# General Decay of a Nonlinear Viscoelastic Wave Equation with Balakrishnân-Taylor Damping and a Delay Involving Variable Exponents

Authors: Jiabin Zuo; Abita Rahmoune; Yanjiao Li
Journal: Journal of Function Spaces
 (2022)

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## Abstract

This paper was aimed at investigating the stability of the following viscoelastic problem with Balakrishnân-Taylor damping and variable-exponent nonlinear time delay termutt−M∇u22Δu+αt∫0tgt−sΔusds+μ1utp.−2ut+μ2utt−τp.−2utt−τ=0inΩ×ℝ+, where Ω is a bounded domain of ℝn, p.:Ω¯⟶ℝ is a measurable function, g>0 is a memory kernel that decays exponentially, α≥0 is the potential, and M∇u22=a+b∇ut22+σ∫Ω∇u∇utdx for some constants a>0, b≥0, and σ>0. Under some assumptions on the relaxation function, we use some suitable Lyapunov functionals to derive the general decay estimate for the energy. The problem considered is novel and meaningful because of the presence of the flutter panel equation and the spillover problem including memory and variable-exponent time delay control. Our result generalizes and improves previous conclusion in the literature.


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## Body

## 1. Introduction

In recent years, much attention has been paid to the study systems with variable exponents of nonlinearities which are models of hyperbolic, parabolic, and elliptic equations. These models may be nonlinear over the gradient of unknown solutions and have nonlinear variable exponents. Researches of these systems usually use the imbedding of Lebesgue and Sobolev spaces with variable exponents (see, e.g., [1, 2]). Or see [3–14] and the references therein for more details of relevant problems.In this paper, we concentrate on the asymptotic behavior of weak solutions for the following weakly damped viscoelastic wave equation with Balakrishnân-Taylor damping and variable-exponent nonlinear time delay term(1)utt−M∇u22Δu+αt∫0tgt−sΔusds+μ1utpx−2ut+μ2utt−τpx−2utt−τ=0,inΩ×0,∞,ux,0=u0x,utx,0=u1x,inΩ,utx,t=j0x,t−τ,inΩ×0,τ,ux,t=0,on∂Ω×0,∞,where u:Ω¯×0,∞⟶ℝ is unknown function, μ1≥0,μ2 is a real number, τ>0 is the time delay, g>0 is a memory kernel, and α≥0 is the potential.Much attention has been paid to the simulation of phenomena such as the vibration of elastic strings and elastic plates, wheng=0, and μ1=μ2=0; equation (1)) degrades into the Kirchhoff’s original equation
(2)ρh∂2u∂t2=p0+Eh2L∫0L∂u∂x2dx∂2u∂x2+f,0≤x≤L,t≥0,which was first introduced to study the oscillations of stretched strings and plates in [15]. In addition, equation (2) is also said to be the wave equation of Kirchhoff type, where the unknown function u=ux,t represents lateral deflection and E,ρ,h,L,p0, and f, respectively, denote Young’s modulus, mass density, cross-section area, length, initial axial tension, and external force. The Kirchhoff equation has been investigated in a lot of articles due to its abundant physical background. At the present paper, we try to mention some considerable efforts on this topic.There are many important results, such as the local solutions in time, well-posedness, and solvability; for the Kirchhoff type, equation (2) in general dimensions and domains has been obtained in lots of articles (see, e.g., [16–24] and the references therein).Whenp>1 identically equals to a constant, problem (1) with the Balakrishnân-Taylor damping term σ>0 is related to the flutter panel equation and the spillover problem involving time delay term. Balakrishnân and Taylor in [25] and Bass and Zes in [26] introduced Balakrishnân-Taylor damping, which arises from a wind tunnel experiment at supersonic speeds (see, e.g., [22, 27–32]).On damping terms, we point out several excellent works: Lian and Xu in [33] studied a class of nonlinear wave equations with weak and strong damping terms, and they established the existence of weak solutions and related blow-up results under three different initial energy levels and different conditions. Yang et al. [34] investigated the exponential stability of a system with locally distributed damping. Lian et al. [35] were interested in a fourth-order wave equation with strong and weak damping terms; they obtained the local solution, the global existence, asymptotic behavior, and blow-up of solutions under some condition.Time delays are common phenomena in many physical, chemical, biological, thermal, and so on (see [36–38] for more details). Several authors have investigated existence and stability of the solutions to the viscoelastic wave equation involving delay term under some appropriate conditions on μ1,μ2, and g (see, e.g., [39]). For other related problems, one can also refer to [40–44]. The terminology variable exponents mean that p. is a measurable function and not a constant. This term μ1utp.−2ut+μ2utt−τp.−2utt−τ is a generalization of μ1ut+μ2utt−τ, which corresponds to p.>1. In fact, (1) is also an extension of the second-order viscoelastic wave equation under variable growth conditions
(3)utt−M∇u22Δu+αt∫0tgt−sΔusds+μ1ut+μ2utt−τ=0inΩ×ℝ+,which is obtained when considering μ1utp.−2ut+μ2utt−τp.−2utt−τ. Equation (3) is a well-known electrorheological fluid model that appears in fluid dynamic treatment (see in [45]). However, the researches related to the viscoelastic wave equation possessing delay terms, Balakrishnân-Taylor damping, and variable growth conditions are not sufficient, and the results about these equations are relatively rare (see [46]). In particular, in [40], the authors considered this class of equations under some suitable assumptions; they use suitable Lyapunov functionals to derive general energy decay results, and one see similar work in [44]. Mingione and Rădulescu [47] were concerned with the regularity theory of elliptic variational problems under nonstandard growth conditions.This paper devotes to generalize some previous results. In particular, in this case, we will use the relaxation function, the specified initial data, and a special Lyapunov functional, which depends on the behavior of the relation function and is not necessary to decay in some polynomial or exponential form, to get a general decay estimate of the energy.In addition to the introduction, this paper is divided into two parts. In Section2, we review some basic definitions about Lebesgue and Sobolev spaces with variable exponentials and give some related properties. At the end of this section, we present our main results. In Section 3, we prove our results, showing that a solution of (1) possesses a general decay with small initial values u0,u1.


## 2. Functional Setting and Main Results

In this section, we will give some preliminaries and our main results.Without loss of generality, hereinafter, we supposeΩ⊆ℝn (n≥1) is a bounded domain with smooth boundary Γ. Moreover, let p:Ω¯⟶1,+∞ be a measurable function and denote
(4)p−≔essinfx∈Ωpx,p+≔esssupx∈Ωpx.As in [1, 48, 49], we define the following variable-exponent Lebesgue spaces and Sobolev spaces. The first one is the variable-exponent space Lp.Ω:(5)Lp.Ω=ψ:Ω⟶ℝmeasurableϱp.,Ωψ≔∫Ωψxpxdx<+∞,and it is obvious a Banach space with the following Luxemburg norm
(6)ψp.,Ω≔infν>0∫Ωuxνpxdx≤1.Actually, in many respects, variable-exponent Lebesgue spaces are very similar to classical Lebesgue spaces (see [49]). In particular, from the above definition of the norm, we can directly get the following results:
(7)minup.p−,up.p+≤ϱp.,Ωu≤maxup.p−,up.p+.For any measurable functionp:Ω¯⟶p−,p+⊂2,∞, where p± are constants, we define the second space and the variable-exponent Lebesgue space
(8)Lp.Ω=ϕ:Ω⟶ℝ:ϕismeasurablefunctiononΩ,∫Ωϕxpxdx<∞,which is a Banach space with the following Luxemburg norm:
(9)up.=infν>0,∫Ωuνpxdx≤1.We also assume thatp satisfies the following Zhikov-Fan condition for the local uniform continuity: there exist a constant M>0 such that for all points x,y in Ω with x−y<1/2, we have the inequality
(10)px−py≤Mlogx−y.In addition,.q and .H1Ω denote the usual LqΩ norm and H1Ω norm.In order to obtain the main results, we give the following lemma firstly.Lemma 1 (see [1]).
(1)
If(11)2≤p−≔essinfx∈Ωpx≤px≤p+≔esssupx∈Ωpx<∞,then
(12)minup.p−,up.p+≤∫Ωu.pxdx≤maxup.p−,up.p+for any u∈Lp.Ω(2)
Assume thatm,n,p:Ω¯⟶1,+∞ are measurable functions satisfying
(13)1m.=1p.+1n.
Then, for all functionsu∈Lp.Ω and v∈Ln.Ω, we have uv∈Lm.Ω with
(14)uvm.≤Cup.vn..Lemma 2.
Suppose thatp:Ω⟶p−,p+⊂1,+∞ is a measurable function satisfying
(15)esssupx∈Ωpx<p∗≤2nn−2withp∗=npxesssupx∈Ωn−px.Then, the embeddingH01Ω=W01,2Ω↪Lp.Ω is continuous and compact, and there is a constant c∗=c∗Ω,p± such that
(16)ϕp.≤c∗∇ϕ2forϕ∈H01Ω.We assume that the relaxation functiong and the potential α satisfy the following assumptions:Hypothesisg,α: g,α:ℝ+⟶ℝ+ are nonincreasing differentiable functions such that
(17)gs≥0,l0=∫0∞gsds<∞,αt>0,a−αt∫0tgsds≥l>0Hypothesisξ: there exist a positive differentiable functions ξ satisfying
(18)g′t≤−ξtgt,fort≥0,limt⟶∞−α′tξtαt=0Hypothesisp.: the function p. satisfies
(19)p−≥2,ifn=1,2,2<p−≤px≤p+<n+2n−2ifn≥3Hypothesisμ1 and μ2: the constants μ1 and μ2 satisfy
(20)μ2<p−μ1Calculatingd/dtαtg∘ut with respect to t, it shows that
(21)αt∫0tgt−s∫Ωusdsuttdx=−αt2gtut22−ddtαt2g∘ut−αt2ut22∫0tgsds+αt2g′∘ut+α′t2g∘ut−α′t2ut22∫0tgsds,where
(22)g∘ut=∫0tgt−sut−us22ds.As in [38, 43], we present a new time-dependent variable to deal with the time delay term:
(23)zx,ρ,t=utx,t−τρ,x∈Ω,ρ∈0,1,t>0.Consequently, we have(24)τztx,ρ,t+zρx,ρ,t=0,inΩ×0,1×0,∞.Therefore, problem (1) can be transformed into
(25)utt−M∇u22Δu+αt∫0tgt−sΔusds+μ1utpx−2ut+μ2z1,tpx−2z1,t=0,inΩ×ℝ+,τztρ,t+zρρ,t=0,in0,1×0,∞,z0,t=ut,in0,+∞,zρ,0=j0−ρτ+1,in0,1,ux,0=u0x,utx,0=u1x,x∈Ω.By the standard methods as in Section3 of [50], we can easily prove the well-posedness of problem (1) presented as follows.Theorem 3.
Let (17)–(20) be in force and u0,u1∈H01Ω×L2Ω, j0∈L2Ω×0,1. Then, problem (1) possesses a unique local solution u such that
(26)u∈C0,T;H01Ω∩C10,T;L2Ω,ut∈C0,T;H01Ω∩L20,T×Ω.


## 3. Main Asymptotic Theorem

Next, we will give the proof of Theorem4.The functionalE of problem (25) is as follows:
(27)Et=12utt22+12a−αt∫0tgsds∇u22+b4∇u24+ξ∫Ω1px∫t−τteλs−tutx,spxdsdx+12αtg∘∇ut,where ξ and λ are positive constants and they satisfy
(28)μ1p−−μ2>ξ>μ2p+p+−1p−,λ<1τlnμ2p+p+−1ξp−.The most important key to solve problem (1) is to obtain a result that concerns the asymptotic stability of solutions.The main result is as follows.Theorem 4.
Suppose (17)–(20) and (28) hold. Then, there exists positive constants C0, C, and t1>0 such that
(29)Et≤C0e−C∫t1tv,fort≥t1.To prove this theorem, the following technical lemmas are necessary.Lemma 5.
Ifu is a solution of problem (25). Then,
(30)E′t≤−σ12ddt∇u222+12αtg′∘∇ut−12α′t∇u22∫0tgsds−12αtgt∇u22+12α′tg∘∇ut−μ1−ξp−−μ2p−∫Ωutpxdx−ξp+e−λτ−μ2p+−1p−∫Ωz1,tpxdx−λξ∫Ω1px∫t−τteλs−tutx,spxdsdx.Proof.
Using the same idea as in [50], multiply the first equation in (25) by ut and then integrate in Ω. Similarly, multiply the second equation in (25) by ξze−λτρ and integrate in 0,1×Ω. Summarizing the above, we can obtain
(31)E′t=−σ12ddt∇u222+αt2g′∘∇ut−12α′t∇u22∫0tgsds−αt2gt∇u22+α′t2g∘∇ut−μ1∫Ωutpxdx−ξ∫Ω1pxe−λτutx,t−τpxdx−μ2∫Ωz1,tpx−2z1,tutdx+ξ∫Ω‍1pxutx,tpxdx−λξ∫Ω1px∫t−τteλs−tutx,spxdsdx.
Byz1,t=utt−τ and the Young inequality, we get
(32)−μ2∫Ωz1,tpx−2z1,tutdx≤μ2p+−1p−∫Ωz1,tpxdx+μ2p−∫Ωutpxdx.
From (23), we have
(33)−ξ∫Ω1pxe−λτutx,t−τpxdx≤−ξp+e−λτ∫Ωz1,tpxdx.
Comparing (31) and (32), we obtain
(34)E′t≤−σ12ddt∇u222+αt2g′∘∇ut−12α′t∇u22∫0tgsds−αt2gt∇u22+α′t2g∘∇ut−μ1−ξp−−μ2p−∫Ωutpxdx−ξp+e−λτ−μ2p+−1p−∫Ωz1,tpxdx−λξ∫Ω1px∫t−τteλs−tutx,spxdsdx.
Setting(35)c0=μ1−ξp−−μ2p−,c1=ξp+e−λτ−μ2p+−1p−,by condition (28), we derived the desired inequality (30).Remark 6.
If(36)−12α′t∇ut22∫0tgsds≥0holds, Et may not be nonincreasing.Lemma 7.
Assume thatu be a solution of problem (25). Then,
(37)∇u22≤2E0lel0/lα0,t≥0,where l0 and l as in (17).Proof.
From (27) and (30), we have
(38)E′t≤−12α′t∇u22∫0tgsds≤−12l0α′t∇u22≤−l0lα′tEt.
Integrating the above inequality in0,t, we get
(39)Et≤E0e−l0/lαt+l0/lα0≤E0el0/lα0.From (27), we see that
(40)∇u22≤2lEt.Combining it with (39), it gives (37).
Now, we give a modified functional:(41)Lt=NEt+ε1αtφt+ε2αtψt,(42)φt=∫Ωututtdx+σ4∇u24,(43)ψt=−∫Ωutt∫0tgt−sut−usdsdx,where ε1, ε2, and N are positive constants. In fact, L is equivalent to E by the following lemma.Lemma 8.
There existsC1,C2>0 such that
(44)C1Et≤Lt≤C2Et,t≥0.Proof.
By the Poincaré theorem and Young inequality, we have the following results through integrating by parts:(45)Lt−NEt=ε1αt∫Ωututtdx+ε1αtσ4∇u24+ε2αtψt≤ε1αt∫Ωututtdx+ε1σ4αt∇u24+ε212αtut22+ε212αtc∗2a−lg∘∇ut≤ε1α02c∗2∇u22+ε1α02ut22+ε1σα04∇u24+ε212α0ut22+ε212α0c∗2a−lg∘∇ut≤Cε1+ε2Et,where c∗ as in Lemma 1, taking C1=N−Cε1+ε2 and C2=N+Cε1+ε2, provided ε1 and ε2 are sufficiently small, and the proof is completed.Lemma 9.
There existscε,Cε>0 fulfilling
(46)φ′t≤ut22−l2∇u22−b∇u24+αta2lg∘∇ut+cε∫Ωutpxdx+∫Ωz1,tpxdx+Cε∫Ωupxdx.Proof.
By the first equation of (25), we differentiate (42), and then we have
(47)φ′t=ut22+∫Ωv+σ∇u22∫Ω∇u∇utdx=ut22−a∇u22−b∇u24+αt∫Ω∫0tgt−s∇usds∇utdx−μ1∫Ωutpx−2utudx−μ2∫Ωz1,tpx−2z1,tudx=ut22−a∇u22−bv24+I1+I2+I3.
By the Hölder inequality, Sobolev-Poincaré inequalities, and (17), we estimate the second part of the right-hand side in (47).
(48)I1=αt∫Ω∫0tgt−s∇usds∇utdx≤αt∫Ω∇u2dx1/2∫Ω∫0t‍gt−s∇usds2dx1/2≤αt∫Ω∇u2dx1/2∫Ω∫0tgsds∫0tgt−s∇us2dsdx1/2≤αt∫Ω∇u2dx∫0tgsds1/2∫Ω∫0tgt−s∇us2dsdx1/2≤αt2∫Ω∇u2dx∫0tgsds+αt2∫Ω∫0tgt−s∇us2dsdx≤αt2∫Ω∇u2dx∫0tgsds+αt2∫Ω∫0tgt−s∇us−∇ut+∇ut2dsdx.
For everyη>0, using the Young inequality and (17), we deduce
(49)αt2∫Ω∫0tgt−s∇us−∇ut+∇ut2dsdx≤αt2∫Ω∫0tgt−s∇us−∇ut2+2∇us−∇ut∇u+∇u2dsdx≤αt2∫Ω∫0tgt−s∇us−∇ut2dsdx+αt2∫Ω∫0tgt−s∇u2dsdx+αt∫Ω∫0tgt−s∇us−∇ut∇udsdx≤αt2g∘∇ut+αt2∫0tgsds∫Ω∇u2dx+ηαt2∫0tgsds∫Ω∇u2dx+αt2ηg∘∇ut≤αt21+η∫0tgsds∫Ω∇u2dx+αt21+1ηg∘∇ut≤1+ηa−l2∫Ω∇u2dx+αt21+1ηg∘∇ut.
Summarizing the above estimates, (48) and (49), we obtain
(50)αt∫Ω∫0tgt−s∇usds∇udx≤a−l2∫Ω∇u2dx+a−l21+η∫Ω∇u2dx+αt21+1ηg∘∇ut=2+ηa−l2∇u2+αt21+1ηg∘∇ut.
Settingη=l/a−l, it is easy to obtain
(51)I1≤αt∫Ω∫0tgt−s∇usds∇udx≤a−l2∇u22+a2lαtg∘∇ut,and by means of the Young inequality, we have
(52)I2≤cε∫Ωutpxdx+εmaxμ1p−,μ1p+∫Ωupxdx≔cε∫Ωutpxdx+εc2∫Ωupxdx,(53)I3≤cε∫Ωz1,tpxdx+εmaxμ1p−,μ1p+∫Ωupxdx≔cε∫Ωz1,tpxdx+εc3∫Ωupxdx.
Substituting (51)–(53) into (47), we deduce
(54)φ′t≤ut22−l2∇u22+Cε∫Ωupxdx−b∇u24+a2lαtg∘∇ut+cε∫Ωutpxdx+∫Ωz1,tpxdx,set Cε=εc2+c3>0, for ε sufficiently small.Lemma 10.
There exists positive constantsδ and cδ satisfying
(55)ψ′t≤−∫0tgsds−δut22+δa+2a−l2αt∇u22+δb∇u24+δ2σE0lel0/lα012ddt∇u222+Cδ+2δ+14δa−lαtg∘∇ut+cδ∫Ωutpxdx+∫Ωz1,tpxdx−g0c∗24δg′∘∇ut.Proof.
Similar to Lemma,9 by the first equation (25), we differentiate (43), and it yields
(56)ψ′t=−∫Ωutt∫0tgt−sut−usdsdx−∫Ωut∫0tg′t−sut−usdsdx−∫0tgsdsut22=a+b∇u22∫Ω∇u∫0tgt−s∇ut−∇usdsdx+σ∫Ω∇u∇utdx∫Ω∇u∫0tgt−s∇ut−∇usdsdx−αt∫Ω∫0tgt−s∇usds∫0tgt−s∇ut−∇usdsdx+μ1∫Ωutpx−2ut∫0tgt−sut−usdsdx+μ2∫Ωz1,tpx−2z1,t∫0tgt−sut−usdsdx−∫Ωut∫0tg′t−sut−usdsdx−∫0tgsdsut22=∑i=16Ii−∫0t‍gsdsut22.
By the Hölder inequality, Sobolev-Poincaré inequalities, and (17), we estimate the second part of the right-hand side in (56).
(57)I1≤a+b∇u22δ∇u22+l04δg∘∇ut≤δa∇u22+δb∇u24+al04δ+bl0E02δlel0/lα0g∘∇ut,(58)I2≤δσ∫Ω∇u∇utdx2∇u22+σl04δg∘∇ut≤δ2σE0lel0/lα012ddt∇u222+σl04δg∘∇ut,(59)I3≤δαt∫Ω∫0tgt−s∇ut−∇us+∇utds2dx+14δαt∫Ω∫0tgt−s∇ut−∇usds2dx≤2δl02αt∇u22+2δ+14δl0αtg∘∇ut,(60)I4≤cδ∫Ωutpxdx+δmaxμ1p−,μ1p+∫Ω∫0tgt−sut−usdspxdx≤cδ∫Ωutpxdx+δmaxμ1p−,μ1p+maxl0p+−1,l0p−−1maxc∗p+,c∗p−∫0tgt−s∇ut−∇us2pxds≤cδ∫Ωutpxdx+δmaxμ1p−,μ1p+maxc∗p+,c∗p−max2E0lel0/lα0p+−2/2,2E0lel0/lα0p−−2/2g∘∇ut≔cδ∫Ωutpxdx+δc4g∘∇ut.
Similarly,(61)I5≤cδ∫Ωz1,tpxdx+δc5g∘∇ut,I6≤δut22−g0c∗24δg′∘∇ut.
Comparing these above estimates (57)–(61), we have
(62)ψ′t≤−∫0tgsds−δut22+δa+2l02αt∇u22+δb∇u24+δ2σE0lel0/lα012ddt∇u222+Cδ+2δ+14δl0αtg∘∇ut+cδ∫Ωutpxdx+∫Ωz1,tpxdx−g0c∗24δg′∘∇ut,where Cδ=al0/4δ+bl0E0/2δlel0/lα0+σl0/4δ+δc4+c5.Lemma 11.
There exists positive constantsC3,C4, and t0 satisfying
(63)L′t≤−C3αtEt+C4αtg∘∇ut,t>t0.Proof.
Sinceg>0 and is continuous, then for any t≥t0>0, we get
(64)∫0tgsds≥∫0t0gsds=g0>0.
Differentiate (41), and using Lemmas 9 and 10, we get
(65)L′t=NE′t+ε1α′tφt+ε1αtφ′t+ε2α′tψt+ε2αtψ′t≤−αtε2g0−δ−ε1ut22−αtε1Cε−ε2δa+2l02αt∇u22−αtbε1−ε2δut24−αtσ−ε2δσE0lel0/lα012ddt∇u222+αtε1αt4+ε2Cδ+ε22δ+14δl0αtg∘∇ut+αtN2−ε2g0c∗24δg′∘∇ut−αtc0α0−ε1cε−ε2cδ∫Ωutpxdx−αtc1α0−ε1cε−ε2cδ∫Ωz1,tpxdx−Nα′t2∫0tgsds∇u22+ε1α′t∫Ωuutdx+ε2α′t∫Ωut∫0tgt−sut−usdsdx.
Indeed,(66)α′t∫Ωuutdx+α′t∫Ωut∫0tgt−sut−usdsdx≤−α′tc∗22∇u22−α′tut22−α′tc∗22∫0tgsdsg∘∇ut.
Thus,(67)L′t≤−αtε2g0−δ−ε1+α′tαtut22−αtε1Cε−ε2δa+2l02α0+Nα′t2αt∫0t‍gsds+c∗2α′t2αt∇u22−αtbε1−ε2δ∇u24−αtσε1−ε2δσE0lel0/lα012ddt∇u222+αtε1αt4+ε2Cδ+ε22δ+14δl0αt−c∗2α′t2αt∫0tgsdsg∘∇ut+αtN2−ε2g0c∗24δg′∘∇ut−αtc0α0−ε1cε−ε2cδ∫Ωutpxdx−αtc1α0−ε1cε−ε2cδ∫Ωz1,tpxdx.
Fixδ>0 such that
(68)g0−δ>12g0,δCεa+2l02α0<14g0,and take ε1 and ε2 small enough to satisfy
(69)g04ε2<ε1<ε2g02,c5=ε2g0−δ−ε1>0,c6=ε1Cε−ε2δa+2l02α0>0.
Selectε1 and ε2 small enough to make (44) and (67) hold, and moreover
(70)bε1−ε2δ>0,σε1−ε2δσE0lel0/lα0>0,N2−ε2g0c∗24δ>0c0α0−ε1cε−ε2cδ>0,c1α0−ε1cε−ε2cδ>0.
Hence, for a generic positive constantc, (67) is equal to the following results:
(71)L′t≤−αtc+α′tαtut22−αtc+α′t2αt∫0t‍gsds+c∗22∇u22+αtc−c∗2g0α′t2αtg∘∇ut,∀t≥t0.
Noticing thatlimt⟶∞−α′t/ξtαt=0, so choose t1>t0, we see
(72)L′t≤−αtcut22+C∇u22+cg∘∇ut≤−C3αtEt+C4αtg∘∇ut,∀t≥t1,where C3 and C4 are positive constants.Now, we are in the position to prove Theorem4.Proof of Theorem 4.
According to Lemma5, Lemma 11, and (17), we have
(73)ζtL′t≤−C3αtζtEt+C4αtζtg∘∇ut≤−C3αtζtEt−C4αtg′∘∇ut≤−C3αtζtEt−C42E′t+α′t∫0tgsds∇u22.
Sinceζt is nonincreasing, by assumption (17) and the definition of Et, we get
(74)l2∇u22≤Et,ddtζtLt+2C4Et≤−C3αtζtEt−C4α′t∫0sgsds∇u22,which leads to
(75)ddtζtFt+2C4Et≤−C3αtζtEt−C4α′t∫0tgsds∇u22≤−C3αtζtEt−2C4Etlα′t∫0tgsds≤−αtζtC3+2C4l0α′tlαtζtEt.
Sincelimt⟶∞−α′t/αtζt=0, we can choose t1≥t0 such that C3+2C4l0α′t/lαtζt>0 for t≥t1. Hence, if we let
(76)Lt=ζtLt+2C4Et,then it is obvious that Lt is equivalent to Et and satisfies
(77)L′t≤−kζtαtLtfort≥t1.
Consequently, to integrate (77) over t1,t, it yields
(78)Lt≤Lt1e−C∫t1t‍ζsαsdst≥t0.Thus, the desired result yields from the equivalence relations of Lt,Lt, and Et.



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*Source: 9801331-2022-05-11.xml*