# A Stability Result for a Viscoelastic Wave Equation in the Presence of Finite and Infinite Memories

Authors: Adel M. Al-Mahdi; Mohammad M. Al-Gharabli; Mohammad Kafini; Shadi Al-Omari
Journal: Journal of Mathematics
 (2022)

---

## Abstract

In this study, we consider a viscoelastic wave equation in the presence of finite and infinite memories. Under suitable conditions on the variable coefficients and for a wide class of relaxation functions, we establish an explicit and general decay result. Moreover, we use a weaker boundedness condition, on the initial data than the one used in the literature. The proof of our decay result is based on the multiplier method together with some convexity arguments. This study generalizes and improves previous literature outcomes.


---

## Body

## 1. Introduction

Let us consider ann-dimensional body occupies with a bounded open set Ω⊆ℝnn≥1 with smooth boundary ∂Ω. Let x⟶ux,t be the position of the material particle x at time t so that ∀x∈Ωand∀t>0; the corresponding motion equation is(1)utt−Δu+∫0tg1t−sdiva1x∇usds+∫0+∞g2sdiva2x∇ut−sds=0,ux,t=0,ux,−t=u0x,t,utx,0=u1x.The functionsg1 and g2 are called the relaxation (kernels) functions, and they are positive nonincreasing and defined on ℝ+. The functions a1 and a2 are essentially bounded nonnegative defined on Ω. Here, u0 and u1 are the history and initial data. This model of materials consist of an elastic part (without memory) and a viscoelastic part, where the dissipation given by the memory is effective.In this study, we are concerned with the above viscoelastic wave problem (1) and mainly interested in the asymptotic behavior of the solution u when t tends to infinity. In fact, we prove that the solution of the corresponding viscoelastic model decays to zero and no matter how small is the viscoelastic part of the material. Note that the above model is dissipative, and the dissipation is given by the memory term only and the memory is effective only in a part of the body. For materials with memory, the stress depends not only on the present values but also on the entire temporal history of the motion. Therefore, we have to prescribe the history of u before 0. Here, we assume that ux,−t=u0x,t,∀x∈Ω,∀t>0. Let us mention some other papers related to the problem we address. We start our literature review with the pioneer work of Dafermos [1], in 1970, where the author discussed a certain one-dimensional viscoelastic problem, established some existence results, and then proved that, for smooth monotone decreasing relaxation functions, the solutions go to zero as t goes to infinity. However, no rate of decay has been specified. In [2], a similar result, under a convexity condition on the kernel, has been established. After that, a great deal of attention has been devoted to the study of viscoelastic problems and many existence and long-time behavior results have been established. For example, Hrusa [3] considered a one-dimensional nonlinear viscoelastic equation of the form(2)utt−cuxx+∫0tmt−sψuxx,sxds=fx,t,and proved several global existence results for large data. He also proved an exponential decay for strong solutions when ms=e−s and ψ satisfies certain conditions. Dassios and Zafiropoulos [4] studied a viscoelastic problem in ℝ3 and proved a polynomial decay results for exponentially decaying kernels. After that, a very important contribution by Rivera was introduced. In 1994, Rivera [5] considered equations for linear isotropic homogeneous viscoelastic solids of integral type which occupy a bounded domain or the whole space ℝn in the bounded domain case, and for exponentially decaying memory kernel and regular solutions, he showed that the sum of the first and the second energy decays exponentially. For the whole-space case and for exponentially decaying memory kernels, he showed that the rate of decay of energy is of algebraic type and depends on the regularity of the solution. This result was later generalized to a situation, where the kernel is decaying algebraically but not exponentially by Cabanillas and Rivera [6]. In [6], the authors considered the case of bounded domains as well as the case when the material is occupying the entire space and showed that the decay of solutions is also algebraic, at a rate which can be determined by the rate of decay of the relaxation function. This latter result was later improved by Baretto et al. [7], where equations related to linear viscoelastic plates were treated. Precisely, they showed that the solution energy decays at the same rate of the relaxation function.Decay results for arbitrary growth of the damping term have been considered for the first time in the work of Lasiecka and Tataru [8]. They showed that the energy decays as fast as the solution of an associated differential equation whose coefficients depend on the damping term. Following the method of Lasiecka and Tataru [8], Liu and Zuazua [9] established decay rates for a nonlinear globally distributed damping for the wave equation with no growth assumptions at the origin. The proof of their result is based on constructing a convex function which captures the nonlinearity of the feedback at the origin and combines it, using Jensen’s and Young’s inequalities, with the use of equivalent energy which satisfies a differential equation, which however is not a priori dissipative. Alabau-Boussouira [10] used weighted integral inequalities and some convexity arguments to establish a semiexplicit formula which leads to decay rates of the energy in terms of the behavior of the nonlinear feedback close to the origin, for which the optimal exponential and polynomial decay rate estimates are only special cases. Alabau-Boussouira [11] presented a general approach based on convexity arguments to establish sharp optimal or quasi-optimal upper energy decay rates for finite and infinite dimensional vibrating damped systems.Inspired by the experience with frictional damping initiated in the work of Lasiecka and Tataru [8], another step forward was done by Alabau-Boussouira and Cannaras [12] when they considered relaxation functions satisfying(3)g′t≤−Hgt.They established an explicit rate of decay under some constraints imposed on the functionH.For partially viscoelastic material, Rivera and Salvatierra [13] showed that the energy decays exponentially and provided the relaxation function decays in a similar fashion and the dissipation is acting on a part of the domain near to the boundary. See also, in this direction, the work of Rivera and Oquendo [14]. The uniform decay of solutions for the viscoelastic wave equation,(4)utt−k0Δu+∫0tdivaxgt−τ∇uτdτ+bxhut+fu=0,was investigated by Cavalcanti and Oquendo [15] where they considered the condition ax+bx≥δ>0. They established exponential and polynomial stability results based on some conditions on g and the linearity of the function h. After that, Guesmia and Messaoudi [16] extended the work of [15], and they established a general decay result for (1) under the same conditions on a1 and a2 used in [15] and for some other conditions for the relaxation functions g1 and g2, from which the usual exponential and polynomial decay rates are only special cases. More precisely, they used the conditions g1′t≤−ξtg1t,∀t≥0, and(5)∫0+∞g2sG−1−g2′sds+sups∈ℝ+g2sG−1−g2′s<+∞,such that(6)G0=G′0=0,limt⟶+∞G′t=+∞,where G:ℝ+⟶ℝ+ is an increasing strictly convex function.In the present work, we consider (1) and under the same conditions on a1 and a2 used in [15], and for a large class of the relaxation functions, we prove that problem (1) is stable. Under this class of relaxation functions gi at infinity, we establish a relation between the decay rate of the solution and the growth of gi at infinity. In fact, we extend the works of [15, 16]. More precisely, we assume that the relaxations functions g1 and g2 satisfy(7)gi′t≤−ξitHigit,∀t≥0.In addition, we use a weaker boundedness condition on the history data used in earlier papers such as [17–20].The paper is organized as follows. In Section2, we present some material needed for our work. We establish some essential lemmas in Section 3. Section 4 contains the statement and the proof of our main result. We give some examples in Sections 5 to illustrate our energy decay result. Finally, we introduce conclusions and comparison study in Section 6.


## 2. Preliminaries

In this section, we present some material needed in the proof of our main result. Through this study, we usec to denote a positive generic constant. Now, we start with the following assumptions:(A1)gi:ℝ+⟶ℝ+ are differentiable nonincreasing functions such that, for any constant β0>0 and i=1,2,(8)−β0gis≤gi′s,gi0>0,1−a1∞∫0+∞g1sds−a2∞∫0+∞g2sds=ℓ>0.and there existC1 functions Hi:ℝ+⟶ℝ+ which are linear or strictly increasing and strictly convex C2 functions on 0,r for some r>0 with Hi0=Hi′0=0, lims⟶+∞Hi′s=+∞, s↦sHi′s, and s↦sHi′−1s are convex on 0,r. Moreover, there exist positive nonincreasing differentiable functions ξi:ℝ+⟶ℝ+, such that(9)gi′t≤−ξitHigit,∀t≥0.(A2)ai:Ω¯⟶ℝ+ are C1Ω¯ functions such that, for positive constants δ and a0 and for Γ1,Γ2⊂∂Ω with meas Γi>0,i=1,2,infx∈Ω¯a1x+a2x≥δ and(10)ai=0,orinfΓiaix≥2a0,i=1,2.(A3) We assume that(11)∫0+∞g2s∇u0s2ds<+∞.Remark 1.
The class of relaxation functions satisfyingA1 in the present study is larger than the ones satisfying (5) and (6) used in [16]. In fact, the boundedness of the sup in (5), used in [16], can be interpreted as the inequality in A1 in the present paper (with ξ=1). Conditions (5) and (6) used in [16] ask also the boundedness of the integral. Moreover, the classical of relaxation functions g that we consider in our example gt=e−t+1β,β>0, satisfy both conditions A1 in our paper and the one in [16]. So, it is better to consider A1 used in the present study than the one used in [16].Remark 2.
HypothesisA3 is needed for proving the existence and stability results. For the stability, if A3 holds, then h0 which is defined in (25) is well defined, so inequality (53) makes sense. Moreover, Hypothesis A3 is weaker than the one used in [16, 18]; that is, there exists a positive constant M such that(12)∇u0s2≤M,and our decay rate depends on the size of ∇u0.Remark 3.
Ifai≠0,i=1,2, there exist neighborhoods wi of Γi,i=1,2, such that(13)infΩ∩wi¯aix≥a0>0,i=1,2.
As in [15, 16], let d=mina0,δ and let αi∈C1Ω¯,i=1,2, be such that(14)0≤αix≤aix,αix=0,ifaix≤d4,αix=aix,ifaix≥d2.Lemma 1.
The functionsαi,i=1,2, are not identically zero and satisfy(15)α1x+α2x≥d2.Proof.
(1)
Forx∈Ω∩wi, we have aix≥a0≥d, which implies, by (14), that αix=aix≥d. Thus, αi is not identically zero.(2)
Ifa1x≥d/2, then α1x=a1x. Consequently, α1x+α2x≥a1x≥d/2. If a1x<d/2, then a2x>d/2 which implies, by (14), α2x=a2x>d/2. Consequently, α1x+α2x>d/2. This completes the proof.
The existence and uniqueness of the solution of problem (1) is given by the following theorem.Theorem 1.
Ifu0,u1∈H01Ω×L2Ω and hypotheses A1−A3 hold, then problem (1) has a unique solution in the class(16)u∈C00,∞;H01Ω∩C10,∞;L2Ω.The proof of the above theorem can be obtained, making use, for instance, of the Faedo–Galerkin method and considering standard arguments of density (see [15]).We define the “modified” energy functional of the weak solution of problem (1) by(17)Et=12∫Ωut2dx+12∫Ω1−a1x∫0tg1sds−a2x∫0+∞g2sds∇u2dx+12g1∘∇ut+12g2∘∇ut,where(18)g1∘∇ut=∫Ωa1x∫0tg1t−s∇ut−∇us2dsdx,g2∘∇ut=∫Ωa2x∫0+∞g2s∇ut−∇ut−s2dsdx.Lemma 2.
The “modified” energy functional satisfies, along the solution of problem (1), the following:(19)E′t=−12g1t∫Ω∇ut2dx+12g1′∘∇ut+12g2′∘∇ut≤0,where(20)g1′∘∇ut=∫Ωa1x∫0tg1′t−s∇ut−∇us2dsdx,g2′∘∇ut=∫Ωa2x∫0+∞g2′s∇ut−∇ut−s2dsdx.Proof.
By multiplying the first equation in problem (1) by ut and integrating over Ω, using integration by parts, hypotheses A1 and A2 and some manipulations as in [6,21] and others, we obtain (19) for regular solutions. This inequality remains valid for weak solutions by a simple density argument.


## 3. Technical Lemmas

In this section, we present and prove additional lemmas that are pivotal for the principal result.Lemma 3 (see [22]).
Fori=1,2, we have(21)∫01∫0tgit−s∇us−∇ut−sds2dx≤Cαhi∘∇ut,where, for any 0<α<1,(22)Cα=∫0∞gi2sαgis−gi′sds,hit=αgit−gi′t.Furthermore, using the fact that(23)αgi2sαgis−gi′s<gis,and recalling the Lebesgue dominated convergence theorem, one can deduce that(24)αCα=∫0∞αgi2sαgis−gi′sds⟶0asα⟶0.Lemma 4.
Assume thatA1−A3 hold. There exists a positive constant M1 such that(25)∫Ωa2x∫t+∞g2s∇ut−∇ut−s2ds≤M1h0t,where h0t=∫0+∞g2t+s1+∇u0s2ds.Proof.
The proof is identical to the one in [23]. Indeed, we have(26)∫Ωa2x∫t+∞g2s∇ut−∇ut−s2ds≤2a2∞∇ut2∫t+∞g2sds+2a2∞∫t+∞g2s∇ut−s2ds≤2a2∞sups≥0∇us2∫0+∞g2t+sds+2a2∞∫0+∞g2t+s∇u−s2ds≤4a2∞Esℓ∫0+∞g2t+sds+2a2∞∫0+∞g2t+s∇u0s2ds≤4a2∞E0ℓ∫0+∞g2t+sds+2a2∞∫0+∞g2t+s∇u0s2ds≤M1∫0+∞g2t+s1+∇u0s2ds,where M1=max2a2∞,4a2∞E0/ℓ.Lemma 5 (see [16]).
Assume thatA1−A3 hold; then, the functionals,(27)ψ1t=∫Ωuutdx,ψ2t=−∫Ωα1xut∫0tg1t−sut−usdsdx,ψ3t=−∫Ωα2xut∫0+∞g2sut−ut−sdsdx,satisfy, along the solution of (1) and for any ε1,ε2,ε3>0 and ∀t≥0; the following estimates(28)ψ1′t≤∫Ωut2dx−1−ε1−a1∞∫0+∞g1sds−a2∞∫0+∞g2sds∫Ω∇u2dx+cCαε1h1∘∇ut+h2∘∇ut,(29)ψ2′t≤−∫0tg1sds−ε2∫Ωα1xut2dx+ε32∫Ω∇u2dx−cε2g1′∘∇ut+cCαε3h1∘∇ut+h2∘∇ut,and(30)ψ3′t≤−∫0+∞g2sds−ε2∫Ωα2xut2dx+ε32∫Ω∇u2dx−cε2g2′∘∇ut+cCαε3h1∘∇ut+h2∘∇ut.Lemma 6.
Assume thatA1−A3 hold; then, for a suitable choice of N,M,m>0 and for all t≥0, the functional,(31)ℒt≔NEt+Mψ1t+ψ2t+ψ3t,satisfies, for any t>0(32)ℒ′t≤−mEt+14g1∘∇u+g2∘∇ut.Proof.
Let, fort0>0 fixed, g0:=min∫0t0g1sds,∫0+∞g2sds. Then, direct differentiation of ℒ and using (19), (28), (29), and (30) leads to(33)ℒ′t≤N2−cε2g1′∘∇ut+g2′∘∇ut−∫Ωg0−ε2α1+α2−Mut2dx+cCαε3+McCαε1h1∘∇ut+h2∘∇ut−ℓ−ε1M−ε3∫Ω∇u2dx,∀t≥t0.
Recalling thatgi′=αgi−hi, where i=1,2. We obtain(34)ℒ′t≤αN2−cε2g1∘∇u+g2∘∇ut−N2−cε2h1∘∇u+h2∘∇ut−∫Ωg0−ε2α1+α2−Mut2dx×cCαε3+McCαε1h1∘∇ut+h2∘∇ut−ℓ−ε1M−ε3∫Ω∇u2dx,∀t≥t0.
Simplifying the above estimate, we arrive at(35)ℒ′t≤αN2−cε2g1∘∇u+g2∘∇ut−N2−cε2−Cαcε3+Mcε1h1∘∇u+h2∘∇ut−∫Ωg0−ε2α1+α2−Mut2dx−ℓ−ε1M−ε3∫Ω∇u2dx,t≥t0.
By using the fact thatα1+α2≥d/2 and choosing(36)ε1=ℓ2,ε2=g02,M=dg08,ε3<dℓ2dg016,we obtain(37)ℓ−ε1M−ε3≥41−ℓ.
As a consequence of (24), there exists 0<α0<1 such that if α<α0, then(38)αCα<18c/ε3+Mc/ε1.
It is clear that (38) gives(39)CαN2−cε2−Cαcε3+Mcε1<N2.
Choosingα=1/2N<α0, then, for some m>0, we have(40)ℒ′t≤−mEt+14g1∘∇u+g2∘∇ut.
Then, (32) is established. Moreover, one can choose N large enough so that ℒ∼E.Lemma 7.
Assume thatA1−A3 hold. Then, for all t∈ℝ+ and fixed positive constant m0, we have the following estimates:(41)∫0tEsds<m01+∫0th0sds,where h0 is defined in (25).Proof.
The proof of this lemma can be established by following the same arguments in [23, 24].Lemma 8.
IfA1−A3 are satisfied, then we have, for all t>0 and for i=1,2, the following estimates:(42)∫Ωaix∫0tgis∇ut−∇ut−s2dsdx≤1ΛitHi−1Λitμitξit,where Λit=λi/1+∫0th0sds, λi∈0,1, Hi and ξi are introduced in A1, and(43)μit≔−∫Ωaix∫0tgi′s∇ut−∇ut−s2dsdx≤−cE′t.Proof.
To establish (42), we introduce the following functional:(44)χit≔Λit∫Ωaix∫0t∇ut−∇ut−s2dsdx.
Then, using (17) and the fact that E is nonincreasing to get,(45)χit≤2Λit∫Ωaix∫0t∇ut2+∫Ωaix∫0t∇ut−s2dsdx≤4Λitai∞ℓ∫0tEt+Et−sds≤8Λitai∞ℓ∫0tEsds.
Using (41) and the definition of Λit, (45) becomes(46)χit≤8m0λiai∞ℓ1+∫0thsds1+∫0thsds.
Therefore, we can choose0<λi<ℓ/8m0ai∞ such that, for all t>0,(47)χit<1.
Without loss of the generality, for allt>0, we assume that χit>0; otherwise, we get an exponential decay from (32). The use of Jensen’s inequality and using (43) and (47) give(48)μ1t=1Λit∫0tΛit−gi′s∫Ωaix∇ut−∇ut−s2dxds≥1Λit∫0tΛitξisHigis∫Ωaix∇ut−∇ut−s2dxds≥ξitΛit∫0tHiΛitgis∫Ωaix∇ut−∇ut−s2dxds≥ξitΛitHiΛit∫0tgis∫Ωaix∇ut−∇ut−s2dxds=ξitΛiHi¯Λit∫0tgis∫Ωa1x∇ut−∇ut−s2dxds.
Hence, (42) is established.


## 4. Decay Result

In this section, we state and prove our main result and provide some examples to illustrate our decay results. Let us start introducing some functions and then establishing several lemmas needed for the proof of our main result. Now, fori=1,2, let us take(49)ξ=minξi,μ=maxμi,λ=maxλi,G=H1−1+H2−1−1.As in [23], we introduce the following functions:(50)G1t≔∫tr1sG′sds,G5t=G1−1c1∫0tξsds,(51)G2t=tG′t,G3t=tG′−1t,G4t=G3∗t,where G3∗ is the convex conjugate of G3 in the sense of Young (see [25]). One can easily verify that G1 is decreasing function over 0,r, and G2,G3, and G4 are convex and increasing functions on 0,r. Furthermore, we introduce the class S of functions ϑ:ℝ+⟶0,r satisfying for fixed c1,c2>0:(52)ϑ∈C1ℝ+,ϑ≤1,ϑ′≤0,and(53)c2G4cdΛth0t≤c1G2G5sϑs−G2G5tϑt,where h0 is defined in (25) and d>0.Theorem 2.
Assume thatA1−A3 hold; then, there exists a strictly positive constant C such that the solution of (1) satisfies, for all t≥0,(54)Et≤CG5tϑtΛt,where G5 and ϑ are defined in (50) and (52), respectively.Proof.
Using (25), (32), and (42), then, for some positive constant m and any t≥0, we obtain(55)L′t≤−mEt+cΛtG−1Λtμtξt+ch0t.
Without loss of generality, one can assume thatE0>0. For ε0<r, let the functional ℱ be defined by(56)ℱt≔G′ε0EtΛtE0Lt,which satisfies ℱ∼E. By noting that G″≥0, Λ′≤0, and E′≤0, we obtain(57)ℱ′t=ε0qE′tE0G″ε0EtΛtE0Lt+G′ε0EtΛtE0L′t≤−mEtG′ε0EtΛtE0+cΛtG′ε0EtΛtE0G−1Λtμtξt+ch0tG′ε0EtΛtE0.
LetG∗ be the convex conjugate of G in the sense of Young (see [25]); then,(58)G∗s=sG′−1s−GG′−1s,ifs∈0,∞,and G∗ satisfies the following generalized Young inequality:(59)AB≤G∗A+GB,ifA,B∈0,∞.
So, withA=G′ε0EtΛt/E0 and B=G−1Λtμt/ξt and using (57)–(59), we arrive at(60)ℱ′t≤−mEtG′ε0EtΛtE0+cΛtG∗G′ε0EtΛtE0+cμtξt+ch0tG′ε0EtΛtE0≤−mEtG′ε0EtΛtE0+cε0EtE0G′ε0EtΛtE0+cμtξt+ch0tG′ε0EtΛtE0.
So, multiplying (60) by ξt and using (43) and the fact that ε0EtΛt/E0<r give(61)ξtℱ′t≤−mξtEtG′ε0EtΛtE0+cξtε0EtE0G′ε0EtΛtE0+cμtΛt+cξth0tG′ε0EtΛtE0≤−ε0mE0ε0−cξtEtE0G′ε0EtΛtE0−cE′t+cξth0tG′ε0EtΛtE0.
Consequently, recalling the definition ofG2 and choosing ε0 so that k=mE0/ε0−c>0, we obtain, for all t∈ℝ+,(62)ℱ1′t≤−kξtEtE0G′ε0EtΛtE0+cξth0tG′ε0EtΛtE0=−kξtΛtG2EtΛtE0+cξth0tG′ε0EtΛtE0,where ℱ1=ξℱ+cE∼E and satisfies for some α1,α2>0:(63)α1ℱ1t≤Et≤α2ℱ1t.
SinceG2′t=G′t+tG″t and G is strictly increasing and strictly convex on 0,r, we find that G2′t,G2t>0 on 0,r. Using the general Young inequality (59) on the last term in (62) with B=G′ε0EtΛt/E0 and A=c/dh0t, we have, for d>0,(64)ch0tG′ε0EtΛtE0=dΛtcdΛth0tG′ε0EtΛtE0≤dΛtG3G′ε0EtΛtE0+dΛtG3∗cdΛth0t≤dΛtε0EtΛtE0G′ε0EtΛtE0+dΛtG4cdΛth0t≤dΛtG2ε0EtΛtE0+dΛtG4cdΛth0t.
Now, combining (62) and (64) and choosing d small enough so that k0=k−d>0, we arrive at(65)ℱ1′t≤−kξtΛtG2ε0EtΛtE0+dξtΛtG2ε0EtΛtE0+dξtΛtG4cdΛth0t≤−k1ξtΛtG2ε0EtΛtE0+dξtΛtG4cdΛth0t.
Using the equivalent property in (63) and the increasing of G2, we have(66)G2ε0EtΛtE0≥G2d0ℱ1tΛt.
Lettingℱ2t≔d0ℱ1tΛt and recalling Λ′≤0, then we arrive at, for some c1,c2>0,(67)ℱ2′t≤−c1ξtG2ℱ2t+c2ξtG4cdΛth0t.
Sinced0Λt is nonincreasing, using the equivalent property ℱ1∼E implies that there exists b0>0 such that ℱ2t≥b0EtΛt. Let t∈ℝ+ and ϑt satisfying (52) and (53). If(68)b0ΛtEt≤2G5tϑt,then, we have(69)Et≤2b0G5tϑtΛt.
If(70)b0ΛtEt>2G5tϑt,then, for any 0≤s≤t, we obtain(71)b0qsEs>2G5tϑt,since ΛtEt is a nonincreasing function. Therefore, for any 0≤s≤t, we have(72)ℱ2s>2G5tϑt.
Using (72), recalling the definition of G2, the fact that G2 is convex, and 0<ϑ≤1, we have, for any 0≤s≤t and 0<ϵ1≤1,(73)G2ε1ϑsℱ2s−ε1G5s≤ε1ϑsG2ℱ2s−G5sϑs≤ε1ϑsℱ2sG′ℱ2s−G5sϑs−ε1ϑsG5sϑsG′ℱ2s−G5sϑs≤ε1ϑsℱ2sG′ℱ2s−ε1ϑsG5sϑsG′G5sϑs.
Now, we let(74)ℱ3s=ε1ϑsℱ2s−ε1G5s,where ε1 small enough so that ℱ30≤1. Then, (73) becomes, for any 0≤s≤t,(75)G2ℱ3s≤ε1ϑsG2ℱ2s−ε1ϑsG2G5sϑs.
Furthermore, we have(76)ℱ3′t=ε1ϑ′tℱ2t+ε1ϑsℱ2′t−ε1G5′t.
Sinceϑ′≤0 and using (67), then, for any 0≤s≤t, 0<ε1≤1, we obtain(77)ℱ3′t≤ε1ϑtℱ2′t−ε1G5′t≤−c1ε1ξtϑtG2ℱ2t+c2ε1ξtϑsG4cdΛth0t−ε1G5′t.
Then, using (53) and (75), we obtain(78)ℱ3′t≤−c1ξtG2ℱ3t+c2ε1ξtϑtG4cdΛth0t−c1ε1ξtϑtG2G5tϑt−ε1G5′t.
From the definition ofG1 and G5, we have(79)G1G5s=c1∫0sξτdτ.
Hence,(80)G5′s=−c1ξsG2G5s.
Now, we have(81)c2ε1ξtϑtG4cdΛth0t−c1ε1ξtϑtG2G5tϑt−ε1G5′t=c2ε1ξtϑtG4cdΛth0t−ε1ξtϑtG2G5tϑt+cε1ξtG2G5t=ε1ξtϑtc2G4cdΛth0t−c1G2G5tϑt+c1G2G5tϑt.
Then, according to (53), we obtain(82)ε1ξtϑtc2G4cdΛth0t−c1G2G5tϑt−c1G2G5tϑt≤0.
Then, (78) gives(83)ℱ3′t≤−c1ξtG2ℱ3t.
Thus, from (83) and the definition of G1 and G2 in (50) and (51), we obtain(84)G1ℱ3t′≥c1ξt.
Integrating (84) over 0,t, we obtain(85)G1ℱ3t≥c1∫0tξsds+G1ℱ30.
SinceG1 is decreasing, ℱ30≤1 and G11=0, then(86)ℱ3t≤G1−1c1∫0tξsds=G5t.
Recalling thatℱ3t=ε1ϑtℱ2t−ε1G5t, we have(87)ℱ2t≤1+ε1ε1G5tϑt.
Similarly, recall thatℱ2t≔d0ℱ1tΛt; then,(88)ℱ1t≤1+ε1d0ε1G5tϑtΛt.
Sinceℱ1∼E, then, for some b>0, we have Et≤bℱ1, which gives(89)Et≤b1+ε1d0ε1G5tϑtΛt.
From (69) and (89), we obtain the following estimate:(90)Et≤c3G5tϑtΛt,where c3=max2/b0,b1+ε1/d0ε1.


## 5. Examples

Letgt=a/1+tβ, where β>1 and 0<a<β−1 so that A1 is satisfied. In this case ξt=βa−1/β and Gt=tβ+1/β. Then, there exist positive constants aii=0,…,3 depending only on a,β such that(91)G4t=a0tβ+1/β,G2t=a1tβ+1/β,G1t=a2t−1/β−1,G5t=a3t+1−β.We will discuss two cases.Case 1.
If(92)m01+tr≤1+∇u02≤m11+tr,where 0<r<β−1 and m0,m1>0, then we have, for some positive constants aii=4,…,7, depending only on a,β,m0,m1,r, the following:(93)a41+t−β+1+r≤h0t≤a51+t−β+1+r,(94)λΛt≥a61+ln1+t,β−r=2,2,β−r>2,1+t−β+r+2,1<β−r<2,(95)λΛt≤a71+ln1+t,β−r=2,2,β−r>2,1+t−β+r+2,1<β−r<2.
We notice that condition (53) is satisfied if(96)t+1βΛth0tϑt≤a81−ϑ1/ββ/β+1,where a8>0 depending on a,β,c1, and c2. Choosing ϑt as the following,(97)ϑt=λ1+t−p,p=r+1β−r≥2,1+t−p,p=β−1,1<β−r<2,so that (52) is valid. Moreover, using (93) and (94), we see that (96) is satisfied if 0<λ≤1 is small enough, and then, (53) is satisfied. Hence, (54) and (95) imply that, for any t∈ℝ+,(98)Et≤a91+ln1+t1+t−β−r−1,β−r=2,1+t−β−r−1,β−r>2or1<β−r<2.
Thus, estimate (98) gives limt⟶+∞Et=0.Case 2.
Ifm0≤1+∇u02≤m1. That is, r=0 in (92), as it was assumed in many papers in the literature. So, it is clear that estimate (98) gives limt⟶+∞Et=0 when r=0.


## 6. Conclusion

Comparing our result with some earlier results in literature, we find that the result in this study improves and extends some earlier decay results in the literature as follows:(i)
Our decay result is obtained with more general relaxation function that is used in the literature such as the one in [15,16].(ii)
Ifm01+tr≤1+∇u02≤m11+tr, where 0<r<β−1 and m0,m1>0, our estimate (98) extends and improves the decay rate 1+tβ˜ obtained in [17] where 0<β˜is small enough and β>2.(iii)
Ifm0≤1+∇u02≤m1, our estimate (98) gives a better decay rate than the ones 1+tβ˜, where 0<β˜<q−1/2 is obtained in [18], and also better than the decay rate 1+tβ˜, where 0<β˜is small enough and obtained in [19]. Moreover, our estimate (98) provides a better decay than 1+t−q2−q−1/q obtained in [20].



---

*Source: 2697155-2022-06-17.xml*