General Decay for the Degenerate Equation with a Memory Condition at the Boundary

Abstract

We consider a degenerate equation with a memory condition at the boundary. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.

1. Introduction

The main purpose of this paper is to investigate the asymptotic behavior of the solutions of the degenerate equation with a memory condition at the boundary 𝐾(𝑥)𝑢^′′+Δ²𝑢+𝑓(𝑢)=0in𝑄=Ω×(0,∞),𝑢=𝜕𝑢𝜕𝜈=0onΓ₀×(0,∞),−𝑢+^𝑡₀𝑔₁(𝑡−𝑠)ℬ₂𝑢(𝑠)𝑑𝑠=0onΓ₁×(0,∞),𝜕𝑢+𝜕𝜈^𝑡₀𝑔₂(𝑡−𝑠)ℬ₁𝑢(𝑠)𝑑𝑠=0onΓ₁×(),𝑢()0,∞0,𝑥=𝑢₀(𝑥),𝑢^′(0,𝑥)=𝑢₁(𝑥)inΩ, (1)(2)(3)(4)(5) where Ω is a bounded domain of ℝ^𝑛 with a smooth boundary Γ and let us assume that Γ, can be divided into two nonnull parts Γ=Γ₀∪Γ₁ and Γ₀∩Γ₁=∅ and 𝐾∈𝐶¹(Ω) and 𝐾(𝑥)≥0 for all 𝑥∈Ω which satisfies some appropriate conditions. 𝜈 is the unit outward normal to Γ and 𝜏=(−𝜈₂,𝜈₁) the corresponding unit tangent vector. Here, the relaxation functions 𝑔_𝑖 (𝑖=1,2) are positive and nondecreasing, the function 𝑓∈𝐶¹(ℝ) and ℬ₁𝑢=Δ𝑢+1−𝜇𝐵₁𝑢,ℬ₂𝑢=𝜕Δ𝑢+𝜕𝜈1−𝜇𝜕𝐵₂𝑢,𝐵𝜕𝜏₁𝑢=2𝜈₁𝜈₂𝑢_𝑥𝑦−𝜈²₁𝑢_𝑦𝑦−𝜈²₂𝑢_𝑥𝑥,𝐵₂𝑢=𝜈²₁−𝜈²₂𝑢_𝑥𝑦+𝜈₁𝜈₂𝑢_𝑦𝑦−𝑢_𝑥𝑥, (6) and the constant 𝜇, 0<𝜇<1/2, represents Poisson’s ratio.

From the physical point of view, we know that the memory effect described in integral equations (3) and (4) can be caused by the interaction with another viscoelastic element. In fact, the boundary conditions (3) and (4) mean that Ω is composed of a material which is clamped in a rigid body in Γ₀ and is clamped in a body with viscoelastic properties in the complementary part of its boundary named Γ₁. Problems related to (1)–(5) are interesting not only from the point of view of PDE general theory, but also due to its applications in mechanics.

The existence of global solutions and exponential decay to the degenerate equation with 𝜕Ω=Γ₀ has been investigated by several authors. See Cavalcanti et al. [1] and Menezes et al. [2]. For instance, when 𝐾(𝑥) is equal to 1, (1) describes the transverse deflection 𝑢(𝑥,𝑡) of beams. There exists a large body of literature regarding viscoelastic problems with the memory term acting in the domain or at the boundary (see [3–17]). Santos et al. [18] studied the asymptotic behavior of the solutions of a nonlinear wave equation of Kirchhoff type with boundary condition of memory type. Cavalcanti et al. [19] proved the uniform decay rates of solutions to a degenerate system with a memory condition at the boundary. Santos and Junior [20] investigated the stability of solutions for Kirchhoff plate equations with a boundary memory condition. Rivera et al. [21] showed the asymptotic behavior to a von Karman plate with boundary memory conditions. Park and Kang [22] studied the exponential decay for the Kirchhoff plate equations with nonlinear dissipation and boundary memory condition. They proved that the energy decays uniformly exponentially or algebraically with the same rate of decay as the relaxation functions. In the present work, we generalize the earlier decay results of the solution of (1)–(5). More precisely, we show that the energy decays at the rate similar to the relaxation functions, which are not necessarily decaying like polynomial or exponential functions. In fact, our result allows a larger class of relaxation functions. Recently, Messaoudi and Soufyane [23], Mustafa and Messaoudi [24], and Santos and Soufyane [25] proved the general decay for the wave equation, Timoshenko system, and von Karman plate system with viscoelastic boundary conditions, respectively.

The organization of this paper is as follows. In Section 2, we present some notations and material needed for our work and state the existence result to system (1)–(5). In Section 3, we prove the general decay of the solutions to the degenerate equation with a memory condition at the boundary.

2. Preliminaries

In this section, we introduce some notations and establish the existence of solutions of the problem (1)–(5).

Note that, because of condition (2), the solution of system (1)–(5) must belong to the following space: 𝑊=𝑣∈𝐻²(Ω)∶𝑣=𝜕𝑣𝜕𝜈=0onΓ₀. (7) Let us define the bilinear form 𝑎(⋅,⋅) as follows: 𝑎(𝑢,𝑣)=_Ω𝑢_𝑥𝑥𝑣_𝑥𝑥+𝑢_𝑦𝑦𝑣_𝑦𝑦+𝜇𝑢_𝑥𝑥𝑣_𝑦𝑦+𝑢_𝑦𝑦𝑣_𝑥𝑥+21−𝜇𝑢_𝑥𝑦𝑣_𝑥𝑦𝑑𝑥𝑑𝑦. (8) Since Γ₀≠∅, we know that 𝑎(𝑢,𝑢) is equivalent to the 𝐻²(Ω) norm on 𝑊; that is, 𝑐₀‖𝑢‖²_𝐻2(Ω)≤𝑎(𝑢,𝑢)≤𝐶₀‖𝑢‖²_𝐻2(Ω), (9) and here and in the sequel, we denote by 𝑐₀ and 𝐶₀ generic positive constants. Simple calculation, based on the integration by parts formula, yields Δ²𝑢,𝑣=𝑎(𝑢,𝑣)+ℬ₂𝑢,𝑣_Γ−ℬ₁𝑢,𝜕𝑣𝜕𝜈_Γ. (10) We assume that there exists 𝑥₀∈ℝ^𝑛 such that Γ₀=𝑥∈Γ∶𝜈(𝑥)⋅𝑥−𝑥₀,Γ≤0₁=𝑥∈Γ∶𝜈(𝑥)⋅𝑥−𝑥₀>0. (11)(12) If we denote the compactness of Γ₁ by 𝑚(𝑥)=𝑥−𝑥₀, condition (12) implies that there exists a small positive constant 𝛿₀ such that 0<𝛿₀≤𝑚(𝑥)⋅𝜈(𝑥), for all 𝑥∈Γ₁.

Next, we will use (3) and (4) to estimate the values ℬ₁ and ℬ₂ on Γ₁. Denoting by 𝑔∗𝜑(𝑡)=^𝑡₀𝑔(𝑡−𝑠)𝜑(𝑠)𝑑𝑠 (13) the convolution product operator and differentiating (3) and (4), we arrive at the following Volterra equations: ℬ₂1𝑢+𝑔₁(0)𝑔^′₁∗ℬ₂1𝑢=𝑔₁(0)𝑢^′,ℬ₁1𝑢+𝑔₂(0)𝑔^′₂∗ℬ₁1𝑢=−𝑔₂(0)𝜕𝑢^′.𝜕𝜈 (14) Applying the Volterra inverse operator, we get ℬ₂1𝑢=𝑔₁(0)𝑢^′+𝑘₁∗𝑢^′,ℬ₁1𝑢=−𝑔₂(0)𝜕𝑢^′𝜕𝜈+𝑘₂∗𝜕𝑢^′,𝜕𝜈 (15) where the resolvent kernels satisfy 𝑘_𝑖+1𝑔_𝑖(0)𝑔^′_𝑖∗𝑘_𝑖1=−𝑔_𝑖(0)𝑔^′_𝑖,∀𝑖=1,2. (16) Denoting that 𝜏₁=1/𝑔₁(0) and 𝜏₂=1/𝑔₂(0), we have ℬ₂𝑢=𝜏₁𝑢^′+𝑘₁(0)𝑢−𝑘₁(𝑡)𝑢₀+𝑘^′₁,ℬ∗𝑢₁𝑢=−𝜏₂𝜕𝑢^′𝜕𝜈+𝑘₂(0)𝜕𝑢𝜕𝜈−𝑘₂(𝑡)𝜕𝑢₀𝜕𝜈+𝑘^′₂∗𝜕𝑢.𝜕𝜈 (17) Therefore, we use (17) instead of the boundary conditions (3) and (4).

Let us denote that 𝑔⋄𝑣(𝑡)∶=^𝑡₀𝑔(𝑡−𝑠)|𝑣(𝑡)−𝑣(𝑠)|²𝑑𝑠. (18) The following lemma states an important property of the convolution operator.

Lemma 1. For 𝑔,𝑣∈𝐶¹([0,∞)∶ℝ), one has 𝑔∗𝑣𝑣^′1=−2𝑔(𝑡)|𝑣(𝑡)|²+12𝑔^′1⋄𝑣−2𝑑×𝑑𝑡𝑔⋄𝑣−^𝑡₀𝑔(𝑠)𝑑𝑠|𝑣|². (19)

The proof of this lemma follows by differentiating the term 𝑔⋄𝑣.

Lemma 2 (see [26]). Suppose that 𝑓∈𝐿²(Ω),  𝑔∈𝐻^1/2(Γ₁) and ℎ∈𝐻^3/2(Γ₁); then, any solution of 𝑎(𝑣,𝑤)=_Ω𝑓𝑤𝑑𝑥+_Γ1𝑔𝑤𝑑Γ+_Γ1ℎ𝜕𝑤𝜕𝜈𝑑Γ,∀𝑤∈𝑊, (20) satisfies 𝑣∈𝐻⁴(Ω) and also Δ²𝑣=𝑓,𝑣=𝜕𝑣𝜕𝜈=0𝑜𝑛Γ₀,ℬ₁𝑣=ℎ,ℬ₂𝑣=𝑔𝑜𝑛Γ₁. (21)

We formulate the following assumptions.(A1) Let 𝑓∈𝐶¹(ℝ) satisfy 𝑓(𝑠)𝑠≥0,∀𝑠∈ℝ. (22)Additionally, we suppose that 𝑓 is superlinear; that is, 𝑓(𝑠)𝑠≥2+𝜂𝐹(𝑠),𝐹(𝑧)=^𝑧₀𝑓(𝑠)𝑑𝑠,∀𝑠∈ℝ, (23) for some 𝜂>0 with the following growth condition: 𝑓(𝑥)−𝑓𝑦≤𝑐1+|𝑥|^𝜌−1+𝑦^𝜌−1𝑥−𝑦,∀𝑥,𝑦∈ℝ (24) for some 𝑐>0 and 𝜌≥1 such that (𝑛−2)𝜌≤𝑛.(A2)𝐾∈𝐶¹(Ω);  𝐻²₀(Ω)∩𝐿^∞(Ω) with 𝐾(𝑥)≥0, for all 𝑥∈Ω, and satisfy the following condition ∇𝐾⋅𝑚≥0inΩ. (25)

The well-posedness of system (1)–(5) is given by the following theorem.

Theorem 3 (see [27]). Consider assumptions (A1)-(A2) and let 𝑘_𝑖∈𝐶²(ℝ⁺) be such that 𝑘_𝑖,−𝑘^′_𝑖,𝑘_𝑖^′′≥0(𝑖=1,2). (26) If 𝑢₀∈𝑊∩𝐻⁴(Ω), 𝑢₁∈𝑊, satisfying the compatibility condition ℬ₁𝑢₀=−𝜏₂𝜕𝑢₁𝜕𝜈,ℬ₂𝑢₀=𝜏₁𝑢₁𝑜𝑛Γ₁, (27) then there is only one solution 𝑢 of the system (1)–(5) satisfying 𝑢∈𝐿^∞0,𝑇∶𝑊∩𝐻⁴(Ω),𝑢^′∈𝐿^∞(),𝑢0,𝑇∶𝑊_′′∈𝐿^∞0,𝑇∶𝐿²(Ω). (28)

3. General Decay

In this section, we show that the solution of system (1)–(5) may have a general decay not necessarily of exponential or polynomial type. For this we consider that the resolvent kernels satisfy the following hypothesis.

(H) 𝑘_𝑖∶ℝ₊→ℝ₊ is twice differentiable function such that 𝑘_𝑖(0)>0,lim_𝑡→∞𝑘_𝑖(𝑡)=0,𝑘^′_𝑖(𝑡)≤0, (29) and there exists a nonincreasing continuous function 𝜉_𝑖∶ℝ₊→ℝ₊ satisfying 𝑘_𝑖^′′(𝑡)≥−𝜉_𝑖(𝑡)𝑘^′_𝑖(𝑡),𝑖=1,2,∀𝑡≥0. (30)

The following identity will be used later.

Lemma 4 (see [26]). For every 𝑣∈𝐻⁴(Ω) and for every 𝜇∈ℝ, one has _Ω(𝑚⋅∇𝑣)Δ²()+1𝑣𝑑𝑥=𝑎𝑣,𝑣2_Γ𝑚⋅𝜈𝑣²_𝑥𝑥+𝑣²_𝑦𝑦+2𝜇𝑣_𝑥𝑥𝑣_𝑦𝑦+21−𝜇𝑣²_𝑥𝑦𝑑Γ+_Γℬ₂𝑣𝑚⋅∇𝑣−ℬ₁𝑣𝜕(𝜕𝜈𝑚⋅∇𝑣)𝑑Γ. (31)

Let us introduce the energy function 𝐸(𝑡)=12_Ω𝐾(𝑥)𝑢^′21𝑑𝑥+2𝑎(𝑢,𝑢)+_Ω𝐹(𝑢)𝜏𝑑𝑥+¹2_Γ1𝑘₁(𝑡)|𝑢|²−𝑘^′₁+𝜏⋄𝑢𝑑Γ₂2_Γ1𝑘₂(𝑡)𝜕𝑢𝜕𝜈²−𝑘^′₂⋄𝜕𝑢𝜕𝜈𝑑Γ. (32) Now, we establish some inequalities for the strong solution of system (1)–(5).

Lemma 5. The energy functional 𝐸 satisfies, along the solution of (1)–(5), the estimate 𝐸^′(𝑡)𝜏≤−¹2_Γ1𝑢^′2−𝑘²₁(𝑡)𝑢₀²−𝑘^′₁(𝑡)|𝑢|²+𝑘₁^′′−𝜏⋄𝑢𝑑Γ₂2_Γ1𝜕𝑢^′𝜕𝜈²−𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²−𝑘^′₂(𝑡)𝜕𝑢𝜕𝜈²+𝑘₂^′′⋄𝜕𝑢𝜕𝜈𝑑Γ. (33)

Proof. Multiplying (1) by 𝑢^′, integrating over Ω, and using (10), we get 12𝑑𝑑𝑡_Ω𝐾𝑢^′2𝑑𝑥+𝑎(𝑢,𝑢)+2_Ω𝐹(𝑢)𝑑𝑥=−_Γ1ℬ₂𝑢𝑢^′𝑑Γ+_Γ1ℬ₁𝑢𝜕𝑢^′𝜕𝜈𝑑Γ. (34) Substituting the boundary terms by (17) and using Lemma 1 and the Young inequality, our conclusion follows.

Let us consider the following binary operator: (𝑘∘𝑢)(𝑡)∶=^𝑡₀𝑘(𝑡−𝑠)(𝑢(𝑡)−𝑢(𝑠))𝑑𝑠. (35) Then applying the Holder inequality for 0≤𝛼≤1 we have |(𝑘∘𝑢)(𝑡)|²≤^𝑡₀|𝑘(𝑠)|^2(1−𝛼)𝑑𝑠|𝑘|^2𝛼⋄𝑢(𝑡). (36) Let us define the functional 𝜓(𝑡)=_Ω𝑛𝑚⋅∇𝑢+2−𝜃𝑢𝐾𝑢^′𝑑𝑥. (37) The following lemma plays an important role in the construction of the desired functional.

Lemma 6. Suppose that the initial data (𝑢₀,𝑢₁)∈(𝐻⁴(Ω)∩𝑊)×𝑊, satisfying the compatibility condition (27). Then, the solution of system (1)–(5) satisfies 𝜓^′(𝑡)≤12_Γ1𝑚⋅𝜈𝐾𝑢^′2𝑑Γ−𝜃_Ω𝐾𝑢^′2𝑛𝑑𝑥−1+2−𝜃−𝜖𝜆₀𝑎()−𝑢,𝑢𝑛𝜂2−2𝜃−𝜂𝜃_Ω𝐹(𝑢)𝑑𝑥+2𝜏₁²𝜖×_Γ1𝑢^′2+𝑘²₁(𝑡)|𝑢|²+𝑘²₁(𝑡)𝑢₀²+𝑘^′₁∘𝑢²+𝑑Γ2𝜏₂²𝜖_Γ1𝜕𝑢^′𝜕𝜈²+𝑘²₂(𝑡)𝜕𝑢𝜕𝜈²+𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²+𝑘^′₂∘𝜕𝑢𝜕𝜈²−1𝑑Γ2−𝜖𝜆₀𝛿_0Γ1𝑚⋅𝜈𝑢²_𝑥𝑥+𝑢²_𝑦𝑦+2𝜇𝑢_𝑥𝑥𝑢_𝑦𝑦+21−𝜇𝑢²_𝑥𝑦𝑑Γ. (38)

Proof. Differentiating 𝜓 using (1) and Lemma 4, we get 𝜓^′(𝑡)=_Ω𝑚⋅∇𝑢^′+𝑛2−𝜃𝑢^′𝐾𝑢^′+𝑑𝑥_Ω𝑛𝑚⋅∇𝑢+2−𝜃𝑢𝐾𝑢^′′=1𝑑𝑥2_Γ1𝑚⋅𝜈𝐾𝑢^′2𝑑Γ−𝜃_Ω𝐾𝑢^′2−1𝑑𝑥2_Ω∇𝐾⋅𝑚𝑢^′2𝑛𝑑𝑥−1+2−𝜃𝑎(𝑢,𝑢)+𝑛_Ω𝐹(𝑢)𝑛𝑑𝑥−2−𝜃_Ω𝑓(𝑢)−1𝑢𝑑𝑥2_Γ𝑚⋅𝜈𝑢²_𝑥𝑥+𝑢²_𝑦𝑦+2𝜇𝑢_𝑥𝑥𝑢_𝑦𝑦+21−𝜇𝑢²_𝑥𝑦−𝑑Γ_Γℬ₂𝑢)+𝑛(𝑚⋅∇𝑢2𝑢+−𝜃𝑑Γ_Γℬ₁𝑢𝜕()+𝑛𝜕𝜈𝑚⋅∇𝑢2−𝜃𝜕𝑢𝜕𝜈𝑑Γ. (39) Let us next examine the integrals over Γ₀ in (39). Since 𝑢=𝜕𝑢/𝜕𝜈=0 on Γ₀, we have 𝐵₁𝑢=𝐵₂𝑢=0 on Γ₀ and 𝜕()=()𝑢𝜕𝜈𝑚⋅∇𝑢𝑚⋅𝜈Δ𝑢,_2𝑥𝑥+𝑢²_𝑦𝑦+2𝜇𝑢_𝑥𝑥𝑢_𝑦𝑦+21−𝜇𝑢²_𝑥𝑦=(Δ𝑢)²onΓ₀, (40) since 𝑢_𝑥𝑥𝑢_𝑦𝑦−𝑢²_𝑥𝑦=0onΓ₀. (41) Therefore, from (39) and (40), we have 𝜓^′(𝑡)≤12_Γ1𝑚⋅𝜈𝐾𝑢^′2𝑑Γ−𝜃_Ω𝐾𝑢^′2−1𝑑𝑥2_Ω∇𝐾⋅𝑚𝑢^′2𝑛𝑑𝑥−1+2−𝜃𝑎(𝑢,𝑢)+𝑛_Ω𝐹(𝑢)𝑛𝑑𝑥−2−𝜃_Ω𝑓(𝑢)+1𝑢𝑑𝑥2_Γ0𝑚⋅𝜈(Δ𝑢)²−1𝑑Γ2_Γ1𝑚⋅𝜈𝑢²_𝑥𝑥+𝑢²_𝑦𝑦+2𝜇𝑢_𝑥𝑥𝑢_𝑦𝑦+21−𝜇𝑢²_𝑥𝑦𝑑Γ−_Γ1ℬ₂𝑢)+𝑛(𝑚⋅∇𝑢2𝑢+−𝜃𝑑Γ_Γ1ℬ₁𝑢𝜕()+𝑛𝜕𝜈𝑚⋅∇𝑢2−𝜃𝜕𝑢𝜕𝜈𝑑Γ. (42) Using the Young inequality, we get _Γ1ℬ₂𝑢)+𝑛(𝑚⋅∇𝑢2𝑢≤1−𝜃𝑑Γ2𝜖_Γ1ℬ₂𝑢²𝑑Γ+𝜖_Γ1|𝑚⋅∇𝑢|²+𝑛2−𝜃²|𝑢|²𝑑Γ,_Γ1ℬ₁𝑢𝜕()+𝑛𝜕𝜈𝑚⋅∇𝑢2−𝜃𝜕𝑢≤1𝜕𝜈𝑑Γ2𝜖_Γ1ℬ₁𝑢²𝑑Γ+𝜖_Γ1𝜕(𝜕𝜈𝑚⋅∇𝑢)²+𝑛2−𝜃²𝜕𝑢𝜕𝜈²𝑑Γ, (43)(44) where 𝜖 is a positive constant. Since the bilinear form 𝑎(𝑢,𝑢) is strictly coercive on 𝑊, using the trace theory, we obtain _Γ1|𝑚⋅∇𝑢|²+𝑛2−𝜃²|𝑢|²+𝑑Γ_Γ1𝜕(𝜕𝜈𝑚⋅∇𝑢)²+𝑛2−𝜃²𝜕𝑢𝜕𝜈²𝑑Γ≤𝜆₀𝑎()+𝜆𝑢,𝑢⁰𝛿₀×_Γ1𝑚⋅𝜈𝑢²_𝑥𝑥+𝑢²_𝑦𝑦+2𝜇𝑢_𝑥𝑥𝑢_𝑦𝑦+21−𝜇𝑢²_𝑥𝑦𝑑Γ, (45) where 𝜆₀ is a constant depending on Ω,  𝜇,  𝜃 and 𝑛. Substituting inequalities (43)–(45) into (42) and taking into account that 𝑚⋅𝜈≤0 on Γ₀, as well as (23) and (25), we have 𝜓^′(𝑡)≤12_Γ1𝑚⋅𝜈𝐾𝑢^′2𝑑Γ−𝜃_Ω𝐾𝑢^′2−𝑛𝑑𝑥1+2−𝜃−𝜖𝜆₀𝑎()−𝑢,𝑢𝑛𝜂2−2𝜃−𝜂𝜃_Ω𝐹(𝑢)+1𝑑𝑥2𝜖_Γ1ℬ₁𝑢²+ℬ₂𝑢²−1𝑑Γ2−𝜖𝜆₀𝛿_0Γ1𝑚⋅𝜈𝑢²_𝑥𝑥+𝑢²_𝑦𝑦+2𝜇𝑢_𝑥𝑥𝑢_𝑦𝑦+21−𝜇𝑢²_𝑥𝑦𝑑Γ. (46) Since the boundary conditions (17) can be written as ℬ₂𝑢=𝜏₁𝑢^′+𝑘₁(𝑡)𝑢−𝑘₁(𝑡)𝑢₀−𝑘^′₁,ℬ∘𝑢₁𝑢=−𝜏₂𝜕𝑢^′𝜕𝜈+𝑘₂(𝑡)𝜕𝑢𝜕𝜈−𝑘₂(𝑡)𝜕𝑢₀𝜕𝜈−𝑘^′₂∘𝜕𝑢,𝜕𝜈 (47) our conclusion follows.

Let us introduce the Lyapunov functional ℒ(𝑡)=𝑁𝐸(𝑡)+𝜓(𝑡), (48) with 𝑁>0. Now, we are in a position to show the main result of this paper.

Theorem 7. Let (𝑢₀,𝑢₁)∈𝑊×𝐿²(Ω). Suppose that the resolvent kernels 𝑘₁,  𝑘₂ satisfy the condition (H). Then, there exist constants 𝜔,𝐶>0 such that, for some 𝑡₀ large enough, the solution of (1)–(5) satisfies 𝐸(𝑡)≤𝐶𝐸(0)𝑒^{−𝜔∫𝑡0𝜉(𝑠)𝑑𝑠},∀𝑡≥𝑡₀,if𝑢₀=𝜕𝑢₀𝜕𝜈=0𝑜𝑛Γ₁. (49) Otherwise, 𝐸(𝑡)≤𝐶𝐸(0)+^𝑡₀𝑘₀(𝑠)𝑒^{𝜔∫𝑠𝑡0𝜉(𝜏)𝑑𝜏}𝑑𝑠𝑒^{−𝜔∫𝑡0𝜉(𝑠)𝑑𝑠}, (50) for all 𝑡≥𝑡₀, where 𝜉(𝑡)=min𝜉₁(𝑡),𝜉₂(𝑡,𝑘₀)(𝑡)=_Γ1𝑘²₁(𝑡)𝑢₀²𝑑Γ+_Γ1𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²𝑑Γ. (51)

Proof. Applying inequality (36) with 𝛼=1/2 in Lemma 6 and from Lemma 5, we obtain ℒ^′(𝑡)≤−_Ω𝜃𝐾𝑢^′2−𝜏𝑑𝑥₁𝑁2_Γ1𝑢^′2−𝑘²₁(𝑡)𝑢₀²−𝑘^′₁(𝑡)|𝑢|²+𝑘₁^′′−𝜏⋄𝑢𝑑Γ₂𝑁2_Γ1𝜕𝑢^′𝜕𝜈²−𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²−𝑘^′₂(𝑡)𝜕𝑢𝜕𝜈²+𝑘₂^′′⋄𝜕𝑢−𝑛𝜕𝜈𝑑Γ1+2−𝜃−𝜖𝜆₀𝑎(𝑢,𝑢)−𝑛𝜂2×−2𝜃−𝜂𝜃_Ω𝐹(𝑢)𝑑𝑥+2𝜏₁²𝜖_Γ1𝑢^′2+𝑘²₁(𝑡)|𝑢|²+𝑘²₁(𝑡)𝑢₀²−𝑘₁(0)𝑘^′₁+⋄𝑢𝑑Γ2𝜏₂²𝜖_Γ1𝜕𝑢^′𝜕𝜈²+𝑘²₂(𝑡)𝜕𝑢𝜕𝜈²+𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²−𝑘₂(0)𝑘^′₂⋄𝜕𝑢+1𝜕𝜈𝑑Γ2_Γ1𝑚⋅𝜈𝐾𝑢^′21𝑑Γ−2−𝜖𝜆₀𝛿₀×_Γ1𝑚⋅𝜈𝑢²_𝑥𝑥+𝑢²_𝑦𝑦+2𝜇𝑢_𝑥𝑥𝑢_𝑦𝑦+21−𝜇𝑢²_𝑥𝑦𝑑Γ. (52) We take 𝜃 and 𝜖 so small such that 𝑛𝜂2𝑛−2𝜃−𝜂𝜃>0,1+2−𝜃−𝜖𝜆₀1>0,2−𝜖𝜆₀𝛿₀>0. (53) Since 𝐾∈𝐿^∞(Ω) and then choosing 𝑁 large enough, we obtain ℒ^′(𝑡)≤−𝑐₀𝐸(𝑡)+𝑐_Γ1𝑘²₁(𝑡)𝑢₀²𝑑Γ+𝑐_Γ1𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²𝑑Γ−𝑐_Γ1𝑘^′₁⋄𝑢𝑑Γ−𝑐_Γ1𝑘^′₂⋄𝜕𝑢𝜕𝜈𝑑Γ,∀𝑡≥𝑡₀. (54) On the other hand, we can choose 𝑁 even larger so that ℒ(𝑡)∼𝐸(𝑡). (55) If 𝜉(𝑡)=min{𝜉₁(𝑡),𝜉₂(𝑡)},  𝑡≥𝑡₀, then, using (30) and (33), we have 𝜉(𝑡)ℒ^′(𝑡)≤−𝑐₀𝜉(𝑡)𝐸(𝑡)+𝑐𝜉(𝑡)_Γ1𝑘²₁(𝑡)𝑢₀²(𝑡)𝑑Γ+𝑐𝜉_Γ1𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²𝑑Γ−𝑐𝜉₁(𝑡)_Γ1𝑘^′₁⋄𝑢𝑑Γ−𝑐𝜉₂(𝑡)_Γ1𝑘^′₂⋄𝜕𝑢𝜕𝜈𝑑Γ≤−𝑐₀𝜉(𝑡)𝐸(𝑡)+𝑐𝜉(𝑡)_Γ1𝑘²₁(𝑡)𝑢₀²(𝑡)𝑑Γ+𝑐𝜉_Γ1𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²𝑑Γ+𝑐_Γ1𝑘₁^′′⋄𝑢𝑑Γ+𝑐_Γ1𝑘₂^′′⋄𝜕𝑢𝜕𝜈𝑑Γ≤−𝑐₀𝜉(𝑡)𝐸(𝑡)+𝑐𝜉(𝑡)_Γ1𝑘²₁(𝑡)𝑢₀²(𝑡)𝑑Γ+𝑐𝜉_Γ1𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²𝑑Γ−𝑐𝐸^′(𝑡),∀𝑡≥𝑡₀, (56) which gives 𝜉(𝑡)ℒ^′(𝑡)+𝑐𝐸^′(𝑡)≤−𝑐₀𝜉(𝑡)𝐸(𝑡)+𝑐𝜉(𝑡)_Γ1𝑘²₁(𝑡)𝑢₀²(𝑡)𝑑Γ+𝑐𝜉_Γ1𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²𝑑Γ,∀𝑡≥𝑡₀. (57) Using the fact that 𝜉 is a nonincreasing continuous function as 𝜉₁ and 𝜉₂ are nonincreasing, and so 𝜉 is differentiable, with 𝜉^′(𝑡)≤0, for a.e. 𝑡, then we infer that (𝜉ℒ+𝑐𝐸)^′(𝑡)≤𝜉(𝑡)ℒ^′(𝑡)+𝑐𝐸^′(𝑡)≤−𝑐₀𝜉(𝑡)𝐸(𝑡)+𝑐𝜉(𝑡)_Γ1𝑘²₁(𝑡)𝑢₀²(𝑡)𝑑Γ+𝑐𝜉_Γ1𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²𝑑Γ,∀𝑡≥𝑡₀. (58) Since using (55), 𝐹=𝜉ℒ+𝑐𝐸∼𝐸, (59) we obtain, for some positive constant 𝜔, 𝐹^′(𝑡)≤−𝜔𝜉(𝑡)𝐹(𝑡)+𝑐_Γ1𝑘²₁(𝑡)𝑢₀²𝑑Γ+𝑐_Γ1𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²𝑑Γ,∀𝑡≥𝑡₀. (60)
Case  1. If 𝑢₀=𝜕𝑢₀/𝜕𝜈=0 on Γ₁, then (60) reduces to 𝐹^′(𝑡)≤−𝜔𝜉(𝑡)𝐹(𝑡),∀𝑡≥𝑡₀. (61) A simple integration over (𝑡₀,𝑡) yields 𝐹(𝑡)≤𝐹𝑡₀𝑒^{−𝜔∫𝑡𝑡0𝜉(𝑠)𝑑𝑠},∀𝑡≥𝑡₀. (62) By using (33) and (59), we then obtain for some positive constant 𝐶𝐸(𝑡)≤𝐶𝐸(0)𝑒^{−𝜔∫𝑡0𝜉(𝑠)𝑑𝑠},∀𝑡≥𝑡₀. (63) Thus, estimate (49) is proved.
Case  2. If (𝑢₀,(𝜕𝑢₀/𝜕𝜈))≠(0,0) on Γ₁, then (60) gives 𝐹^′(𝑡)≤−𝜔𝜉(𝑡)𝐹(𝑡)+𝑐𝑘₀(𝑡),∀𝑡≥𝑡₀, (64) where 𝑘₀(𝑡)=_Γ1𝑘²₁(𝑡)𝑢₀²𝑑Γ+_Γ1𝑘²₂(𝑡)𝜕𝑢₀𝜕𝜈²𝑑Γ. (65) In this case, we introduce 𝐺(𝑡)∶=𝐹(𝑡)−𝑐𝑒^{−𝜔∫𝑡𝑡0𝜉(𝑠)𝑑𝑠𝑡}_𝑡0𝑘₀(𝑠)𝑒^{𝜔∫𝑠𝑡0𝜉(𝜏)𝑑𝜏}𝑑𝑠. (66) A simple differentiation of 𝐺, using (64), leads to 𝐺^′(𝑡)=𝐹^′(𝑡)+𝜔𝜉(𝑡)𝑐𝑒^{−𝜔∫𝑡𝑡0𝜉(𝑠)𝑑𝑠}×^𝑡_𝑡0𝑘₀(𝑠)𝑒^{𝜔∫𝑠𝑡0𝜉(𝜏)𝑑𝜏}𝑑𝑠−𝑐𝑘₀(𝑡)≤−𝜔𝜉(𝑡)𝐺(𝑡),∀𝑡≥𝑡₀. (67) Again, a simple integration over (𝑡₀,𝑡) yields 𝐺(𝑡)≤𝐺𝑡₀𝑒^{−𝜔∫𝑡𝑡0𝜉(𝑠)𝑑𝑠},∀𝑡≥𝑡₀, (68) which implies, for all 𝑡≥𝑡₀, 𝐹(𝑡)≤𝐹𝑡₀+𝑐^𝑡_𝑡0𝑘₀(𝑠)𝑒^{𝜔∫𝑠𝑡0𝜉(𝜏)𝑑𝜏}𝑑𝑠𝑒^{−𝜔∫𝑡𝑡0𝜉(𝑠)𝑑𝑠}. (69) By using (59), we deduce that 𝐸(𝑡)≤𝐶𝐸(0)+^𝑡₀𝑘₀(𝑠)𝑒^{𝜔∫𝑠𝑡0𝜉(𝜏)𝑑𝜏}𝑑𝑠𝑒^{−𝜔∫𝑡𝑡0𝜉(𝑠)𝑑𝑠},∀𝑡≥𝑡₀. (70) Consequently, by the boundedness of 𝜉, (50) is established.

Remark 8. Note that the exponential and polynomial decay estimates are only particular cases of (49) and (50). More precisely, we have exponential decay for 𝜉₁(𝑡)≡𝑐₁ and 𝜉₂(𝑡)≡𝑐₂ and polynomial decay for 𝜉₁(𝑡)=𝑐₁(1+𝑡)⁻¹ and 𝜉₂(𝑡)≡𝑐₂, where 𝑐₁ and 𝑐₂ are positive constants.

Example 9. As in [24], we give some examples to illustrate the energy decay rates given by (49).(1)If 𝑘₁(𝑡)=𝑘₂(𝑡)=𝑎𝑒^{−𝑏(1+𝑡)𝑝},  0<𝑝≤1, then, for 𝑖=1,2,𝑘_𝑖^′′(𝑡)≥−𝜉(𝑡)𝑘^′_𝑖(𝑡), where 𝜉(𝑡)=𝑏𝑝(1+𝑡)^𝑝−1. For suitably chosen positive constants 𝑎 and 𝑏, 𝑘_𝑖 satisfies (H) and (49) gives 𝐸(𝑡)≤𝑐𝑒^{−𝜔𝑏(1+𝑡)𝑝}. (71)(2)If 𝑘₁(𝑡)=𝑎₁/(1+𝑡)^𝑞,  𝑞>0, and 𝑘₂(𝑡)=𝑎₂𝑒^{−𝑏(1+𝑡)𝑝},  0<𝑝≤1, then, for 𝑖=1,2,𝑘_𝑖^′′(𝑡)≥−𝜉(𝑡)𝑘^′_𝑖(𝑡), where 𝜉(𝑡)=𝑞(1+𝑡)⁻¹. Then 𝐸(𝑡)≤𝑐(1+𝑡)^𝜔𝑞. (72) The aforementioned two examples are included in the following more general one.(3)For any nonincreasing functions 𝑘_𝑖(𝑡),  𝑖=1,2, which satisfy (H), 𝜉_𝑖=−𝑘^′/𝑘 are also nonincreasing differentiable functions, and 𝑐𝜉₁(𝑡)≤𝜉₂(𝑡), for some 0<𝑐≤1, and (49) gives 𝐸(𝑡)≤𝑐𝑘₁(𝑡)^𝜔. (73)

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 (Grant no. 2012R1A1A3011630).