# Global Existence and Decay of Solutions for Coupled Nondegenerate Kirchhoff System with a Time Varying Delay Term

Authors: Nadia Mezouar; Salah Mahmoud Boulaaras; Sultan Alodhaibi; Salem Alkhalaf
Journal: Complexity
 (2020)

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## Abstract

This paper deals with the global existence of solutions in a bounded domain for nonlinear viscoelastic Kirchhoff system with a time varying delay by using the energy and Faedo–Galerkin method with respect to the delay term weight condition in the feedback and the delay speed. Furthermore, by using some convex functions properties, we prove a uniform stability estimate.


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## Body

## 1. Introduction

### 1.1. Model

Consider the following viscoelastic Kirchhoff system:(1)utlutt+αv−M∇u2Δu−Δutt+∫0th1t−sΔusds−μ1Δutx,t−τt=0,inΩ×0,+∞,vtlvtt+αu−M∇v2Δv−Δvtt+∫0th2t−sΔvsds−μ2Δvtx,t−τt=0,inΩ×0,+∞,ux,t=vx,t=0,on∂Ω×0,+∞,ux,0,vx,0=u0x,v0x,utx,0,vtx,0=u1x,v1x,inΩ,utx,t−τ0,vtx,t−0=f0x,t−τ0,g0x,t−τ0,inΩ×0,τ0,where Ω is a bounded domain in IRn, n ϵ IN∗, with a smooth boundary ∂Ω, l>0, α, μ1 and μ2 are positive real numbers, h1 and h2 are positive functions which decay exponentially, τt>0 is a time varying delay, and the initial data u0,v0,u1,v1,f0,g0 are in a suitable function space. Mr=a+brγ is a C1-function for r≥0, with a,b>0, and γ≥1.Time delay is often present in applications and practical problems. In last few years, the control of PDEs with time delay effects has become an active area of research (see, for example, [1–4] and the references therein). In [5], the authors showed that a small delay in a boundary control could turn a well-behaved hyperbolic system into a wild one and therefore delay becomes a source of instability. However, sometimes it can also improve the performance of the system.By using the Faedo–Galerkin method, Wu in [6] proved the result of local existence and established the decay result by suitable Lyapunov functionals according to appropriate conditions on μ1, μ2 and on the kernel h.Daewook [7] studied the following viscoelastic Kirchhoff equation with nonlinear source term and varying time delay:(2)utt−Mx,t,∇u2Δu+∫0tht−sdivax∇usds+umu+μ1utx,t+μ2utx,t−τt=0,inΩ×0,+∞,which is a description of axially moving viscoelastic materials. According to the smallness condition taking into account of Kirchhoff coefficient and the relaxation function and by summing 0≤m≤2/n−2 if n>2 or 0≤m if n≤2, he got the uniform decay rate of the Kirchhoff type energy.Very recently, in [1], we have proved the global existence and energy decay of solutions of the following viscoelastic nondegenerate Kirchhoff equation:(3)utlutt−M‖∇u‖2Δu−Δutt+∫0tht−sΔusds+μ1g1utx,t+μ2g2utx,t−τt=0,inΩ×0,+∞,ux,t=0,on∂Ω×0,+∞,ux,0=u0x,utx,0=u1x,inΩ,utx,t−τ0=f0x,t−τ0,inΩ×0,τ0,with respect to some proposed assumptions. Under assumption setting on g1, g2, σ, and τ, the authors have obtained the global existence of solution and the decay rate of energy.Recently, Mezouar and Boulaaras [1] have studied viscoelastic nondegenerate Kirchhoff equation with varying delay term in the internal feedback.In the present paper, we extent our recent published paper in [1] for a coupled system (3). The famous technique using the presence of delay in PDE’s problem is to set a new variable defined by velocity depending on delay, which will give us new problem equivalent to our studied problem but the last one is a coupled system without delay. After this, we can prove the existence of global solutions in suitable Sobolev spaces by combining the energy method with the Faedo–Galerkin procedure, and under a choice of a suitable Lyapunov functional, we establish an exponential decay result.The outline of the paper is as follows. In Section2, some hypotheses related to problem are given and we state our main result. Then, in Section 3, the global existence of weak solutions is proven. Finally, in Section 4, we give the uniform energy decay.


### 1.2. Preliminaries and Assumptions

We denote by (.,.) the inner product inL2Ω.Now, we introduce, as in [8], the new variables(4)z1x,ρ,t=utx,t−ρτt,x∈Ω,ρ∈0,1,t>0,z2x,ρ,t=vtx,t−ρτt,x∈Ω,ρ∈0,1,t>0.Then, we have(5)τtz1′x,ρ,t+1−ρτ′t∂∂ρz1x,ρ,t=0,inΩ×0,1×0,+∞.Similarly, we have(6)τtz2′x,ρ,t+1−ρτ′t∂∂ρz2x,ρ,t=0,inΩ×0,1×0,+∞.Therefore, problem (3) is equivalent to(7)utlutt+αv−M∇u2Δu−Δutt+∫0th1t−sΔusds−μ1Δz1x,1,t=0,inΩ×0,+∞,vtlutt+αu−M∇v2Δv−Δvtt+∫0th2t−sΔvsds−μ2Δz2x,1,t=0,inΩ×0,+∞,τtz1′x,ρ,t+1−ρτ′t∂∂ρz1x,ρ,t=0,inΩ×0,1×0,+∞,τtz2′x,ρ,t+1−ρτ′t∂∂ρz2x,ρ,t=0,inΩ×0,1×0,+∞,ux,t=vx,t=0,on∂Ω×0,∞,z1x,0,t,z2x,0,t=utx,t,vtx,t,onΩ×0,∞,ux,0,vx,0=u0x,v0x,utx,0,vtx,0=u1x,v1x,inΩ,z1x,ρ,0,z2x,ρ,0=f0x,−ρτ0,g0x,−ρτ0,inΩ×0,1.To state and prove our result, we need some assumptions.Assumption 1.
Assume that0<l≤γ satisfies(8)γ≤2n−2,ifn>2,γ<∞,ifn≤2.Assumption 2.
For the relaxation functions,hi:IR+⟶IR+ are bounded C1 functions such that(9)a−∫0∞hisds≥k>0,and suppose that there exist positive constants ζi satisfying(10)hi′t≤−ζihit,for i=1,2.Assumption 3.
τ is a function in W2,∞0,T, T>0, such that(11)0<τ0≤τ′t≤τ1,∀t>0,τ′t≤d<1,∀t>0.
We define the energy associated to the solution of system (7) by(12)Et=1l+2utl+2l+2+vtl+2l+2+2α∫Ωux,tvx,tdx+b2γ+1∇u2γ+1+∇v2γ+1+12a−∫0th1sds∇u2+12a−∫0th2sds∇v2+12∇ut2+∇vt2+12h1o∇ut+12h2o∇vt+ξτt∫01∇z1x,ρ,t2+∇z2x,ρ,t2dρ,where ξ is a positive constant such that(13)maxμ1,μ221−d<ξ,(14)hiowt=∫0thit−sw.,t−w.,s2ds,fori=1,2.Theorem 1 (global existence).
Letu0,v0∈H2Ω∩H01Ω2, s, and f0,g0∈H01Ω,H10,12 satisfy the compatibility condition(15)f0.,0,g0.,0=u1,v1.
Assume that Assumptions1–3 hold. Then, problem (3) admits a weak solution such that u,v∈L∞0,∞;H2Ω∩H01Ω, ut,vt∈L∞0,∞;H01Ω, and utt,vtt∈L20,∞,H01Ω.Theorem 2 (decay rates of energy).
Assume that Assumptions1–3 hold. Then, for every t0>0 there exist positive constants K and c′ such that the energy defined by (12) possesses the following decay:(16)Et≤Ke−c′t,∀t≥t0.


## 1.1. Model

Consider the following viscoelastic Kirchhoff system:(1)utlutt+αv−M∇u2Δu−Δutt+∫0th1t−sΔusds−μ1Δutx,t−τt=0,inΩ×0,+∞,vtlvtt+αu−M∇v2Δv−Δvtt+∫0th2t−sΔvsds−μ2Δvtx,t−τt=0,inΩ×0,+∞,ux,t=vx,t=0,on∂Ω×0,+∞,ux,0,vx,0=u0x,v0x,utx,0,vtx,0=u1x,v1x,inΩ,utx,t−τ0,vtx,t−0=f0x,t−τ0,g0x,t−τ0,inΩ×0,τ0,where Ω is a bounded domain in IRn, n ϵ IN∗, with a smooth boundary ∂Ω, l>0, α, μ1 and μ2 are positive real numbers, h1 and h2 are positive functions which decay exponentially, τt>0 is a time varying delay, and the initial data u0,v0,u1,v1,f0,g0 are in a suitable function space. Mr=a+brγ is a C1-function for r≥0, with a,b>0, and γ≥1.Time delay is often present in applications and practical problems. In last few years, the control of PDEs with time delay effects has become an active area of research (see, for example, [1–4] and the references therein). In [5], the authors showed that a small delay in a boundary control could turn a well-behaved hyperbolic system into a wild one and therefore delay becomes a source of instability. However, sometimes it can also improve the performance of the system.By using the Faedo–Galerkin method, Wu in [6] proved the result of local existence and established the decay result by suitable Lyapunov functionals according to appropriate conditions on μ1, μ2 and on the kernel h.Daewook [7] studied the following viscoelastic Kirchhoff equation with nonlinear source term and varying time delay:(2)utt−Mx,t,∇u2Δu+∫0tht−sdivax∇usds+umu+μ1utx,t+μ2utx,t−τt=0,inΩ×0,+∞,which is a description of axially moving viscoelastic materials. According to the smallness condition taking into account of Kirchhoff coefficient and the relaxation function and by summing 0≤m≤2/n−2 if n>2 or 0≤m if n≤2, he got the uniform decay rate of the Kirchhoff type energy.Very recently, in [1], we have proved the global existence and energy decay of solutions of the following viscoelastic nondegenerate Kirchhoff equation:(3)utlutt−M‖∇u‖2Δu−Δutt+∫0tht−sΔusds+μ1g1utx,t+μ2g2utx,t−τt=0,inΩ×0,+∞,ux,t=0,on∂Ω×0,+∞,ux,0=u0x,utx,0=u1x,inΩ,utx,t−τ0=f0x,t−τ0,inΩ×0,τ0,with respect to some proposed assumptions. Under assumption setting on g1, g2, σ, and τ, the authors have obtained the global existence of solution and the decay rate of energy.Recently, Mezouar and Boulaaras [1] have studied viscoelastic nondegenerate Kirchhoff equation with varying delay term in the internal feedback.In the present paper, we extent our recent published paper in [1] for a coupled system (3). The famous technique using the presence of delay in PDE’s problem is to set a new variable defined by velocity depending on delay, which will give us new problem equivalent to our studied problem but the last one is a coupled system without delay. After this, we can prove the existence of global solutions in suitable Sobolev spaces by combining the energy method with the Faedo–Galerkin procedure, and under a choice of a suitable Lyapunov functional, we establish an exponential decay result.The outline of the paper is as follows. In Section2, some hypotheses related to problem are given and we state our main result. Then, in Section 3, the global existence of weak solutions is proven. Finally, in Section 4, we give the uniform energy decay.


## 1.2. Preliminaries and Assumptions

We denote by (.,.) the inner product inL2Ω.Now, we introduce, as in [8], the new variables(4)z1x,ρ,t=utx,t−ρτt,x∈Ω,ρ∈0,1,t>0,z2x,ρ,t=vtx,t−ρτt,x∈Ω,ρ∈0,1,t>0.Then, we have(5)τtz1′x,ρ,t+1−ρτ′t∂∂ρz1x,ρ,t=0,inΩ×0,1×0,+∞.Similarly, we have(6)τtz2′x,ρ,t+1−ρτ′t∂∂ρz2x,ρ,t=0,inΩ×0,1×0,+∞.Therefore, problem (3) is equivalent to(7)utlutt+αv−M∇u2Δu−Δutt+∫0th1t−sΔusds−μ1Δz1x,1,t=0,inΩ×0,+∞,vtlutt+αu−M∇v2Δv−Δvtt+∫0th2t−sΔvsds−μ2Δz2x,1,t=0,inΩ×0,+∞,τtz1′x,ρ,t+1−ρτ′t∂∂ρz1x,ρ,t=0,inΩ×0,1×0,+∞,τtz2′x,ρ,t+1−ρτ′t∂∂ρz2x,ρ,t=0,inΩ×0,1×0,+∞,ux,t=vx,t=0,on∂Ω×0,∞,z1x,0,t,z2x,0,t=utx,t,vtx,t,onΩ×0,∞,ux,0,vx,0=u0x,v0x,utx,0,vtx,0=u1x,v1x,inΩ,z1x,ρ,0,z2x,ρ,0=f0x,−ρτ0,g0x,−ρτ0,inΩ×0,1.To state and prove our result, we need some assumptions.Assumption 1.
Assume that0<l≤γ satisfies(8)γ≤2n−2,ifn>2,γ<∞,ifn≤2.Assumption 2.
For the relaxation functions,hi:IR+⟶IR+ are bounded C1 functions such that(9)a−∫0∞hisds≥k>0,and suppose that there exist positive constants ζi satisfying(10)hi′t≤−ζihit,for i=1,2.Assumption 3.
τ is a function in W2,∞0,T, T>0, such that(11)0<τ0≤τ′t≤τ1,∀t>0,τ′t≤d<1,∀t>0.
We define the energy associated to the solution of system (7) by(12)Et=1l+2utl+2l+2+vtl+2l+2+2α∫Ωux,tvx,tdx+b2γ+1∇u2γ+1+∇v2γ+1+12a−∫0th1sds∇u2+12a−∫0th2sds∇v2+12∇ut2+∇vt2+12h1o∇ut+12h2o∇vt+ξτt∫01∇z1x,ρ,t2+∇z2x,ρ,t2dρ,where ξ is a positive constant such that(13)maxμ1,μ221−d<ξ,(14)hiowt=∫0thit−sw.,t−w.,s2ds,fori=1,2.Theorem 1 (global existence).
Letu0,v0∈H2Ω∩H01Ω2, s, and f0,g0∈H01Ω,H10,12 satisfy the compatibility condition(15)f0.,0,g0.,0=u1,v1.
Assume that Assumptions1–3 hold. Then, problem (3) admits a weak solution such that u,v∈L∞0,∞;H2Ω∩H01Ω, ut,vt∈L∞0,∞;H01Ω, and utt,vtt∈L20,∞,H01Ω.Theorem 2 (decay rates of energy).
Assume that Assumptions1–3 hold. Then, for every t0>0 there exist positive constants K and c′ such that the energy defined by (12) possesses the following decay:(16)Et≤Ke−c′t,∀t≥t0.


## 2. Preliminaries

Lemma 1Sobolev−Poincare inequality.
Letq be a number with(17)2≤q<+∞n=1,2or2≤q≤2nn−2n≥3.
Then, there exists a constantCs=CsΩ,q such that(18)uq≤Cs∇u2foru∈H01Ω.The following lemma states an important property of the convolution operator.Lemma 2 (see [9]).
Forh,φ∈C10,+∞,IR, we have(19)∫Ωh∗φφtdx=−12htφt2+12h′oφt−12ddthoφt−∫0thsdsφ2.Lemma 3.
Letu,v,z1,z2 be a solution of problem (7). Then, the energy functional defined by (12) satisfies(20)Et≤−β∇z1x,1,t2+∇z2x,1,t2+λ∇utx,t2+∇vtx,t2+12h1′o∇ut+h2′o∇vt,where λ=ξ+μ/2, β=ξ1−d−μ/2, and μ=maxμ1,μ2 are positive.Proof.
Multiplying the first equation in (7) by ut, integrating over Ω, and using integration by parts, we get(21)ddt1l+2utl+2l+2+b2γ+1∇u2γ+1+12a∇u2+12∇ut2−∫Ω∫0th1t−s∇us∇uttdsdx+α∫Ωutx,tvx,tdx+μ1∫Ω∇utx,t∇z1x,1,tdx=0.
Consequently, by applying Lemma2, equation (21) becomes(22)ddt1l+2utl+2l+2+b2γ+1∇u2γ+1+12a−∫0th1sds∇u2+12∇ut2+12h1o∇ut+12h1t∇ut2−12h1′o∇ut+α∫Ωutx,tvx,tdx+μ1∫Ω∇utx,t∇z1x,1,tdx=0.
Similarly by multiplying the second equation in (7) by vt, integrating over Ω, and using integration by parts, we get(23)ddt1l+2vtl+2l+2+b2γ+1∇v2γ+1+12a−∫0th2sds∇v2+12∇vt2+12h2o∇vt+12h2t∇vt2−12h2′o∇vt+α∫Ωux,tvtx,tdx+μ2∫Ω∇vtx,t∇z2x,1,tdx=0.
Multiply the third equation in (7) by ξΔz1 and integrate the result over Ω×0,1 to obtain(24)ξτt∫Ω∫01z1′x,ρ,tΔz1x,ρ,tΩ×0,1dρdx=−ξ∫Ω∫011−ρτ′t∂∂ρz1x,ρ,tΔz1x,ρ,tdρdx.
Consequently,(25)ddtξτt∫Ω∫01∇z1x,ρ,t2dρdx=ξτ′t∫Ω∫01∇z1x,ρ,t2dρdx−ξ∫Ω∫011−ρτ′t∂∂ρ∇z1x,ρ,t2dρdx=−ξ∫Ω∫01∂∂ρ1−ρτt∇z1x,ρ,t2dρdx=−ξ1−τ′t∫Ω∇z1x,1,t2dx+ξ∫Ω∇utx,t2dx.
Similarly, we get(26)ddtξτt∫Ω∫01∇z2x,ρ,t2dρdx=−ξ1−τ′t∫Ω∇z2x,1,t2dx+ξ∫Ω∇vtx,t2dx.
Combining (22)–(26), we obtain(27)E′t=−ξ1−τ′t∫Ω∇z1x,1,t2+∇z2x,1,t2dx+ξ∫Ω∇utx,t2+∇vtx,t2dx−12h1t∇ut2+h2t∇vt2+12h1′o∇ut+h2′o∇vt−μ1∫Ω∇utx,t∇z1x,1,tdx−μ2∫Ω∇vtx,t∇z2x,1,tdx.
From Assumption3, we get(28)E′t≤−ξ1−d−μ12∫Ω∇z1x,1,t2dx−ξ1−d−μ22∫Ω∇z2x,1,t2dx−12h1t∇utt2+ξ+μ12∇utt2−12h2t∇vt2+ξ+μ22∇vtt2+12h1′o∇ut+h2′o∇vt.
Using (13), this completes the proof.


## 3. Global Existence (Proof of Theorem 1)

Throughout this section we assumeu0,v0∈H2Ω∩H01Ω,u1,v1∈H01Ω, and f0,g0∈H01Ω,H10,1. We will use the Faedo–Galerkin method to prove the existence of global solutions. Let T>0 be fixed and let wk,k∈IN be a basis of H2Ω∩H01Ω and Vk be the space generated by wk. Now, we define, for 1≤j≤k, the sequence ϕjx,ρ as follows:(29)ϕjx,0=wj.Then, we may extendϕjx,0 by ϕjx,ρ over L2Ω×0,1 such that ϕjj forms a basis of L2Ω,H10,1 and denote Zk the space generated by ϕk. We construct approximate solutions uk,vk,z1k,z2k, k=1,2,3,…, in the form(30)ukt=∑j=1kajktwj,vkt=∑j=1kbjktwj,z1kt=∑j=1kcjktϕj,z2kt=∑j=1kdjktϕj,where ajk, bjk, cjk, and djkj=1,2,…,k are determined by the following ordinary differential equations:(31)utkluttk,wj+αvk,wj+M∇ukt2∇uk,∇wj+∇uttk,∇wj−∫0th1t−s∇uks,∇wjds+μ1∇z1k.,1,∇wj=0,1≤j≤k,vtklvttk,wj+αuk,wj+M∇vkt2∇vk,∇wj+∇vttk,∇wj−∫0th2t−s∇vks,∇wjds+μ2∇z2k.,1,∇wj=0,1≤j≤k,z1kx,0,t=utkx,t,z2kx,0,t=vtkx,t,(32)uk0=u0k=∑j=1ku0,wjwj⟶u0,inH2Ω∩H01Ωask⟶+∞,(33)vk0=v0k=∑j=1kv0,wjwj⟶v0,inH2Ω∩H01Ωask⟶+∞,(34)utk0=u1k=∑j=1ku1,wjwj⟶u1,inH01Ωask⟶+∞,(35)+vtk0=v1k=∑j=1kv1,wjwj⟶v1,inH01Ωask⟶+∞,(36)τt∂∂tz1k+1−ρτ′t∂∂ρz1k,ϕj=0,1≤j≤k,τt∂∂tz2k+1−ρτ′t∂∂ρz2k,ϕj=0,1≤j≤k,(37)z1kρ,0=∑j=1kf0,ϕjϕj⟶f0,inH01Ω,H10,1ask⟶+∞,(38)z2kρ,0=∑j=1kg0,ϕjϕj⟶g0,inH01Ω,H10,1ask⟶+∞.Noting thatl/2l+1+1/2l+1+1/2=1, from the generalized Hölder inequality, we obtain(39)utkluttk,wj=∫Ωutkluttkwjdx≤∫Ωutk2l+1dxl/2l+1uttk2l+1wj2.Since Assumption2 holds, according to Sobolev embedding, the nonlinear terms utkluttk,wj and vtklvttk,wj in (31) make sense.The standard theory of ODE guarantees that systems (31)–(38) have a unique solution in 0,tk, with 0<tk<T. In the next step, we obtain a priori estimates for the solution of systems (31)–(38), so that it can be extended outside 0,tk to obtain one solution defined for all t>0, using a standard compactness argument for the limiting procedure.


### 3.1. First Estimate

Since the sequencesu0k,v0k,u1k,v1k,z1kρ,0, and z2kρ,0 converge and from Lemma 3 with employing Gronwall’s lemma, we can find a positive constant C1 independent of k such that(40)Ekt+β∫0t∇z1kx,1,s2+∇z2kx,1,s2ds≤C1,where(41)Ekt=1l+2utkl+2l+2+vtkl+2l+2+2α∫Ωukvkdx+b2γ+1∇uk2γ+1+∇vk2γ+1+12a−∫0th1sds∇uk2+12a−∫0th2sds∇vk2+12∇utk2+∇vtk2+12h1o∇ukt+h2o∇vkt+ξτt∫01∇z1kx,ρ,t2+∇z2kx,ρ,t2dρ.Noting Assumption1 and estimate (40) yields that(42)uk,vkare bounded inLloc∞0,∞,H01Ω,utk,vtkare bounded inLloc∞0,∞,H01Ω,z1kx,ρ,t,z2kx,ρ,tare bounded inLloc∞0,∞,L10,1,H01Ω.


### 3.2. Second Estimate

Multiplying the first equation (respectively, the second equation) in (31) by attjk (respectively, by bttjk ) and summing over j from 1 to k, it follows that (43)∫Ωutkluttk2dx−α∫Ωvtkutkdx+∫ΩM∇uk2∇uk∇uttkdx+∇uttk2=∫0th1t−s∫Ω∇uks∇uttktdxds−μ1∫Ω∇uttk∇z1kx,1,tdx,∫Ωvtklvttk2dx−α∫Ωvtkutkdx+∫ΩM∇vk2∇vk∇vttkdx+∇vttk2=∫0th2t−s∫Ω∇vks∇vttktdxds−μ2∫Ω∇vttk∇z2kx,1,tdx.Differentiating (36) with respect to t, we get(44)τt1−ρτ′t′∂∂tz1k+τt1−ρτ′t∂2∂t2z1k+∂2∂t∂ρz1k,ϕj=0,τt1−ρτ′t′∂∂tz2k+τt1−ρτ′t∂2∂t2z2k+∂2∂t∂ρz2k,ϕj=0.Multiplying the first equation byctjk (respectively, the second equation by dtjk) and summing over j from 1 to k, it follows that(45)τt1−ρτ′t′∂∂tz1k2+12τt1−ρτ′tddt∂∂tz1k2+12ddρ∂∂tz1k2=0,τt1−ρτ′t′∂∂tz2k2+12τt1−ρτ′tddt∂∂tz2k2+12ddρ∂∂tz2k2=0,and then we have(46)12τt1−ρτ′t′∂∂tz1k2+12ddtτt1−ρτ′t∂∂tz1k2+12ddρ∂∂tz1k2=0,12τt1−ρτ′t′∂∂tz2k2+12ddtτt1−ρτ′t∂∂tz2k2+12ddρ∂∂tz2k2=0.Integrating over0,1 with respect to ρ, we obtain(47)12∫01τt1−ρτ′t′∂∂tz1k2dρ+12ddt∫01τt1−ρτ′t∂∂tz1k2dρ+12∂∂tz1kx,1,t2−12uttkx,t2=0,12∫01τt1−ρτ′t′∂∂tz2k2dρ+12ddt∫01τt1−ρτ′t∂∂tz2k2dρ+12∂∂tz2kx,1,t2−12vttkx,t2=0.Summing (43) and (47) and as Mr≥a, we get(48)∫Ωutkluttk2dx+∇uttk2+12ddt∫01τt1−ρτ′t∂∂tz1k2dρ+12∂∂tz1kx,1,t2≤−a∫Ω∇uk∇uttkdx+12uttkx,t2+α∫Ωvtkutkdx−12∫01τt1−ρτ′t′∂∂tz1k2dρ+∫0th1t−s∫Ω∇uks∇uttktdxds−μ1∫Ω∇uttk∇z1kx,1,tdx,∫Ωvtklvttk2dx+∇vttk2+12ddt∫01τt1−ρτ′t∂∂tz2k2dρ+12∂∂tz2kx,1,t2≤−a∫Ω∇vk∇vttkdx+12vttkx,t2+α∫Ωvtkutkdx−12∫01τt1−ρτ′t′∂∂tz2k2dρ+∫0th2t−s∫Ω∇vks∇vttktdxds−μ2∫Ω∇vttk∇z2kx,1,tdx.By Young’s inequality, the right hand side of (48) can be estimated as follows:(49)∫Ωa∇uk∇uttkdx≤η∇uttk2+a24η∇uk2,∫Ωa∇vk∇vttkdx≤η∇vttk2+a24η‖∇vk2.Using Cauchy–Schwarz inequality and Sobolev–Poincare inequality, we obtain(50)α∫Ωvtkutkdx≤α2vtk2+utk2≤Csα2∇vtk2+∇utk2,(51)∫0th1t−s∫Ω∇uks∇uttktdxds≤η∇uttk2+14η∫Ω∫0th1t−s∇uksdsdx2≤η∇uttk2+14η∫0thsds∫Ω∫0th1t−s∇uks2dsdx≤η∇uttk2+14ηa−k∫Ω∫0th1t−s∇uks2dsdx≤η∇uttk2+14ηa−k∫0th1t−s∇uks2ds≤η∇uttk2+14ηa−kh10∫0t∇uks2ds.Similarly,(52)∫0th2t−s∫Ω∇vks∇vktttdxds≤η∇vktt2+14ηa−kh20∫0t∇vks2ds,(53)μ1∫Ω∇uttk∇z1kx,1,tdx≤ημ12∇uttk2+14η∇z1kx,1,t2,μ2∫Ω∇vttk∇z2kx,1,tdx≤ημ22∇vttk2+14η∇z2kx,1,t2.By using (49)–(53) in (48), we deduce(54)∫Ωutkluttk2dx+∇uttk2+12ddt∫01τt1−ρτ′t∂∂tz1k2dρ+12∂∂tz1kx,1,t2≤ημ12+2+Cs22∇uttk2+Csα2∇vtk2+∇utk2+a24η∇uk2+14η∇z1kx,1,t2+14ηa−kh10∫0t∇uks2ds−12∫01τt1−ρτ′t′∂∂tz1k2dρ,∫Ωvtklvttk2dx+∇vttk2+12ddt∫01τt1−ρτ′t∂∂tz2k2dρ+12∂∂tz2kx,1,t2≤ημ22+2+Cs22∇vttk2+Csα2∇vtk2+∇utk2+a24η∇vk2+14η∇z2kx,1,t2+14ηa−kh20∫0t∇vks2ds−12∫01τt1−ρτ′t′∂∂tz2k2dρ.By using Assumption3 and taking the first estimate (40) into account, we infer(55)∫Ωutkluttk2dx+1−ημ12+2+Cs22∇uttk2+12ddt∫01τt1−ρτ′t∂∂tz1k2dρ+12∂∂tz1kx,1,t2≤C2+14ηa−k1h10C1T+a1∫01τt1−ρτ′t∂∂tz1k2dρ,∫Ωvtklvttk2dx+1−ημ22+2+Cs22∇vttk2+12ddt∫01τt1−ρτ′t∂∂tz2k2dρ+12∂∂tz2kx,1,t2≤C2+14ηa−k2h20C1T+a2∫01τt1−ρτ′t∂∂tz2k2dρ,where C2 is a positive constant which depends on η,a,C1,Cs,α.Integrating (55) over (0, t), we obtain(56)∫0tutkluttk2dxdt+1−ημ12+2+Cs22∫0t∇uttks2ds+∫01τt1−ρτ′t∂∂tz1k2dρ+12∫01∂∂tz1kx,1,t2dt≤C2T+14ηa−kh10C1T2+a1∫0t∫01τt1−ρτ′t∂∂sz1k2dρds,∫0t∫Ωvtklvttk2dxdt+1−ημ22+2+Cs22∫0t∇vttks2ds+∫01τt1−ρτ′t∂∂tz2k2dρ+12∫0t∂∂tz2kx,1,t2dt≤C2T+14ηa−kh20C1T2++a2∫0t∫01τs1−ρτ′s∂∂sz2k2dρds.For a suitableη>0 such that 1−ημi2+2+Cs2/2>0 for i=1,2 and using Gronwall’s lemma, we obtain the second estimate(57)∫0t∇uktts2+∇vktts2ds+∫01τt1−ρτ′t∂∂tz1k2+∂∂tz2k2dρ≤C3.We observe from estimates (40) and (57) that there exist subsequences um of uk and vm of vk such that(58)um,vm⇀u,vweakly star inL∞0,T,H01Ω,(59)utm,vtm⇀ut,vtweakly star inL∞0,T,H01Ω,(60)uttm,vttm⇀utt,vttweakly inL20,T,H01Ω,(61)z1m,z1m⇀z1,z2weakly star inL∞0,T,H01Ω,L20,1,(62)∂∂tz1m,∂∂tz2m⇀∂∂tz1,∂∂tz2weakly star inL∞0,T,L2Ω×0,1,In the following, we will treat the nonlinear term. From the first estimate (40) and Lemma 1, we deduce(63)utklutkL20,T,L2Ω=∫0Tutk2l+12l+1dt≤Cs2l+1∫0T∇utk22l+1dt≤Cs2l+1C1l+1T.On the other hand, from Aubin–Lions theorem (see [10]), we deduce that there exists a subsequence of um, still denoted by um such that(64)utm⟶utstrongly inL20,T,L2Ω,which implies(65)utm⟶utalmost everywhere inA.Hence,(66)utmlutm⟶utlutalmost everywhere inA,where A=Ω×0,T. Thus, using (64) and (66) and Lions lemma, we derive(67)utmlutm⇀utlutweakly inL20,T,L2Ω.Similarly,(68)vtmlvtm⇀vtlvtweakly inL20,T,L2Ω,(69)z1m,z2m⟶z1,z2strongly inL20,T,L2Ω,which implies z1m,z2m⟶z1,z2 almost everywhere in A.By multiplying (31) and (36) by θt∈D0,T and by integrating over 0,T, it follows that(70)−1l+1∫0Tutktlutkt,wjθ′tdt+∫0TM∇ukt2∇ukt,∇wjθtdt+∫0T∇uttk,∇wjθtdt−∫0T∫0th1t−s∇uks,∇wjθtdsdt+μ1∫0T∇z1k.,1,∇wjθtdt=0,−1l+1∫0Tvtktlvtkt,wjθ′tdt+∫0TM∇vkt2∇vkt,∇wjθtdt+∫0T∇vttk,∇wjθtdt−∫0T∫0th2t−s∇vks,∇wjθtdsdt+μ2∫0T∇z2k.,1,∇wjθtdt=0,∫0T∫01∫Ωτt∂∂tz1k+1−ρτ′t∂∂ρz1kϕjθtdxdρdt=0,∫0T∫01∫Ωτt∂∂tz2k+1−ρτ′t∂∂ρz2kϕjθtdxdρdt=0,for all j=1…k.The convergence condition in (58)–(62), (68) and (67) are sufficient, thus we can pass to the limit in (70). Then, we have(71)∫0T∫Ωutlutt−M∇ut2Δu−Δutt+∫0th1t−sΔusds−μ1Δz1.,1wjθtdxdt=0,∫0T∫Ωvtlvtt−M∇vt2Δv−Δvtt+∫0th2t−sΔusds−μ2Δz2.,1wjθtdxdt=0,∫0T∫01∫Ωτt∂∂tz1+1−ρτ′t∂∂ρz1ϕjθtdxdρdt=0,∫0T∫01∫Ωτt∂∂tz2+1−ρτ′t∂∂ρz2ϕjθtdxdρdt=0,for all j=1…k. This completes the proof of Theorem 1.


## 3.1. First Estimate

Since the sequencesu0k,v0k,u1k,v1k,z1kρ,0, and z2kρ,0 converge and from Lemma 3 with employing Gronwall’s lemma, we can find a positive constant C1 independent of k such that(40)Ekt+β∫0t∇z1kx,1,s2+∇z2kx,1,s2ds≤C1,where(41)Ekt=1l+2utkl+2l+2+vtkl+2l+2+2α∫Ωukvkdx+b2γ+1∇uk2γ+1+∇vk2γ+1+12a−∫0th1sds∇uk2+12a−∫0th2sds∇vk2+12∇utk2+∇vtk2+12h1o∇ukt+h2o∇vkt+ξτt∫01∇z1kx,ρ,t2+∇z2kx,ρ,t2dρ.Noting Assumption1 and estimate (40) yields that(42)uk,vkare bounded inLloc∞0,∞,H01Ω,utk,vtkare bounded inLloc∞0,∞,H01Ω,z1kx,ρ,t,z2kx,ρ,tare bounded inLloc∞0,∞,L10,1,H01Ω.


## 3.2. Second Estimate

Multiplying the first equation (respectively, the second equation) in (31) by attjk (respectively, by bttjk ) and summing over j from 1 to k, it follows that (43)∫Ωutkluttk2dx−α∫Ωvtkutkdx+∫ΩM∇uk2∇uk∇uttkdx+∇uttk2=∫0th1t−s∫Ω∇uks∇uttktdxds−μ1∫Ω∇uttk∇z1kx,1,tdx,∫Ωvtklvttk2dx−α∫Ωvtkutkdx+∫ΩM∇vk2∇vk∇vttkdx+∇vttk2=∫0th2t−s∫Ω∇vks∇vttktdxds−μ2∫Ω∇vttk∇z2kx,1,tdx.Differentiating (36) with respect to t, we get(44)τt1−ρτ′t′∂∂tz1k+τt1−ρτ′t∂2∂t2z1k+∂2∂t∂ρz1k,ϕj=0,τt1−ρτ′t′∂∂tz2k+τt1−ρτ′t∂2∂t2z2k+∂2∂t∂ρz2k,ϕj=0.Multiplying the first equation byctjk (respectively, the second equation by dtjk) and summing over j from 1 to k, it follows that(45)τt1−ρτ′t′∂∂tz1k2+12τt1−ρτ′tddt∂∂tz1k2+12ddρ∂∂tz1k2=0,τt1−ρτ′t′∂∂tz2k2+12τt1−ρτ′tddt∂∂tz2k2+12ddρ∂∂tz2k2=0,and then we have(46)12τt1−ρτ′t′∂∂tz1k2+12ddtτt1−ρτ′t∂∂tz1k2+12ddρ∂∂tz1k2=0,12τt1−ρτ′t′∂∂tz2k2+12ddtτt1−ρτ′t∂∂tz2k2+12ddρ∂∂tz2k2=0.Integrating over0,1 with respect to ρ, we obtain(47)12∫01τt1−ρτ′t′∂∂tz1k2dρ+12ddt∫01τt1−ρτ′t∂∂tz1k2dρ+12∂∂tz1kx,1,t2−12uttkx,t2=0,12∫01τt1−ρτ′t′∂∂tz2k2dρ+12ddt∫01τt1−ρτ′t∂∂tz2k2dρ+12∂∂tz2kx,1,t2−12vttkx,t2=0.Summing (43) and (47) and as Mr≥a, we get(48)∫Ωutkluttk2dx+∇uttk2+12ddt∫01τt1−ρτ′t∂∂tz1k2dρ+12∂∂tz1kx,1,t2≤−a∫Ω∇uk∇uttkdx+12uttkx,t2+α∫Ωvtkutkdx−12∫01τt1−ρτ′t′∂∂tz1k2dρ+∫0th1t−s∫Ω∇uks∇uttktdxds−μ1∫Ω∇uttk∇z1kx,1,tdx,∫Ωvtklvttk2dx+∇vttk2+12ddt∫01τt1−ρτ′t∂∂tz2k2dρ+12∂∂tz2kx,1,t2≤−a∫Ω∇vk∇vttkdx+12vttkx,t2+α∫Ωvtkutkdx−12∫01τt1−ρτ′t′∂∂tz2k2dρ+∫0th2t−s∫Ω∇vks∇vttktdxds−μ2∫Ω∇vttk∇z2kx,1,tdx.By Young’s inequality, the right hand side of (48) can be estimated as follows:(49)∫Ωa∇uk∇uttkdx≤η∇uttk2+a24η∇uk2,∫Ωa∇vk∇vttkdx≤η∇vttk2+a24η‖∇vk2.Using Cauchy–Schwarz inequality and Sobolev–Poincare inequality, we obtain(50)α∫Ωvtkutkdx≤α2vtk2+utk2≤Csα2∇vtk2+∇utk2,(51)∫0th1t−s∫Ω∇uks∇uttktdxds≤η∇uttk2+14η∫Ω∫0th1t−s∇uksdsdx2≤η∇uttk2+14η∫0thsds∫Ω∫0th1t−s∇uks2dsdx≤η∇uttk2+14ηa−k∫Ω∫0th1t−s∇uks2dsdx≤η∇uttk2+14ηa−k∫0th1t−s∇uks2ds≤η∇uttk2+14ηa−kh10∫0t∇uks2ds.Similarly,(52)∫0th2t−s∫Ω∇vks∇vktttdxds≤η∇vktt2+14ηa−kh20∫0t∇vks2ds,(53)μ1∫Ω∇uttk∇z1kx,1,tdx≤ημ12∇uttk2+14η∇z1kx,1,t2,μ2∫Ω∇vttk∇z2kx,1,tdx≤ημ22∇vttk2+14η∇z2kx,1,t2.By using (49)–(53) in (48), we deduce(54)∫Ωutkluttk2dx+∇uttk2+12ddt∫01τt1−ρτ′t∂∂tz1k2dρ+12∂∂tz1kx,1,t2≤ημ12+2+Cs22∇uttk2+Csα2∇vtk2+∇utk2+a24η∇uk2+14η∇z1kx,1,t2+14ηa−kh10∫0t∇uks2ds−12∫01τt1−ρτ′t′∂∂tz1k2dρ,∫Ωvtklvttk2dx+∇vttk2+12ddt∫01τt1−ρτ′t∂∂tz2k2dρ+12∂∂tz2kx,1,t2≤ημ22+2+Cs22∇vttk2+Csα2∇vtk2+∇utk2+a24η∇vk2+14η∇z2kx,1,t2+14ηa−kh20∫0t∇vks2ds−12∫01τt1−ρτ′t′∂∂tz2k2dρ.By using Assumption3 and taking the first estimate (40) into account, we infer(55)∫Ωutkluttk2dx+1−ημ12+2+Cs22∇uttk2+12ddt∫01τt1−ρτ′t∂∂tz1k2dρ+12∂∂tz1kx,1,t2≤C2+14ηa−k1h10C1T+a1∫01τt1−ρτ′t∂∂tz1k2dρ,∫Ωvtklvttk2dx+1−ημ22+2+Cs22∇vttk2+12ddt∫01τt1−ρτ′t∂∂tz2k2dρ+12∂∂tz2kx,1,t2≤C2+14ηa−k2h20C1T+a2∫01τt1−ρτ′t∂∂tz2k2dρ,where C2 is a positive constant which depends on η,a,C1,Cs,α.Integrating (55) over (0, t), we obtain(56)∫0tutkluttk2dxdt+1−ημ12+2+Cs22∫0t∇uttks2ds+∫01τt1−ρτ′t∂∂tz1k2dρ+12∫01∂∂tz1kx,1,t2dt≤C2T+14ηa−kh10C1T2+a1∫0t∫01τt1−ρτ′t∂∂sz1k2dρds,∫0t∫Ωvtklvttk2dxdt+1−ημ22+2+Cs22∫0t∇vttks2ds+∫01τt1−ρτ′t∂∂tz2k2dρ+12∫0t∂∂tz2kx,1,t2dt≤C2T+14ηa−kh20C1T2++a2∫0t∫01τs1−ρτ′s∂∂sz2k2dρds.For a suitableη>0 such that 1−ημi2+2+Cs2/2>0 for i=1,2 and using Gronwall’s lemma, we obtain the second estimate(57)∫0t∇uktts2+∇vktts2ds+∫01τt1−ρτ′t∂∂tz1k2+∂∂tz2k2dρ≤C3.We observe from estimates (40) and (57) that there exist subsequences um of uk and vm of vk such that(58)um,vm⇀u,vweakly star inL∞0,T,H01Ω,(59)utm,vtm⇀ut,vtweakly star inL∞0,T,H01Ω,(60)uttm,vttm⇀utt,vttweakly inL20,T,H01Ω,(61)z1m,z1m⇀z1,z2weakly star inL∞0,T,H01Ω,L20,1,(62)∂∂tz1m,∂∂tz2m⇀∂∂tz1,∂∂tz2weakly star inL∞0,T,L2Ω×0,1,In the following, we will treat the nonlinear term. From the first estimate (40) and Lemma 1, we deduce(63)utklutkL20,T,L2Ω=∫0Tutk2l+12l+1dt≤Cs2l+1∫0T∇utk22l+1dt≤Cs2l+1C1l+1T.On the other hand, from Aubin–Lions theorem (see [10]), we deduce that there exists a subsequence of um, still denoted by um such that(64)utm⟶utstrongly inL20,T,L2Ω,which implies(65)utm⟶utalmost everywhere inA.Hence,(66)utmlutm⟶utlutalmost everywhere inA,where A=Ω×0,T. Thus, using (64) and (66) and Lions lemma, we derive(67)utmlutm⇀utlutweakly inL20,T,L2Ω.Similarly,(68)vtmlvtm⇀vtlvtweakly inL20,T,L2Ω,(69)z1m,z2m⟶z1,z2strongly inL20,T,L2Ω,which implies z1m,z2m⟶z1,z2 almost everywhere in A.By multiplying (31) and (36) by θt∈D0,T and by integrating over 0,T, it follows that(70)−1l+1∫0Tutktlutkt,wjθ′tdt+∫0TM∇ukt2∇ukt,∇wjθtdt+∫0T∇uttk,∇wjθtdt−∫0T∫0th1t−s∇uks,∇wjθtdsdt+μ1∫0T∇z1k.,1,∇wjθtdt=0,−1l+1∫0Tvtktlvtkt,wjθ′tdt+∫0TM∇vkt2∇vkt,∇wjθtdt+∫0T∇vttk,∇wjθtdt−∫0T∫0th2t−s∇vks,∇wjθtdsdt+μ2∫0T∇z2k.,1,∇wjθtdt=0,∫0T∫01∫Ωτt∂∂tz1k+1−ρτ′t∂∂ρz1kϕjθtdxdρdt=0,∫0T∫01∫Ωτt∂∂tz2k+1−ρτ′t∂∂ρz2kϕjθtdxdρdt=0,for all j=1…k.The convergence condition in (58)–(62), (68) and (67) are sufficient, thus we can pass to the limit in (70). Then, we have(71)∫0T∫Ωutlutt−M∇ut2Δu−Δutt+∫0th1t−sΔusds−μ1Δz1.,1wjθtdxdt=0,∫0T∫Ωvtlvtt−M∇vt2Δv−Δvtt+∫0th2t−sΔusds−μ2Δz2.,1wjθtdxdt=0,∫0T∫01∫Ωτt∂∂tz1+1−ρτ′t∂∂ρz1ϕjθtdxdρdt=0,∫0T∫01∫Ωτt∂∂tz2+1−ρτ′t∂∂ρz2ϕjθtdxdρdt=0,for all j=1…k. This completes the proof of Theorem 1.


## 4. Uniform Decay of the Energy (Proof of Theorem 2)

In this section we study the solution’s asymptotic behavior of system (3).To prove our main result, we construct a Lyapunov functional equivalent toE. For this, we define some functionals which allow us to obtain the desired estimate.Lemma 4.
Letu,v,z1,z2 be a solution of problem (7). Then, the functional(72)It=τt∫01e−2τtρ∇z1x,ρ,t2+∇z1x,ρ,t2dρ,satisfies the estimate.(i)
It≤1/ξEt.(ii)
I′t≤−2τte−2τ1∫01∇z1x,ρ,t2+∇z2x,ρ,t2dρ−1−de−2τ1∇z1x,1,t2+∇z2x,1,t2+∇utx,t2+∇vtx,t2.Proof.
(ii) Differentiating (72) with respect to t and using (5)-(6) and Assumption 3, we get(73)ddtIt=τ′t∫01e−2τtρ∇z1x,ρ,t2+∇z2x,ρ,t2dρ+τt∫Ω∫01e−2τtρ∂∇z1x,ρ,t2+∇z2x,ρ,t2∂t−2τ′tρe−2τtρ∇z1x,ρ,t2+∇z2x,ρ,t2dρdx=∫Ω∫01e−2τtρτ′t∇z1x,ρ,t2+∇z2x,ρ,t2+τt∂∇z1x,ρ,t2+∇z2x,ρ,t2∂tdρdx−2∫Ω∫01τtτ′tρe−2τtρ∇z1x,ρ,t2+∇z2x,ρ,t2dρdx=−∫01e−2τtρ∂∂ρ1−ρτ′t∇z1x,ρ,t2+∇z2x,ρ,t2dρ−2∫01τtτ′tρe−2τtρ∇z1x,ρ,t2+∇z2x,ρ,t2dρ=−∫01∂∂ρe−2τtρ1−τ′tρ∇z1x,ρ,t2+∇z2x,ρ,t2−2τte−2τtρ1−τ′tρ∇z1x,ρ,t2+∇z2x,ρ,t2dρ−2τtτ′t∫01ρe−2τtρ∇z1x,ρ,t2+∇z2x,ρ,t2dρ=∇z1x,ρ,t2+∇z2x,ρ,t2−e−2τtρ1−τ′t∇z1x,ρ,t2+∇z2x,ρ,t2−2τt∫Ω∫011−τ′tρ+τ′tρe−2τtρ∇z1x,ρ,t2+∇z2x,ρ,t2dρdx≤−2It+∇utx,t2+∇vtx,t2−e−2τtρ1−d∇z1x,ρ,t2+∇z2x,ρ,t2.
Sincee−2τtρ is a decreasing function for ρ∈0,1 and τt∈τ0,τ1, we deduce(74)It≥τt∫01e−2τ1∇z1x,ρ,t2+∇z2x,ρ,t2dρ.
Thus, our proof is completed.Lemma 5.
Letu,v,z1,z2 be a solution of problem (7). Then, the functional(75)ϕt=1l+1∫Ωutlutu+vtlvtvdx+∫Ω∇ut∇udx+∫Ω∇vt∇vdx,satisfies the estimate(i)
ϕt≤1/l+2utl+2l+2+vtl+2l+2+l+1−1/l+2csl+2+c/2∇ul+2+∇vl+2+1/2∇u2+∇v2.(ii)
ϕ′t≤1/l+1utl+2l+2+vtl+2l+2+ηa−k+1+αCs−k∇u2+∇v2+1/4ηh1o∇ut+h1o∇ut+μ12/4η∇z1x,1,t2+μ22/4η∇z2x,1,t2+∇ut2+∇vt2.Proof.
(i)
From Young’s inequality, the Sobolev embedding, andLl+2⟶L2, we deduce
(76)
ϕ
t
≤
1
l
+
2
u
t
l
+
2
l
+
2
+
l
+
1
−
1
l
+
2
u
l
+
2
l
+
2
+
1
l
+
2
v
t
l
+
2
l
+
2
+
l
+
1
−
1
l
+
2
v
l
+
2
l
+
2
+
1
2
∇
u
t
2
+
1
2
∇
u
2
+
1
2
∇
v
t
2
+
1
2
∇
v
2
≤
1
l
+
2
u
t
l
+
2
l
+
2
+
l
+
1
−
1
l
+
2
c
s
l
+
2
∇
u
l
+
2
+
1
l
+
2
v
t
l
+
2
l
+
2
+
l
+
1
−
1
l
+
2
c
s
l
+
2
∇
v
l
+
2
+
1
2
∇
u
t
2
+
1
2
∇
u
2
+
1
2
∇
v
t
2
+
1
2
∇
v
2
≤
1
l
+
2
u
t
l
+
2
l
+
2
+
v
t
l
+
2
l
+
2
+
l
+
1
−
1
l
+
2
c
s
l
+
2
+
c
2
∇
u
l
+
2
+
∇
v
l
+
2
+
1
2
∇
u
t
2
+
∇
v
t
2
.
(i)
Differentiatingϕt with respect to t and using the first and second equations of (7), we get
(77)
ϕ
′
t
=
1
l
+
1
∫
Ω
u
t
l
u
t
′
u
d
x
+
1
l
+
1
∫
Ω
u
t
l
+
2
d
x
+
1
l
+
1
∫
Ω
v
t
l
v
t
′
vd
x
+
1
l
+
1
∫
Ω
v
t
l
+
2
d
x
+
∫
Ω
∇
u
t
t
∇
u
d
x
+
∫
Ω
∇
u
t
∇
u
t
d
x
+
∫
Ω
∇
v
t
t
∇
v
d
x
+
∫
Ω
∇
v
t
∇
v
t
d
x
=
∫
Ω
u
t
l
u
t
u
d
x
+
1
l
+
1
‖
u
t
‖
l
+
2
l
+
2
+
∫
Ω
v
t
l
v
t
vd
x
+
1
l
+
1
v
t
l
+
2
l
+
2
−
∫
Ω
∇
u
t
t
u
d
x
+
∇
u
t
2
−
∫
Ω
∇
v
t
t
∇
u
d
x
+
∇
v
t
2
=
1
l
+
1
u
t
l
+
2
l
+
2
+
u
t
l
+
2
l
+
2
+
∫
Ω
u
t
l
u
t
−
Δ
u
t
t
u
d
x
+
∫
Ω
v
t
l
v
t
−
Δ
v
t
t
u
d
x
+
∇
u
t
2
+
∇
v
t
2
=
1
l
+
1
u
t
l
+
2
l
+
2
+
u
t
l
+
2
l
+
2
+
∫
Ω
−
α
v
+
M
∇
u
2
Δ
u
−
∫
0
t
h
1
t
−
s
Δ
u
s
d
s
+
μ
1
Δ
z
1
x
,
1
,
t
u
d
x
+
∫
Ω
−
α
u
+
M
∇
v
2
Δ
v
−
∫
0
t
h
1
t
−
s
Δ
v
s
d
s
+
μ
2
Δ
z
2
x
,
1
,
t
u
d
x
+
∇
u
t
2
+
∇
v
t
2
=
1
l
+
1
u
t
l
+
2
l
+
2
+
u
t
l
+
2
l
+
2
−
M
∇
u
t
2
∇
v
t
2
+
∫
Ω
∇
u
t
∫
0
t
h
1
t
−
s
Δ
u
s
d
s
d
x
−
μ
1
∫
Ω
∇
z
1
x
,
1
,
t
∇
u
d
x
−
M
∇
v
2
∇
v
2
+
∫
Ω
∇
v
t
∫
0
t
h
2
t
−
s
Δ
v
s
d
s
d
x
−
μ
2
∫
Ω
∇
z
2
x
,
1
,
t
∇
v
d
x
+
∇
u
t
2
+
∇
v
t
2
−
2
α
∫
Ω
u
v
d
x
.
AsMr≥a and making use Young’s inequality, we obtain(78)ϕ′t≤1l+1utl+2l+2+vtl+2l+2−a∇u2+∫Ω∇ut∫0th1t−s∇usdsdx+μ124η∇z1x,1,t2+η∇u2−a∇v2+∫Ω∇vt∫0th2t−s∇vsdsdx+μ224η∇z2x,1,t2+η∇v2+∇ut2+∇vt2.
By use of Young’s inequality, we can estimate the third term in the right side as follows:(79)∫Ω∇ut∫0th1t−s∇usdsdx≤∫0tht−s∫Ω∇ut∇us−∇utdxds+∇ut2∫0th1t−sds≤η∇ut2∫0th1sds+14η∫0th1t−s∇us−∇ut2ds+∇ut2∫0th1sds≤1+ηa−k∇ut2+14ηh1o∇ut.
Similarly,(80)∫Ω∇vt∫0th2t−s∇vsdsdx≤1+ηa−k∇vt2+14ηh2o∇vt,−2α∫Ωuvdx≤αCs∇u2+∇v2.
Thus, our proof is completed.Lemma 6.
Let (u, z) be a solution of problem (7). Then, the functional(81)ψt=∫ΩΔut−1l+1utlut∫0th1t−sut−usdsdx+∫ΩΔvt−1l+1vtlvt∫0th2t−svt−vsdsdx,satisfies the estimates(i)
ψt≤1/2∇ut2+∇vt2+1/2a−k1+l+1−1/l+2a−klcsl+2h1o∇ut+h2o∇vt+l+1−1/l+2a−kl+2csl+222l+1∇u2l+1+∇v2l+1+1/l+2utl+2l+2+vtl+2l+2.(ii)
ψt≤δa−k+l+1−1/l+2h10l+2csl+222l+1M∇u2∇u2+αCs/2+2δa−k2∇u2+M∇u2/4δ+αCs/2+2δ+1/2δa−kh1o∇vt−h10/4δ1+l+1−1/l+2h10lcsl+2h1′o∇vt+δ−∫0th1sds∇ut2+μ12δ∇z1x,1,t2+1/l+11−∫0th1sdsutl+2l+2+δa−k+l+1−1/l+2h20l+2csl+222l+1M∇v2∇v2+αCs/2+2δa−k2∇v2+M∇u2/4δ+αCs/2+2δ+1/2δa−kh2o∇vt−h20/4δ1+l+1−1/l+2h20lcsl+2h2′o∇vt+δ−∫0th2sds∇vt2+μ22δ∇z2x,1,t2+1/l+11−∫0th2sdsvtl+2l+2,where δ>0 and cs is the Sobolev embedding constant.Proof.
(i)
ψ
t
=
−
∫
Ω
∇
u
t
∫
0
t
h
2
t
−
s
∇
u
t
−
∇
u
s
d
s
d
x
−
∫
Ω
1
/
l
+
1
u
t
l
u
t
∫
0
t
h
2
t
−
s
u
t
−
u
s
d
s
d
x
−
∫
Ω
∇
v
t
∫
0
t
h
2
t
−
s
∇
v
t
−
∇
v
s
d
s
d
x
−
∫
Ω
1
/
l
+
1
v
t
l
v
t
∫
0
t
h
2
t
−
s
v
t
−
v
s
d
s
d
x.
We use Young’s inequality with the conjugate exponentsp=l+2/l+1 and q=l+2, and the second term in the right hand side can be estimated as
(82)
−
∫
Ω
1
l
+
1
u
t
l
u
t
∫
0
t
h
1
t
−
s
u
t
−
u
s
d
s
d
x
≤
1
l
+
1
∫
Ω
u
t
l
u
t
∫
0
t
h
1
t
−
s
u
t
−
u
s
d
s
d
x
≤
1
l
+
1
1
p
∫
Ω
u
t
l
u
t
P
d
x
+
1
q
∫
Ω
∫
0
t
h
1
t
−
s
u
t
−
u
s
d
s
q
d
x
≤
1
l
+
1
1
p
∫
Ω
u
t
l
+
1
P
d
x
+
1
q
∫
Ω
∫
0
t
h
1
t
−
s
u
t
−
u
s
d
s
q
d
x
≤
1
l
+
2
u
t
l
+
2
l
+
2
+
l
+
1
−
1
l
+
2
∫
Ω
∫
0
t
h
1
t
−
s
l
+
1
l
+
2
h
1
t
−
s
1
l
+
2
u
t
−
u
s
d
s
l
+
2
d
x
.
We have by Hölder’s inequality
(83)
∫
Ω
∫
0
t
h
1
t
−
s
l
+
1
/
l
+
2
h
1
t
−
s
1
/
l
+
2
u
t
−
u
s
d
s
l
+
2
d
x
≤
∫
Ω
∫
0
t
h
1
t
−
s
l
+
1
/
l
+
2
P
d
s
1
/
P
∫
0
t
h
1
t
−
s
1
/
l
+
2
u
t
−
u
s
q
d
s
1
/
q
l
+
2
d
x
≤
∫
Ω
∫
0
t
h
1
t
−
s
d
s
l
+
1
/
l
+
2
∫
0
t
h
1
t
−
s
d
s
u
t
−
u
s
l
+
2
d
s
1
/
l
+
2
l
+
2
d
x
≤
∫
0
t
h
1
t
−
s
d
s
l
+
1
∫
0
t
h
1
t
−
s
u
t
−
u
s
l
+
2
l
+
2
d
s
≤
a
−
k
l
+
1
c
s
l
+
2
∫
0
t
h
1
t
−
s
h
1
t
−
s
∇
u
t
−
∇
u
s
l
+
1
∇
u
t
−
∇
u
s
d
s
≤
a
−
k
l
+
1
c
s
l
+
2
1
2
∫
0
t
h
1
t
−
s
∇
u
t
−
∇
u
s
2
l
+
2
d
s
+
1
2
∫
0
t
h
1
t
−
s
∇
u
t
−
∇
u
s
2
d
s
≤
a
−
k
l
+
1
c
s
l
+
2
1
2
∫
0
t
h
1
t
−
s
2
∇
u
t
2
l
+
2
d
s
+
1
2
h
1
o
∇
u
t
≤
a
−
k
l
+
1
c
s
l
+
2
2
2
l
+
1
a
−
k
∇
u
t
2
l
+
1
+
1
2
h
1
o
∇
u
t
.
Combining (83) with (82), we obtain
(84)
−
∫
Ω
1
l
+
1
u
t
l
u
t
∫
0
t
h
1
t
−
s
u
t
−
u
s
d
s
d
x
≤
1
l
+
2
u
t
l
+
2
l
+
2
+
l
+
1
−
1
l
+
2
a
−
k
l
+
1
c
s
l
+
2
2
2
l
+
1
a
−
k
∇
u
t
2
l
+
1
+
1
2
h
1
o
∇
u
t
.
Similarly, we get
(85)
−
∫
Ω
∇
u
t
∫
0
t
h
1
t
−
s
∇
u
t
−
∇
u
s
d
s
d
x
≤
1
2
∇
u
t
2
+
1
2
∫
Ω
∫
0
t
h
1
t
−
s
∇
u
t
−
∇
u
s
d
s
2
d
x
≤
1
2
∇
u
t
2
+
1
2
a
−
k
h
1
o
∇
u
t
.
Similarly,
(86)
−
∫
Ω
1
l
+
1
v
t
l
v
t
∫
0
t
h
2
t
−
s
v
t
−
v
s
d
s
d
x
≤
1
l
+
2
v
t
l
+
2
+
l
+
1
−
1
l
+
2
a
−
k
l
+
1
c
s
l
+
2
2
2
l
+
1
a
−
k
∇
v
t
2
l
+
1
+
1
2
h
2
o
∇
v
t
−
∫
Ω
Ω
∇
v
t
∫
0
t
h
2
t
−
s
∇
v
t
−
∇
v
s
d
s
d
x
≤
1
2
∇
v
t
2
+
1
2
a
−
k
h
2
o
∇
v
t
.
Combining (84)–(86), we deduce (i).
(ii)
Using the Leibniz formula and the first and second equations of (7), we have
(87)
ψ
′
t
=
∫
Ω
Δ
u
t
t
−
u
t
l
u
t
t
∫
0
t
h
1
t
−
s
u
t
−
u
s
d
s
d
x
+
∫
Ω
Δ
u
t
−
1
l
+
1
u
t
l
u
t
∫
0
t
h
1
′
t
−
s
u
t
−
u
s
+
h
1
t
−
s
u
t
t
d
s
d
x
+
∫
Ω
Δ
v
t
t
−
v
t
l
v
t
t
∫
0
t
h
2
t
−
s
v
t
−
v
s
d
s
d
x
+
∫
Ω
Δ
v
t
−
1
l
+
1
v
t
l
v
t
∫
0
t
h
2
′
t
−
s
v
t
−
v
s
+
h
2
t
−
s
v
t
t
d
s
d
x
=
−
α
∫
Ω
v
t
∫
0
t
h
1
t
−
s
u
t
−
u
s
d
s
d
x
+
∫
Ω
M
∇
u
2
∇
u
t
∫
0
t
h
1
t
−
s
∇
u
t
−
∇
u
s
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,where(88)I1=∫ΩM∇u2∇ut∫0th1t−s∇ut−∇usdsdx+∫ΩM∇v2∇vt∫0th2t−s∇vt−∇vsdsdxI2=−∫Ω∫0th1t−s∇usds∫0th1t−s∇ut−∇usdsdx−∫Ω∫0th2t−s∇vsds∫0th2t−s∇vt−∇vsdsdxI3=μ1∫Ω∇z1x,1,t∫0th1t−s∇ut−∇usdsdx+μ2∫Ω∇z2x,1,t∫0th2t−s∇vt−∇vsdsdxI4=−∫Ω∇ut∫0th1′t−s∇ut−∇usdsdx−∫Ω∇vt∫0th2′t−s∇vt−∇vsdsdxI5=−1l+1∫Ωutlut∫0th1′t−s∇ut−∇usdsdx−1l+1∫Ωvtlvt∫0th2′t−s∇vt−∇vsdsdxI6=−α∫Ωvt∫0th1′t−s∇ut−∇usdsdx−α∫Ωut∫0th2′t−s∇vt−∇vsdsdx.
In what follows we will estimateI1,…,I6.
ForI1, we use Hölder and Young’s inequalities with p=q=2, and we get(89)I1≤M∇u2∫Ω∇ut∫0th1sds1/2∫0th1s∇ut−∇us2ds1/2dx+M∇v2∫Ω∇vt∫0th2sds1/2∫0th2t−s∇vt−∇vs2ds1/2dx≤M∇u2δ∫Ω∇ut2∫0th1sdsdx+14δ∫Ω∫0th1t−s∇ut−∇us2dsdx+M∇v2δ∫Ω∇vt2∫0th2sdsdx+14δ∫Ω∫0th2t−s∇vt−∇vs2dsdx≤M∇u2δa−k∇ut2+14δh1o∇ut+M∇v2δa−k∇vt2+14δh1o∇vt.
Similarly,(90)I2≤δ∫Ω∫0th1t−s∇usds2dx+14δ∫Ω∫0th1t−s∇ut−∇usds2dx+δ∫Ω∫0th2t−s∇vsds2dx+14δ∫Ω∫0th2t−s∇vt−∇vsds2dx≤δ∫Ω∫0th1t−s∇us−∇ut|+|∇utds2dx+14δ∫0th1sdsh1o∇ut+δ∫Ω∫0th2t−s∇vs−∇vt|+|∇vtds2dx+14δ∫0th2sdsh2o∇vt≤2δ∇ut2∫0th1tds2dx+2δ+14δ∫0th1sdsh1o∇ut+2δ∇vt2∫0th2tds2dx+2δ+14δ∫0th2sdsh1o∇vt≤2δ∇ut2a−k2+2δ+14δa−kh1o∇ut+2δ∇vt2a−k2+2δ+14δa−kh2o∇vt,(91)I3≤δμ12∇z1x,1,t2+μ22∇z2x,1,t2+a−k4δh1o∇ut+a−k4δh2o∇vt,(92)I4≤δ∫Ω∇ut2dx+14δ∫Ω∫0th1′t−s∇ut−∇usds2dx+δ∫Ω∇vt2dx+14δ∫Ω∫0th2′t−s∇vt−∇vsds2dx≤δ∇ut2+14δ∫0t−h1′t−sds∫Ω∫0t−h1′t−s∇ut−∇us2dsdx+δ∇vt2+14δ∫0t−h2′t−sds∫Ω∫0t−h2′t−s∇vt−∇vs2dsdx≤δ∇ut2−h104δh1′o∇ut+δ∇vt2−h204δh2′o∇ut,and since l≤γ, we follow almost the same steps to obtain ψt)(93)I5≤1l+2utl+2l+2+vtl+2l+2+l+1−1l+2h10l+1∫0t−h1′t−sut−usl+2l+2ds+h20l+1∫0t−h2′t−svt−vsl+2l+2ds≤1l+2utl+2l+2+vtl+2l+2+l+1−1l+2csl+2h10l+1∫0t−h1′t−s∇ut−∇usl+2ds+h20l+1∫0t−h2′t−s∇ut−∇usl+2ds≤1l+2utl+2l+2+vtl+2l+2+l+1−1l+2csl+2h10l+1δ22l+1h10∇ut2l+1−14δh1′o∇ut+l+1−1l+2csl+2h20l+1δ22l+1h20∇vt2l+1−14δh2′o∇vt≤1l+2utl+2l+2+vtl+2l+2+l+1−1l+2csl+2h10l+1δ22l+1h10M∇ut2∇ut2−14δh1′o∇ut+l+1−1l+2csl+2h20l+1δ22l+1h20M∇vt2∇vt2−14δh2′o∇vt,(94)I6≤αCs2∇ut2+∇vt2+a−kh1o∇ut+h2o∇vt.
Combining (87) and (89)–(94), we finish the proof.Proof of Theorem 1.2.
Now, forM,ε1>0, we introduce the following functional:(95)Ft=MEt+ε1ϕt+ψt+It∼Et.
Indeed, to proveFt∼Et, we show that there exist two positive constants κ1 and κ2 such that(96)κ1Et≤Ft≤κ2Et.
From (i) of Lemmas4–6 and recalling the fact that l≤γ, we get(97)ε1ϕt+ψt+It≤ε1+1l+2utl+2l+2+vtl+2l+2+ε1+12∇ut2+∇vt2+ε1c2+l+1−1l+2csl+2ε1+22l+1a−kl+2∇u2γ+1+∇v2γ+1+a−k21+l+1−1l+2a−klcsl+2h1o∇ut+h2o∇vt+1ξEt≤κEt,where κ>0 depending on ε1,a,b,l,c,cs,k,ξ . If we choose M=κ+ϵ, we can obtain our result.
By recalling Lemmas3–6 and by Assumption 2, we deduce that for t≥t0>0,(98)F′t=ME′t+ε1ϕ′t+ψ′t+I′t≤μ2δ+ε1μ24η−1−de−2τ1−Mβ∇z1x,1,t2+∇z2x,1,t2−2τte−2τ1∫01∇z1x,p,t2+∇z2x,p,t2dρ−ε1k−ηa−k+1−αCS2−2δa−k2−δa−k+l+1−1l+2h2csl+222l+1M0∇u2+∇v2+M2−h14δ1+l+1−1l+2h1lcsl+2−1ζε14η+M04δ+αCS2+2δ+12δa−kh1′o∇ut+h2′o∇ut−1l+1h0−1−ε1utl+2l+2+vtl+2l+2−h0−δ−Mλ−1−ε1∇ut2+∇vt2,where M0=maxM∇u2,M∇v2, h0=min∫0t0h1sds,∫0t0h2sds, h1=minh10,h20, h2=minh10,h20, and ζ=maxζ1,ζ1.
Letϵ>0 be sufficiently small so M is fixed; we take h0−Mλ−1>ε1 and δ small enough such that(99)a3=h0−1−ε1>0,a4=h0−δ−Mλ−1−ε1.
Further, we chooseη,α small enough such that(100)a1=μ12δ+ε1μ24η−1−de−2τ1−Mβ>0,a2=ε1k−ηa−k+1−αCS2−2δa−k2−δa−k+l+1−1l+2h2csl+222l+1M0>0,a5=M2−h14δ1+l+1−1l+2h1lcsl+2−1ζε14η+M04δ+αCS2+2δ+12δa−k<0.
Thus,(101)F′t≤−a31l+2utl+2l+2+vtl+2l+2−a2∇u2+∇v2−2τte−2τ1∫01∇z1x,ρ,t2+∇z2x,ρ,t2dρ+a1∇z1x,1,t2+∇z2x,1,t2−a4∇ut2+∇vt2+a5h1′o∇ut+h2′o∇vt≤−mEt−cE′t,where m=min2e−2τ1/ξ,2a2/a,a3 and a1/β≤c≤mina4/λ,−2a5.
LetLt=Ft+cEt∼Et. From (101), we get(102)L′t≤−c′Lt,∀t≥t0,for some c′>0. A simple integration over t0,t yields(103)Lt≤Lt0e−c′t−t0,∀t≥t0.
Thanks to equivalence betweenL and E, we obtain (16).



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*Source: 6324971-2020-05-09.xml*