# A New General Decay Rate of Wave Equation with Memory-Type Boundary Control

Authors: Sheng Fan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Of interest is a wave equation with memory-type boundary oscillations, in which the forced oscillations of the rod is given by a memory term at the boundary. We establish a new general decay rate to the system. And it possesses the character of damped oscillations and tends to a finite value for a large time. By assuming the resolvent kernel that is more general than those in previous papers, we establish a more general energy decay result. Hence the result improves earlier results in the literature.


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## Body

## 1. Introduction

It is well-known that if we add a damping to a system, the amplitude of the oscillations can be reduced very fast. The memory term can be as a damping (viscoelastic damping) which is weaker than frictional damping. For viscoelastic materials, Boltzmann theory gives us that the stress-strain viscoelastic law depending on a relaxation measure, see Prüss [1] and Eden et al. [2]. Based on the Boltzmann principle, the viscoelastic stress-strain relations can be generally given by a convolution term, which can be regarded as a lower order perturbation and can also be regarded as a kind of memory effect, for instance, g∗u. And we call the function gt memory kernel. One can find a detail derivation on some systems with memory in [3].To motivate our work, we start with some known results on wave equation with memory-type oscillations. The general wave equation with viscoelastic term in the internal feedback(1)utt−Δu+∫0tgt−sΔusds=ℱu.Messaoudi and Messaoudi [4, 5] studied ℱu=0 and ℱu=uρu, by introducing the assumption g′t≤−ξtgt, and obtained the energy decays exponentially (polynomially) as g decays exponentially (polynomially), respectively.Lasiecka et al. [6] considered the general assumption on g: g′t≤−Hgt to establish general decay of energy. Here H, which was introduced by Alabau-Boussouira and Cannarsa [7], is strictly convex and increasing function. Cavalcanti et al. [8, 9], Lasiecka and Wang [10], Mustafa and Messaoudi [11], and Xiao and Liang [12] also used this assumption to obtain some general decay rates of corresponding models. In recent papers [13–15], the authors investigated three classes of viscoelastic wave equation as in [4, 5] and established optimal and explicit decay results of energy by adopting the assumption on g: g′t≤−ξtHgt.In this paper, we considered the following wave equation with boundary oscillations of memory type:(2)utt−Δu=0,inΩ×ℝ+,u=0,onΓ0×ℝ+,u+∫0tgt−s∂u∂νsds=0,onΓ1×ℝ+,ux,0=u0x,utx,0=u1x,x∈Ω,where Ω⊂ℝnn≥1 is a bounded domain with smooth boundary Γ, Γ=Γ0∪Γ1, and Γ0 and Γ1 are closed and disjoint with measure Γ0>0. ν is the unit outward normal to Γ.For wave equation with memory-type boundary oscillations, it can be regarded as a wave equation with viscoelastic damping at the boundary. Santos [16] considered a one-dimensional wave equation with memory conditions at the boundary, respectively. He proved that the energy of solutions decays exponentially (polynomially) as k and k′ decay exponentially (polynomially). Here k is the resolvent kernel of −g′/g0. Santos et al. [17] extended the results in [16] to an n-dimensional wave equation of Kirchhoff type with memory-type boundary. They proved the global existence of solutions and obtained that the energy of solution decays uniformly with the same rate of decay k under the same conditions on k and k′, which improves the results in [18] by Park et al. Santos and Junior [19] obtained a similar result for plate equation with memory-type boundary. We also mention the work of Cavalcanti et al. [20], where the authors showed the global existence and the uniform decay of solutions to a semilinear wave equation with memory-type boundary condition and a nonlinear boundary source. Messaoudi and Soufyane [21] considered a general assumption on k′: k″≥−ξtk′t and established a general decay result. Wu [22] used this assumption to study a wave Kirchhoff-type wave equation with a boundary control of memory type. For nonlinear wave equations with memory-type boundary condition, we refer to Cavalcanti and Guesmia [23], Feng [24], Feng et al. [25–27], Muñoz Rivera and Andrade [28], and Zhang [29].Concerning the system (2), Mustafa [30], by assuming the function k: k″t≥H−k′t, where k is the resolvent kernel of −g′/g0, established a general decay of solutions of the form(3)Et≤k3H1−1k1t+k2,∀t≥0.Here(4)H1t=∫t11sH0′ε0sds,H0t=HDt,and D is a positive C1 function with D0=0, and H0 is strictly increasing and strictly convex C2 function on 0,r. In particular, for Ht=tp, i.e., k″≥c−k′p, the author proved the energy decay holds for 1≤p<3/2. Whether can the range be extended to a more larger range? In this paper, we give a positive answer to study problem (2) and extend the result to get a more general decay rate. In particular, we obtain that the energy result holds for Ht=tp with the full admissible range 1≤p<2. More exactly, by assuming the relaxation function k with minimal conditions on L10,∞, i.e., k″t≥ηtH−k′t, where H is linear or strictly increasing and strictly convex functions of class C2ℝ+, we establish an optimal explicit and general energy decay result. In particular, the energy result holds for Ht=tp with the range p∈1,2 instead of p∈1,3/2 in [30]. Hence our results extend and improve the stability results in [30] and also in [16–18, 21]. We mainly adopt the idea of [14, 15, 31] and some properties of convex function developed in [7, 32].The remaining of the paper is organized as follows: in Section2, we propose some preliminaries. In Section 3, main results are given. Section 4 is devoted to proving the general decay result.


## 2. Preliminaries

Taking the derivative of (2) with respect to t, we shall see that(5)∂u∂ν=−1g0ut+g2′∗∂u∂ν.We denote the resolvent kernel of−g′/g0 by k satisfying for t≥0:(6)kt+1g0g′∗kt=−1g0g′t.Using Volterra’s inverse operator and takingα=1/g0, we have(7)∂u∂ν=−αut+k2∗ut.Assumeu0=0 on Γ1 in this paper, we get(8)∂u∂ν=−αut+k0+k∗u,onΓ1×ℝ+.In the following, we use boundary conditions (8) instead of (2).As in [30], we consider the following assumption:(A1)
There exists a fixed pointx0∈ℝ2 and some constant δ0>0 such that for mx=x−x0,(9)Γ0=x∈Γ:mx⋅νx≤0,Γ1=x∈Γ:mx⋅νx≥δ0.For the kernelk, we assume(A2)
k:ℝ+⟶ℝ+ is nonincreasing and twice differentiable function satisfying for any t≥0,(10)k0>0,k′t≤0.(A3)
There exist aC1 function H:ℝ+⟶ℝ+, with H0=H′0=0, which is linear or is strictly increasing and strictly convex function of class C2ℝ+ on 0,r, r≤−k′0 such that(11)k″t≥ηtH−k′t,∀t≥0,whereηt is C1 nonincreasing continuous function.Remark 2.1.
If assuming furtherlimt⟶∞kt=0, since limt⟶∞kt=0 and −k′t is nonincreasing and nonnegative, we can get(12)limt⟶∞−k′t=0.
Then for somet1≥0 large,(13)−k′t1=r⇒−k′t≤r,∀t≥t1.
Noting that−k′ is nonincreasing, −k′0>0, and −k′t1>0, we have −k′t1>0 for any t∈0,t1, and for any t∈0,t1,(14)0<−k′t1≤−k′t≤−k′0,0<ηt1≤ηt≤η0.
Therefore we obtain that there exist two positive constantsa and b such that for any t∈0,t1,(15)a≤ηtH−k′t≤b.
Then for anyt∈0,t1,(16)k″t≥ηtH−k′t≥a=ak′0k′0≥ak′0k′t.
This implies that there exists a constantd>0 such that for any t∈0,t1,(17)k″t≥−dk′t.
The proof is done.


## 3. Main Results

The well-posedness result is given in [30] proved by using the Faedo–Galerkin method as in [17].Theorem 1.
Assume that (A1) and (A2) hold. Letu0,u1∈H2Ω∩V×V, and then problem (2) admits a unique solution u satisfying(18)u∈L∞0,T;H2Ω∩V∩W1,∞0,T;V∩W2,∞0,T;L2Ω,where V=v∈H1Ω:v=0onΓ0.The total energy of the system is defined by(19)Et=12ut2+12∇u2+α2ktuΓ12−∫Γ1k′°udΓ,where(20)k∘ut=∫0tkt−sut−us2ds.We can get the following stability result.Theorem 2.
Assumek satisfies (A1)–(A3) and further limt⟶∞kt=0. Then there exist λ1,λ2>0 such that(21)Et≤λ2H4−1λ1∫K−1rtηsds,∀t>K−1r,where(22)H4t=∫tr1sH0sds,H0t=H′t,and Kt=−k′t. In particular, if Ht=tp, then for any t>0,(23)Et≤c1e−c2∫0tηsds,ifp=1,c31+∫0tηsds−1/p−1,if1<p<2,where c1,c3, and c2≤1 are positive constants.Remark 3.1.
From (23), the energy result holds for Ht=tp with the full admissible range p∈1,2 instead of p∈1,3/2. If the viscoelastic term is as internal feedback, Lasiecka and Wang [10] provided the proof for optimal decay rates of second-order systems in the full admissible range 1,2.At last, we show two examples to illustrate explicit formulas for the decay rates of the energy, which can be found in the studies of Mustafa and Mustafa [14, 15].Example 1.
Takek′t=−e−tq with 0<q<1, we get k″t=H−k′t, where Ht=qt/ln1/t1/q−1. Since(24)H′t=1−q+qln1/tln1/t1/q,H″t=1−qln1/t+1/qln1/t1/q+1,we can deduce that the function H satisfies (A3) on 0,r for any 0<r<1. Then,(25)Et≤c1e−c2tq.Example 2.
Considerk′t=−1/t+elnt+ep with p>1, we get k″t=lnt+e+p/t+e2lnt+ep+1. Clearly,(26)k″t=lnt+e+pt+elnt+e−k′t.
By part 1 of (23), we get(27)Et≤c1exp−c2∫0tlnt+e+pt+elnt+eds=c1t+elnt+epc2.
Asc2≤1, this is slower rate than −k′t. In addition,(28)k″t=lnt+e+pt+e1−1/p−k′t1+1/p.
From part 2 of (23), we infer that for large t(29)Etc31+∫0tlnt+e+pt+e1−1/pds−p≤c3t+elnt+ep,which is the same rate as −k′t.


## 4. Proof of Main Result

To prove Theorem2, we need the following lemmas.


### 4.1. Technical Lemmas

Lemma 1.
The total energy functionalEt satisfies for any t≥0,(30)ℰ′t≤−α2utΓ12+∫Γ1k″°udΓ≤0.Proof.
See [30].As in [31], for 0<δ<1, we introduce(31)Cδ=∫0∞k′s2k″s−δk′sds,ht=k″s−δk′s.Lemma 2.
Define the functionalΦt by(32)Φt≔∫Ω2m⋅∇u+n−1uutdx.Then we can get for anyt≥t1,(33)Φ′t≤−ut2−12∇u2+cutΓ12+Cδ∫Γ1h∘udΓ.Proof.
From the same arguments as in the study of Mustafa [30], we can obtain(34)Φ′t≤−ut2−∇u2−δ0∇uΓ12+∫Γ1m⋅νut2dΓ+∫Γ12m⋅∇u∂u∂νdΓ+n−1∫Γ1u∂u∂νdΓ.
It follows from Young’s inequality that for anyε>0,(35)∫Γ12m⋅∇u∂u∂νdΓ+n−1∫Γ1u∂u∂νdΓ≤δ0∇uΓ12+εuΓ12+c∂u∂νΓ12.
Recallingk′∗u=−k′⊙u+kt−k0u, where k⊙u=∫0tkt−sut−usds; then we have from (8),(36)∂u∂νt=−αutt+ktut+−k′⊙ut.
By using Young’s inequality, we obtain(37)∂u∂νtΓ12≤4α2utΓ12+k2tuΓ12+∫Γ1−k′⊙u2dΓ.
Hölder’s inequality implies(38)−k′⊙u2=∫0t−k′t−sut−usds2=∫0t−k′t−sk″t−s−δk′t−sk″t−s−δk′t−sut−usds2≤∫0tk′s2k″s−δk′sds∫0tk″t−s−δk′t−sut−us2ds≤Cδh∘u,which, together with (37), gives us that(39)∂u∂νtΓ12≤4α2utΓ12+k2tuΓ12+Cδ∫Γ1h∘udΓ.
Inserting (39) into (35), we obtain for any ε>0,(40)∫Γ12m⋅∇u∂u∂νdΓ+n−1∫Γ1u∂u∂νdΓ≤δ0∇uΓ12+ε+4α2k2tuΓ12+4α2cutΓ12+Cδ∫Γ1h∘udΓ.
Noting that(41)uΓ12≤c∇u2,using limt⟶∞kt=0 and taking ε>0 small enough, we can get (33) from (34) and (40). The proof is done.To get the optimal energy decay, we need the following estimate.Lemma 3.
The functionalΨt is defined by(42)Ψt≔∫0tkt−susΓ12ds,which satisfies for any t>0,(43)Ψ′t≤12∫Γ1k′°udΓ+3k0utΓ12.Proof.
DifferentiatingΨt with respect to t, we get(44)Ψ′t=k20utΓ12+∫0tk2′t−susΓ12ds=∫0tk′t−s∫Γ1us−ut2dΓds+ktutΓ22+2∫Γ1ut∫0tk′t−sus−utdsdΓ.
In view of Young’s and Hölder’s inequalities, we obtain(45)2∫Γ1ut∫0tk′t−sus−uta2+b2dsdΓ≤2k0∫Γ1u2tdΓ+12k0∫Γ1∫0t−k′t−s−k′t−sus−utds2dΓ≤2k0utΓ12+∫0tk′sds2k0∫0tk′t−sus−utΓ12ds.
Then we can get (43) following from the fact(46)kt≤k0,∫0tk′sds2k0≥−12.
The proof is complete.


### 4.2. Proof of Theorem2

Proof.
Define the functionalLt by(47)Lt≔NEt+Φt,where N>0 is a constant that will be taken later. Clearly we can take N a large value to get(48)Lt∼Et.
Recallingk″=δk′+h, combining (30) and (33), we conclude that for any t>t1,(49)L′t≤−α2N−cutΓ12−ut2−12∇u2−α2Nδ∫Γ1k′°udΓ−α2N−cCδ∫Γ1h°udΓ.
Noting−k′>0 and k″>0, for each s∈0,∞, we shall see below,(50)limδ⟶0δk′s2k″s−δk′sds=0,δk′s2k″s−δk′s<−k′s.
It follows from Lebesgue dominated convergence theorem that(51)limδ⟶0δCδ=limδ⟶0l∫0∞δk′s2k″s−δk′sds=0.
Therefore there exist0<γ<1 such that if δ<γ, then(52)δCδ<14c.
And then we chooseN a larger value that(53)α2N−c>4k0,and take δ>0 satisfying(54)δ=12αN<γ.
This implies(55)α2N−cCδ>0.
Then there exists a positive constantβ such that for large t1>0,(56)L′t≤−βut2+∇u2−4k0utΓ12−14∫Γ1k′°udΓ,∀t≥t1.
By (17) and (30), we get(57)∫0t1−k′s∫Γ1ut−ut−s2dΓds≤1d∫0t1k″s∫Γ1ut−ut−s2dΓds≤−cℰ′t.
Then from (56), we infer that there exists a constant χ>0 such that(58)L′t≤−χEt−c∫0tk′s∫Γ1ut−ut−s2dΓds≤−χEt−cℰ′t−c∫t1tk′s∫Γ1ut−ut−s2dΓds.
DenotingFt≔Lt+cEt∼Et, and using (58), we know that(59)F′t≤−χEt−c∫t1tk2′s∫Γ1ut−ut−s2dΓds.In the sequel, we consider two cases.Case 1.
The particular caseHt=tp.(I) p=1.Multiplying (59) by ηt, and using (19) and (A2)-(A3), we have(60)ηtF′t≤−χηtEt−cℰ′t,∀t≥t1.Sinceηt is a nonincreasing continuous function and η′t≤0 for a.e. t, then(61)ηF+cE′t≤ηtF′t+cℰ′t≤−mηtEt,a.e.t≥t1.In view ofηF+cE∼E, we obtain that there exist two positive constants c1,c2>0,(62)Et≤c1e−c2∫t1tηsds.(II) 1<p<2.DefineGt by(63)Gt=Lt+Ψt.It follows from (43) and (56) that Et≥0, and for any t≥t1,(64)G′t≤−βut2+∇u2−k0utΓ12+14∫Γ1k′°udΓ.Then there exists a certain constantβ1>0,(65)G′t≤−β1Et,∀t≥t1.This gives us(66)β1∫t1tEsds≤Gt1−Gt≤Gt1.Hence(67)∫0∞Esds<∞.Define(68)It=∫0t∫Γ1ut−ut−s2dΓds,we know that(69)It≤c∫0tEsds.Without loss of generality assumingt1 so large that It1>0, then(70)0<It1≤It<∞,∀t≥t1.Using Jensen’s inequality and by (30) and (A2)-(A3), we can derive from (56) that for some constant q>0,(71)L′t≤−qEt+cItIt∫Γ1−k′p⋅1/p∘udΓ≤−qEt+cIt1It∫Γ1−k′p∘udΓ1/p≤−qEt+cI1−1/pt∫Γ1k″η∘udΓ1/p≤−qEt+cηt1/p∫Γ1k″∘udΓ1/p≤−qEt+cηt1/p−ℰ′t1/p.We multiply (71) by Ep−1t and use (19) to deduce(72)LEp−1′t≤L′tEp−1t≤−qEpt+c−ℰ′tηt1/pEp−1t.By Young’s inequality, we have for anyε1>0,(73)LEp−1′t≤−qEpt+ε1Ept+cε1−ℰ′tηt.Takingε1<1/2q, we conclude(74)LEp−1′t≤−q2Ept−cℰ′tηt.DefineFt=ηLEp−1+cE∼E. Multiplying (74) by ηt, we have(75)F′t≤−q2ηtEpt.Then there exists a certain constantq0>0 such that(76)F′t≤−q0ηtFpt,from which we obtain(77)Et≤c31+∫0tηsds−1/p−1,wherec3 is a positive constant.Combining (I) and (II) and using the boundedness of ηt and Et, we can get (23).Case 2.
The general case.
Define(78)It≔q∫t1t∫Γ1ut−ut−s2dΓds.
In view of (67), we can take 0<q<1 such that(79)It<1,∀t≥t1.
Without loss of generality, we assume thatIt>0 for all t≥t1. On the other hand, we define(80)πt≔∫t1tk″s∫Γ1ut−ut−s2dΓds.
From (30), we can easily get πt≤−cΕ′t. As Ht is strictly convex on 0,r and H0=0, we see that(81)Hλx≤λHx,i=1,2,0≤λ≤1,x∈0,r.
It follows from Jensen’s inequality and (11) and (79) that(82)π1t=1qIt∫t1tItk″sq∫Γ1ut−ut−s2dΓds≥1qIt∫t1tItηsH−k′sq∫Γ1ut−ut−s2dΓds≥ηtqIt∫t1tHIt−k′sq∫Γ1ut−ut−s2dΓds≥ηtqH1It∫t1tIt−k′sq∫Γ1ut−ut−s2dΓds=ηtqHq∫t1t−k′s∫Γ1ut−ut−s2dΓds=ηtqH¯q∫t1t−k′s∫Γ1ut−ut−s2dΓds,where H¯, which is strictly convex and increasing function on 0,∞ of class C2, is called the extension of H. We infer from (82) that(83)∫t1t−k′s∫Γ1ut−ut−s2dΓds≤1qH¯−1qπtηt.
Then we can get from (59) that for any t≥t1,(84)F′t≤−χEt+cH¯−1qπtηt.
Denote(85)H0t=H¯′t.
Forr0<r, we define K1t by(86)K1t=H0r0EtE0Ft+Et∼Et.
SinceE′t≤0, H¯′>0, and H¯″>0, we get from (84) that(87)K1′t=r0ℰ′tE0H0′r0EtE0Ft+H0r0EtE0F′t+ℰ′t≤−mEtH0r0EtE0+cH0r0EtE0H¯−1qπtηt.
We denote byH¯∗ the conjugate function of the convex function H¯ (see Arnold [33]), and then(88)H¯∗s=sH¯′−1s−H¯H¯′−1ssatisfies Young’s inequality,(89)AB≤H¯∗A+H¯B.
TakingA=H¯0′r0Εt/Ε0 and B=H¯−1qπt/ηt, and using H¯∗s≤sH¯′−1s and (87), we have(90)K1′t≤−χEtH0r0EtE0+cH¯∗H0r0EtE0+cqπtηt≤−χEtH0r0EtE0+cH0r0EtE0H¯′−1H0r0EtE0+cqπtηt≤−χEtH0r0EtE0+cH0r0EtE0H¯′−1H¯′r0EtE0+cqπtηt≤−χE0−cr0EtE0H0r0EtE0+cqπtη1t.
We multiply (90) by ηt to arrive at(91)ηtK1′t≤−χE0−cr0ηtEtE0H0r0EtE0+cqπt≤−χE0−cr0ηtEtE0H0r0EtE0−cℰ′t.
The functionalK2t is defined by(92)K2t=ηtK1t+cEt.
Then we can easily obtain that there exist constantsβ5>0 and β6>0 such that(93)β5K2t≤Εt≤β6K2t.
Choosing a suitabler0>0, and defining H3t=tH0r0t, from (91), we infer that for a constant γ>0,(94)K2′t≤−γηtEtE0H0r0EtE0≔−γηtH3EtE0.
It follows from0≤r0Et/E0<r that for any t>0,(95)H0r0EtE0=H¯′r0EtE0=H′r0EtE0.
Using (93), we have(96)Rt≔β5K2tE0∼Et.
SinceH3′t=H0r0t+r0tH0′r0t, then, noting the strict convexity of H0 on 0,r, we know H3′t,H3t>0 on 0,1. By (94), we conclude that there exists γ1>0 such that for any t≥t1,(97)R′t≤−γ1ηtH3Rt.
Integrating (97) over t1,t, we see that(98)∫t1t−R′sH3Rsds≥γ1∫t1tηsds⇒∫r0Rtr0Rt11sH0sds≥γ1∫t1tηsds.
Define(99)H4t=∫tr1sH0sds.
It is to verify thatH4 is strictly decreasing on 0,r and limt⟶0H4t=+∞. It follows that(100)Rt≤1r0H4−1ζ1∫t1tηsds.
Combining (96) and (100), we can obtain (21). This finishes the proof of Theorem 2


## 4.1. Technical Lemmas

Lemma 1.
The total energy functionalEt satisfies for any t≥0,(30)ℰ′t≤−α2utΓ12+∫Γ1k″°udΓ≤0.Proof.
See [30].As in [31], for 0<δ<1, we introduce(31)Cδ=∫0∞k′s2k″s−δk′sds,ht=k″s−δk′s.Lemma 2.
Define the functionalΦt by(32)Φt≔∫Ω2m⋅∇u+n−1uutdx.Then we can get for anyt≥t1,(33)Φ′t≤−ut2−12∇u2+cutΓ12+Cδ∫Γ1h∘udΓ.Proof.
From the same arguments as in the study of Mustafa [30], we can obtain(34)Φ′t≤−ut2−∇u2−δ0∇uΓ12+∫Γ1m⋅νut2dΓ+∫Γ12m⋅∇u∂u∂νdΓ+n−1∫Γ1u∂u∂νdΓ.
It follows from Young’s inequality that for anyε>0,(35)∫Γ12m⋅∇u∂u∂νdΓ+n−1∫Γ1u∂u∂νdΓ≤δ0∇uΓ12+εuΓ12+c∂u∂νΓ12.
Recallingk′∗u=−k′⊙u+kt−k0u, where k⊙u=∫0tkt−sut−usds; then we have from (8),(36)∂u∂νt=−αutt+ktut+−k′⊙ut.
By using Young’s inequality, we obtain(37)∂u∂νtΓ12≤4α2utΓ12+k2tuΓ12+∫Γ1−k′⊙u2dΓ.
Hölder’s inequality implies(38)−k′⊙u2=∫0t−k′t−sut−usds2=∫0t−k′t−sk″t−s−δk′t−sk″t−s−δk′t−sut−usds2≤∫0tk′s2k″s−δk′sds∫0tk″t−s−δk′t−sut−us2ds≤Cδh∘u,which, together with (37), gives us that(39)∂u∂νtΓ12≤4α2utΓ12+k2tuΓ12+Cδ∫Γ1h∘udΓ.
Inserting (39) into (35), we obtain for any ε>0,(40)∫Γ12m⋅∇u∂u∂νdΓ+n−1∫Γ1u∂u∂νdΓ≤δ0∇uΓ12+ε+4α2k2tuΓ12+4α2cutΓ12+Cδ∫Γ1h∘udΓ.
Noting that(41)uΓ12≤c∇u2,using limt⟶∞kt=0 and taking ε>0 small enough, we can get (33) from (34) and (40). The proof is done.To get the optimal energy decay, we need the following estimate.Lemma 3.
The functionalΨt is defined by(42)Ψt≔∫0tkt−susΓ12ds,which satisfies for any t>0,(43)Ψ′t≤12∫Γ1k′°udΓ+3k0utΓ12.Proof.
DifferentiatingΨt with respect to t, we get(44)Ψ′t=k20utΓ12+∫0tk2′t−susΓ12ds=∫0tk′t−s∫Γ1us−ut2dΓds+ktutΓ22+2∫Γ1ut∫0tk′t−sus−utdsdΓ.
In view of Young’s and Hölder’s inequalities, we obtain(45)2∫Γ1ut∫0tk′t−sus−uta2+b2dsdΓ≤2k0∫Γ1u2tdΓ+12k0∫Γ1∫0t−k′t−s−k′t−sus−utds2dΓ≤2k0utΓ12+∫0tk′sds2k0∫0tk′t−sus−utΓ12ds.
Then we can get (43) following from the fact(46)kt≤k0,∫0tk′sds2k0≥−12.
The proof is complete.


## 4.2. Proof of Theorem2

Proof.
Define the functionalLt by(47)Lt≔NEt+Φt,where N>0 is a constant that will be taken later. Clearly we can take N a large value to get(48)Lt∼Et.
Recallingk″=δk′+h, combining (30) and (33), we conclude that for any t>t1,(49)L′t≤−α2N−cutΓ12−ut2−12∇u2−α2Nδ∫Γ1k′°udΓ−α2N−cCδ∫Γ1h°udΓ.
Noting−k′>0 and k″>0, for each s∈0,∞, we shall see below,(50)limδ⟶0δk′s2k″s−δk′sds=0,δk′s2k″s−δk′s<−k′s.
It follows from Lebesgue dominated convergence theorem that(51)limδ⟶0δCδ=limδ⟶0l∫0∞δk′s2k″s−δk′sds=0.
Therefore there exist0<γ<1 such that if δ<γ, then(52)δCδ<14c.
And then we chooseN a larger value that(53)α2N−c>4k0,and take δ>0 satisfying(54)δ=12αN<γ.
This implies(55)α2N−cCδ>0.
Then there exists a positive constantβ such that for large t1>0,(56)L′t≤−βut2+∇u2−4k0utΓ12−14∫Γ1k′°udΓ,∀t≥t1.
By (17) and (30), we get(57)∫0t1−k′s∫Γ1ut−ut−s2dΓds≤1d∫0t1k″s∫Γ1ut−ut−s2dΓds≤−cℰ′t.
Then from (56), we infer that there exists a constant χ>0 such that(58)L′t≤−χEt−c∫0tk′s∫Γ1ut−ut−s2dΓds≤−χEt−cℰ′t−c∫t1tk′s∫Γ1ut−ut−s2dΓds.
DenotingFt≔Lt+cEt∼Et, and using (58), we know that(59)F′t≤−χEt−c∫t1tk2′s∫Γ1ut−ut−s2dΓds.In the sequel, we consider two cases.Case 1.
The particular caseHt=tp.(I) p=1.Multiplying (59) by ηt, and using (19) and (A2)-(A3), we have(60)ηtF′t≤−χηtEt−cℰ′t,∀t≥t1.Sinceηt is a nonincreasing continuous function and η′t≤0 for a.e. t, then(61)ηF+cE′t≤ηtF′t+cℰ′t≤−mηtEt,a.e.t≥t1.In view ofηF+cE∼E, we obtain that there exist two positive constants c1,c2>0,(62)Et≤c1e−c2∫t1tηsds.(II) 1<p<2.DefineGt by(63)Gt=Lt+Ψt.It follows from (43) and (56) that Et≥0, and for any t≥t1,(64)G′t≤−βut2+∇u2−k0utΓ12+14∫Γ1k′°udΓ.Then there exists a certain constantβ1>0,(65)G′t≤−β1Et,∀t≥t1.This gives us(66)β1∫t1tEsds≤Gt1−Gt≤Gt1.Hence(67)∫0∞Esds<∞.Define(68)It=∫0t∫Γ1ut−ut−s2dΓds,we know that(69)It≤c∫0tEsds.Without loss of generality assumingt1 so large that It1>0, then(70)0<It1≤It<∞,∀t≥t1.Using Jensen’s inequality and by (30) and (A2)-(A3), we can derive from (56) that for some constant q>0,(71)L′t≤−qEt+cItIt∫Γ1−k′p⋅1/p∘udΓ≤−qEt+cIt1It∫Γ1−k′p∘udΓ1/p≤−qEt+cI1−1/pt∫Γ1k″η∘udΓ1/p≤−qEt+cηt1/p∫Γ1k″∘udΓ1/p≤−qEt+cηt1/p−ℰ′t1/p.We multiply (71) by Ep−1t and use (19) to deduce(72)LEp−1′t≤L′tEp−1t≤−qEpt+c−ℰ′tηt1/pEp−1t.By Young’s inequality, we have for anyε1>0,(73)LEp−1′t≤−qEpt+ε1Ept+cε1−ℰ′tηt.Takingε1<1/2q, we conclude(74)LEp−1′t≤−q2Ept−cℰ′tηt.DefineFt=ηLEp−1+cE∼E. Multiplying (74) by ηt, we have(75)F′t≤−q2ηtEpt.Then there exists a certain constantq0>0 such that(76)F′t≤−q0ηtFpt,from which we obtain(77)Et≤c31+∫0tηsds−1/p−1,wherec3 is a positive constant.Combining (I) and (II) and using the boundedness of ηt and Et, we can get (23).Case 2.
The general case.
Define(78)It≔q∫t1t∫Γ1ut−ut−s2dΓds.
In view of (67), we can take 0<q<1 such that(79)It<1,∀t≥t1.
Without loss of generality, we assume thatIt>0 for all t≥t1. On the other hand, we define(80)πt≔∫t1tk″s∫Γ1ut−ut−s2dΓds.
From (30), we can easily get πt≤−cΕ′t. As Ht is strictly convex on 0,r and H0=0, we see that(81)Hλx≤λHx,i=1,2,0≤λ≤1,x∈0,r.
It follows from Jensen’s inequality and (11) and (79) that(82)π1t=1qIt∫t1tItk″sq∫Γ1ut−ut−s2dΓds≥1qIt∫t1tItηsH−k′sq∫Γ1ut−ut−s2dΓds≥ηtqIt∫t1tHIt−k′sq∫Γ1ut−ut−s2dΓds≥ηtqH1It∫t1tIt−k′sq∫Γ1ut−ut−s2dΓds=ηtqHq∫t1t−k′s∫Γ1ut−ut−s2dΓds=ηtqH¯q∫t1t−k′s∫Γ1ut−ut−s2dΓds,where H¯, which is strictly convex and increasing function on 0,∞ of class C2, is called the extension of H. We infer from (82) that(83)∫t1t−k′s∫Γ1ut−ut−s2dΓds≤1qH¯−1qπtηt.
Then we can get from (59) that for any t≥t1,(84)F′t≤−χEt+cH¯−1qπtηt.
Denote(85)H0t=H¯′t.
Forr0<r, we define K1t by(86)K1t=H0r0EtE0Ft+Et∼Et.
SinceE′t≤0, H¯′>0, and H¯″>0, we get from (84) that(87)K1′t=r0ℰ′tE0H0′r0EtE0Ft+H0r0EtE0F′t+ℰ′t≤−mEtH0r0EtE0+cH0r0EtE0H¯−1qπtηt.
We denote byH¯∗ the conjugate function of the convex function H¯ (see Arnold [33]), and then(88)H¯∗s=sH¯′−1s−H¯H¯′−1ssatisfies Young’s inequality,(89)AB≤H¯∗A+H¯B.
TakingA=H¯0′r0Εt/Ε0 and B=H¯−1qπt/ηt, and using H¯∗s≤sH¯′−1s and (87), we have(90)K1′t≤−χEtH0r0EtE0+cH¯∗H0r0EtE0+cqπtηt≤−χEtH0r0EtE0+cH0r0EtE0H¯′−1H0r0EtE0+cqπtηt≤−χEtH0r0EtE0+cH0r0EtE0H¯′−1H¯′r0EtE0+cqπtηt≤−χE0−cr0EtE0H0r0EtE0+cqπtη1t.
We multiply (90) by ηt to arrive at(91)ηtK1′t≤−χE0−cr0ηtEtE0H0r0EtE0+cqπt≤−χE0−cr0ηtEtE0H0r0EtE0−cℰ′t.
The functionalK2t is defined by(92)K2t=ηtK1t+cEt.
Then we can easily obtain that there exist constantsβ5>0 and β6>0 such that(93)β5K2t≤Εt≤β6K2t.
Choosing a suitabler0>0, and defining H3t=tH0r0t, from (91), we infer that for a constant γ>0,(94)K2′t≤−γηtEtE0H0r0EtE0≔−γηtH3EtE0.
It follows from0≤r0Et/E0<r that for any t>0,(95)H0r0EtE0=H¯′r0EtE0=H′r0EtE0.
Using (93), we have(96)Rt≔β5K2tE0∼Et.
SinceH3′t=H0r0t+r0tH0′r0t, then, noting the strict convexity of H0 on 0,r, we know H3′t,H3t>0 on 0,1. By (94), we conclude that there exists γ1>0 such that for any t≥t1,(97)R′t≤−γ1ηtH3Rt.
Integrating (97) over t1,t, we see that(98)∫t1t−R′sH3Rsds≥γ1∫t1tηsds⇒∫r0Rtr0Rt11sH0sds≥γ1∫t1tηsds.
Define(99)H4t=∫tr1sH0sds.
It is to verify thatH4 is strictly decreasing on 0,r and limt⟶0H4t=+∞. It follows that(100)Rt≤1r0H4−1ζ1∫t1tηsds.
Combining (96) and (100), we can obtain (21). This finishes the proof of Theorem 2



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*Source: 5571072-2021-02-13.xml*