# On the System of Coupled Nondegenerate Kirchhoff Equations with Distributed Delay: Global Existence and Exponential Decay

Authors: Abdelbaki Choucha; Salah Mahmoud Boulaaras; Djamel Ouchenane; Salem Alkhalaf; Ibrahim Mekawy; Mohamed Abdalla
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper studies the system of coupled nondegenerate viscoelastic Kirchhoff equations with a distributed delay. By using the energy method and Faedo-Galerkin method, we prove the global existence of solutions. Furthermore, we prove the exponential stability result.


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## Body

## 1. Introduction

LetH=Ω×τ1,τ2×0,∞, in this work, we consider
(1)ulutt−M∇u2Δu−Δutt+∫0tg1t−sΔusds−∫τ1τ2μ1ϱΔutx,t−ϱdϱ+f1u,v=0,vlvtt−M∇v2Δv−Δvtt+∫0tg2t−sΔvsds−∫τ1τ2μ2ϱΔvtx,t−ϱdϱ+f2u,v=0,where
(2)x,ϱ,t∈H,under the initial and boundary conditions
(3)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,∞,where Ω be a bounded domain in ℝn with smooth boundary ∂Ω, l>0 and Δ is the Laplacian operator, and the functions μ1,μ2:τ1,τ2⟶ℝ are bounded, with 0≤τ1<τ2, and the relaxation functions are denoted by g1,g2. The function M is given by
(4)M:ℝ+⟶ℝ+,r↦Mr=a+brγ,with a,b>0, and γ≥1, and the functions f1,f2 will be defined later.In 1976, Kirchhoff developed an equation describing the vibrations produced by a fixed series at its end, since it is considered a generalization of the d’Alembert equation, and it belongs to the wave equation models. Over time, many researchers and authors addressed these issues and problems with their continuous and rapid development, for example, see [1–4].As for viscoelasticity, it is possible to delve into the following works for further clarification [3–10].Also, the time or delay recorded in many natural and physical phenomena, especially problems resulting from vibrations, is an important factor for stability in general. And it has been studied extensively by many authors, including [5–7, 11–21]. Recently, in the presence of the varying delay, Mezouar and Boularrass studied system (1); for more information, see [22]. Based on these works, we in this work expand the results in [22] by adding the term of distributed delay.We, under appropriate conditions, obtained the global existence of solutions, and we proved the exponential stability result of the system.And we divided the paper into the following: in the second part, we set out the necessary hypotheses and the main result; in the third part, we prove the global existence of solutions, while in the fourth part, we present our result for exponential stability.


## 2. Preliminaries

In this section, we set the necessary hypotheses for proving the main result.We need the following assumptions:(A1)gi:ℝ+⟶ℝ+,i=1,2 are C1 functions satisfying
(5)g0>0,a−∫0∞gisds≥k>0,i=1,2.(A2)∃ξi>0 satisfying
(6)gi′t≤−ξigit,i=1,2,t≥0.(A3) The numberl satisfying 0<l≤γ and
(7)≤2n−2ifn>2,γ<∞ifn≤2.(A4)(8)f1u,v=a1v+b1vq+1vp−1u,f2u,v=a1u+b2uq+1vp−1v,where a1>0,b1=p+1p+q,b2=q+1p+q such that pand q are onjugate 1/p+1/q=1,p,q<γ−1/2 and satisfy
(9)2≤p,q≤n2n−2ifn>2,∞ifn≤2.We set the notations(10)g∘Ψt≔∫0tgt−sΨt−Ψs2ds.As in [17], we introduce the new variables
(11)utx,t−ϱρ=Zx,ρ,ϱ,t,vtx,t−ϱρ=Yx,ρ,ϱ,t.We have
(12)ϱZtx,ρ,ϱ,t+Zρx,ρ,ϱ,t=0,utx,t=Zx,0,ϱ,t,ϱYtx,ρ,ϱ,t+Yρx,ρ,ϱ,t=0,vtx,t=Yx,0,ϱ,t.Consequently, problem (1) is equivalent to
(13)ulutt−M∇u2Δu−Δutt+∫0tg1t−sΔusds−∫τ1τ2μ1ϱΔZx,1,ϱ,tdϱ+f1u,v=0,vlvtt−M∇v2Δv−Δvtt+∫0tg2t−sΔvsds−∫τ1τ2μ2ϱΔYx,1,ϱ,tdϱ+f2u,v=0,ρZtx,ρ,ϱ,t+Zρx,ρ,ϱ,t=0,ρYtx,ρ,ϱ,t+Yρx,ρ,ϱ,t=0,where
(14)x,ρ,s,t∈Ω×0,1×τ1,τ2×0,∞,with the initial and boundary conditions
(15)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=f0x,ρϱ,inΩ×0,1×0,τ2,Yx,ρ,ϱ,0=g0x,ρϱ.We need the following lemma.Lemma 1.
The energy functionalE, given by
(16)Et=1l+2∥ut∥l+2l+2+∥vt∥l+2l+2+b2γ+2∥∇u∥2γ+2+∥∇v∥2γ+2+12a−∫0tg1sds∥∇u∥2+12a−∫0tg2sds∥∇v∥2+12∥∇ut∥2+∥∇vt∥2+12g1∘∇ut+12g2∘∇vt+12∫01∫τ1τ2ρμ1ρ∥∇Z∥2+μ2ρ∥∇Y∥2dρdρ+α∫Ωuvdx+p+q∫Ωup+1vq+1dx,satisfies
(17)E′t≤−β∫τ1τ2μ1ϱ∥∇Zx,1,ϱ,t∥2+μ2ϱ∥∇Yx,1,ϱ,t∥2dϱ+λ∥∇ut∥2+∥∇vt∥2+12g1′∘∇ut+12g2′∘∇vt−12g1t∥∇ut∥2−12g2t∥∇vt∥2,where β=1−δ1/2>0,
andλ=maxλ1=δ1+1/2∫τ1τ2μ1ρdρ,λ2=δ1+1/2∫τ1τ2μ2ρdρ,δ1<1.Proof.
Multiplying equation (13)1,2 by ut,vt, and we use (15), one gets
(18)ddt1l+2∥ut∥l+2l+2+b2γ+1∥∇ut∥2γ+1+12a−∫0tg1sds∥∇u∥2+12∥∇ut∥2+12g1∘∇ut−12g1′∘∇ut+12g1t∥∇ut∥2+∫Ωut∫τ1τ2μ1ϱΔZx,1,ϱ,tdϱdx+∫Ωutvdx+b1∫Ωutuup−1vq+1dx+ddt1l+2∥vt∥l+2l+2+b2γ+1∥∇vt∥2γ+1+12a−∫0tg2sds∥∇v∥2+12∥∇vt∥2+12g2∘∇vt−12g2′∘∇vt+12g2t∥∇vt∥2+∫Ωvt∫τ1τ2μ2ϱΔYx,1,ϱ,tdϱdx+∫Ωvtudx+b2∫Ωvtvvp−1uq+1dx.
And multiplying equation (13)3 by −ΔZμ1ϱ, and integrating the result over Ω×0,1×τ1,τ2, one gets
(19)ddt12∫Ω∫01∫τ1τ2ϱμ1ϱ∇Z2dϱdρdx=−∫Ω∫01∫τ1τ2∣μ1ϱ∣∇Z∇Zρdϱdρdx=−12∫Ω∫01∫τ1τ2μ1ϱddρ∇Z2dϱdρdx=12∫Ω∫τ1τ2μ1ϱ∇Zx,0,ϱ,t2−∇Zx,1,ϱ,t2dϱdx=12∫τ1τ2μ1ϱdρ∫Ω∇ut2dx−12∫Ω∫τ1τ2μ1ϱ∇Zx,1,ϱ,t2dϱdx=12∫τ1τ2μ1ϱdϱ∥∇ut∥2−12∫τ1τ2∣μ1ϱ∣∥∇Zx,1,ϱ,t∥2dϱ.
Similarly, multiplying equation (13)4 by −ΔY∣μ2ρ∣, we find
(20)ddt12∫Ω∫01∫τ1τ2ϱμ2ϱ∇Y2dϱdρdx=12∫τ1τ2μ2ϱdϱ∥∇vt∥2−12∫τ1τ2μ2ϱ∥∇Yx,1,ϱ,t∥2dϱ,by using the inequalities of Young and Cauchy-Schwartz for δ1>0, we have
(21)∫Ω∇ut∫τ1τ2μ1ϱ∇Zx,1,ϱ,tdϱdx≤δ12∫τ1τ2μ1ϱdϱ∥∇ut∥2+δ12∫τ1τ2μ1ϱ∇Zx,1,ϱ,t2dϱ.
Similarly, we get(22)∫Ω∇vt∫τ1τ2μ2ϱ∇Yx,1,ϱ,tdϱdx≤δ12∫τ1τ2μ2ϱdϱ∥∇vt∥2+δ12∫τ1τ2μ2ϱ∥∇Yx,1,ϱ,t∥2dϱ.
By summing (18)–(20) and using (21) and (22), and choosing δ1 such that δ1<1, we find (16) and (17). This completes the proof.


## 3. Global Existence

Theorem 2.
Suppose that (5)–(8) hold. Then, given u0,v0∈H2Ω∩H01Ω2, u1,v1∈H01Ω2, and f0,g0∈H1Ω,0,1,τ1,τ22, there exists a weak solution u,v,Z,Y of problem (13)–(15) such that
(23)u,v,Z,Y∈L∞ℝ+,H1,ut,vt∈L∞ℝ+,H01Ω,utt,vtt∈L2ℝ+,H01Ω,where
(24)H1=H2Ω∩H01Ω2×H1Ω,0,1,τ1,τ22.Proof.
Let the Galerkin basisuj,vj,Zj,Yj, for n≥1, we set
(25)Wn=spanu1,u2,.⋯,un,Kn=spanv1,v2,.⋯,vn.
The sequencesZjx,τ,p,Yjx,τ,p are defined for 1≤j≤n by
(26)Zjx,0,p=ujx,Yjx,0,p=vjx.
Then, takingZjx,0,p,Yjx,0,p by over L20,1×0,1×τ1,τ2 and denoting
(27)Zn=spanZ1,Z2,.⋯,Zn,Yn=spanY1,Y2,.⋯,Yn.
Given initial datau0,v0∈H2Ω∩H01Ω, u1,v1∈H01Ω, and f0,g0∈L2Ω×0,1×τ1,τ2, we define the approximations
(28)um=∑j=1ngjmtujx,vm=∑j=1nhjmtvjx,Zm=∑j=1nfjmtZjx,τ,p,Ym=∑j=1nkjmtYjx,τ,p.
It investigates the following problem:(29)umtlumtt,uj+M∥∇umt∥∇um,∇uj+∇umtt,∇uj+f1um,vm,uj−∫0tg1t−s∇ums,∇ujds+∫τ1τ2μ1ϱ∇Zmx,1,ϱ,t,∇ujdϱ=0,vmtlvmtt,vj+M∥∇vmt∥∇vm,∇vj+∇vmtt,∇vj+f2um,vm,vj−∫0tg2t−s∇vms,∇vjds+∫τ1τ2μ2ϱ∇Ymx,1,ϱ,t,∇vjdϱ=0,ϱZmtx,ρ,ϱ,t,Zj+Zmρx,ρ,ϱ,t,Zj=0,ϱYmtx,ρ,ϱ,t,Yj+Ymρx,ρ,ϱ,t,Yj=0,with initial conditions
(30)um0=u0m,umt0=u1m,vm0=v1m,vmt0=v1m,Zm0=Z0m,Ym0=Y0m,which satisfies
(31)u0m⟶u0,inH2Ω∩H01Ω,u1m⟶u1,inH01Ω,v0m⟶v0,inH2Ω∩H01Ω,v1m⟶v1,inH01Ω,Z0m⟶Z0,inL2Ω×0,1×τ1,τ2,Y0m⟶Y0,inL2Ω×0,1×τ1,τ2.
Noting thatl/2l+1+1/2l+1+1/2=1, by using Hölder’s inequality, we get
(32)umtlumtt,uj=∫Ωumtlumttujdx≤∫Ω∣umt2l+1dx1/2l+1∥umtt∥2l+1∥uj∥2.
As (8) holds, using the embedding of Sobolev, the terms umtlumtt,uj and vmtlvmtt,vj in (29) make sense (see [22]).
First estimate.
As the sequencesu0m,v0m,u1m,v1m,Z0m.,.,0 and Y0m.,.,0 converge and from (17) and Gronwall’s lemma, we get C1>0 independent of m such that
(33)Emt+β∫τ1τ2ϱμ1ϱ∥∇Zmx,1,ϱ,t∥2+μ2ϱ∥∇Ymx,1,ϱ,t∥2dϱ≤C1,where
(34)Emt=1l+2∥umt∥l+2l+2+∥vmt∥l+2l+2+b2γ+2∥∇um∥2γ+2+∥∇vm∥2γ+2+12a−∫0tg1sds∥∇um∥2+12a−∫0tg2sds∥∇vm∥2+12∥∇umt∥2+∥∇vmt∥2+12g1∘∇umt+12g2∘∇vmt+12∫01∫τ1τ2ϱμ1ϱ∥∇Zm∥2+μ2ϱ∥∇Ym∥2dϱdρ+α∫Ωumvmdx+p+q∫Ωump+1vmq+1dx,using (33) and (8), one gets
(35)um,vmareboundedinLloc∞ℝ+,H01Ω,umt,vmtareboundedinLloc∞ℝ+,H01Ω,Zmx,ρ,ϱ,t,Ymx,ρ,ϱ,tareboundedinLloc∞ℝ+,H01Ω×0,1×τ1,τ2.
The second estimate.
We multiply equation (29)1,2 by gjmtt,hjmtt; by summing j from 1 to n, one gets
(36)∫Ωumtlumtt2dx+∫ΩM∥∇umt∥∇um∇umttdx+∫Ω∇umtt2dx+∫Ωf1um,vmumttdx−∫Ω∫0tg1t−s∇ums∇umttdsdx+∫Ω∫τ1τ2μ1ρ∇Zmx,1,ρ,t∇umttdρdx=0,∫Ωvmtlvmtt2dx+∫ΩM∥∇vmt∥∇vm∇vmttdx+∫Ω∇vmtt2dx+∫Ωf2um,vmvmttdx−∫Ω∫0tg2t−s∇vms∇vmttdsdx+∫Ω∫τ1τ2μ2ϱ∇Ymx,1,ϱ,t∇vmttdϱdx=0.
By differentiating (29)3,4, we get
(37)ϱZmttx,ρ,ϱ,t+Zmtρx,ρ,ϱ,t,Zj=0,ϱYmttx,ρ,ϱ,t+Ymtρx,ρ,ϱ,t,Yj=0.
And we multiply (37)1 by Zjmt and (37)2 by Yjmt; by summing j from 1 to n, we have
(38)12ϱddt∥Zmt∥2+12ddρ∥Zmt∥2=0,12ϱddt∥Ymt∥2+12ddρ∥Ymt∥2=0.
Integrating the result (38) over 0,1 with respect to ρ, we obtain
(39)12ddt∫01ϱ∥Zmt∥2dρ+12∥Zmtx,1,ϱ,t∥2−12∥umttx,t∥2=0,12ddt∫01ϱ∥Ymt∥2dρ+12∥Ymtx,1,ϱ,t∥2−12vmttx,t2=0.
Summing (36) and (39) and using Mr≥a, we get
(40)∫Ωumtlumtt2dx+∥∇umtt∥2+12ddt∫01ϱ∥Zmt∥2dρ+12∥Zmtx,1,ρ,t∥2≤12∥umtt∥2−∫Ωa∇um∇umttdx−∫Ωf1um,vmumttdx+∫Ω∫0tg1t−s∇ums∇umttdsdx−∫Ω∫τ1τ2μ1ϱ∇Zmx,1,ϱ,t∇umttdϱdx,∫Ωvmtlvmtt2dx+∥∇vmtt∥2+12ddt∫01ϱ∥Ymt∥2dρ+12∥Ymtx,1,ϱ,t∥2≤12∥vmttx,t∥2−∫Ωa∇vm∇vmttdx−∫Ωf2um,vmvmttdx+∫Ω∫0tg2t−s∇vms∇vmttdsdx−∫Ω∫τ1τ2∣μ2ϱ∣∇Ymx,1,ϱ,t∇vmttdρdx.
At this point, we estimate the RHS of (40).
Integrating by parts, and using Young’s and Poincare’ inequalities, one gets(41)−∫Ωf1um,vmumttdx≤C∗2α2∇umt2+∇vmt2+b1ηC∗4qq+12Ωq−12q∥∇vm∥4qq+1+b1C∗2p28η∥∇um∥2p2+b1C∗22∥∇umtt∥2.
Similarly, we get(42)−∫Ωf2um,vmvmttdx≤C∗2α2∥∇umt∥2+∥∇vmt∥2+b1ηC∗4pp+12Ωp−1/2p∥∇um∥4pp+1+b1C∗2q28η∥∇vm∥2q2+b1C∗22∥∇vmtt∥2.
And, by using the inequality of Young, we get(43)∫Ωa∇um∇umttdx≤η∥∇umtt∥2+a24η∥∇um∥2,∫Ωa∇vm∇vmttdx≤η∥∇vmtt∥2+a24η∥∇vm∥2,we have
(44)∫Ω∫0tg1t−s∇ums∇umttdsdx≤η∥∇umtt∥2+a−kg104η∫0t∥∇ums∥2ds,∫Ω∫0tg2t−s∇vms∇vmttdsdx≤η∥∇vmtt∥2+a−kg204η∫0t∥∇vms∥2ds.
Similarly, we get(45)∫Ω∫τ1τ2μ1ϱ∇Zmx,1,ϱ,t∇umttdϱdx≤ηλ1∇umtt2+14η∫τ1τ2μ1ϱ∇Zmx,1,ϱ,t2dϱ∫Ω∫τ1τ2μ2ϱ∇Ymx,1,ϱ,t∇vmttdϱdx≤ηλ2∥∇vmtt∥2+14η∫τ1τ2∣μ2ϱ∣∥∇Ymx,1,ϱ,t∥2dϱ,substituting (41)–(45) into (40), and using (17), one gets
(46)∫Ωumtlumtt2dx+1−ηλ1+2+1+b1C∗22∇umtt2+12ddt∫01ϱZmt2dϱ+12Zmtx,1,ϱ,t2≤C2+14ηa−kg10C1T,∫Ωvmtlvmtt2dx+1−ηλ2+2+1+b2C∗22∇vmtt2+12ddt∫01ϱYmt2dϱ+12Ymtx,1,ϱ,t2≤C2+14ηa−kg20C1T,where C2>0 depends on η,α,a,C∗,b1,b2,p,q,C1.
Integrating (41) over 0,t, we get
(47)∫0t∫Ωumtσlumttσ2dxdσ+1−ηλ1+2+1+b1C∗22∫0t∇umttσ2dσ+12∫01ϱZmt2dϱ+12∫0tZmtx,1,ϱ,σ2dσ≤C2+14ηa−kg10C1TT,∫0t∫Ωvmtσlvmttσ2dxdσ+1−ηλ2+2+1+b2C∗22∫0t∇vmttσ2dσ+12∫01ϱYmt2dϱ+12∫0tYmtx,1,ϱ,σ2dσ≤C2+14ηa−kg20C1TT.
At this stage, choosingη>0 such that
(48)1−ηλi+2+1+biC∗22>0,fori=1,2,we find
(49)∫0t∇umttσ2+∇vmttσ2dσ+12∫01ρZmt2+Ymt2dρ≤C3.
We have from (17) and (49) that there exist subsequences uk of um and vk of vm such that
(50)uk,vk⇀u,vweaklystarinL∞0,T,H01Ω,ukt,vkt⇀ut,vtweaklystarinL∞0,T,H01Ω,uktt,vktt⇀utt,vttweaklystarinL20,T,H01Ω,Zk,Yk⇀Z,YweaklystarinL∞0,T,L2Ω×0,1×τ1,τ2,Zkt,Ykt⇀Zt,YtweaklystarinL∞0,T,L2Ω×0,1×τ1,τ2.
We work now with the nonlinear term. From (17), we find
(51)uktluktL20,T,L2Ω=∫0Tukt2l+12l+1dt≤C∗2l+1∫0Tukt22l+1dt≤C4,where C4 depends only on C∗,C1,T,l.
And from the theorem of Aubin-Lions (see Lions [23]), we deduce that there exists a subsequence of uk, given by uk, such that
(52)ukt⟶utstonglyinL20,T,L2Ω,we get
(53)ukt⟶utalmosteverywhereinΩ×ℝ+.
Hence,(54)uktlukt⟶utlutalmosteverywhereinΩ×ℝ+.
Thus, using (46) and (48) and the Lions lemma, we derive
(55)uktlukt⇀utlutweaklyinL20,T,L2Ω.
Similarly,(56)vktlvkt⇀vtlvtweaklyinL20,T,L2Ω,(57)Zk,Yk⟶Z,YstonglyinL20,T,L2Ω×0,1×τ1,τ2,which implies
(58)Zk,Yk⟶z,yalmosteverywhereinΩ×0,1×τ1,τ2×ℝ+.
The sequencesuk and vk satisfy
(59)f1uk,vk⟶f1u,vstonglyinL20,T,L2Ω,f2uk,vk⟶f2u,vstonglyinL20,T,L2Ω.We have
(60)f1uk,vk−f1u,v2=∫Ωvmq+1umpum−vq+1upu2dx.
Noting thatl/2p+1/2q+1/2=1, by applying the generalized Hölder’s and Young’s inequalities, and (8), we get
(61)f1uk,vk−f1u,v2≤C∇um−u2+∇vm−v2.
Asuk and vk are Cauchy sequences in L∞0,T,H01Ω (prove it as in [1]), then we get (59)1. Similarly, we get the convergence (59)2.
Multiplying (29) by Ψt∈D0,T and integrating the result over 0,T, we get
(62)−1l+1∫0Tumtlumtt,ujΨ′tdt+∫0TM∇umt∇um,∇ujΨtdt+∫0T∇umtt,∇ujΨtdt+∫0Tf1um,vm,ujΨtdt−∫0T∫0tg1t−s∇ums,∇ujΨtdsdt+∫0T∫τ1τ2μ1ϱ∇Zmx,1,ϱ,t,∇ujΨtdϱdt=0,−1l+1∫0Tvmtlvmtt,vjΨ′tdt+∫0TM∇vmt∇vm,∇vjΨtdt+∫0T∇vmtt,∇vjΨtdt+∫0Tf2um,vm,vjΨtdt−∫0T∫0tg2t−s∇vms,∇vjΨtdsdt+∫0T∫τ1τ2μ2ϱ∇Ymx,1,ϱ,t,∇vjΨtdϱdt=0,∫0TϱZmtx,ρ,ϱ,t+Zmρx,ρ,ϱ,t,ZjΨtdt=0,∫0TρYmtx,ρ,ϱ,t+Ymρx,ρ,ϱ,t,YjΨtdt=0,∀j=1,⋯,m.
We obtain (62) by the convergence of (50), (54), (56), and (59). This completes the proof.


## 4. Exponential Decay

In this section, the stability result of the system (13)–(15) is proved.We need the following lemmas.Lemma 3.
The functional(63)F1t≔1l+1∫Ωutlutu+vtlvtvdx+∫Ω∇ut∇u+∇vt∇vdx,satisfies
(64)F1t≤1l+2utl+2l+2+vtl+2l+2+12∇ut2+∇vt2+l+1−1l+2C∗l+2+c2∇ul+2+∇vl+2,(65)F1′t≤1l+1utl+2l+2+vtl+2l+2+∇ut2+∇vt2+ε1a−k+λ−k+b1+b22+αC∗2∇u2+∇v2+14ε1∫τ1τ2μ1ϱ∇Zx,1,ϱ,t2+μ2ϱ∇Yx,1,ϱ,t2dϱ+14ε1g1∘∇u+g2∘∇v.Proof.
(1)
By applying the inequalities of Young and Poincare’, we find(66)F1t≤1l+2utl+2l+2+l+1−1l+2ul+2+1l+2vtl+2l+2+l+1−1l+2vl+2+12∇ut2+∇u2+12∇vt2+∇v2≤1l+2utl+2l+2+vtl+2l+2+12∇ut2+∇vt2+l+1−1l+2C∗l+2+c2∇ul+2+∇vl+2(2)
Direct computation using integration by parts, we get(67)F1′t=∫Ωutluttudx+1l+1utl+2l+2+∫Ωvtlvttvdx+1l+1vtl+2l+2−∫ΩΔuttudx+∇ut2−∫ΩΔvttvdx+∇vt2=1l+1utl+2l+2+vtl+2l+2−M∇u2∇u2−M∇v2∇v2+∫Ω∇u∫0tg1t−s∇usdsdx+∫Ω∇v∫0tg2t−s∇vsdsdx−∫Ω∇u∫τ1τ2μ1ϱ∇Zx,1,ϱ,tdϱdx+∇ut2−∫Ω∇v∫τ1τ2μ2ϱ∇Yx,1,ϱ,tdϱdx+∇vt2−b1+b2∫Ωvq+1up+1dx−2α∫Ωuvdx,estimate (65) easily follows by using Mr≥a, Young’s inequality for ε1>0, and (8).Lemma 4.
The functional(68)F2t≔∫ΩΔut−1l+1utlut∫0tg1t−sut−usdsdx+∫ΩΔvt−1l+1vtlvt∫0tg2t−svt−vsdsdx,satisfies
(69)F2t≤1l+2utl+2l+2+vtl+2l+2+12∇ut2+∇vt2+l+1−1l+2a−kl+2C∗l+222l+1∇u2l+1+∇v2l+1+12a−k1+l+1−1l+2a−klC∗l+2g1∘∇u+g2∘∇v,and for any ε2>0,
(70)F2′t≤1l+11−∫0tg1sdsutl+2l+2+1−∫0tg2sdsvtl+2l+2+2ε2a−k2+αC∗22∇u2+∇v2+ε2a−k+l+1−1l+2g10l+2C∗l+222l+1+b2C∗4p+12+b1C∗2p2M∇u2∇u2+ε2a−k+l+1−1l+2g20l+2C∗l+222l+1+b1C∗4q+12+b2C∗2q2M∇v2∇v2+ε2−∫0tg1sds∇ut2+ε2∫τ1τ2μ1ρ∇Zx,1,ρ,t2dρ+ε2−∫0tg2sds∇vt2+ε2∫τ1τ2μ2ρ∇Yx,1,ρ,t2dρ+M∥∇u∥24ε2+2ε2+λ14ε2+αC∗22a−kg1∘∇u+M∥∇v∥24ε2+2ε2+λ24ε2+αC∗22a−kg2∘∇v−g104ε21+l+1−1l+2g10lC∗l+2g1′∘∇u−g204ε21+l+1−1l+2g20lC∗l+2g2′∘∇v.Proof.
(1)
By using Young’s inequality and the conjugate exponentsp′=l+2/l+1, q′=l+2, and Hölder’s inequality, we obtain
(71)−∫Ω1l+1utlut∫0tg1t−sut−usdsdx≤1l+2utl+2l+2+l+1−1l+2a−kl+1C∗l+222l+1a−k∇u2l+1+12g1∘∇u,(72)−∫Ω∇ut∫0tg1t−s∇ut−∇usdsdx≤12∇ut2+12a−kg1∘∇ut.
Similarly, we get(73)−∫Ω1l+1vtlvt∫0tg2t−svt−vsdsdx≤1l+2vtl+2l+2+l+1−1l+2a−kl+1C∗l+222l+1a−k∇v2l+1+12g2∘∇v,(74)−∫Ω∇vt∫0tg2t−s∇vt−∇vsdsdx≤12∇vt2+12a−kg2∘∇vt
By combining (71)–(74), we find (69).
(2)
By derivation ofF2, and integrating by parts and (15), we find
(75)F2′t=∫ΩM∇u2∇u∫0tg1t−s∇ut−∇usdsdx−∫Ω∫0tg1t−s∇usds∫0tg1t−s∇ut−∇usdsdx+∫Ω∫τ1τ2μ1ρ∇Zx,1,ρ,t∫0tg1t−s∇ut−∇usdsdρdx+∫Ωf1u,v∫0tg1t−sut−usdsdx−∫Ω∇ut∫0tg1′t−s∇ut−∇usdsdx−∫Ω1l+1utlut∫0tg1′t−sut−usdsdx−∇ut2+1l+1utl+2l+2∫0tg1sds+∫ΩM∇v2∇v∫0tg2t−s∇vt−∇vsdsdx−∫Ω∫0tg2t−s∇vsds∫0tg2t−s∇vt−∇vsdsdx+∫Ω∫τ1τ2μ2ρ∇Yx,1,ρ,t∫0tg2t−s∇vt−∇vsdsdρdx+∫Ωf2u,v∫0tg2t−svt−vsdsdx−∫Ω∇vt∫0tg2′t−s∇vt−∇vsdsdx−∫Ω1l+1vtlvt∫0tg2′t−svt−vsdsdx−∇vt2+1l+1vtl+2l+2∫0tg2sds
Using Young’s, Cauchy-Schwarz, Hölder’s, and Poincaré’s inequalities, andl≤γ, we obtain (70).At this point, let us introduce the functional given byLemma 5.
The functional(76)F3t≔∫Ω∫01∫τ1τ2ρe−ρρμ1ρZ2+μ2ρY2dρdρdx,satisfies
(77)F3t≤∫Ω∫01∫τ1τ2ρμ1ρZ2+μ2ρY2dρdρdx,(78)F3′t≤−η1∫01∫τ1τ2ρμ1ρZ2+μ2ρY2dρdρ+λ∇ut2+∇vt2−η1∫τ1τ2μ1ρZx,1,ρ,t2+μ2ρYx,1,ρ,t2dρdρ,where η1>0.Proof.
By derivation ofF3, and using equations (13)3 and (13)4, we get
(79)F3′t=−2∫Ω∫01∫τ1τ2e−ρρμ1ρ∇Z∇Zρx,ρ,ρ,tdρdρdx−2∫Ω∫01∫τ1τ2e−ρρμ2ρ∇Y∇Yρx,ρ,ρ,tdρdρdx=−∫Ω∫01∫τ1τ2ρe−ρρμ1ρ∇Z2dρdρdx−∫Ω∫τ1τ2μ1ρe−ρ∇Z2x,1,ρ,t−∇Z2x,0,ρ,tdρdx−∫Ω∫01∫τ1τ2ρe−ρρμ2ρ∇Y2dρdρdx−∫Ω∫τ1τ2μ2ρe−ρ∇Y2x,1,ρ,t−∇Y2x,0,ρ,tdρdx.Applying the equality Zx,0,ρ,t=utx,t, Yx,0,ρ,t=vtx,t, and e−ρ≤e−ρρ≤1, for any 0<ρ<1, we get
(80)F3′t=−∫01∫τ1τ2ρe−ρρμ1ρ∇Z2+μ2ρ∇Y2dρdρ−∫τ1τ2e−ρμ1ρ∇Zx,1,ρ,t2+μ2ρ∇Yx,1,ρ,t2dρ+∫τ1τ2μ1ρdρ∇ut2+∫τ1τ2μ2ρdρ∇vt2.
As–e−ρ is an increasing function, we have –e−ρ≤−e−τ2, for any ρ∈τ1,τ2.Then, settingη1=e−τ2, we find (78).Theorem 6.
Assume (5)–(8) hold, then ∃ζ1,ζ2>0 such that the energy functional (16) satisfies
(81)Et≤ζ2e−ζ1t,∀t≥t0.Proof.
We define the functional of Lyapunov(82)Lt≔NEt+F1t+N2F2t+F3t,where N,N2>0.
First, if we let(83)Kt=F1t+N2F2t+F3t,then, by (64), (69), and (77), we get
(84)Kt≤cEt.
Consequently,(85)Kt=Lt−NEt≤cEt,which yields
(86)N−cEt≤Lt≤N+cEt.
By derivation (82) and applying (17), (65), (70), (78), and (6), one gets
(87)L′t≤1l+11−h0+N1utl+2l+2+vtl+2l+2+λ1+N+N1+ε2−h0∇ut2+∇vt2+ε2M0a−k+l+1−1l+2h2C∗l+222l+1+R1+N1ε1a−k+λ−k+b1+b22+αC∗2+2ε2a−k2+αC∗22∇u2+∇v2+−1ξM04ε2+2ε2+λ4ε2+αC∗22a−k+N14ε1+N2−h14ε21+l+1−1l+2h1lC∗l+2g1′∘∇u+g2′∘∇v−η1∫01∫τ1τ2ρμ1ρZ2+μ2ρY2dρdρ−η1+Nβ−ε2−N14ε1∫τ1τ2μ1ρZx,1,ρ,t2+μ2ρYx,1,ρ,t2dρdρ,where h0=min∫0t0g1sds,∫0t0g2sds, M0=maxM∇u2,M∇v2., h1=ming10,g20, h2=maxg10,g20,ξ=maxξ1,ξ2, and R1=minb1C∗4q+1/2+b2C∗4q/2,b2C∗4p+1/2+b1C∗4p/2.
At this stage, choosing two fixed numbersN, N1, such that N−c>0, and
(88)h1−λ1+N−N1>0,α1=h1−1−N1>0,we choose ε2 small enough such that
(89)α2=h1−λ1+N−N1−ε2>0.
After that, we chooseε1 small enough such that
(90)α3=η1+Nβ−ε2−N14ε1<0,α4=−ε2M0a−k+l+1−1l+2h2C∗l+222l+1+R1+N1k−ε1a−k+λ−b1+b22+αC∗2−2ε2a−k2+αC∗22>0,α5=1ξN14ε1+M04ε2+2ε2+λ4ε2+αC∗22a−k−N2+h14ε21+l+1−1l+2h1lC∗l+2>0.Thus, we get
(91)L′t≤−1l+1α1utl+2l+2+vtl+2l+2−α2∇ut2+∇vt2−α4∇u2+∇v2−α5g1′∘∇u+g2′∘∇v−η1∫01∫τ1τ2ρμ1ρZ2+μ2ρY2dρdρ+α3∫τ1τ2μ1ρZx,1,ρ,t2+μ2ρYx,1,ρ,t2dρdρ,(92)c1Et≤Lt≤c2Et,∀t≥0,using (16), estimates (91) and (86), respectively, we get
(93)L′t≤−k1Et−k2E′t,∀t≥t0,for some k1,k2,c1,c2>0.
By the combination of (93) with (92), we obtain
(94)R′t≤−λ1Rt,where
(95)Rt=Lt+k2Et~Et.
Integrating the result (94) over t0,t, we find
(96)Rt≤Rt0e−ζ1t−t0,∀t0≥t.
It follows from (95) that (81) holds. This completes the proof.



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