Exponential Decay for Nonlinear von Kármán Equations with Memory

Abstract

We consider the nonlinear von Kármán equations with memory term. We show the exponential decay result of solutions. Our result is established without imposing the usual relation between 𝑔 and its derivative. This result improves on earlier ones concerning the exponential decay.

1. Introduction

In this paper we consider the exponential decay rate of solutions for the nonlinear von Kármán equations with memory term: 𝑢^′𝜌𝑢^′′−ℎΔ𝑢^′′+Δ²𝑢−^𝑡₀𝑔(𝑡−𝑠)Δ²𝑢(𝑠)[],(),Δ𝑑𝑠=𝑢,𝑣inΩ×0,∞₂[](),𝑢𝑣=−𝑢,𝑢,inΩ×0,∞𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑢^′𝑥,𝑦,0=𝑢₁𝑥,𝑦,inΩ, (1)(2)(3) and the boundary conditions 𝑣=𝜕𝑣(),𝜕𝜈=0,onΓ×0,∞𝑢=𝜕𝑢𝜕𝜈=0,onΓ₀×(),ℬ0,∞₁𝑢−ℬ₁^𝑡₀𝑔(𝑡−𝑠)𝑢(𝑠)𝑑𝑠=0,onΓ₁×(),ℬ0,∞₂𝑢−ℎ𝜕𝑢^′′𝜕𝜈−ℬ₂^𝑡₀𝑔(𝑡−𝑠)𝑢(𝑠)𝑑𝑠=0,onΓ₁×(0,∞), (4) where Ω is an open bounded set of ℝ², with a sufficiently smooth boundary Γ=Γ₀∪Γ₁. Here, Γ₀ and Γ₁ are closed and disjoint. The constants ℎ,𝜌>0. Let us denote by 𝜈=(𝜈₁,𝜈₂) the external unit normal to Γ and by 𝜂=(−𝜈₂,𝜈₁) the corresponding unit tangent vector. The von Kármán bracket is given by [𝑢,𝑣]=𝑢_𝑥𝑥𝑣_𝑦𝑦+𝑢_𝑦𝑦𝑣_𝑥𝑥−2𝑢_𝑥𝑦𝑣_𝑥𝑦. (5)

Here, we are denoting by ℬ₁, ℬ₂ the following differential operators: ℬ₁𝑢=Δ𝑢+1−𝜇𝐵₁ℬ𝑢,₂𝑢=𝜕Δ𝑢+𝐵𝜕𝜈1−𝜇₂𝑢, (6) where 𝐵₁ and 𝐵₂ are given by 𝐵₁𝑢=2𝜈₁𝜈₂𝑢_𝑥𝑦−𝜈²₁𝑢_𝑦𝑦−𝜈²₂𝑢_𝑥𝑥,𝐵₂𝜕𝑢=𝜈𝜕𝜂²¹−𝜈²₂𝑢_𝑥𝑦+𝜈₁𝜈₂𝑢_𝑦𝑦−𝑢_𝑥𝑥, (7) and the constant 𝜇(0<𝜇<1/2) represents Poisson’s ratio.

This system describes the transversal displacement 𝑢(𝑥,𝑦,𝑡) and the Airy stress function 𝑣(𝑥,𝑦,𝑡) of a vibrating plate. The dissipation in (1) is due to the term −𝑔∗Δ²𝑢, where 𝑔 is positive real function and the convolution product ∗ is given by (𝑔∗𝑢)(𝑡)=∫^𝑡₀𝑔(𝑡−𝑠)𝑢(𝑠)𝑑𝑠. A material whose contained term is −𝑔∗Δ²𝑢 is called viscoelastic and is said to be “endowed with long-range memory” since the stress at any instant depends on the complete history of strain that the material has undergone.

Problems related to 𝑓𝑢^′𝑢^′′−Δ𝑢−Δ𝑢^′′=0 (8) are interesting not only from the point of view of PDE general theory, but also due to its applications in Mechanics. For instance, when the material density, 𝑓(𝑢^′), is equal to 1, (8) describes the extensional vibrations of thin rods; see Love [1] for the physical details. When the material density 𝑓(𝑢^′) is not constant, we are dealing with a thin rod which possesses a rigid surface and whose interior is somehow permissive to slight deformations such that the material density varies according to the velocity.

On the other hand, the problem of stability of the solutions to the following wave equation with memory was studied by many authors [2–6]: 𝑢^′𝜌𝑢^′′−Δ𝑢−Δ𝑢^′′+^𝑡₀𝑔(𝑡−𝑠)Δ𝑢(𝑠)𝑑𝑠+𝐹𝑢,𝑢^′=0,𝑥∈Ω,𝑡>0. (9) Cavalcanti et al. [2] showed an exponential and polynomial decay for the viscoelastic wave equation (9) with 𝐹(𝑢,𝑢^′)=−𝛾Δ𝑢^′ under the usual conditions −𝑐₁𝑔(𝑡)≤𝑔^′(𝑡)≤−𝑐₂𝑔(𝑡),0≤𝑔^′′(𝑡)≤𝑐₃𝑔(𝑡) (10) for some 𝑐_𝑖,𝑖=1,2,3. Han and Wang [3] proved the uniform decay for the nonlinear viscoelastic equation under condition 𝑔^′(𝑡)≤−𝑐𝑔(𝑡), (11) where 𝑐>0. Park and Kang [7] studied the uniform decay for a nonlinear viscoelastic problem with damping. They obtained the exponential decay estimate under condition (11). Later, this assumption was relaxed by several authors. Messaoudi and Tatar [6] investigated exponential and polynomial decay for a quasilinear viscoelastic equation under condition on 𝑔 such as 𝑔^′(𝑡)≤−𝜉𝑔^𝑝(𝑡)3,for1≤𝑝<2,𝑡≥0, (12) where 𝜉>0, by choosing a suitable perturbed energy. Liu [5] showed exponential and polynomial decay for the system of two coupled quasilinear viscoelastic equation, under condition (12). Messaoudi and Tatar [8] proved the exponential decay rate for a quasilinear viscoelastic equation under the conditions 𝑔^′(𝑡)≤0,_∞0𝑔(𝑡)𝑒^𝛼𝑡𝑑𝑡<+∞forsomelarge𝛼>0. (13) They improved some earlier results concerning the exponential decay. Han and Wang [4] studied the general decay rate for the nonlinear viscoelastic equations under the more general conditions on 𝑔 such as 𝑔^′(𝑡)(𝑡)𝑔(𝑡),𝜉≤−𝜉^′(𝑡)𝜉(𝑡𝜉(𝑡))≤𝑘,>0,𝜉^′(𝑡)≤0,∀𝑡>0. (14) When 𝜌=0, the problem of stability of the solutions to the viscoelastic system with memory has been studied by many authors. In [9, 10], the authors proved exponential and polynomial decay for the viscoelastic wave equation under conditions (10). Berrimi and Messaoudi [11] studied exponential and polynomial decay rates under condition (12). Messaoudi [12] investigated the general decay rate for the viscoelastic equations under general conditions (14). Guesmia and Messaoudi [13] obtained general stability for the Timoshenko system under weaker condition on 𝑔 such as 𝑔^′(𝑡)≤−𝜉(𝑡)𝑔(𝑡), (15) where 𝜉 is a nonincreasing and positive function. As for problem of stability of the solutions to a viscoelastic system under condition (15), we also refer the reader to [14–16] and references therein. These general decay estimates extended and improved on some earlier results—exponential or polynomial decay rates.

The problem of stability of the solutions to a von Kármán system with dissipative effects has been studied by several authors. For example, in [17, 18] the authors studied the von Kármán equation in the presence of thermal effects. In [19–23] the authors considered the von Kármán system with frictional dissipations effective in the boundary. It is shown in these works that these dissipations produce uniform rate of decay of the solution when 𝑡 goes to infinity. Rivera and Menzala [24] and Rivera et al. [25] studied the stability of the solutions to a von Kármán system for viscoelastic plates with memory and boundary memory conditions. They proved that the energy decays uniformly exponentially or algebraically with the same rate of decay as the relaxation function. Later, Santos and Soufyane [26] generalized the decay result of [24]. Raposo and Santos [27] considered the general decay of the solutions to a von Kármán plate model (1)–(4) for 𝜌=0. They showed that the energy decays with a similar rate of decay of the relaxation function, which is not necessarily decaying in a polynomial or exponential fashion. Kang [28] investigated the general decay of the solution to a von Kármán system with memory and boundary damping. Recently, Kang [29] proved that solutions for a von Kármán plate with memory decay exponentially to zero as time goes to infinity in case 𝑔^′(𝑡)+𝛾𝑔(𝑡)≥0 for all 𝑡≥0 provided that [𝑔^′(𝑡)+𝛾𝑔(𝑡)]𝑒^𝛼𝑡∈𝐿¹(0,∞) for some 𝛼>0.

In this paper, we establish an exponential decay of the solutions to the nonlinear von Kármán plate model (1)–(4) without assumption (15), which is the usual relation between 𝑔 and its derivative. Instead of (15), we require the function 𝑒^𝛼𝑡𝑔(𝑡) to have sufficiently small 𝐿¹-norms on (0,∞) for some 𝛼>0. This result improves on earlier ones concerning the exponential decay of the solutions to the von Kármán equations.

The organization of this paper is as follows. In Section 2, we give some notations and introduce the relative results of Airy stress function and von Kármán bracket. In Section 3, we prove that the energy decreases exponentially. The construction of the Lyapunov function is inspired in multiplier techniques that was used in [8].

2. Preliminaries

In this section, we present some material needed in the proof of our result and state the main result. Throughout this paper we denote (𝑢,𝑣)=_Ω𝑢𝑥,𝑦𝑣𝑥,𝑦𝑑Ω (16) and define 𝑉=𝑣∈𝐻¹(Ω)∣𝑣=0onΓ₀,𝑈=𝑢∈𝐻²(Ω)∣𝑢=𝜕𝑢𝜕𝜈=0onΓ₀. (17)

For a Banach space 𝑋, ‖⋅‖_𝑋 denotes the norm of 𝑋. For simplicity, we denote ‖⋅‖_𝐿²_(Ω) by ‖⋅‖. We define for all 1≤𝑝<∞‖𝑢‖^𝑝_𝑝=_Ω𝑢𝑥,𝑦^𝑝𝑑Ω. (18)

A simple calculation, based on the integration by parts formula, yields Δ²𝑢,𝑣=𝑎(𝑢,𝑣)+ℬ₂𝑢,𝑣_Γ−ℬ₁𝑢,𝜕𝑣𝜕𝜈_Γ, (19) where the bilinear symmetric form 𝑎(𝑢,𝑣) is given by 𝑎(𝑢,𝑣)=_Ω𝑢_𝑥𝑥𝑣_𝑥𝑥+𝑢_𝑦𝑦𝑣_𝑦𝑦+𝜇𝑢_𝑥𝑥𝑣_𝑦𝑦+𝑢_𝑦𝑦𝑣_𝑥𝑥+21−𝜇𝑢_𝑥𝑦𝑣_𝑥𝑦𝑑Ω, (20) where 𝑑Ω=𝑑𝑥𝑑𝑦. Since Γ₀≠∅, we know that 𝑎(𝑢,𝑢) is equivalent to the 𝐻²(Ω) norm; that is, 𝑐₀‖𝑢‖²_𝐻2(Ω)≤𝑎(𝑢,𝑢)≤𝑐₁‖𝑢‖²_𝐻2(Ω), (21) where 𝑐₀ and 𝑐₁ are generic positive constants. This and Sobolev embedding theorem imply that for some positive constants 𝐶_𝑝 and 𝐶_𝑠‖𝑢‖²≤𝐶_𝑝𝑎(𝑢,𝑢),‖∇𝑢‖²≤𝐶_𝑠𝑎(𝑢,𝑢),∀𝑢∈𝑈. (22)

We establish the following hypotheses on the relaxation function 𝑔 (see [8]). The relaxation function 𝑔∶ℝ₊→ℝ₊ is nonincreasing 𝐶¹ function satisfying 𝑔(0)>0,𝑙∶=^∞₀𝑔(𝑠)𝑔𝑑𝑠<1,_′(𝑡)≤0,^∞₀𝑒^𝛼𝑡𝑔(𝑡)𝑑𝑡<+∞,forsome𝛼>0. (23)(24)

To simplify calculation in our analysis, we introduce the following notation: 𝑔□𝑢∶=^𝑡₀𝑔(𝑡−𝑠)‖𝑢(⋅,𝑡)−𝑢(⋅,𝑠)‖²𝑑𝑠,𝑔□𝜕²𝑢∶=^𝑡₀𝑔(𝑡−𝑠)𝑎(𝑢(⋅,𝑡)−𝑢(⋅,𝑠),𝑢(⋅,𝑡)−𝑢(⋅,𝑠))𝑑𝑠. (25)

From the symmetry of 𝑎(⋅,⋅), we have that, for any 𝑣∈𝐶¹(0,𝑇;𝐻²(Ω)), 𝑎𝑔∗𝑣,𝑣^′1=−2𝑔(𝑡)𝑎()+1𝑣,𝑣2𝑔^′□𝜕²1𝑣−2𝑑×𝑑𝑡𝑔□𝜕²𝑣−^𝑡₀𝑔(𝑠)𝑑𝑠𝑎(𝑣,𝑣). (26)

Now, we introduce the relative results of the Airy stress function and von Kármán bracket [⋅,⋅].

Lemma 1 (see [30]). Let 𝑢,𝑤 be functions in 𝐻²(Ω) and 𝑣 in 𝐻²₀(Ω), where Ω is an open bounded and connected set of ℝ² with regular boundary. Then, _Ω𝑤[𝑣,𝑢]𝑑Ω=_Ω𝑣[𝑤,𝑢]𝑑Ω. (27)

Lemma 2 (see [20, 31]). If 𝑢,𝑣∈𝐻²(Ω), then [𝑢,𝑣]∈𝐿²(Ω) and satisfies ‖[𝑢,𝑣]‖≤𝑐‖𝑢‖_𝐻²_(Ω)‖𝑣‖_𝑊^2,∞_(Ω),‖𝑣‖_𝑊^2,∞_(Ω)≤𝑐‖𝑢‖²_𝐻2(Ω). (28)

The energy of problem (1)–(4) is given by 𝐸(𝑡)=1𝑢𝜌+2^′(𝑡)^𝜌+2_𝜌+2+12𝑎()+ℎ𝑢,𝑢2∇𝑢^′(𝑡)²+14‖Δ𝑣(𝑡)‖². (29)

The existence of solutions can be proved by the Faedo-Galerkin method; see [2, 9].

Theorem 3. Assume that the kernel 𝑔 is a positive continuous function satisfying (23). Let (𝑢₀,𝑢₁)∈𝐻⁴(Ω)×𝐻²(Ω). Then, the system (1)–(4) has a unique weak solution 𝑢 such that 𝑢∈𝐿^∞0,∞;𝑈∩𝐻⁴(Ω,𝑢_′)∈𝐿^∞0,∞;𝑉∩𝐻²(Ω,𝑢_′′)∈𝐿²0,∞;𝐻¹₀(Ω). (30)

3. Exponential Decay of the Energy

In this section we will prove the exponential decay rates. To demonstrate the stability of the system (1)–(4), the lemmas below are essential. The following result shows the dissipative property of the system (1)–(4). Multiplying (1) by 𝑢^′(𝑡), we get the identity 𝐸^′(𝑡)=𝑎𝑔∗𝑢,𝑢^′. (31)

Define the modified energy by 𝐹(𝑡)=1𝑢𝜌+2^{′𝜌+2𝜌+2}+ℎ2∇𝑢^′2+121−^𝑡₀𝑔(𝑠)𝑎()+1𝑑𝑠𝑢,𝑢2𝑔□𝜕²1𝑢+4‖Δ𝑣‖², (32) and applying (26) to (31), we have 𝐹^′(𝑡)1=−2𝑔(𝑡)𝑎()+1𝑢,𝑢2𝑔^′□𝜕²𝑢. (33)

This implies that 𝐹(𝑡) is nonincreasing, and one easily sees that 𝐸(𝑡)≤1𝐹(𝑡)1−𝑙,∀𝑡≥0. (34)

Therefore, it is enough to obtain the desired decay for the modified energy 𝐹(𝑡), which will be done below. The key point for showing our desired result is finding a Lyapunov functional 𝐿 which is equivalent to 𝐹(𝑡). First, we introduce three functionals and establish several lemmas. So, let Φ₁(𝑡)=^𝑡₀𝐺(𝛼;𝑡−𝑠)𝑎(𝑢(𝑠),𝑢(𝑠))𝑑𝑠 (35) with 𝐺(𝛼;𝑡)=𝑒^−𝛼𝑡_𝑡^+∞𝑒^𝛼𝑠𝑔(𝑠)Φ𝑑𝑠,₂(𝑡)=1𝜌+1_Ω𝑢^′𝜌𝑢^′𝑢𝑑Ω+ℎ_Ω∇𝑢^′Φ∇𝑢𝑑Ω,₃(𝑡)1=−𝜌+1_Ω𝑢^′𝜌𝑢^′𝑡₀𝑔(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠𝑑Ω−ℎ_Ω∇𝑢^′𝑡₀𝑔(𝑡−𝑠)(∇𝑢(𝑡)−∇𝑢(𝑠))𝑑𝑠𝑑Ω. (36)

We define the modified energy by 𝐿(𝑡)=𝑁𝐹(𝑡)+³_𝑖=1𝛾_𝑖Φ_𝑖(𝑡),𝑡≥0; (37) for some positive constants 𝛾_𝑖 is to be specified later.

Lemma 4. Assume that 𝑔 satisfies (23) and (24). For 𝑁>0 large enough, there exist 𝛼₁>0 and 𝛼₂>0 such that 𝛼₁𝐹(𝑡)≤𝐿(𝑡)≤𝛼₂𝐹(𝑡)+Φ₁(𝑡),∀𝑡≥0. (38)

Proof. From Young inequality, we deduce Φ₂(𝑡≤1)𝑢𝜌+2^{′𝜌+2𝜌+2}+1‖𝑢‖𝜌+1𝜌+2^𝜌+2𝜌+2+ℎ2∇𝑢^′2+ℎ2‖∇𝑢‖². (39)
Considering the embedding 𝐻¹₀(Ω)↪𝐿^𝜌+2(Ω) and taking (22) into account, it holds that Φ₂(𝑡≤1)𝑢𝜌+2^{′𝜌+2𝜌+2}+𝐶^𝜌+2‖‖𝜌+1𝜌+2∇𝑢^𝜌+2+ℎ2∇𝑢^′2+ℎ2‖∇𝑢‖²≤1𝑢𝜌+2^{′𝜌+2𝜌+2}+𝐶^𝜌+2𝐶_𝑠𝜌+1𝜌+22𝐶_𝑠𝐸(0)^𝜌/2+ℎ𝐶_𝑠2()+ℎ×𝑎𝑢,𝑢2∇𝑢^′2, (40) where 𝐶 comes from the inequality ‖𝑢‖_𝜌+2≤𝐶||∇𝑢|| for all 𝑢∈𝐻¹₀(Ω). On the other hand, by Young inequality, Hölder inequality and (22) can be estimated as Φ₃(𝑡≤ℎ)2∇𝑢^′2+ℎ2×_Ω^𝑡₀𝑔(𝑡−𝑠)|∇𝑢(𝑡)−∇𝑢(𝑠)|𝑑𝑠²+1𝑑Ω𝑢𝜌+2^{′𝜌+2𝜌+2}+1×𝜌+1𝜌+2_Ω𝑡0𝑔(𝑡−𝑠)|𝑢(𝑡)−𝑢(𝑠)|𝑑𝑠^𝜌+2≤ℎ𝑑Ω2∇𝑢^′2+ℎ𝑙𝐶_𝑠2𝑔□𝜕²1𝑢+𝑢𝜌+2^{′𝜌+2𝜌+2}+1𝜌+1𝜌+2^𝑡₀𝑔(𝑠)𝑑𝑠^𝜌+1×_Ω^𝑡₀𝑔(𝑡−𝑠)|𝑢(𝑡)−𝑢(𝑠)|^𝜌+2≤ℎ𝑑𝑠𝑑Ω2∇𝑢^′2+1𝑢𝜌+2^{′𝜌+2𝜌+2}+ℎ𝑙𝐶_𝑠2+𝑙^𝜌+1𝐶^𝜌+2𝐶_𝑠𝜌+1𝜌+22𝐶_𝑠𝐸(0)^𝜌/2𝑔□𝜕²𝑢. (41)
Thus, from (40) and (41) we obtain 𝐿(𝑡)−𝑁𝐹(𝑡)−𝛾₁Φ₁(𝑡≤1)𝛾𝜌+2²+𝛾₃𝑢^′𝜌+2_𝜌+2+ℎ2𝛾₂+𝛾₃∇𝑢^′2+𝐶^𝜌+2𝐶_𝑠𝜌+1𝜌+22𝐶_𝑠𝐸(0)^𝜌/2+ℎ𝐶_𝑠2𝛾₂𝑎()+𝑢,𝑢ℎ𝑙𝐶_𝑠2+𝑙^𝜌+1𝐶^𝜌+2𝐶_𝑠𝜌+1𝜌+22𝐶_𝑠𝐸(0)^𝜌/2×𝛾₃𝑔□𝜕²𝑢≤𝑐₀𝐹(𝑡), (42) where 𝑐₀ is a positive constant depending on 𝛾₂,𝛾₃,ℎ,𝜌,𝑙,𝐶, and 𝐶_𝑠. Choosing 𝑁>0 large, we complete the proof of Lemma 4.

Lemma 5. For each 𝑡₀>0 and sufficiently large 𝑁>0, there exists positive constant 𝑐₂ such that 𝐿^′(𝑡)≤−𝑐₂𝐹(𝑡)+Φ₁(𝑡),∀𝑡≥𝑡₀. (43)

Proof. By differentiating Φ₁(𝑡) and using Young inequality, we get Φ^′₁(𝑡)=𝑔_𝛼𝑎(𝑢,𝑢)−𝛼Φ₁(𝑡)−^𝑡₀𝑔()𝑎(𝑢(𝑠)(𝑠=𝑡−𝑠,𝑢))𝑑𝑠,𝑔_𝛼𝑎(𝑢,𝑢)−𝛼Φ₁(𝑡)−𝑔□𝜕²𝑢+^𝑡₀𝑔(𝑠)𝑑𝑠𝑎(𝑢,𝑢)−2^𝑡₀𝑔(𝑡−𝑠)𝑎(𝑢(𝑡),𝑢(𝑠))𝑑𝑠≤−𝛼Φ₁(𝑡)−𝑔□𝜕²𝑢+𝑔_𝛼+1−4𝛿^𝑡₀𝑔(𝑠)𝑑𝑠𝑎(𝑢,𝑢)+𝛿𝑙𝑔□𝜕²𝑢, (44) where 𝑔_𝛼=∫^∞₀𝑒^𝛼𝑠|𝑔(𝑠)|𝑑𝑠,𝛼>0, and 𝛿>0. Using (1)–(4), we have Φ^′₂(𝑡)()+])+1=−𝑎𝑢,𝑢+𝑎𝑔∗𝑢,𝑢([𝑢,𝑣,𝑢𝑢𝜌+1^{′𝜌+2𝜌+2}+ℎ∇𝑢^′2. (45)
We use the following inequality: 𝑎𝑔∗𝑢,𝑢=^𝑡₀𝑔()𝑎(𝑢(𝑠)(𝑡)(𝑡+𝑡−𝑠−𝑢,𝑢))𝑑𝑠_𝑡0𝑔(𝑠)()≤𝑑𝑠𝑎𝑢,𝑢𝜂+^𝑡₀𝑔(𝑠)𝑎()+𝑙𝑑𝑠𝑢,𝑢4𝜂𝑔□𝜕²𝑢; (46) then we obtain Φ^′₂(𝑡)≤1𝑢𝜌+1^{′𝜌+2𝜌+2}+ℎ∇𝑢^′2−1−𝜂−^𝑡₀𝑔(𝑠)𝑑𝑠𝑎(𝑢,𝑢)−‖Δ𝑣‖²+𝑙4𝜂𝑔□𝜕²𝑢, (47) where 𝜂>0. Similarly we deduce Φ^′₃(𝑡)=^𝑡₀𝑔()𝑎(𝑢(𝑡)(𝑠)(𝑡−𝑡−𝑠−𝑢,𝑢))𝑑𝑠_𝑡0𝑔(𝑢(𝑡)(𝑠),[−𝑡−𝑠)(−𝑢𝑢,𝑣])𝑑𝑠_𝑡0𝑔(𝑡−𝑠)𝑎𝑢(𝑡)−𝑢_𝑡0(𝑠),𝑔(𝑡−𝜏)𝑢(𝜏)𝑑𝜏𝑑𝑠−ℎ^𝑡₀𝑔^′(𝑡−𝑠)∇𝑢(𝑡)−∇𝑢(𝑠),∇𝑢^′(𝑡)𝑑𝑠−ℎ^𝑡₀𝑔(𝑠)𝑑𝑠∇𝑢^′2−1×𝜌+1_𝑡0𝑔^′(𝑡−𝑠)𝑢(𝑡)−𝑢(𝑠),𝑢^′𝜌𝑢^′−1𝑑𝑠𝜌+1^𝑡₀𝑔(𝑠)𝑑𝑠𝑢^′𝜌+2_𝜌+2=1−^𝑡₀𝑔(𝑠)𝑑𝑠^𝑡₀𝑔()(𝑢(𝑡)(𝑠)(𝑡−𝑡−𝑠×𝑎−𝑢,𝑢))𝑑𝑠_𝑡0𝑔(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠),[𝑢,𝑣])𝑑𝑠+^𝑡₀𝑔()𝑢(𝑡)(𝑠),𝑡−𝑠×𝑎−𝑢_𝑡0𝑔(𝑡−𝜏)(𝑢(𝑡)−𝑢(𝜏))𝑑𝜏𝑑𝑠−ℎ^𝑡₀𝑔^′(𝑡−𝑠)∇𝑢(𝑡)−∇𝑢(𝑠),∇𝑢^′(𝑡)𝑑𝑠−ℎ^𝑡₀𝑔(𝑠)𝑑𝑠∇𝑢^′2−1×𝜌+1_𝑡0𝑔^′(𝑡−𝑠)𝑢(𝑡)−𝑢(𝑠),𝑢^′𝜌𝑢^′−1𝑑𝑠𝜌+1^𝑡₀𝑔(𝑠)𝑑𝑠𝑢^′𝜌+2_𝜌+2∶=𝐼₁+𝐼₂+⋯+𝐼₅−ℎ^𝑡₀𝑔(𝑠)𝑑𝑠∇𝑢^′2−1𝜌+1^𝑡₀𝑔(𝑠)𝑑𝑠𝑢^′𝜌+2_𝜌+2. (48)
Now, we estimate the terms in the right hand side of (48). The Young and Hölder inequalities and (22) give that 𝐼₁+𝐼₃≤1−^𝑡₀𝑔(𝑠)()+𝑙𝑑𝑠𝜂𝑎𝑢,𝑢4𝜂𝑔□𝜕²𝑢+𝑙𝑔□𝜕²𝐼𝑢,₄≤ℎ𝜂∇𝑢^′2+ℎ4𝜂_Ω^𝑡₀𝑔^′(𝑡−𝑠)|∇𝑢(𝑡)−∇𝑢(𝑠)|𝑑𝑠²𝑑Ω≤ℎ𝜂∇𝑢^′2−𝑔(0)𝐶_𝑠ℎ𝑔4𝜂^′□𝜕²𝐼𝑢,₅≤𝜂𝑢𝜌+1^{′22(𝜌+1)𝜌+1}+1×4𝜂𝜌+1_Ω𝑡0𝑔^′(𝑡−𝑠)|𝑢(𝑡)−𝑢(𝑠)|𝑑𝑠²≤𝑑Ω𝜂𝐶^2(𝜌+1)𝜌+1∇𝑢^′2(𝜌+1)−𝑔(0)×4𝜂𝜌+1_Ω𝑡0𝑔^′(𝑡−𝑠)|𝑢(𝑡)−𝑢(𝑠)|²≤𝑎𝑑𝑠𝑑Ω₀𝜂𝜌+1∇𝑢^′2−𝑔(0)𝐶_𝑝𝑔4𝜂𝜌+1^′□𝜕²𝑢, (49) where 𝑎₀=𝐶^2(𝜌+1)(2ℎ⁻¹𝐸(0))^𝜌>0. From Lemmas 1 and 2 and (22), we obtain 𝐼₂=^𝑡₀𝑔(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠,[𝑢,𝑣]≤𝜂‖[𝑢,𝑣]‖²+14𝜂^𝑡₀𝑔(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠²≤𝜂𝑐‖𝑢‖_𝐻²_(Ω)‖𝑣‖_𝑊^2,∞_(Ω)²+𝑙𝐶_𝑝4𝜂𝑔□𝜕²𝑢≤𝜂𝐶₀𝑎(𝑢,𝑢)+𝑙𝐶_𝑝4𝜂𝑔□𝜕²𝑢. (50)
Summarizing these estimates with (48), we deduce that Φ^′₃(𝑡)≤𝑎ℎ𝜂+⁰𝜂𝜌+1−ℎ^𝑡₀𝑔(𝑠)𝑑𝑠∇𝑢^′2+𝑙𝑙++4𝜂𝑙𝐶_𝑝4𝜂𝑔□𝜕²𝑢+𝜂1+𝐶₀𝑎()−𝑔(0)𝑢,𝑢𝐶4𝜂^𝑠𝐶ℎ+^𝑝𝑔𝜌+1^′□𝜕²𝑢−1𝜌+1^𝑡₀𝑔(𝑠)𝑑𝑠𝑢^′𝜌+2_𝜌+2. (51)
Since 𝑔 is continuous and positive, for any 𝑡≥𝑡₀>0 we have ^𝑡₀𝑔(𝑠)𝑑𝑠≥^𝑡0₀𝑔(𝑠)𝑑𝑠∶=𝑔₀>0. (52)
Thus, making use of (52) and combining (33), (37), (44), (47), and (51), we obtain 𝐿^′(𝑡)≤−ℎ𝑔₀−ℎ𝜂−𝜂𝑎₀𝛾𝜌+1³−ℎ𝛾₂∇𝑢^′2+𝑁2−𝑔(0)𝐶4𝜂^𝑠𝐶ℎ+^𝑝𝛾𝜌+1³𝑔^′□𝜕²𝑢−𝑁2𝑔(𝑡)+1−𝜂−^𝑡₀𝑔(𝑠)𝑑𝑠𝛾₂−𝑔_𝛼+14𝛿−𝑔₀𝛾₁−𝜂1+𝐶₀𝛾₃𝑎(𝑢,𝑢)−𝛼𝛾₁Φ₁(𝑡)−𝛾₂‖Δ𝑣‖²−(1−𝛿𝑙)𝛾₁−𝑙𝛾₂−𝑙4𝜂𝑙++4𝜂𝑙𝐶_𝑝𝛾4𝜂³𝑔□𝜕²𝑢−1𝑔𝜌+1⁰𝛾₃−𝛾₂𝑢^′𝜌+2_𝜌+2,∀𝑡≥𝑡₀. (53)
We first take 𝛾₂>0 and 𝛿>0 so small that 𝑔₀𝛾₃−𝛾₂>0,1−𝛿𝑙>0, (54) respectively. And then, we choose 𝜂>0 and 𝛾₃>0 so small that 𝑔₀−𝜂−𝜂𝑎₀ℎ𝛾𝜌+1³−𝛾₂>0,1−𝜂−^𝑡₀𝑔(𝑠)𝑑𝑠>0, (55) respectively. We then pick 𝛾₁ large enough so that (1−𝛿𝑙)𝛾₁−𝑙𝛾₂−𝑙4𝜂𝑙++4𝜂𝑙𝐶_𝑝𝛾4𝜂³>0. (56)
Finally, taking 𝑁>0 large enough and by (53), we conclude that 𝐿^′(𝑡)≤−𝑐₂𝐹(𝑡)+Φ₁(𝑡),∀𝑡≥𝑡₀, (57) for some 𝑐₂>0.

Our main result reads as follows.

Theorem 6. Suppose that 𝑔 satisfies (23) and (24). Then, for each 𝑡₀>0, there exist two positive constants 𝐶₁ and 𝛽 such that 𝐸(𝑡)≤𝐶₁𝑒^−𝛽𝑡,∀𝑡≥𝑡₀. (58)

Proof. From (38) and (43), we have 𝐿^′(𝑡)𝑐≤−²𝛼₂𝐿(𝑡),∀𝑡≥𝑡₀. (59)
Integrating this over (𝑡₀,𝑡), we obtain 𝐿(𝑡)≤𝐿𝑡₀𝑒^{−𝛽(𝑡−𝑡0)},∀𝑡≥𝑡₀, (60) with 𝛽=𝑐₂/𝛼₂. Consequently, (34), (38), and (60) yield the result in Theorem 6.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (2012R1A1A3011630).