# Optimal Decay Rate Estimates of a Nonlinear Viscoelastic Kirchhoff Plate

Authors: Baowei Feng; Mostafa Zahri
Journal: Complexity
 (2020)

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## Abstract

This paper is concerned with a nonlinear viscoelastic Kirchhoff plateuttt−σΔuttt+Δ2ut−∫0tgt−sΔ2usds=divF∇ut. By assuming the minimal conditions on the relaxation function g: g′t≤ξtGgt, where G is a convex function, we establish optimal explicit and general energy decay results to the system. Our result holds for Gt=tp with the range p∈1,2, which improves earlier decay results with the range p∈1,3/2. At last, we give some numerical illustrations and related comparisons.


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## Body

## 1. Introduction

In this paper, we consider the following nonlinear viscoelastic Kirchhoff plate equation:(1)uttt−σΔuttt+Δ2ut−∫0tgt−sΔ2usds=divF∇ut,inΩ×ℝ+,together with simply supported boundary condition(2)ut=Δut=0,on∂Ω×ℝ+,and initial conditions(3)ux,0=u0x,utx,0=u1x,x∈Ω,where Ω⊆ℝnn≥1 is a regular and bounded domain; σ > 0 is a constant; and F:ℝn⟶ℝn is a vector field which will be assumed later. The relaxation function gt is a real function. From the physical point of view, systems (1)–(3) are related to classical theory for beams/plates appearing from materials with viscoelastic structures.For viscoelastic wave equation,(4)utt−σΔutt−Δu+∫0tgt−sΔusds=ℱu,the problems are truly overworked. For example, for specific behavior of the relaxation function g, it is showed that the energy decays exponentially (polynomially) if g decays exponentially (polynomially) (see Cabanillas and Muñoz Rivera[1], Muñoz Rivera et al. [2, 3], Santos [4], Cavalcanti and Oquendo [5], and so on). Messaoudi [6, 7] considered two classes of system (4), by first introducing the assumption g′t≤−ξtgt, and established general decay results. Han and Wang [8], Liu [9, 10], Messaoudi and Mustafa [11], Mustafa [12], and Park and Park [13] also used this assumption on g to get general decay of energy for problems related to (4). In [14], Lasiecka and Tataru introduced a more general assumption on g, which satisfies g′t≤−Hgt, where H is strictly increasing and convex function. There are also so many stability results established by using this condition. We refer the reader to Cavalcanti et al. [15, 16], Lasiecka et al. [17, 18], Mustafa [19], Mustafa and Messaoudi [20], and Xiao and Liang [21]. Very recently, in [22, 23], Mustafa considered two classes of wave equations and proved general and explicit decay results of energy by using a new general assumption on g: g′t≤−ξtHgt.For viscoelastic plate equation, Rivera et al. [24] studied the following equation:(5)utt−γΔutt+Δ2u−∫0tgt−sΔ2usds=0,together with initial and dynamical boundary conditions. They proved the energy decays exponentially (resp. polynomially) if the relaxation function g decays exponentially (resp. polynomially). Alabau-Boussouira et al. [25] considered(6)utt+Δ2u−∫0tgt−sΔ2usds=fu,together with Dirichlet–Neumann boundary conditions and established exponential and polynomial decay results for sufficiently small initial data. When f (u) = 0 in (6), Cavalcanti [26] considered the equation subject to nonlinear boundary conditions and established exponential decay of energy by assuming a nonlinear and nonlocal feedback acting on the boundary and provided that the relaxation function decays exponentially. In Andrade et al. [27], the authors considered a viscoelastic plate equation with p-Laplacian:(7)utt+Δ2u−Δpu+∫0tgt−sΔusds−Δut+fu=0,where Δpu = div(|∇u|p−1∇u), and they established an exponential decay of energy under the assumption g′t≤−k1gt When Δpu is replaced by div∇up⟶x,t−1∇u in (7), Ferreira and Messaoudi [28] proved a general decay result of energy. Feng [29] established a general decay result of a plate equation with time delay. Recently, Gomes Tavares et al. [30] considered a class of nonlinear plate equations with memory; they proved, by using the methods developed by Lasiecka and Wang [18], the decay rates are expressed in terms of the solution to a given nonlinear dissipative ODE. In [31], the authors proved the well-posedness of solutions to problem (1)–(3) with the case σ > 0 and the case σ = 0. Under the assumptions on gt:(8)g′t≤−ξtgt,∀t>0,ξt>0,ξ′t≤0,ξ′tξt≤ξ0,∀t>0,they established the general decay rates of energy of the form(9)Et≤ce−γ∫0tξsds.If the memory term is infinite, one can find some results on plate equation with history memory in [32–38] and so on.In this paper, we continue to study (1)–(3), in which we consider σ = 1 for simplicity, with minimal conditions on the L1(0, ∞) relaxation function g (see (12)). We establish explicit and general energy decay results of systems (1)–(3) by using the idea of Mustafa [22, 23] and some properties of convex functions developed in [18, 39]. We point out that the decay results established here are optimal exponential and polynomial rates for 1 ≤ p < 2 when G(t) = tp, which improved the previous known results for 1≤p<3/2. Under this level of generality, the decay rates we get are optimal, and our results improve the stability results in previous works. At last, we give some numerical illustrations.The rest of this paper is as follows. In Section2, we give some assumptions and results. The general decay result of the energy will be established in Section 3. In Section 4, we give some numerical illustrations.


## 2. Assumptions and Results

In the following, for simplicity, we write ‖⋅‖ instead of ‖⋅‖2.c > 0 is used to denote a generic constant. The positive constants λ1 and λ2 represent the embedding constants(10)λ1u2≤Δu2,λ2∇u2≤Δu2,for u∈H2Ω∩H01Ω.For relaxation functiong, we assume the following:(A1)
gt:0,∞⟶ℝ+ are nonincreasing C1 functions satisfying(11)g0>0,1−∫0∞gsds=l>0.In addition, there exists aC1 function G:ℝ+⟶ℝ+ satisfying G(0) = G′(0) = 0. The function G(t) is linear or it is an increasing strictly convex function of class C2ℝ+ on (0, r], r≤g0, such that(12)g′t≤−ξtGgt,∀t≥0,whereξ(t) is a C1 function satisfying(13)ξt>0,ξ′t≤0,∀t≥0.With respect toF, we assume that(A2)
F:ℝn⟶ℝn is a C1−vector field given by F = (F1, F2, …, Fn) such that(14)∇Fju≤kj1+upj−1/2,∀u∈ℝn,where forj = 1, 2, …, n, the constants kj > 0 and pj satisfy(15)pj≥1,ifn=1,2,1≤pj≤n+2n−2,ifn≥3.Moreover, we also assume that(16)0≤fu≤Fu⋅u,∀u∈ℝn,whereF is a conservation field with F = ∇f and f:ℝn⟶ℝ is a real valued function.Remark 1 (see [31]).
Condition (14) implies that there exists a positive constant K = K(kj, pj, n), j = 1, 2, …, n, such that(17)Fu−Fv≤K∑j=1n1+upj−1/2+vpj−1/2u−v,∀u,v∈ℝn.
The existence of global solutions has been proved in [31].Theorem 1.
Let (11) and (A2) hold. If the initial data u0,u1∈H2Ω∩H01Ω×H01Ω, then problem (1)–(3) has a unique weak solution satisfying that for any T > 0,(18)ut∈L∞0,T;H2Ω∩H01Ω,utt∈L∞0,T;H01Ω.The total energy of problem (1)–(3) is given by(19)Et=12utt2+∇utt2+1−∫0tgsdsΔut2+g∘Δut+∫Ωf∇utdx,where(20)g∘vt=∫0tgt−svt−vs2ds.Now, we give the stability result of energy to problem (1)–(3).Theorem 2.
Let (A1) and (A2) hold. Ifu0,u1∈H2Ω∩H01Ω×H01Ω, then the energy E(t) satisfies(21)Et≤k2G1−1k1∫g−1rtξsds,∀t>g−1r,where the positive constant k1 depends on the size of initial data, k2 > 0, and(22)G1t=∫tr1sG′sds.In particular, forG(t) = tp in (12), we can get(23)Et≤kexp−k1∫0tξsds,ifp=1,k^1+∫0tξsds−1/p−1,if1<p<2,where k,k^, and k1 are positive constants.Remark 2.
(1)
Since the constantcCα appearing within the estimate (47) depends on E(0), which leads to the final constant which will depend on the initial data, the constant k1 > 0 depends on the size of initial data.
(2)
Here,G1 is strictly convex and decreasing on (0, r] with limt⟶0 G1(t) = +∞.Remark 3.
It follows from (A1) thatlimt⟶0+∞gt=0. We know that there exists some t1 ≥ 0 large enough such that(24)gt1=r⟹gt≤r,∀t≥t1.
Then, we can get for everyt ∈ [0, t1],(25)0<gt1≤gt≤g0,0<ξt1≤ξt≤ξ0.
Therefore, there exist positive constantsa and b,(26)a≤ξtGgt≤b,which yields for every t ∈ [0, t1],(27)g′t≤−ξtGgt≤−ag0g0≤−ag0gt.
We end this section by giving three examples.Example 1.
Letgt=ae−bt, (b > 0). We take a > 0 satisfying (11). If we choose Gt=t, then g′t=−bGgt. From (23)1, we find(28)Et≤ke−bk1t.Example 2.
Letgt=a/1+tb, (b > 1). We take a > 0 satisfying (11). For a fixed positive constant ρ, we have g′t=−ρGgt with Gt=tp, p=1+b/b∈ (1, 2). From (23)2, we get(29)Et≤k^1+tb.Example 3.
Letgt=aexp−11+tb, (0 < b < 1). We take a > 0 satisfying (11). Then, g′t=−ξtGgt with Gt=t and ξ(t) = b(1 + t)b − 1. Then, we obtain from (23)1 that(30)Et≤kexp−b1+tb−1.


## 3. Optimal Decay

To prove Theorem2, we need some lemmas.


### 3.1. Technical Lemmas

Lemma 1.
It holds that for anyt ≥ 0,(31)E′t=12g′⋅Δut−12gtΔut2.Proof.
MultiplyingL2(Ω) in equation (1) by ut and using integration by parts and boundary conditions, we can get (31).
Let us define the functionals:(32)ϕt=∫Ωutt−Δuttutdx,ψt=−∫Ωutt−Δutt∫0tgt−sut−usdsdx.Lemma 2.
The functionalϕ(t) satisfies for any t ≥ 0,(33)ϕ′t≤−l2Δut2+utt2+∇utt2−∫Ωf∇utdx+Cαlh⋅Δut,for any 0<a<1, where(34)Cα=∫0∞g2sαgsg′sds,0<α<1ht=αgt−g′t,introduced in [22, 23].Proof.
In view of (1), and integration by parts, we can obtain(35)ϕ′t=−Δut2+utt2+∇utt2−∫ΩF∇ut⋅∇utdx+∫ΩΔut∫0tgt−sΔusdsdx=−1−∫0tgsdsΔut2+utt2+∇utt2−∫ΩF∇ut⋅∇utdx+∫ΩΔut∫0tgt−sΔus−Δutdsdx.
It follows from (16) that(36)−∫ΩF∇ut⋅∇utdx≤−∫Ωf∇udx.
Hölder’s inequality gives us(37)∫Ω∫0tgt−sΔus−Δutds2dx=∫Ω∫0tgt−sαgt−s−g′t−sαgt−s−g′t−sΔus−Δutds2dx≤∫0tg2sαgs−g′sds∫Ω∫0tαgt−s−g′t−sΔus−Δut2dsdx=Cαh⋅Δut,which, together with Young’s inequality, implies(38)∫ΩΔut∫0tgt−sΔus−Δutdsdx≤l2Δut2+12l∫Ω∫0tgt−sΔus−Δutds2dx≤l2Δut2+Cαh⋅Δut.
Replacing (36) and (38) in (35), we can get the desired estimate (33).Lemma 3.
Assume that (A1) and (A2) hold, then the functionalψ(t) satisfies for any δ > 0,(39)ψ′t≤−∫0tgsds−δutt2−∫0tgsds−δ∇utt2+δΔut2+cCα+1δh⋅Δut.Proof.
Taking the derivative ofψ(t), using equation (1) and integration by parts, we can derive that(40)ψ′t=1−∫0tgsds∫ΩΔut∫0tgt−sΔut−Δusdsdx︸≔I1+∫Ω∫0tgt−sΔut−Δusds2dx︸≔I2+∫ΩF∇u∫0tgt−s∇ut−∇usdsdx︸≔I3−∫Ωutt∫0tg′t−sut−usdsdx︸≔I4−∫0tgsdsutt2−∫Ω∇utt∫0tg′t−s∇ut−∇usdsdx︸≔I5−∫0tgsds∇utt2.
By Young’s inequality and (37), we shall see below, for any δ > 0,(41)I1≤δ2Δut2+cCαδh⋅Δut,I2≤Cαh⋅Δut,I4=∫Ωut∫0tht−sut−usdsdx−∫Ωut∫0tαgt−sut−usdsdx≤δutt2+12δ∫Ω∫0tht−sut−usds2dx+α22δ∫Ω∫0tgt−sut−usds2dx≤δutt2+∫0thsds2δh⋅ut+α2Cα2δh⋅ut≤δutt2+cδh⋅Δut+cCαδh⋅Δut.
Similarly, we can get for anyδ > 0,(42)I5≤δ∇utt2+cCα+1δh⋅Δut.
With respect toI3, we have(43)I3≤∫0tgt−s∫ΩF∇ut∇ut−∇usdx︸≔I31ds.
By using (A2), we derive that(44)I31≤K∫Ω∑j=1n1+∇utpj−1/2∇ut∇ut−∇usdx≤K∑j=1nΩpj−1/2pj+1+∇utpj+1pj−1/2∇utpj+1∇ut−∇us≤ΔutKλ2∑j=1nμpjΩpj−1/2pj+1+∇utpj+1pj−1/2︸≔I32Δut−Δus.
Following the same method as in [31], we can get(45)I322≤μ1+μ2E0p−1/2,where μ1 and μ2 are two positive constants and(46)p=maxp1,…,pn,ifE0≥1,p=minp1,…,pn,ifE0<1.
In view of Young and Hölder’s inequalities and (37), we have for any δ > 0,(47)I3≤∫0tgt−sΔutKλ2∑j=1nμpjΩpj−1/2pj+1+∇utpj+1pj−1/2Δut−Δus≤δ2Δut2+I3222δ∫0tgt−sΔut−Δusds2≤δ2Δut2+cCαδh⋅Δut.
Thus, we can find (39) from (27) and (40).Lemma 4.
Let (A1) and (A2) hold, the functionalθ(t) defined by(48)θt=∫Ω∫0tσt−sΔus2dsdx,where σt=∫t∞gsds, satisfies(49)θ′t≤−12g∘Δut+31−lΔut2.Proof.
First we know thatσ′(t) = −g(t). Then,(50)θ′t=σ0Δut2−∫Ω∫0tgt−sΔus2dsdx=−∫Ω∫0tgt−sΔus−Δut2dsdx+σtΔut2−2∫ΩΔut∫0tgt−sΔus−Δutdsdx.
By using Young and Hölder’s inequalities, we will show that(51)−2∫ΩΔut∫0tgt−sΔus−Δutdsdx≤21−lΔut2+∫0tgsds21−lg∘Δut.
Sinceσ(t) ≤ σ(0) = 1 − l and ∫0tgsds≤1−l, we can obtain (49).Our argument in the following is based on the choice of a suitable Lyapunov functionℒt by(52)ℒt≔NEt+N1ϕt+N2ψt,where N, N1 and N2 are positive constants. Clearly, for N large, there exist β1 > 0 and β2 > 0 such that(53)β1Et≤ℒt≤β2Et.Lemma 5.
It holds that for anyt ≥ t1,(54)ℒ′t≤−41−lΔut2−utt2−∇utt2−∫Ωf∇utdx+14g∘Δut.Proof.
Letg1=∫0t1gsds>0. Combining (31), (33), and (39), taking δ=1/2N2, and noting g′=ag−h, we can infer that for any t ≥ t1,(55)ℒ′t≤−N12l−12Δut2−N2g1−12−N1utt2−N2g1−12−N1∇utt2+α2Ng∘Δut−N1∫Ωf∇utdx−N2−2cN22−CαN1l+2cN22h⋅Δut.
Firstly, we takeN1 large enough so that(56)N1>1,N12l−12>41−l,and then we choose N1 so large that(57)N2g1−12−N1>1.
Sinceαg2s/αgs−g′s<gs, using the Lebesgue dominated convergence theorem, we can get(58)limα⟶0αCα=limα⟶0∫0∞αg2sαgs−g′sds=0.
Hence, there exist someα0 (0 < α0 < 1) such that if α < α0, then(59)αCα<18N1/l+2cN22.
At last, for any fixedN1 and N2, we choose N large enough and choose α satisfying(60)14N−2cN22>0,α=12N<α0,and then(61)N2−2cN22−CαN1l+2cN22>0.
This ends the proof.


### 3.2. Proof of Theorem2

Taking into account (27) and (31), we find that for any t ≥ t1,(62)∫0t1gs∫ΩΔut−Δut−s2dxds≤−g0a∫0t1g′s∫ΩΔut−Δut−s2dxds≤−cE′t,which, along with (54), gives us for some constant m > 0 and for all t ≥ t1,(63)ℒ′t≤−mEt+cg∘Δut≤−mEt−cE′t+c∫t1tgs∫ΩΔut−Δut−s2dxds.DefineFt=ℒt+cEt∼Et. Then, we find from (63),(64)F′t≤−mEt+c∫t1tgs∫ΩΔut−Δut−s2dxds.We consider the following two cases.Case 1.
The particular caseG(t) = tp.(I)
p=1.Multiplying (64) by ξ(t) and using (31) and (A2)-(A3), we have(65)ξtF′t≤−mξtEt−cE′t,∀t≥t1.Sinceξ(t) is a nonincreasing continuous function and ξ′(t) ≤ 0 for a.e. t, then(66)ξF+cE′t≤ξtF′t+cE′t≤−mξtEt,a.e.t≥t1.In view ofξF + cE ∼ E, we obtain that there exist two positive constants c1, c2 > 0,(67)Et≤c1e−c2∫0tξsds.(II)
1 <p < 2.Defineℰt by(68)ℰt=ℒt+θt.It follows from (49) and (54) that ℰt≥0 and there exists a positive constant β such that for any t ≥ t1,(69)ℰ′t≤−βutt2+Δut2+∇utt2+∫Ωf∇utdx−14g⋅Δut.Then, there exists a certain constantβ1 > 0,(70)ℰ′t≤−β1Et,∀t≥t1.
This gives us(71)β1∫t1tEsds≤ℰt1−ℰt≤ℰt1.
Hence,(72)∫0∞Esds<∞.
Define(73)It=∫0t∫ΩΔt−Δut−s2dxds,and we know that(74)It≤c∫0tEsds.
Without loss of generality assumingt1 so large that I(t1) > 0, then(75)0<It1≤It<∞,∀t≥t1.
Using Jensen’s inequality and (12), we can derive from (64) that for some constant q > 0,(76)F′t≤−qEt+cItIt∫t1tgps1/p∫ΩΔut−Δut−s2dxds≤−qEt+cIt1It∫t1tgps∫ΩΔut−Δut−s2dxds1/p≤−qEt+cIt1It∫t1t−g′sξs∫ΩΔut−Δut−s2dxds1/p≤−qEt+cI1−1pt∫t1t−g′sξs∫ΩΔut−Δut−s2dxds1/p≤−qEt+cξ1/pt∫t1t−g′s∫ΩΔut−Δut−s2dxds1/p≤−qEt+cξ1/pt−E′t1/p.
We multiply (76) by Ep−1(t) and use (31) to deduce(77)FEp−1′t≤F′tEp−1t≤−qEpt+c−E′t/ξt1/pEp−1t.
By Young’s inequality, we have for anyɛ1 > 0,(78)FEp−1′t≤−qEpt+ε1Ept+cε1−E′tξt.
Takingε1<1/2q, we conclude(79)FEp−1′t≤−q2Ept−cEtξt.
Defineℱt=ξFEp−1+cE∼E. Multiplying (79) by ξ(t), we have(80)ℱ′t≤−q2ξtEpt.
Then, there exists a certain constantq0 > 0 such that(81)ℱ′t≤−q0ξtℱpt,from which we obtain(82)Et≤c31+∫0tξsds−1/p−1,where c3 is a positive constant.
Combining (I) and (II), we can get (35).Case 2.
The general case.
DefineI(t) by(83)It=q∫t1t∫ΩΔt−Δut−s2dxds.
It follows from (72) that we can choose a constant 0 < q < 1 so that for all t ≥ t1,(84)It<1.
Without loss of generality, we assume thatI(t) > 0 for all t ≥ t1; otherwise, (64) implies an exponential decay. In addition, we define λ(t) by(85)λt=−∫t1tg′s∫ΩΔut−Δut−s2dxds.
Clearly,λ(t) ≤ −cE′(t). Since G(t) is strictly convex on (0, r] and G(0) = 0, then(86)Gνx≤νGx,provided 0 ≤ ν ≤ 1 and x ∈ (0, r]. By using (12), (84), and Jensen’s inequality, we can obtain(87)λt=1qIt∫t1tIt−g′s∫ΩqΔut−Δut−s2dxds≥1qIt∫t1tItξsGgs∫ΩqΔut−Δut−s2dxds≥ξtqIt∫t1tGItgs∫ΩqΔut−Δut−s2dxds≥ξtqG1It∫t1tItgs∫ΩqΔut−Δut−s2dxds=ξtqGq∫t1tgs∫ΩΔut−Δut−s2dxds=ξtqG¯q∫t1tgs∫ΩΔut−Δut−s2dxds.
Here,G¯ is an extension of G, which is strictly convex and strictly increasing C2 function on (0, ∞). We can get from (87) that(88)∫t1tgs∫ΩΔut−Δut−s2dxds≤1qG¯−1qλtξt,which, together with (64), implies for any t ≥ t1,(89)F′t≤−mEt+cG¯−1qλtξt.
Now, forɛ0 < r, we define the function K1t by(90)K1t=G¯′ε0EtE0Ft+Et,which is equivalent to E(t). In view of E′(t) ≤ 0, G¯′>0, and G¯″>0, and using (89), we conclude that(91)K1′t=ε0E′tE0G¯″ε0EtE0Ft+G¯′ε0EtE0F′t+E′t≤−mEtG¯′ε0EtE0+cG−1qλtξtG¯′ε0EtE0+E′t.
As in Arnold [40], we define the conjugate function of G¯ by G¯∗, which satisfies(92)M1M2≤G¯∗M1+G¯M2.
ForM1=G¯′ε0Et/E0 and M2=G¯−1qλt/ξt, and using (91), we get(93)K1′t≤−mEtG¯′ε0EtE0+cG¯∗G¯′ε0EtE0+cqλtξt+E′t≤−mEtG¯′ε0EtE0+cε0EtE0G¯′ε0EtE0+cqλtξt+E′t.
Multiplying (93) by ξ(t), we see that(94)ξtK1′t≤−mξtEtG′ε0EtE0+cε0EtE0ξtG′ε0EtE0+cqλt+ξtE′t≤−mξtEtG′ε0EtE0+cε0EtE0ξtG′ε0EtE0−cE′t,where we used the fact that as ε0Et/E0<r, G¯′ε0Et/E0=G′ε0Et/E0.
Define the functionalK2t by(95)K2t=ξtK1t+cEt.
It is easy to obtain thatK2t∼Et, i.e., there exist two positive constants β3 and β4 such that(96)β3K2t≤Et≤β4K2t.
We choose suitableɛ0 to get from (94) that for a certain constant k > 0,(97)K2′t≤−kξtEtE0G′ε0EtE0=−kξtG2EtE0,where G2(t) = tG′(ɛ0t). Denote Rt=β3K2t/E0. It follows from (96) that(98)Rt∼Et.
SinceG′2(t) = G′(ɛ0t) + ɛ0tG″(ɛ0t), then, using the strict convexity of G on (0, r], we know that G′2(t), G2(t) > 0 on (0, 1]. By (97), we get for some k1 > 0 and for all t ≥ t1,(99)R′t≤−k1ξtG2Rt.
Integrating (99) over (t1, t), we have(100)∫t1t−R′sG2Rsds≥k1∫t1tξsds⟹∫ε0Rtε0Rt1sG′sds≥k1∫t1tξsds,which, noting G1, defined by(101)G1t=∫TR1sG′sds,is strictly decreasing on 0,r and limt⟶0G1t=+∞, we get(102)Rt≤1ε0G1−1k1∫t1tξsds.
Then, (21) follows from (98) and (102). The proof is done.


## 3.1. Technical Lemmas

Lemma 1.
It holds that for anyt ≥ 0,(31)E′t=12g′⋅Δut−12gtΔut2.Proof.
MultiplyingL2(Ω) in equation (1) by ut and using integration by parts and boundary conditions, we can get (31).
Let us define the functionals:(32)ϕt=∫Ωutt−Δuttutdx,ψt=−∫Ωutt−Δutt∫0tgt−sut−usdsdx.Lemma 2.
The functionalϕ(t) satisfies for any t ≥ 0,(33)ϕ′t≤−l2Δut2+utt2+∇utt2−∫Ωf∇utdx+Cαlh⋅Δut,for any 0<a<1, where(34)Cα=∫0∞g2sαgsg′sds,0<α<1ht=αgt−g′t,introduced in [22, 23].Proof.
In view of (1), and integration by parts, we can obtain(35)ϕ′t=−Δut2+utt2+∇utt2−∫ΩF∇ut⋅∇utdx+∫ΩΔut∫0tgt−sΔusdsdx=−1−∫0tgsdsΔut2+utt2+∇utt2−∫ΩF∇ut⋅∇utdx+∫ΩΔut∫0tgt−sΔus−Δutdsdx.
It follows from (16) that(36)−∫ΩF∇ut⋅∇utdx≤−∫Ωf∇udx.
Hölder’s inequality gives us(37)∫Ω∫0tgt−sΔus−Δutds2dx=∫Ω∫0tgt−sαgt−s−g′t−sαgt−s−g′t−sΔus−Δutds2dx≤∫0tg2sαgs−g′sds∫Ω∫0tαgt−s−g′t−sΔus−Δut2dsdx=Cαh⋅Δut,which, together with Young’s inequality, implies(38)∫ΩΔut∫0tgt−sΔus−Δutdsdx≤l2Δut2+12l∫Ω∫0tgt−sΔus−Δutds2dx≤l2Δut2+Cαh⋅Δut.
Replacing (36) and (38) in (35), we can get the desired estimate (33).Lemma 3.
Assume that (A1) and (A2) hold, then the functionalψ(t) satisfies for any δ > 0,(39)ψ′t≤−∫0tgsds−δutt2−∫0tgsds−δ∇utt2+δΔut2+cCα+1δh⋅Δut.Proof.
Taking the derivative ofψ(t), using equation (1) and integration by parts, we can derive that(40)ψ′t=1−∫0tgsds∫ΩΔut∫0tgt−sΔut−Δusdsdx︸≔I1+∫Ω∫0tgt−sΔut−Δusds2dx︸≔I2+∫ΩF∇u∫0tgt−s∇ut−∇usdsdx︸≔I3−∫Ωutt∫0tg′t−sut−usdsdx︸≔I4−∫0tgsdsutt2−∫Ω∇utt∫0tg′t−s∇ut−∇usdsdx︸≔I5−∫0tgsds∇utt2.
By Young’s inequality and (37), we shall see below, for any δ > 0,(41)I1≤δ2Δut2+cCαδh⋅Δut,I2≤Cαh⋅Δut,I4=∫Ωut∫0tht−sut−usdsdx−∫Ωut∫0tαgt−sut−usdsdx≤δutt2+12δ∫Ω∫0tht−sut−usds2dx+α22δ∫Ω∫0tgt−sut−usds2dx≤δutt2+∫0thsds2δh⋅ut+α2Cα2δh⋅ut≤δutt2+cδh⋅Δut+cCαδh⋅Δut.
Similarly, we can get for anyδ > 0,(42)I5≤δ∇utt2+cCα+1δh⋅Δut.
With respect toI3, we have(43)I3≤∫0tgt−s∫ΩF∇ut∇ut−∇usdx︸≔I31ds.
By using (A2), we derive that(44)I31≤K∫Ω∑j=1n1+∇utpj−1/2∇ut∇ut−∇usdx≤K∑j=1nΩpj−1/2pj+1+∇utpj+1pj−1/2∇utpj+1∇ut−∇us≤ΔutKλ2∑j=1nμpjΩpj−1/2pj+1+∇utpj+1pj−1/2︸≔I32Δut−Δus.
Following the same method as in [31], we can get(45)I322≤μ1+μ2E0p−1/2,where μ1 and μ2 are two positive constants and(46)p=maxp1,…,pn,ifE0≥1,p=minp1,…,pn,ifE0<1.
In view of Young and Hölder’s inequalities and (37), we have for any δ > 0,(47)I3≤∫0tgt−sΔutKλ2∑j=1nμpjΩpj−1/2pj+1+∇utpj+1pj−1/2Δut−Δus≤δ2Δut2+I3222δ∫0tgt−sΔut−Δusds2≤δ2Δut2+cCαδh⋅Δut.
Thus, we can find (39) from (27) and (40).Lemma 4.
Let (A1) and (A2) hold, the functionalθ(t) defined by(48)θt=∫Ω∫0tσt−sΔus2dsdx,where σt=∫t∞gsds, satisfies(49)θ′t≤−12g∘Δut+31−lΔut2.Proof.
First we know thatσ′(t) = −g(t). Then,(50)θ′t=σ0Δut2−∫Ω∫0tgt−sΔus2dsdx=−∫Ω∫0tgt−sΔus−Δut2dsdx+σtΔut2−2∫ΩΔut∫0tgt−sΔus−Δutdsdx.
By using Young and Hölder’s inequalities, we will show that(51)−2∫ΩΔut∫0tgt−sΔus−Δutdsdx≤21−lΔut2+∫0tgsds21−lg∘Δut.
Sinceσ(t) ≤ σ(0) = 1 − l and ∫0tgsds≤1−l, we can obtain (49).Our argument in the following is based on the choice of a suitable Lyapunov functionℒt by(52)ℒt≔NEt+N1ϕt+N2ψt,where N, N1 and N2 are positive constants. Clearly, for N large, there exist β1 > 0 and β2 > 0 such that(53)β1Et≤ℒt≤β2Et.Lemma 5.
It holds that for anyt ≥ t1,(54)ℒ′t≤−41−lΔut2−utt2−∇utt2−∫Ωf∇utdx+14g∘Δut.Proof.
Letg1=∫0t1gsds>0. Combining (31), (33), and (39), taking δ=1/2N2, and noting g′=ag−h, we can infer that for any t ≥ t1,(55)ℒ′t≤−N12l−12Δut2−N2g1−12−N1utt2−N2g1−12−N1∇utt2+α2Ng∘Δut−N1∫Ωf∇utdx−N2−2cN22−CαN1l+2cN22h⋅Δut.
Firstly, we takeN1 large enough so that(56)N1>1,N12l−12>41−l,and then we choose N1 so large that(57)N2g1−12−N1>1.
Sinceαg2s/αgs−g′s<gs, using the Lebesgue dominated convergence theorem, we can get(58)limα⟶0αCα=limα⟶0∫0∞αg2sαgs−g′sds=0.
Hence, there exist someα0 (0 < α0 < 1) such that if α < α0, then(59)αCα<18N1/l+2cN22.
At last, for any fixedN1 and N2, we choose N large enough and choose α satisfying(60)14N−2cN22>0,α=12N<α0,and then(61)N2−2cN22−CαN1l+2cN22>0.
This ends the proof.


## 3.2. Proof of Theorem2

Taking into account (27) and (31), we find that for any t ≥ t1,(62)∫0t1gs∫ΩΔut−Δut−s2dxds≤−g0a∫0t1g′s∫ΩΔut−Δut−s2dxds≤−cE′t,which, along with (54), gives us for some constant m > 0 and for all t ≥ t1,(63)ℒ′t≤−mEt+cg∘Δut≤−mEt−cE′t+c∫t1tgs∫ΩΔut−Δut−s2dxds.DefineFt=ℒt+cEt∼Et. Then, we find from (63),(64)F′t≤−mEt+c∫t1tgs∫ΩΔut−Δut−s2dxds.We consider the following two cases.Case 1.
The particular caseG(t) = tp.(I)
p=1.Multiplying (64) by ξ(t) and using (31) and (A2)-(A3), we have(65)ξtF′t≤−mξtEt−cE′t,∀t≥t1.Sinceξ(t) is a nonincreasing continuous function and ξ′(t) ≤ 0 for a.e. t, then(66)ξF+cE′t≤ξtF′t+cE′t≤−mξtEt,a.e.t≥t1.In view ofξF + cE ∼ E, we obtain that there exist two positive constants c1, c2 > 0,(67)Et≤c1e−c2∫0tξsds.(II)
1 <p < 2.Defineℰt by(68)ℰt=ℒt+θt.It follows from (49) and (54) that ℰt≥0 and there exists a positive constant β such that for any t ≥ t1,(69)ℰ′t≤−βutt2+Δut2+∇utt2+∫Ωf∇utdx−14g⋅Δut.Then, there exists a certain constantβ1 > 0,(70)ℰ′t≤−β1Et,∀t≥t1.
This gives us(71)β1∫t1tEsds≤ℰt1−ℰt≤ℰt1.
Hence,(72)∫0∞Esds<∞.
Define(73)It=∫0t∫ΩΔt−Δut−s2dxds,and we know that(74)It≤c∫0tEsds.
Without loss of generality assumingt1 so large that I(t1) > 0, then(75)0<It1≤It<∞,∀t≥t1.
Using Jensen’s inequality and (12), we can derive from (64) that for some constant q > 0,(76)F′t≤−qEt+cItIt∫t1tgps1/p∫ΩΔut−Δut−s2dxds≤−qEt+cIt1It∫t1tgps∫ΩΔut−Δut−s2dxds1/p≤−qEt+cIt1It∫t1t−g′sξs∫ΩΔut−Δut−s2dxds1/p≤−qEt+cI1−1pt∫t1t−g′sξs∫ΩΔut−Δut−s2dxds1/p≤−qEt+cξ1/pt∫t1t−g′s∫ΩΔut−Δut−s2dxds1/p≤−qEt+cξ1/pt−E′t1/p.
We multiply (76) by Ep−1(t) and use (31) to deduce(77)FEp−1′t≤F′tEp−1t≤−qEpt+c−E′t/ξt1/pEp−1t.
By Young’s inequality, we have for anyɛ1 > 0,(78)FEp−1′t≤−qEpt+ε1Ept+cε1−E′tξt.
Takingε1<1/2q, we conclude(79)FEp−1′t≤−q2Ept−cEtξt.
Defineℱt=ξFEp−1+cE∼E. Multiplying (79) by ξ(t), we have(80)ℱ′t≤−q2ξtEpt.
Then, there exists a certain constantq0 > 0 such that(81)ℱ′t≤−q0ξtℱpt,from which we obtain(82)Et≤c31+∫0tξsds−1/p−1,where c3 is a positive constant.
Combining (I) and (II), we can get (35).Case 2.
The general case.
DefineI(t) by(83)It=q∫t1t∫ΩΔt−Δut−s2dxds.
It follows from (72) that we can choose a constant 0 < q < 1 so that for all t ≥ t1,(84)It<1.
Without loss of generality, we assume thatI(t) > 0 for all t ≥ t1; otherwise, (64) implies an exponential decay. In addition, we define λ(t) by(85)λt=−∫t1tg′s∫ΩΔut−Δut−s2dxds.
Clearly,λ(t) ≤ −cE′(t). Since G(t) is strictly convex on (0, r] and G(0) = 0, then(86)Gνx≤νGx,provided 0 ≤ ν ≤ 1 and x ∈ (0, r]. By using (12), (84), and Jensen’s inequality, we can obtain(87)λt=1qIt∫t1tIt−g′s∫ΩqΔut−Δut−s2dxds≥1qIt∫t1tItξsGgs∫ΩqΔut−Δut−s2dxds≥ξtqIt∫t1tGItgs∫ΩqΔut−Δut−s2dxds≥ξtqG1It∫t1tItgs∫ΩqΔut−Δut−s2dxds=ξtqGq∫t1tgs∫ΩΔut−Δut−s2dxds=ξtqG¯q∫t1tgs∫ΩΔut−Δut−s2dxds.
Here,G¯ is an extension of G, which is strictly convex and strictly increasing C2 function on (0, ∞). We can get from (87) that(88)∫t1tgs∫ΩΔut−Δut−s2dxds≤1qG¯−1qλtξt,which, together with (64), implies for any t ≥ t1,(89)F′t≤−mEt+cG¯−1qλtξt.
Now, forɛ0 < r, we define the function K1t by(90)K1t=G¯′ε0EtE0Ft+Et,which is equivalent to E(t). In view of E′(t) ≤ 0, G¯′>0, and G¯″>0, and using (89), we conclude that(91)K1′t=ε0E′tE0G¯″ε0EtE0Ft+G¯′ε0EtE0F′t+E′t≤−mEtG¯′ε0EtE0+cG−1qλtξtG¯′ε0EtE0+E′t.
As in Arnold [40], we define the conjugate function of G¯ by G¯∗, which satisfies(92)M1M2≤G¯∗M1+G¯M2.
ForM1=G¯′ε0Et/E0 and M2=G¯−1qλt/ξt, and using (91), we get(93)K1′t≤−mEtG¯′ε0EtE0+cG¯∗G¯′ε0EtE0+cqλtξt+E′t≤−mEtG¯′ε0EtE0+cε0EtE0G¯′ε0EtE0+cqλtξt+E′t.
Multiplying (93) by ξ(t), we see that(94)ξtK1′t≤−mξtEtG′ε0EtE0+cε0EtE0ξtG′ε0EtE0+cqλt+ξtE′t≤−mξtEtG′ε0EtE0+cε0EtE0ξtG′ε0EtE0−cE′t,where we used the fact that as ε0Et/E0<r, G¯′ε0Et/E0=G′ε0Et/E0.
Define the functionalK2t by(95)K2t=ξtK1t+cEt.
It is easy to obtain thatK2t∼Et, i.e., there exist two positive constants β3 and β4 such that(96)β3K2t≤Et≤β4K2t.
We choose suitableɛ0 to get from (94) that for a certain constant k > 0,(97)K2′t≤−kξtEtE0G′ε0EtE0=−kξtG2EtE0,where G2(t) = tG′(ɛ0t). Denote Rt=β3K2t/E0. It follows from (96) that(98)Rt∼Et.
SinceG′2(t) = G′(ɛ0t) + ɛ0tG″(ɛ0t), then, using the strict convexity of G on (0, r], we know that G′2(t), G2(t) > 0 on (0, 1]. By (97), we get for some k1 > 0 and for all t ≥ t1,(99)R′t≤−k1ξtG2Rt.
Integrating (99) over (t1, t), we have(100)∫t1t−R′sG2Rsds≥k1∫t1tξsds⟹∫ε0Rtε0Rt1sG′sds≥k1∫t1tξsds,which, noting G1, defined by(101)G1t=∫TR1sG′sds,is strictly decreasing on 0,r and limt⟶0G1t=+∞, we get(102)Rt≤1ε0G1−1k1∫t1tξsds.
Then, (21) follows from (98) and (102). The proof is done.


## 4. Numerical Tests

In this section, we present various tests in order to illustrate our theoretical results proved in Theorem2. We solve problem (1) using the nonlinear Lax–Wendroff method in time and space in the space-time domain [0, 1] × [0, 5]. Moreover, for all partial derivatives of problem (1), we used a second-order discretization in time and space, and we consider the vector field below(103)Fu=∂u∂x2,where divFu=2∂2u∂x2·∂u∂x.For the following values of the parameter σ=0.1,1,5, we simulate six tests of the decay of the energy (19) (for similar constructions, we refer to [41, 42]).Test 1: in the first three tests, we present the decay case using the exponential function g1(t) = e−2t, the vector field (103), and the parameters σ = 0.1, 1, 5 (Test 1.1, 1.2, and 1.3).Test 2: in the second three numerical tests, we examine the energy decay (19) using the polynomial function g2t=1/1+t, the vector field (103), and the parameters σ = 0.1, 1, 5 (Test 1.1, 1.2, and 1.3).In order to ensure the scheme stability, we use Δt = 0.0005 < dx = 0.005 satisfying the stability of the Courant–Friedrichs–Lewy (CFL) inequality, where Δt represents the time step and dx represents the spatial step. The spatial interval [0, 1] is subdivided into 200 subintervals, where the temporal interval [0, 5] is deduced from the stability condition above. We run our code for 10000 time steps using the following initial conditions:(104)ux,0=x1−x;utx,0=x1−xin0,1.In Figure1, we show the results of the first three tests, namely, Test 1.1 for σ = 0.1, Test 1.2 for σ = 5, and Test 1.3 for σ = 5. We present the cross section cuts at x = 0.25, x = 0.5, and x = 0.75. The damping behavior is demonstrated for all experiments. Moreover, it should be stressed that for larger σ, the pseudoperiod decreases within decaying envelope. But for smaller σ (tending to 0), the pseudoperiod increases within same decaying envelope. Under similar initial and boundary conditions, we present in Figure 2 the results obtained for the Test 2.1 for σ = 0.1, Test 2.2 for σ = 5, and Test 2.3 for σ = 5. In Figure 3 we can clearly compare the energy decay obtained in Test 1 and in Test 2. We remarked that the energy decay is not affected by the choice of the memory functions g1t=e−2t and g2t=1/1+t.Figure 1
Test 1: damping behavior using the exponential functiong1t.
(a)
(b)
(c)Figure 2
Test 2: damping behavior using the polynomial functiong2t 2g2t.
(a)
(b)
(c)Figure 3
Energy decay for the exponential and the polynomial functionsg1 and g2. (a) Case exponential function g1t. (b) Case polynomial function g2t.
(a)
(b)Finally, it should be stressed that solving problem (1) using linear vector field functional F leads to similar damping behavior of the waves and similar decay results, either for the choice of the type of the function gi, i = 1, 2 or the positive parameter σ.


## 5. Conclusion

In this paper we investigate a nonlinear Kirchhoff viscoelastic plate. Under suitable assumptions on relaxation function, we establish a more general decay result of energy, by introducing suitable energy and perturbed Lyapunov functionals. The decay results established here are optimal exponential and polynomial rates for 1 ≤p < 2 when G(t) = tp, which improved the previous known results for 1≤p<3/2. Under this level of generality, the decay rates we get are optimal, and our results improve the stability results in previous works. At last, we give some numerical illustrations and related comparisons.



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