# Global Well-Posedness and Stability for a Viscoelastic Plate Equation with a Time Delay

Authors: Baowei Feng
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

A plate equation with a memory term and a time delay term in the internal feedback is investigated. Under suitable assumptions, we establish the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations and some energy estimates. Moreover, by using energy perturbation method, we prove a general decay result of the energy provided that the weight of the delay is less than the weight of the damping.


---

## Body

## 1. Introduction

In this paper, we are concerned with the following plate equation with a memory term and a time delay term in the internal feedback:(1)utt+Δ2u-M∇u2Δu+∫0t‍gt-sΔu(s)ds+μ1utx,t+μ2ut(x,t-τ)+f(u)=0,where Ω⊆Rn(n≥1) is a bounded domain with smooth boundary ∂Ω. Here M(·) is a function satisfying suitable conditions (see below), μ1, μ2 are positive constants, and τ>0 represents the time delay.Equation (1) with the memory term ∫0t‍g(t-s)Δu(s)ds, where the function g is called kernel, can be regarded as a fourth order viscoelastic plate equation with a lower order perturbation, and it can be also regarded as an elastoplastic flow equation with some kind of memory effect.In this paper, we consider the following initial conditions:(2)u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,ut(x,t-τ)=h0(x,t-τ),x∈Ω,t∈(0,τ)and the following boundary conditions: (3)u=Δu=0,on⁡∂Ω×R+.Fourth order equations with lower order perturbation are related to models of elastoplastic microstructure flows. For the single plate equation without delay, that is,μ2=0, as considered by Woinowsky-Krieger [1], the author first proposed the one-dimensional nonlinear equation of vibration of beams, which is given by (4)utt+αuxxxx-β+γ∫0L‍ux2dxuxx=0,where L is the length of the beam and α,β,γ are positive physical constants. The nonlinear part of (4) represents for the extensible effect for the beam whose ends are restrained to remain in a fixed distance apart in its transverse vibrations. A more general equation of (4) reads(5)utt-M∇u2Δu+Δ2u+gut+fu=hx,where M(·) is a function satisfying some conditions. There are so many existing results concerning global existence, stability, and long-time dynamics for (5); we would like to refer the reader to de Brito [2], Cavalcanti et al. [3, 4], Ma [5], Ma and Narciso [6], de Lacerda Oliveira and de Lima [7], J. Y. Park and S. H. Park [8], Patcheu [9], Rivera [10, 11], Tusnal [12], Vasconcellos and Teixeira [13], Yang [14, 15], and the references therein. Very recently, Andrade et al. [16] investigated a viscoelastic plate equation with p-Laplacian and memory terms with strong damping(6)utt+Δ2u-Δpu+∫0t‍gt-sΔu(s)ds-Δut+fu=0,where Δpu=div(|∇u|p-2∇u) is the p-Laplacian operator. Under suitable assumptions on the memory kernel g and a forcing term f, the authors proved the existence of weak solutions by using Faedo-Galerkin approximations, the uniqueness of strong solutions, and the exponential stability of solutions to (6) with initial and boundary value problem. For more results on viscoelastic equations, we can refer to Berrimi and Messaoudi [17], Messaoudi [18], Messaoudi and Tartar [19, 20], and the references therein.In recent years, many mathematical workers studied some systems with time delay effects. Datko et al. [21] studied the following system:(7)utt-Δu=0,x∈Ω,t>0,u(x,t)=0,x∈Γ0,t>0,∂u∂ν=μ1ut(x,t)+μ2ut(x,t-τ),x∈Γ1,t>0,u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,u(x,t-τ)=g0x,t-τ,x∈Ω,t∈[0,τ].By using an observability inequality, they proved the exponential stability for the energy when μ2<μ1. Subsequently, Xu et al. [22] obtained the same result as in [21] for the one space dimension by using the spectral analysis approach. Later on, Kirane and Said-Houari [23] considered a viscoelastic wave equation with a delay term in internal feedback with initial conditions and boundary value conditions of Dirichlet type. Under suitable assumptions on the relaxation function and some restriction on the parameters μ1 and μ2, they established the global well-posedness of the system. Moreover, under the assumption μ2≤μ1 between the weight of the delay term in the feedback and the weight of the term without delay, the authors proved a general decay of the total energy of the system. For more some results concerning the different boundary conditions under an appropriate assumption between μ1 and μ2, one can refer to Nicaise and Pignotti [24], Nicaise et al. [25], Nicaise and Valein [26], and the references therein.Equation (1) is a plate equation with a memory term and a time delay term in the internal feedback. Noting that μ1≠0, we know that it is a plate equation with weak damping. For viscoelastic plate equations, it is well known that one considered a memory of the form ∫0t‍g(t-s)Δ2u(s)ds (see, e.g., [10, 27, 28]). However, because the main dissipation of the system (1)–(3) is given by a weak damping ut, here we consider a weaker memory, acting only on Δu. To the best of our knowledge, the global well-posedness and energy decay for system (1)–(3) were not previously considered. So the objective of this work is to establish the global well-posedness and stability of initial boundary value problem (1)–(3). The main dissipation of the system (1)–(3) is given by a weak damping ut, which makes the analysis in this work different from [16], because the authors considered the case of a strong damping -Δut in [16].The outline of this paper is as follows. In Section2, we give some preparations for our consideration and our main results. In Section 3, we establish the global posedness of the system by using the Faedo-Galerkin approximations and some energy estimates. In Section 4, we will show a general decay result of the energy by using energy perturbation method provided that the weight of the delay is less than the weight of the damping.The notation in this paper will be as follows:Lq, 1≤q≤+∞, Wm,q, m∈N, H1=W1,2, H01=W01,2 denote the usual (Sobolev) spaces on Ω. In addition, ·B denotes the norm in the space B, and we also put ·=·L2(Ω).


## 2. Preliminaries and Main Results

In this section, we give some preparations for our consideration and our main results.(i)
We assume thatM(·):R+→R+ is a C1 function satisfying(8)zM(z)≥M^(z),M^(z)=∫0z‍M(s)ds(ifM(z) is monotone nondecreasing).(ii)
For the memory kernelg(t), we assume that
(G1)
g:R+→R+ is a function satisfying(9)g(t)∈C1(R+)∩L1(R+),g(0)>0,1-λ∫0∞‍g(s)ds=l>0,whereλ>0 is the embedding constant for ∇u2≤λΔu2.(G2)
There exists a positive nonincreasing differentiable functionμ(t) such that (10)g′t≤-μtgt,∀t≥0,(11)∫0∞‍μ(t)dt=∞.(iii)
The nonlinear termf(u) satisfies(12)f0=0,fu-fv≤c1+uρ+vρu-v,∀u,v∈R,wherec>0 is a constant, and ρ satisfies(13)0<ρ<4n-4if≥5,ρ>0if1≤n≤4.We denote f^(z)=∫0z‍f(s)ds and assume that(14)0≤f^u≤fuu,∀u∈R.In order to deal with the delay feedback term, motivated by [24, 26], we introduce the following new dependent variable:(15)zx,ρ,t=ut(x,t-τρ),x∈Ω,ρ∈(0,1),t>0.Then it is easy to verify(16)τztx,ρ,t+zρx,ρ,t=0,inΩ×(0,1)×0,∞.Thus, problem (1)–(3) is transformed into(17)uttx,t+Δ2u-M∇u2Δu+∫0t‍gt-sΔu(s)ds+μ1ut+μ2zx,1,t=0,τzt(x,ρ,t)+zρ(x,ρ,t)=0,with x∈Ω, ρ∈(0,1) and t>0, and the initial and boundary conditions are(18)ux,0=u0,utx,0=u1,x∈Ω,z(x,ρ,0)=h0(x,t-τ),(x,t)∈Ω×(0,τ),u=Δu=0,on⁡∂Ω×R+,z(x,0,t)=utx,tx∈Ω,t>0.Letξ be a positive constant satisfying (19)τμ2<ξ<τ2μ1-μ2.Now we define the weak solutions of (1)–(3): for given initial data (u0,u1)∈H2(Ω)∩H01(Ω)×L2(Ω), we say that a function z=(u,ut)∈C(R+,H2(Ω)∩H01(Ω)×L2(Ω)) is a weak solution to the problem (1)–(3) if z(0)=(u0,u1) and(20)utt,ω+Δu,Δω+M∇u2∇u,∇ω-∫0t‍g(t-s)(∇u(s),∇ω)ds+μ1ut,ω+μ2utt-τ,ω+fu,ω=0,for all ω∈H2(Ω)∩H01(Ω).Next we state the global well-posedness of problem (17)-(18) given in the following theorem.Theorem 1.
Letμ2≤μ1 hold and assume the assumptions (8)–(14) hold.(i)
If the initial data(u0,u1)∈(H2(Ω)∩H01(Ω)×L2(Ω)), h0∈L2(Ω×(0,1)), then problem (17)-(18) has a weak solution such that(21)u∈CR+;H2Ω∩H01Ω∩C1R+;L2Ω,ut∈L2R+;L2Ω.(ii)
If the initial data(u0,u1)∈H∂Ω3(Ω)×H01(Ω), h0∈H1(Ω×(0,1)), where(22)H∂Ω3Ω=u∈H3Ω∣u=Δu=0on⁡∂Ω,then the above weak solution has higher regularity(23)u∈L∞(R+,H3(Ω)),ut∈L∞R+,H01Ω∩L2R+,H01Ω.(iii)
In both cases, we have that the solution(u,ut) depends continuously on the initial data in H2(Ω)∩H01(Ω)×L2(Ω). In particular, problem (17)-(18) has a unique weak solution.We define the energy of problem (17)-(18) by(24)Et=12utt2+12Δut2+12M^∇ut2+∫Ω‍f^utdx+ξ2∫Ω‍∫01‍z2x,ρ,tdρdx.Finally, we give the energy decay of problem (17)-(18).Theorem 2.
Letμ2<μ1 hold and assume the assumptions (8)–(14) hold. In both cases (i) and (ii), there exist two constants α>0 and β>0 such that the energy E(t) defined by (24) satisfies (25)Et≤αexp⁡-β∫0t‍μsds,∀t≥0.


## 3. The Global Well-Posedness

In this section, we will prove the global existence and the uniqueness of the solution of problem (17)-(18) by using the classical Faedo-Galerkin approximations along with some priori estimates. We only prove the existence of solution in (i). For the existence of stronger solution in (ii), we can use the same method as in (i) and one can refer to Andrade e al. [16] and Jorge Silva and Ma [28].


### 3.1. Approximate Problem

Let{wj} be the Galerkin basis given by the eigenfunctions of Δ2 with boundary condition u=Δu=0 on ∂Ω. For any m≥1, let Wm=span⁡{w1,w2,…,wn}.We define for1≤j≤m the sequence ϕj(x,ρ) by (26)ϕj(x,0)=wj(x).Then we can extend ϕj(x,0) by ϕj(x,ρ) over L2(Ω×(0,1)) and denote Vm=span⁡{ϕ1,ϕ2,…,ϕn}.Given initial datau0∈H2(Ω)∩H01(Ω), u1∈L2(Ω), and h0∈L2Ω×0,1, we define the approximations(27)um(t)=∑j=1m‍gjm(t)wj(x),zm(x,ρ,t)=∑j=1m‍hjm(t)ϕj(x,ρ),which satisfy the following approximate problem: (28)umttt,wj+Δumt,Δwj+-M∇um2Δum,wj+fumt,wj+∫0t‍gt-sΔums,wjds+μ1umtt,wj+μ2zmx,1,t,wj=0,τzmtx,ρ,t,ϕj+zmρx,ρ,t,ϕj=0,with initial conditions (29)um(0)=u0m,umt(0)=u1mzm(x,ρ,0)=z0m,which satisfies(30)u0m⟶u0stronglyinH2(Ω)∩H01(Ω),u1m⟶u1stronglyinL2(Ω),z0m⟶h0stronglyinL2Ω×0,1.By using standard ordinary differential equations theory, the problem (28)-(29) has a solution (gjm,hjm)j=1,m defined on [0,tm). The following estimate will give the local solution being extended to [0,T], for any given T>0.


### 3.2. A Priori Estimate

Now multiplying the first approximate equation of (28) by gjm′, we see that(31)ddt12umtt2+12Δumt2+12M^∇umt2∫Ω‍f^umt+∫Ω‍f^umt+μ1untt2+μ2∫Ω‍zmx,1,tumttdx-∫0t‍g(t-s)(∇um(s),∇umt(t))ds=0.Noting the following fact:(32)12ddtg∘∇ut-∫0t‍gsds·∇ut2+∫0t‍∫0t‍gt-s∇us,∇uttds=12g′∘∇ut-12gt∇ut2,where(33)g∘∇ut=∫0t‍gt-s∇us-∇ut2ds,we know that(34)12ddtumtt2+Δumt2+M^∇umt2∫0t‍gsds-∫0t‍gsds∇umt2∫0t‍gsds+2∫Ω‍f^(um(t))dx+g∘∇umt+μ1umtt2+μ2∫Ω‍zmx,1,tumttdx=12g′∘∇umt-12g(t)∇umt2.Multiplying the second approximate equation of (28) by ξ/τhjm′ and then integrating over (0,t)×(0,1), we obtain(35)ξ2∫Ω‍∫01‍zm2x,ρ,tdρdx+ξτ∫0t‍∫Ω‍∫01‍zmρzmx,ρ,sdρdxds=ξ2∫Ω‍∫01‍z0mx,ρdρdx.A straightforward calculation gives(36)∫0t‍∫Ω‍∫01‍zmρzmx,ρ,sdρdxds=12∫0t‍∫Ω‍∫01‍∂∂ρzm2x,ρ,sdρdxds=12∫0t‍∫Ω‍zm2x,1,s-zm2x,0,sdxds.Now integrating (34) and using (35)-(36) and zm2(x,0,s)=umt2(s), we infer that(37)Em(t)+μ1-ξ2τ∫0t‍umtt2ds+ξ2τ∫0t‍∫Ω‍zm2x,1,sdxds+μ2∫0t‍∫Ω‍zmx,1,sumtsdxds+12∫0t‍g(s)∇ums2ds-12∫0t‍g′∘∇umsds=Em(0),with(38)Emt=12umtt2+Δumt2+M^∇umt2∫0t‍gsdsWWWWWw-∫0t‍gsds∇umt2WWWWWw∫0t‍gsds+2∫Ω‍f^umtdx+g∘∇umtWWWW+ξ2∫Ω‍∫01‍zm2x,ρ,tdρdx.Then we have the following cases.(i)
Considerμ2<μ1. Using Young’s inequality, we have(39)μ2∫0t‍∫Ω‍zmx,1,sumtsdxds≥-μ22∫0t‍∫Ω‍zm2x,1,sdxds-μ22∫0t‍umt2(s)ds,which, together with (37), yields(40)Emt+μ1-ξ2τ-μ22∫0t‍umtt2ds+ξ2τ-μ22∫0t‍∫Ω‍zm2x,1,sdxds+12∫0t‍g(s)∇ums2ds-12∫0t‍g′∘∇um(s)ds≤Em(0).It follows from (19) that there exist two constants c1>0 and c2>0 such that(41)Em(t)+c1∫0t‍umtt2ds+c2∫0t‍∫Ω‍zm2x,1,sdxds+12∫0t‍gs∇ums2ds-12∫0t‍g′∘∇umsds≤Em0.(ii)
Considerμ1=μ2. Taking ξ=τμ1=τμ2 and using (37), we know that(42)Em(t)+12∫0t‍gs∇ums2ds-12∫0t‍g′∘∇umsds≤Em(0).Then, in both cases, we infer that there exists a positive constant C independent on m such that (43)Em(t)≤C,t≥0.It follows from (9), (14), and (43) that(44)umtt2+Δumt2+M^∇umt2+∫Ω‍∫01‍zm2x,ρ,tdρdx≤C.Thus we can obtain tm=T, for all T>0.


### 3.3. Passage to Limit

From (44), we conclude that for any m∈N,(45)umisboundedinL∞R+;H2Ω∩H01Ω,(46)umtisboundedinL∞R+;L2Ω,(47)zmisboundedinL∞R+;L2Ω×0,1.Thus we get(48)um⇀uweaklystarinL∞R+;H2Ω∩H01Ω,umt⇀utweaklystarinL2R+;L2Ω,zm⇀zweaklystarinL2R+;L2Ω×0,1.By (45)–(47), we can also deduce that um is bounded in L2(R+;H2(Ω)∩H01(Ω)) and umt is bounded in L2(R+;L2(Ω)). Then from Aubin-Lions theorem [29], we infer that for any T>0, (49)um⟶ustronglyinL∞0,T;H01Ω.We also obtain by Lemma 1.4 in Kim [30] that (50)um⟶ustronglyinC0,T;H01Ω.Then we can pass to limit the approximate problem (28)-(29) in order to get a weak solution of problem (17)-(18).


### 3.4. Continuous Dependence and Uniqueness

Firstly we prove the continuous dependence and uniqueness for stronger solutions of problem (17)-(18).Let(u(t),ut(t),z1(x,ρ,t)) and (v(t),vt(t),z2(x,ρ,t)) be two global solutions of problem (17)-(18) with respect to initial data (u0,u1,h01) and (v0,v1,h02) respectively. Let ω(t)=u(t)-v(t), χ(x,ρ,t)=z1(x,ρ,t)-z2(x,ρ,t). Then (ω(t),χ(x,ρ,t)) verifies(51)ωtt+Δ2ω-M∇u2Δu-M∇v2Δv+∫0t‍gt-sΔωsds+μ1ωt+μ2χ(x,1,t)+f(u)-f(v)=0,τχt+χρ=0,with boundary conditions(52)ω=Δω=0,on⁡∂Ωand initial data(53)ωx,0=ω0,ωtx,0=ω1,χ(x,ρ,0)=χ0=h01-h02.Multiplying (47) by ωt and integrating the result over Ω, we get(54)12ddtωtt2+Δωt2∫0t‍gsds-∫0t‍gsds∇ωt2+g∘∇ωt+μ1ωtt2≤M∇u2Δu-M∇v2Δv,ωt+μ2∫Ω‍χx,1,tωtdx+∫Ω‍fu-fvωtdx.By mean value theorem and Hölder’s inequality, we derive(55)M∇u2Δu-M∇v2Δv,ωt=∫Ω‍M∇u2Δu-M∇u2ΔvWWWWi∫Ω‍+M∇u2Δv-M∇v2Δvωtdx≤∫Ω‍M∇u2Δωωtdx+∫Ω‍M′η∇u2-∇v2Δvωtdx≤∫Ω‍M∇u2Δωωtdx+C1∫Ω‍∇u+∇v∇ωΔvωtdx≤C1Δω2+ωt2+C1∇uL∞+∇vL∞∇ωL∞Δvωt≤C1Δω2+ωt2.It follows from (12)-(13) and Hölder’s inequality that(56)∫Ω‍fu-fvωtdx≤C1∫Ω‍1+uρ+vρωωtdx≤C1u2ρ+1ρ+v2ρ+1ρω2(ρ+1)ωt≤C1Δω2+ωt2.Moreover,(57)μ2∫Ω‍χx,1,tωtdx≤μ22∫Ω‍χ2x,1,tdx+μ22ωt2.Noting that (35)-(36) and combining (54)–(57), we conclude that(58)ddtE(t)+μ1-ξ2τ-μ22ωt2+ξ2τ-μ22∫Ω‍χ2x,1,tdx≤C1Δω2+ωt2,where(59)Et=ωtt2+Δωt2-∫0t‍gsds∇ωt2+g∘∇ωt+ξ2∫Ω‍∫01‍χ2x,ρ,tdρdx.It follows (19) that(60)Et≤E0+C1∫0t‍Δω2+ωt2sds,which, along with (9), gives(61)Δω2+ωt2≤E(0)+C1∫0t‍Δω2+ωt2sds.Applying Gronwall’s inequality to (61), we get(62)Δω2+ωt2≤eC1tE(0).This shows that solution of problem (17)-(18) depends continuously on the initial data. In particular, problem (17)-(18) has a unique stronger solution.We can prove the continuous dependence and uniqueness for weak solutions by using density arguments (see, e.g., Cavalcanti et al. [27]) which also can be found in Lions [29] (Chapter 1, Theorem 1.2) by using a regularization method and in Pata and Zucchi [31] or Giorgi et al. [32] by using the mollifiers.This ends the proof of Theorem1.


## 3.1. Approximate Problem

Let{wj} be the Galerkin basis given by the eigenfunctions of Δ2 with boundary condition u=Δu=0 on ∂Ω. For any m≥1, let Wm=span⁡{w1,w2,…,wn}.We define for1≤j≤m the sequence ϕj(x,ρ) by (26)ϕj(x,0)=wj(x).Then we can extend ϕj(x,0) by ϕj(x,ρ) over L2(Ω×(0,1)) and denote Vm=span⁡{ϕ1,ϕ2,…,ϕn}.Given initial datau0∈H2(Ω)∩H01(Ω), u1∈L2(Ω), and h0∈L2Ω×0,1, we define the approximations(27)um(t)=∑j=1m‍gjm(t)wj(x),zm(x,ρ,t)=∑j=1m‍hjm(t)ϕj(x,ρ),which satisfy the following approximate problem: (28)umttt,wj+Δumt,Δwj+-M∇um2Δum,wj+fumt,wj+∫0t‍gt-sΔums,wjds+μ1umtt,wj+μ2zmx,1,t,wj=0,τzmtx,ρ,t,ϕj+zmρx,ρ,t,ϕj=0,with initial conditions (29)um(0)=u0m,umt(0)=u1mzm(x,ρ,0)=z0m,which satisfies(30)u0m⟶u0stronglyinH2(Ω)∩H01(Ω),u1m⟶u1stronglyinL2(Ω),z0m⟶h0stronglyinL2Ω×0,1.By using standard ordinary differential equations theory, the problem (28)-(29) has a solution (gjm,hjm)j=1,m defined on [0,tm). The following estimate will give the local solution being extended to [0,T], for any given T>0.


## 3.2. A Priori Estimate

Now multiplying the first approximate equation of (28) by gjm′, we see that(31)ddt12umtt2+12Δumt2+12M^∇umt2∫Ω‍f^umt+∫Ω‍f^umt+μ1untt2+μ2∫Ω‍zmx,1,tumttdx-∫0t‍g(t-s)(∇um(s),∇umt(t))ds=0.Noting the following fact:(32)12ddtg∘∇ut-∫0t‍gsds·∇ut2+∫0t‍∫0t‍gt-s∇us,∇uttds=12g′∘∇ut-12gt∇ut2,where(33)g∘∇ut=∫0t‍gt-s∇us-∇ut2ds,we know that(34)12ddtumtt2+Δumt2+M^∇umt2∫0t‍gsds-∫0t‍gsds∇umt2∫0t‍gsds+2∫Ω‍f^(um(t))dx+g∘∇umt+μ1umtt2+μ2∫Ω‍zmx,1,tumttdx=12g′∘∇umt-12g(t)∇umt2.Multiplying the second approximate equation of (28) by ξ/τhjm′ and then integrating over (0,t)×(0,1), we obtain(35)ξ2∫Ω‍∫01‍zm2x,ρ,tdρdx+ξτ∫0t‍∫Ω‍∫01‍zmρzmx,ρ,sdρdxds=ξ2∫Ω‍∫01‍z0mx,ρdρdx.A straightforward calculation gives(36)∫0t‍∫Ω‍∫01‍zmρzmx,ρ,sdρdxds=12∫0t‍∫Ω‍∫01‍∂∂ρzm2x,ρ,sdρdxds=12∫0t‍∫Ω‍zm2x,1,s-zm2x,0,sdxds.Now integrating (34) and using (35)-(36) and zm2(x,0,s)=umt2(s), we infer that(37)Em(t)+μ1-ξ2τ∫0t‍umtt2ds+ξ2τ∫0t‍∫Ω‍zm2x,1,sdxds+μ2∫0t‍∫Ω‍zmx,1,sumtsdxds+12∫0t‍g(s)∇ums2ds-12∫0t‍g′∘∇umsds=Em(0),with(38)Emt=12umtt2+Δumt2+M^∇umt2∫0t‍gsdsWWWWWw-∫0t‍gsds∇umt2WWWWWw∫0t‍gsds+2∫Ω‍f^umtdx+g∘∇umtWWWW+ξ2∫Ω‍∫01‍zm2x,ρ,tdρdx.Then we have the following cases.(i)
Considerμ2<μ1. Using Young’s inequality, we have(39)μ2∫0t‍∫Ω‍zmx,1,sumtsdxds≥-μ22∫0t‍∫Ω‍zm2x,1,sdxds-μ22∫0t‍umt2(s)ds,which, together with (37), yields(40)Emt+μ1-ξ2τ-μ22∫0t‍umtt2ds+ξ2τ-μ22∫0t‍∫Ω‍zm2x,1,sdxds+12∫0t‍g(s)∇ums2ds-12∫0t‍g′∘∇um(s)ds≤Em(0).It follows from (19) that there exist two constants c1>0 and c2>0 such that(41)Em(t)+c1∫0t‍umtt2ds+c2∫0t‍∫Ω‍zm2x,1,sdxds+12∫0t‍gs∇ums2ds-12∫0t‍g′∘∇umsds≤Em0.(ii)
Considerμ1=μ2. Taking ξ=τμ1=τμ2 and using (37), we know that(42)Em(t)+12∫0t‍gs∇ums2ds-12∫0t‍g′∘∇umsds≤Em(0).Then, in both cases, we infer that there exists a positive constant C independent on m such that (43)Em(t)≤C,t≥0.It follows from (9), (14), and (43) that(44)umtt2+Δumt2+M^∇umt2+∫Ω‍∫01‍zm2x,ρ,tdρdx≤C.Thus we can obtain tm=T, for all T>0.


## 3.3. Passage to Limit

From (44), we conclude that for any m∈N,(45)umisboundedinL∞R+;H2Ω∩H01Ω,(46)umtisboundedinL∞R+;L2Ω,(47)zmisboundedinL∞R+;L2Ω×0,1.Thus we get(48)um⇀uweaklystarinL∞R+;H2Ω∩H01Ω,umt⇀utweaklystarinL2R+;L2Ω,zm⇀zweaklystarinL2R+;L2Ω×0,1.By (45)–(47), we can also deduce that um is bounded in L2(R+;H2(Ω)∩H01(Ω)) and umt is bounded in L2(R+;L2(Ω)). Then from Aubin-Lions theorem [29], we infer that for any T>0, (49)um⟶ustronglyinL∞0,T;H01Ω.We also obtain by Lemma 1.4 in Kim [30] that (50)um⟶ustronglyinC0,T;H01Ω.Then we can pass to limit the approximate problem (28)-(29) in order to get a weak solution of problem (17)-(18).


## 3.4. Continuous Dependence and Uniqueness

Firstly we prove the continuous dependence and uniqueness for stronger solutions of problem (17)-(18).Let(u(t),ut(t),z1(x,ρ,t)) and (v(t),vt(t),z2(x,ρ,t)) be two global solutions of problem (17)-(18) with respect to initial data (u0,u1,h01) and (v0,v1,h02) respectively. Let ω(t)=u(t)-v(t), χ(x,ρ,t)=z1(x,ρ,t)-z2(x,ρ,t). Then (ω(t),χ(x,ρ,t)) verifies(51)ωtt+Δ2ω-M∇u2Δu-M∇v2Δv+∫0t‍gt-sΔωsds+μ1ωt+μ2χ(x,1,t)+f(u)-f(v)=0,τχt+χρ=0,with boundary conditions(52)ω=Δω=0,on⁡∂Ωand initial data(53)ωx,0=ω0,ωtx,0=ω1,χ(x,ρ,0)=χ0=h01-h02.Multiplying (47) by ωt and integrating the result over Ω, we get(54)12ddtωtt2+Δωt2∫0t‍gsds-∫0t‍gsds∇ωt2+g∘∇ωt+μ1ωtt2≤M∇u2Δu-M∇v2Δv,ωt+μ2∫Ω‍χx,1,tωtdx+∫Ω‍fu-fvωtdx.By mean value theorem and Hölder’s inequality, we derive(55)M∇u2Δu-M∇v2Δv,ωt=∫Ω‍M∇u2Δu-M∇u2ΔvWWWWi∫Ω‍+M∇u2Δv-M∇v2Δvωtdx≤∫Ω‍M∇u2Δωωtdx+∫Ω‍M′η∇u2-∇v2Δvωtdx≤∫Ω‍M∇u2Δωωtdx+C1∫Ω‍∇u+∇v∇ωΔvωtdx≤C1Δω2+ωt2+C1∇uL∞+∇vL∞∇ωL∞Δvωt≤C1Δω2+ωt2.It follows from (12)-(13) and Hölder’s inequality that(56)∫Ω‍fu-fvωtdx≤C1∫Ω‍1+uρ+vρωωtdx≤C1u2ρ+1ρ+v2ρ+1ρω2(ρ+1)ωt≤C1Δω2+ωt2.Moreover,(57)μ2∫Ω‍χx,1,tωtdx≤μ22∫Ω‍χ2x,1,tdx+μ22ωt2.Noting that (35)-(36) and combining (54)–(57), we conclude that(58)ddtE(t)+μ1-ξ2τ-μ22ωt2+ξ2τ-μ22∫Ω‍χ2x,1,tdx≤C1Δω2+ωt2,where(59)Et=ωtt2+Δωt2-∫0t‍gsds∇ωt2+g∘∇ωt+ξ2∫Ω‍∫01‍χ2x,ρ,tdρdx.It follows (19) that(60)Et≤E0+C1∫0t‍Δω2+ωt2sds,which, along with (9), gives(61)Δω2+ωt2≤E(0)+C1∫0t‍Δω2+ωt2sds.Applying Gronwall’s inequality to (61), we get(62)Δω2+ωt2≤eC1tE(0).This shows that solution of problem (17)-(18) depends continuously on the initial data. In particular, problem (17)-(18) has a unique stronger solution.We can prove the continuous dependence and uniqueness for weak solutions by using density arguments (see, e.g., Cavalcanti et al. [27]) which also can be found in Lions [29] (Chapter 1, Theorem 1.2) by using a regularization method and in Pata and Zucchi [31] or Giorgi et al. [32] by using the mollifiers.This ends the proof of Theorem1.


## 4. General Decay

In this section, we will establish the decay property of the solution for problem (17)-(18) in the case μ2<μ1. Motivated by [27, 33], we use a perturbed energy method and suitable Lyapunov functionals.We first consider stronger solutions. Define the modified energy by(63)Ft=12utt2+12Δut2+12M^∇ut2+∫Ω‍f^utdx-12∫0t‍gsds∇ut2+12g∘∇ut+ξ2∫Ω‍∫01‍z2x,ρ,tdρdx,where ξ is a positive constant satisfying (19).It follows from (9) and (14) that(64)Ft=E(t)-12∫0t‍gsds∇ut2+12g∘∇ut≥lE(t),that is,(65)E(t)≤1lF(t).Lemma 3.
Under the assumptions in Theorem2, the modified energy functional defined by (63) satisfies that there exists a constant c>0 such that, for any t≥0,(66)ddtFt≤-c∫Ω‍ut2tdx-c∫Ω‍z2x,1,tdx+12g′∘∇ut-12g(t)∇ut2.Proof.
For the same argument as (41) in Section 3.2, we can easily get (66). Here we omit the detailed proof.Now we define the following functional:(67)Φ(t)=∫Ω‍ut(t)u(t)dx.Then we have the following lemma.Lemma 4.
Under the assumptions in Theorem2, the functional Φ(t) defined in (67) satisfies that, for any η>0,(68)ddtΦt≤1+μ14ηutt2-l-ηλ-2ηλλ1μ1+μ2Δut2-M^∇ut2+μ24η∫Ω‍z2x,1,tdx+1-l4ηλ(g∘∇u)(t),where λ1 is the Poincaré constant.Proof.
By taking a derivative of (67) and using the first equation of (17), we conclude that(69)ddtΦt=∫Ω‍utt(t)u(t)dx+utt2=utt2+∫Ω‍-Δ2u(t)+M∇ut2Δu(t)∫0t‍-∫0t‍gt-sΔusds∫0t‍-μ2utt-μ2zx,1,t-fu·u(t)dx≤utt2-Δut2-M^∇ut2+∫0t‍gt-s∇us,∇utds-μ1∫Ω‍uttutdx-μ2∫Ω‍zx,1,tutdx-∫Ω‍fututdx.Using Hölder’s inequality, we know that, for any η>0,(70)∫0t‍gt-s∇us,∇utds=∫0t‍gt-s∫Ω‍∇us-∇ut+∇ut·∇utdxds≤∫0t‍gt-s∫Ω‍∇ut-∇us∇utdxds+∫0t‍g(s)ds·∇ut2≤∇ut2∫0t‍gt-s∇ut-∇us2ds+∫0t‍g(s)ds·∇ut2≤η∇ut2+14ηgtL1R+g∘∇ut+∫0t‍g(s)ds·∇ut2≤ηλΔut2+14ηgtL1R+g∘∇ut+∫0t‍g(s)ds·∇ut2≤1-l+ηλΔut2+14ηgtL1(R+)(g∘∇u)(t).By using Young’s inequality and Poincaré’s inequality and noting ∇u2≤λΔu2, we infer that, for any η>0, (71)∫Ω‍ut(t)u(t)dx≤ηλλ1Δut2+14ηutt2,(72)∫Ω‍z(x,1,t)u(t)dx≤ηλλ1Δut2+14η∫Ω‍z2(x,1,t)dx.Combining (69)–(72) and noting (14), we complete the proof.In order to handle the termz(x,ρ,t), we introduce the functional (73)Ψ(t)∶=∫Ω‍∫01‍e-2τρz2x,ρ,tdρdx.Then we have the following estimate.Lemma 5.
Under the assumptions in Theorem2, the functional Ψ(t) defined in (73) satisfies that(74)ddtΨt≤-ρΨ(t)-c12τ∫01‍z2(x,1,t)dx+12τ∫01‍ut2(t)dx,where c1 is a positive constant.Proof.
Differentiating (73) with respect to t and using the second equation (17), we obtain(75)ddtΨt=-1τ∫Ω‍∫01‍e-2τρzx,ρ,tzρx,ρ,tdρdx=-∫Ω‍∫01‍e-2τρz2x,ρ,tdρdx-12τ∫Ω‍∫01‍∂∂ρe-2τρz2x,ρ,tdρdx.Then it is easy to verify that there exists a constant c1>0 satisfying (74).Proof of Theorem2.
We define the Lyapunov functional(76)G(t)∶=F(t)+ϵΦ(t)+ϵΨ(t),where ϵ>0 is a real number which will be taken later.
First, we claim that there exist two positive constantsβ1 and β2 such that, for any t≥0, (77)β1F(t)≤G(t)≤β2F(t).Indeed, it is easy to get(78)Φt+Ψt≤12utt2+12λ′Δut2+∫Ω‍∫01‍z2x,ρ,tdρdx≤1lmax⁡1,1λ′,ξF(t),where λ′>0 is the first eigenvalue of Δ2u=λu in Ω with u=Δu=0 on ∂Ω. Choosing C1=1/lmax⁡1,1/λ′,ξ, we know that(79)Gt-Ft=ϵΦt+Ψt≤ϵC1F(t).Now putting ϵ>0 small enough and choosing β1∶=1-ϵC1>0 and β2=1+ϵC1>0, we see that (77) holds.
Next, combining (66), (68), and (74), we arrive at(80)ddtGt≤-c-ϵ1+μ14η-ϵ2τutt2-ϵM^∇ut2-ρϵΨ(t)-ϵl-ηλ-2ηλλ1μ1+μ2Δut2-c+ϵc12τ-ϵ4η∫Ω‍z2(x,1,t)dx+1-l4ηϵg∘∇ut+12g′∘∇ut-12g(t)∇ut2.Now we choose η>0 and ϵ>0 so small that we can take two positive constants α1 and α2 such that, for any t≥0, (81)G′t≤-α1F(t)+α2(g∘∇u)(t).Multiplying (81) by μ(t), we have, for any t≥0, (82)μtG′t≤-α1μ(t)F(t)+α2μtg∘∇ut,which, along with (10) and (66), implies(83)μtG′t≤-α1μtFt-α2g′∘∇ut≤-α1μ(t)F(t)-2α2E′t,∀t≥0,that is,(84)μtGt+2α2Ft′-μ′tGt≤-α1μtFt,∀t≥0.Denote F(t)=μ(t)G(t)+2α2F(t), and then F(t) is equivalent to F(t); that is,(85)F(t)~F(t).Thus we conclude that, for any t≥0,(86)F′t≤-α2μtFt≤-α3μtFt.Integrating (86) over (0,t), we will see the following:(87)F(t)≤F(0)exp⁡-α3∫0t‍μ(s)ds,∀t≥0,which, together with (65), (77), and (85), gives (25).
This proves the general decay for regular solutions. We can extend the result to weak solutions by using a standard density argument; one can refer to Cavalcanti et al. [27]. The proof is hence complete.Remark 6.
There are some open problems concerning our present work, and here we give some of them.(1)
It is obvious that the weak damping termμ1ut plays a crucial role in our proofs. It is still an open problem when μ1=0.(2)
We only obtain the general decay forμ1>μ2. Whether the stability property holds for μ1=μ2 is still open.(3)
It is interesting to study that the weight of the delay is bigger than the weight of the damping; that is,μ1<μ2.



---

*Source: 585021-2015-03-31.xml*