣μ1ρ∣∇zx,1,ρ,t2dρdx+cCζ,1δh1∘∇ut≤δk1∫Ω∫τ1τ2∣μ1ρ∣∇zx,1,ρ,t2dρdx+cCζ,1δh1∘∇ut.
Finally, Young’s inequality gives(63)k1∫Ω∇ut∫0tg1t−sus−utdsdx≤δk1∇ut2+k14δ∫Ω∫0tg1t−sus−utds2dx≤δk1∇ut+cCζ,1δh1∘∇ut.
Then(64)ψ1′t≤δ−∫0tg1sdsutt2+δ+δΛ3l1∇u2+δdivu2+δΛ3l2∇v2+cCζ,1+1δh1∘∇ut+δk1∇ut2+δk1∫Ω∫τ1τ2∣μ1ρ∣∇zx,1,ρ,t2dρdx.
The same steps can be taken to get the next estimate forψ2′t,(65)ψ2′t≤δ−∫0tg2sdsvtt2+δ+δΛ4l2∇v2+δdivv2+δΛ4l1∇u2+cCζ,2+1δh2∘∇vt+δk2∇vt2+δk2∫Ω∫τ1τ2∣μ2ρ∣∇yx,1,ρ,t2dρdx,where Λ4=Λ22p+1/p−1E0p−1.Lemma 7.
Letu,v,z,y be a solution of the problem (19). Then, the functional
(66)It=∫Ω∫01∫τ1τ2ρe−ρρμ1ρ∇zx,ρ,ρ,t2+μ2ρ∇yx,ρ,ρ,t2dxdρdρ,satisfies the estimate
(67)I′t≤−e−τ2∫Ω∫τ1τ2μ1ρ∇zx,1,ρ,t2dxdρ−e−τ2∫Ω∫τ1τ2μ2ρ∇yx,1,ρ,t2dxdρ+k1∇utt2+k2∇vtt2−It.Proof.
Differentiating (66) with respect to t, we get
(68)ddtIt=2∫Ω∫01∫τ1τ2ρe−ρρμ1ρ∇zx,ρ,ρ,t∇ztx,ρ,ρ,t+μ2ρ∇yx,ρ,ρ,t∇ytx,ρ,ρ,tdxdρdρ.
By using (17) and (18), we have
(69)ddtIt=−∫Ω∫01∫τ1τ2e−ρρμ1ρ∂∂ρ∇zx,ρ,ρ,t2+μ2ρ∂∂ρ∇yx,ρ,ρ,t2dxdρdρ=−∫Ω∫01∫τ1τ2μ1ρ∂∂ρe−ρρ∇zx,ρ,ρ,t2+μ2ρ∂∂ρe−ρρ∇yx,ρ,ρ,t2dxdρdρ−∫Ω∫01∫τ1τ2ρe−ρρμ1ρ∇zx,ρ,ρ,t2+μ2ρ∇yx,ρ,ρ,t2dxdρdρ.
Thus,(70)ddtIt=−∫Ω∫τ1τ2e−ρμ1ρ∇zx,1,ρ,t2dxdρ+∫τ1τ2μ1ρdρ∇utx,ρ,t2−∫Ω∫τ1τ2e−ρμ2ρ∇yx,1,ρ,t2dxdρ+∫τ1τ2μ2ρdρ∇vtx,ρ,t2−∫Ω∫01∫τ1τ2ρe−ρρμ1ρ∇zx,ρ,ρ,t2+μ2ρ∇yx,ρ,ρ,t2dxdρdρ.
Sincee−ρ is deceasing function over τ1,τ2, the desired estimate (67) follows immediately from (25).The following lemmas are needed to prove the general decay when the functionsGiti=1,2 are nonlinear. The proof can be found in Mustafa [19].Lemma 8.
The functional(71)θ1t=∫Ω∫0tσ1t−s∇us2dsdx,where σ1t=∫t∞g1sds satisfies
(72)θ1′t≤−12g1∘∇ut+3μ−l1∇u2.Lemma 9.
The functional(73)θ2t=∫Ω∫0tσ2t−s∇vs2dsdx,where σ2t=∫t∞g2sds satisfies
(74)θ2′t≤−12g2∘∇vt+3μ−l2∇v2.Now, we define the following functional(75)Ft=NEt+N1ϕt+N2ψt+It,whereN,N1, and N2 are positive constants. It is easy to prove Ft and Et are equivalent, namely, there exist two positive constants κ1 and κ2 such that
(76)κ1Et≤Ft≤κ2Et.By Young’s inequality, we get(77)Ft≤N2+N12+N22ut2+vt2+N2μ−∫0tg1sds+cN12+cN22∇u2+N2μ−∫0tg2sds+cN12+cN22∇v2+N2+cN22∫0tg1sdsg1∘∇ut+N2+cN22∫0tg2sdsg2∘∇ut+N2η+C∫Ω∫0t∫τ1τ2ρμ1ρ∇zx,ρ,ρ,t2dρdρdx+N2η+C∫Ω∫0t∫τ1τ2ρμ2ρ∇yx,ρ,ρ,t2dρdρdx+N2λ+μdivu2+divv2−N∫ΩFu,vdx.Then, for anyN, there exists κ1>0 such that
(78)F≤κ1Et.On the other hand, we can find(79)Ft≥N2−N12−N22ut2+vt2+N2μ−∫0tg1sds−cN12−cN22∇u2+N2μ−∫0tg2sds−cN12−cN22∇v2+N2−cN22∫0tg1sdsg1∘∇ut+N2−cN22∫0tg2sdsg2∘∇ut+N2η+c∫Ω∫0t∫τ1τ2ρμ1ρ∇zx,ρ,ρ,t2dρdρdx+N2η+c∫Ω∫0t∫τ1τ2ρμ2ρ∇yx,ρ,ρ,t2dρdρdx+N2λ+μdivu2+divv2−N∫ΩFu,vdx.We chooseN large enough so that
(80)N2−N12−N22>0,N2μ−∫0tgisds−cN12−cN22>0,N2−cN22∫0tgisds>0,i=1,2.Then, there existκ2>0 such that
(81)Ft≥κ2Et.Lemma 10.
The functionalFt satisfies for any t≥t1,
(82)F′t≤−4μ−l1∇ut2−4μ−l2∇vt2−utt2−vtt2−divut2−divvt2+14g1∘∇ut+14g2∘∇vt+c∫ΩFut,vtdx−∫Ω∫01∫τ1τ2ρμ1ρ∇zx,ρ,ρ,t2dρdρdx−∫Ω∫01∫τ1τ2ρμ2ρ∇yx,ρ,ρ,t2dρdρdx.Proof.
Let(83)g0=min∫0t1g1sds,∫0t1g2sds.
From Lemmas5, 6, and 7, noting that gi′=ζgi−hi we have for any t≥t1,(84)F′t≤−l12N1−N2δ1+Λ3l1−N2δΛ4l1∇ut2−l22N1−N2δ1+Λ4l2−N2δΛ3l2∇vt2−g0N2−δN2−N1utt2+vtt2+ζN2g1∘∇ut+g2∘∇vt+N1p+1∫ΩFut,vtdx−N2−N2cCζ,1+1δ−3N1Cζ,12l1h1∘∇ut−N2−N2cCζ,2+1δ−3N1Cζ,22l2h2∘∇vt−λ+μN1−δN2divut2+divvt2−Nσ+e−τ2−N132l1k1−δN2k1∫Ω∫τ1τ2∣μ1ρ∣∇zx,1,ρ,t2dρdx−Nσ+e−τ2−N132l2k2−δN2k2∫Ω∫τ1τ2∣μ2ρ∣∇yx,1,ρ,t2dρdx−Nσ1−N13k122l1−δN2k1−k1∇utt2−Nσ2−N13k222l2−δN2k2−k2∇vtt2−It,where
(85)σ1=k1−η+12∫τ1τ2μ1ρdρ,σ2=k2−η+12∫τ1τ2μ2ρdρ,σ=η−12.
Takingδ=1/2N2, we can get
(86)F′t≤−l12N1−121+Λ3l1−12Λ4l1∇ut2−l22N1−121+Λ4l2−12Λ3l2∇vt2−g0N2−12−N1utt2+vtt2+ζN2g1∘∇ut+g2∘∇vt+N1p+1∫ΩFut,vtdx−N2−2cN22−Cζ,12cN22+3N12l1h1∘∇ut−N2−2cN22−Cζ,22cN22+3N12l2h2∘∇vt−λ+μN1−12divut2+divvt2−Nσ+e−τ2−N132l1k1−k12∫Ω∫τ1τ2∣μ1ρ∣∇zx,1,ρ,t2dρdx−Nσ+e−τ2−N132l2k2−k22∫Ω∫τ1τ2∣μ2ρ∣∇yx,1,ρ,t2dρdx−Nσ1−N13k122l1−3k12∇utt2−Nσ2−N13k222l2−3k22∇vtt2−It.
First, we takeN1>0 large such that
(87)λ+μN1−12>0,l12N1−121+Λ3l1−12Λ4l1>4μ−l1,l22N1−121+Λ3l2−12Λ4l2>4μ−l2.
We chooseN2>0 large enough so that
(88)g0N2−12−N1>1.
Note that(89)0<ζgi2sζgis−gi′s<ζgi2s−gi′s,i=1,2.
Then, for anys∈0,∞, we get
(90)limζ→0ζgi2sζgis−gi′s=0,i=1,2.
By using the factζgi2s/ζgis−gi′s<gis,i=1,2 and using Lebesgue-dominated convergence theorem, we can get
(91)limζ→0ζCζ,i=limζ→0∫0∞ζgi2sζgis−gi′s=0,i=1,2.
Thus, there exist someζ00<ζ0<1 such that if ζ<ζ0, then
(92)ζCζ,2<18N1/2l1+2cN22andζCζ,2<18N1/2l2+2cN22.
At last, we chooseN large enough and choose ζ satisfying
(93)14N−2cN22>0andζ=12N>ζ0,
so, we arrive at(94)N2−2cN22−Cζ,12cN22+3N12l1>0andN2−2cN22−Cζ,12cN22+3N12l1>0.
Therefore, we chooseN even larger (if needed) so that
(95)Nn1+e−τ2−N132l1k1−k12>0,Nn1+e−τ2−N132l2k2−k22>0,Nm−N13k122l1−3k12>0andNm−N13k222l2−3k22>0.
Thus, (82) is established.Proof of Theorem 11.
Taking into account (43) and (26), we obtain that for any t≥t1,
(96)∫0t1g1s∫Ω∇ut−∇ut−s2dxds≤−g10a1∫0t1g1′s∫Ω∇ut−∇ut−s2dxds≤−cE′t,∫0t1g2s∫Ω∇vt−∇vt−s2dxds≤−g20a2∫0t1g2′s∫Ω∇vt−∇vt−s2dxds≤−cE′t.
Noting (82), we shall see that there exists a constant m>0 such that for all t≥t1,
(97)F′t≤−mEt−cE′+c∫t1tg1s∫Ω∇ut−∇ut−s2dxds+c∫t1tg2s∫Ω∇vt−∇vt−s2dxds.
DenoteLt=Ft+cEt. It is obvious that Lt is equivalent to Et. It follows from (97) that
(98)L′t≤−mEt+c∫t1tg1s∫Ω∇ut−∇ut−s2dxds+c∫t1tg2s∫Ω∇vt−∇vt−s2dxds.
We consider two cases.Case 11.
Gt is linear: By multiplying (98) by ξt and using (A2) and (26), we obtain
(99)ξtL′t≤−mξtEt+cξt∫t1tg1s∫Ω∇ut−∇ut−s2dxds+cξt∫t1tg2s∫Ω∇vt−∇vt−s2dxds≤−mξtEt+c∫t1tξ1tg1s∫Ω∇ut−∇ut−s2dxds+c∫t1tξ2tg2s∫Ω∇vt−∇vt−s2dxds≤−mξtEt−c∫t1tg1′s∫Ω∇ut−∇ut−s2dxds−c∫t1tg2′s∫Ω∇vt−∇vt−s2dxds≤−mξtEt−cE′t,
which gives asξt is nonincreasing,
(100)ξtLt+cEt′≤ξtL′t+cE′t≤−mξtEt∀t≥t1.
DenoteKt=ξtLt+cEt. We get
(101)Kt′≤−mξtEt.
Hence, using the fact thatKt~Et, we easily obtain
(102)Et≤c1exp−c2∫t1tξsds.Case 12.
Gt is nonlinear: First, we use Lemmas 8 and 9 to deduce that
(103)Jt=Ft+θ1t+θ2t,is nonnegative, and it satisfies for some positive constant k and for any t≥t1,
(104)J′t≤−μ−l1∇u2−μ−l2∇v2−ut2−vt2−divu2−divv2−14g1∘∇u−14g2∘∇v+c∫ΩFu,vdx−∫Ω∫01∫τ1τ2ρμ1ρ∇zx,ρ,ρ,t2dρdρdx−∫Ω∫01∫τ1τ2ρμ2ρ∇yx,ρ,ρ,t2dρdρdx≤−kEt≤0.
Therefore,(105)k∫t1tEsds≤Jt1−Jt≤Jt1,this implies that
(106)∫0∞Esds<∞.Now, we defineIit by
(107)I1t≔q∫t1t∫Ω∇ut−∇ut−s2dxds,I2t≔q∫t1t∫Ω∇vt−∇vt−s2dxds,where (106) allows for a constant 0<q<1 chosen so that for all t≥t1(108)Iit<1,i=1,2.We also assume without loss of generality thatIit>0 for all t≥t1; otherwise, (98) yields an exponential decay. Also, we define λ1t and λ2t by
(109)λ1t≔−∫t1tg1′s∫Ω∇ut−∇ut−s2dxds,λ2t≔−∫t1tg2′s∫Ω∇vt−∇vt−s2dxds.We observe thatλit≤−cE′ti=1,2. Noting that Git is strictly convex on 0,r and Gi0=0, then
(110)Giνx≤νGix,i=1,2,provided 0≤ν≤1 and x∈0,r. By using (A2), (108), and Jensen’s inequality, we can obtain
(111)λ1t=1qI1t∫t1tI1t−g1′s∫Ωq∇ut−∇ut−s2dxds≥1qI1t∫t1tI1tξ1sG1g1s∫Ωq∇ut−∇ut−s2dxds,≥ξ1tqI1t∫t1tG1I1tg1s∫Ωq∇ut−∇ut−s2dxds,≥ξ1tqG11I1t∫t1tI1tg1s∫Ωq∇ut−∇ut−s2dxds=ξ1tqG1q∫t1tg1s∫Ω∇ut−∇ut−s2dxds=ξ1tqG1¯q∫t1tg1s∫Ω∇ut−∇ut−s2dxds,whereG1¯ is an extension of G1 such that G1¯ is strictly increasing and strictly convex C2 function on 0,+∞, see [19]. We have from (111)
(112)∫t1tg1s∫Ω∇ut−∇ut−s2dxds≤1qG1¯−1qλ1tξ1t.Similarly, we have(113)∫t1tg2s∫Ω∇vt−∇vt−s2dxds≤1qG2¯−1qλ2tξ2t.We infer from (98), (112), and (113) that for any t≥t1(114)L′t≤−mEt+cG1¯−1qλ1tξ1t+cG2¯−1qλ2tξ2t.Let us denote(115)G0t=minG1¯−1,G2¯−1.Forε0<r, the functional K1t defined by:
(116)K1t=G0ε0EtE0Lt+Et,is equivalent to E, and by using the fact that E′≤0,Gi¯′>0, and Gi¯′′>0, we infer from (114) that
(117)K1′t=ε0E′tE0G0′ε0EtE0Lt+G0ε0EtE0L′t+E′t≤−mEtG0ε0EtE0+cG0ε0EtE0G1¯−1qλ1tξ1t+cG0ε0EtE0G2¯−1qλ2tξ2t.LetG¯∗ be the convex conjugate of G¯ in the sense of Young (see Arnold [21]), then
(118)Gi¯∗s=sGi¯′−1s−Gi¯Gi¯′−1s,i=1,2,and G¯∗ satisfies the following Young’s inequality
(119)MDi≤Gi¯∗M+Gi¯Di,i=1,2.WithM=G0ε0Et/E0 and Di=Gi¯−1qλit/ξit and noting Gi¯∗t≤tGi¯′−1t and (117), we conclude
(120)K1′t≤−mEtG0ε0EtE0+cG1¯∗G0ε0EtE0+cqλ1tξ1t+cG2¯∗G0ε0EtE0+cqλ2tξ2t≤−mEtG0ε0EtE0+cG0ε0EtE0G1¯′−1G0ε0EtE0+cqλ1tξ1t+cG0ε0EtE0G2¯′−1G0ε0EtE0+cqλ2tξ2t≤−mEtG0ε0EtE0+cG0ε0EtE0G1¯′−1G1¯′ε0EtE0+cqλ1tξ1t+cG0ε0EtE0G2¯′−1G2¯′ε0EtE0+cqλ2tξ2t≤−mE0−cε0EtE0G0ε0EtE0+cqλ1tξ1t+λ2tξ2t.Multiplying byξt, we get
(121)ξtK1′t≤−mE0−cε0ξtEtE0G0ε0EtE0+cqλ1t+λ2t≤−mE0−cε0ξtEtE0G0ε0EtE0−cE′t.Consequently, withK2t=ξtK1t+cEt, which satisfies for some β1,β2>0,
(122)β1K2t≤Et≤β2K2t.Choosing a suitableε0, we can get from (121) that there exists a constant k>0,
(123)K2′t≤−kξtEtE0G0ε0EtE0≔−kξtG3ε0EtE0,where G3t=tG0ε0t.From0≤ε0Et/E0<r, we conclude that for any t>0,
(124)G0ε0EtE0=minG1¯′ε0EtE0,G2¯′ε0EtE0=minG1′ε0EtE0,G2′ε0EtE0.DenoteRt=β1K2t/E0. Using (122), then
(125)Rt~Et.SinceG3′t=G0ε0t+ε0tG0′ε0t, then, using the strict convexity of G0 on 0,r, we arrive at G3′t,G3t>0 on 0,1. We infer from (123) that there exists a constant b1>0 such that for all t≤t1,
(126)R′t≤−b1ξtG3Rt.Then, by integration overt1,t, we have
(127)∫t1t−R′sG3Rsds≥b1∫t1tξsds∫ε0Rtε0Rt11sG0sds≥b1∫t1tξsdsRt≤1ε0G4−1b1∫t1tξsds,where G4t=∫tr1/sG0sds is strictly decreasing on 0,r and limt→0G4t=+∞. A combination of (125) and (127), estimate (38), is established.


## 4. Conclusion

In this work, we have proved a general energy decay of a coupled Lamé system with distributed time delay. This result is an natural extension of Feng’s work in [18]. In order to complete this work, the study of the global existence and the blow-up of the solutions of (1) and (2) will be the subject of forthcoming works.



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