Stability for the Kirchhoff Plates Equations with Viscoelastic Boundary Conditions in Noncylindrical Domains

Abstract

We study Kirchhoff plates equations with viscoelastic boundary conditions in a noncylindrical domain. This work is devoted to proving the global existence, uniqueness of solutions, and decay of the energy of solutions for Kirchhoff plates equations in a non-cylindrical domain.

1. Introduction

Let Ω be an open bounded domain of ℝ² containing the origin and having 𝐶² boundary. Let 𝛾∶[0,∞[→ℝ be a continuously differentiable function. Consider the family of subdomains {Ω_𝑡}_0≤𝑡<∞ of ℝ² given by Ω_𝑡=𝑇(Ω),𝑇∶𝑦∈Ω⟶𝑥=𝛾(𝑡)𝑦, (1) whose boundaries are denoted by Γ_𝑡, and let 𝑄 be the noncylindrical domain of ℝ³ given by 𝑄=_0≤𝑡<∞Ω_𝑡×{𝑡} (2) with boundary Σ=_0≤𝑡<∞Γ_𝑡×{𝑡}. (3) In this paper, we consider the following Kirchhoff plates equations with viscoelastic boundary conditions: 𝑢^′′+Δ²𝑢=0inΩ_𝑡×(0,∞),𝑢=𝜕𝑢𝜕𝜈=0onΓ_0,𝑡×(0,∞),−𝑢+^𝑡₀𝑔₁(𝑡−𝑠)ℬ₂𝑢(𝑠)𝑑𝑠=0onΓ_1,𝑡×(0,∞),𝜕𝑢+𝜕𝜈^𝑡₀𝑔₂(𝑡−𝑠)ℬ₁𝑢(𝑠)𝑑𝑠=0onΓ_1,𝑡×(),𝑢()0,∞0,𝑥=𝑢₀(𝑥),𝑢^′(0,𝑥)=𝑢₁(𝑥)inΩ₀, (4)(5)(6)(7)(8) where 𝜈=(𝜈₁,𝜈₂) is the unit normal at (𝜎,𝑡)∈Σ directed towards the exterior of 𝑄. We divide the boundary into two parts: Γ_𝑡=Γ_0,𝑡∪Γ_1,𝑡withΓ_0,𝑡∩Γ_1,𝑡=∅,Γ_0,𝑡≠∅. (9) We are denoting by ℬ₁ and ℬ₂ the following differential operators: ℬ₁𝑢=Δ𝑢+1−𝜇𝐵₁𝑢,ℬ₂𝑢=𝜕Δ𝑢+𝜕𝜈1−𝜇𝜕𝐵₂𝑢,𝜕𝜂 (10) where 𝐵₁ and 𝐵₂ are given by 𝐵₁𝑢=2𝜈₁𝜈₂𝑢_𝑥1𝑥2−𝜈²₁𝑢_𝑥2𝑥2−𝜈²₂𝑢_𝑥1𝑥1,𝐵₂𝑢=𝜈²₁−𝜈²₂𝑢_𝑥1𝑥2+𝜈₁𝜈₂𝑢_𝑥2𝑥2−𝑢_𝑥1𝑥1, (11) and the constant 𝜇,0<𝜇<1/2, represents Poisson’s ratio. From the physics point of view, system (4) describes the small transversal vibrations of a thin plate with a moving boundary device. The integral equations (6) and (7) describe the memory effects which can be caused, for example, by the interaction with another viscoelastic element. The relaxation functions 𝑔₁,𝑔₂∈𝐶¹(0,∞) are positive and nondecreasing.

The uniform stabilization of plates equations with linear or nonlinear boundary feedback in cylindrical domain was investigated by several authors; see for example [1–3] among others. The uniform decay for viscoelastic plates with memory was studied by [4, 5] and the references therein. Santos et al. [6] studied the asymptotic behavior of the solutions of a nonlinear wave equation of Kirchhoff type with boundary condition of memory type. Santos and Junior [7] investigated the stability of solutions for Kirchhoff plate equations with boundary memory condition. Park and Kang [8] 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. But the existence of solutions and decay of energy for the Kirchhoff plate equations with viscoelastic boundary conditions in noncylindrical domain are not studied yet. In a moving domain, the transverse deflection 𝑢(𝑥,𝑡) of the thin plate which changes its configuration at each instant of time increases its deformation and hence increases its tension. Moreover, the horizontal movement of the boundary yields nonlinear terms involving derivatives in the space variables. To control these nonlinearities, we add in the boundary a memory type. This term will play an important role in the dissipative nature of the problem.

In [9–17], the authors considered the global existence and the uniform decay of solution in noncylindrical domains. Dal Passo and Ughi [15] investigated a certain class of parabolic equations in noncylindrical domains. Benabidallah and Ferreira [9] proved the existence of solutions for the nonlinear beam equation in noncylindrical domains. Santos et al. [17] studied the global solvability and asymptotic behavior for the nonlinear coupled system of viscoelastic waves with memory in noncylindrical domains. Park and Kang [14] investigated the global existence and stability for von Karman equations with memory in noncylindrical domains. Motivated by these results, we prove the exponential decay of the energy to the Kirchhoff plate equations with viscoelastic boundary conditions in noncylindrical domains.

This paper is organized as follows. In Section 2, we recall notations and hypotheses. In Section 3, we prove the existence and uniqueness of solutions by employing Faedo-Galerkin’s method. In Section 4, we establish the exponential decay rate of the solution.

2. Notations and Hypotheses

We begin this section introducing notations and some hypotheses. Throughout this paper we use standard functional spaces and denote that ‖⋅‖_𝑝,‖⋅‖_𝑝,𝑡 are 𝐿^𝑝(Ω) norm and 𝐿^𝑝(Ω_𝑡) norm. We define the inner product (𝑢,𝑣)=_Ω𝑢(𝑥)𝑣(𝑥)𝑑𝑥,(𝑢,𝑣)_𝑡=_Ω𝑡𝑢(𝑥)𝑣(𝑥)𝑑𝑥. (12) Also, let us assume that there exists 𝑥₀∈ℝ² such that Γ_0,𝑡=𝑥∈Γ_𝑡∶𝜈(𝑥)⋅𝑥−𝑥₀,Γ≤0_1,𝑡=𝑥∈Γ∶𝜈(𝑥)⋅𝑥−𝑥₀>0. (13) The method used to prove the result of existence and uniqueness is based on the transformation of our problem into another initial boundary value problem defined over a cylindrical domain whose sections are not time dependent. This is done using a suitable change of variable. Then we show the existence and uniqueness for this new problem. Our existence result on noncylindrical domains will follow by using the inverse transformation. That is, by using the diffeomorphism 𝜏∶𝑄⟶𝑄,(𝑥,𝑡)∈Ω_𝑡×{𝑡}⟶=𝑥𝑦,𝑡𝛾(𝑡),𝑡 (14) and 𝜏⁻¹∶𝑄→𝑄 defined by 𝜏⁻¹𝑦,𝑡=(𝑥,𝑡)=𝛾(𝑡)𝑦,𝑡. (15) For each function 𝑢 we denote by 𝑣 the function 𝑣𝑦,𝑡=𝑢∘𝜏⁻¹𝑦,𝑡=𝑢(𝑥,𝑡), (16) the initial boundary value problem (4)–(8) becomes 𝑣^′′+𝛾⁻⁴Δ²𝑣+𝐴(𝑡)𝑣+𝑏𝑦,𝑡⋅∇𝑣+𝑐𝑦,𝑡⋅∇𝑣^′(),=0inΩ×0,∞𝑣=𝜕𝑣𝜕𝜈=0onΓ₀×(0,∞),−𝑣+^𝑡₀𝑔₁(𝑡−𝑠)𝛾⁻²(𝑠)ℬ₂𝑣(𝑠)𝑑𝑠=0onΓ₁×(0,∞),𝜕𝑣+𝜕𝜈^𝑡₀𝑔₂(𝑡−𝑠)𝛾⁻²(𝑠)ℬ₁𝑣(𝑠)𝑑𝑠=0onΓ₁×(),𝑣0,∞𝑦,0=𝑣₀𝑦,𝑣^′𝑦,0=𝑣₁𝑦inΩ, (17)(18)(19)(20)(21) where 𝐴(𝑡)𝑣=²_𝑖,𝑗=1𝜕_𝑦𝑖𝑎_𝑖𝑗𝜕_𝑦𝑗𝑣,𝑎_𝑖𝑗=𝛾^′𝛾⁻¹²𝑦_𝑖𝑦_𝑗,𝑏𝑖,𝑗=1,2𝑦,𝑡=−𝛾⁻²𝛾^′′𝛾+𝛾^′2𝑐𝑦,𝑦,𝑡=−2𝛾^′𝛾⁻¹𝑦. (22) The above method was introduced by Dal Passo and Ughi [15] for studying a certain class of parabolic equations in noncylindrical domains. This idea was used in [11, 13, 14, 16, 17].

We will use (19) and (20) to estimate the values ℬ₁ and ℬ₂ on Γ₁. Denoting by 𝑔∗𝑣(𝑡)=^𝑡₀𝑔(𝑡−𝑠)𝑣(𝑠)𝑑𝑠 (23) the convolution product operator and differentiating (19) and (20) we arrive at the following Volterra equations: ℬ₂𝑣𝛾²+1𝑔₁(0)𝑔^′₁∗ℬ₂𝑣𝛾²=1𝑔₁(0)𝑣^′,ℬ₁𝑣𝛾²+1𝑔₂(0)𝑔^′₂∗ℬ₁𝑣𝛾²1=−𝑔₂(0)𝜕𝑣^′.𝜕𝜈 (24) Applying Volterra’s inverse operator, we get ℬ₂𝑣𝛾²=1𝑔₁(0)𝑣^′+𝑘₁∗𝑣^′,ℬ₁𝑣𝛾²1=−𝑔₂(0)𝜕𝑣^′𝜕𝜈+𝑘₂∗𝜕𝑣^′,𝜕𝜈 (25) where the resolvent kernels of −𝑔^′_𝑖/𝑔_𝑖(0) satisfy 𝑘_𝑖+1𝑔_𝑖(0)𝑔^′_𝑖∗𝑘_𝑖1=−𝑔_𝑖(0)𝑔^′_𝑖,∀𝑖=1,2. (26) Denoting by 𝜏₁=1/𝑔₁(0) and 𝜏₂=1/𝑔₂(0), we obtain ℬ₂𝑣𝛾²=𝜏₁𝑣^′+𝑘₁(0)𝑣−𝑘₁(𝑡)𝑣₀+𝑘^′₁,ℬ∗𝑣₁𝑣𝛾²=−𝜏₂𝜕𝑣^′𝜕𝜈+𝑘₂(0)𝜕𝑣𝜕𝜈−𝑘₂(𝑡)𝜕𝑣₀𝜕𝜈+𝑘^′₂∗𝜕𝑣.𝜕𝜈 (27)(28) Therefore, we use (27) and (28) instead of the boundary conditions (19) and (20).

Let us define the bilinear form 𝑎(⋅,⋅) as follows: 𝑎(𝑤,𝑣)=𝑤_𝑥1𝑥1𝑣_𝑥1𝑥1+𝑤_𝑥2𝑥2𝑣_𝑥2𝑥2+𝜇𝑤_𝑥1𝑥1𝑣_𝑥2𝑥2+𝑤_𝑥2𝑥2𝑣_𝑥1𝑥1+21−𝜇𝑤_𝑥1𝑥2𝑣_𝑥1𝑥2. (29)

Since Γ₀≠∅ we know that ∫_Ω𝑎(𝑣,𝑣)𝑑𝑦 is equivalent to the 𝐻²(Ω) norm, that is, 𝑐₀‖𝑣‖²_𝐻2(Ω)≤_Ω𝑎(𝑣,𝑣)𝑑𝑦≤𝐶₀‖𝑣‖²_𝐻2(Ω), (30) where 𝑐₀ and 𝐶₀ are generic positive constants.

Let us denote that 𝑔∘𝑣(𝑡)∶=^𝑡₀𝑔(𝑣(𝑡)(𝑠(𝑡)𝑡−𝑠)(−𝑣))𝑑𝑠,𝑔□𝑣∶=^𝑡₀𝑔(𝑡−𝑠)|𝑣(𝑡)−𝑣(𝑠)|²𝑑𝑠. (31) 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□𝑣2𝑑𝑑𝑡𝑔□𝑣−^𝑡₀𝑔(𝑠)𝑑𝑠|𝑣|². (32)

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

We state the following lemma which will be useful in what follows.

Lemma 2 (see [7]). Let 𝑤 and 𝑣 be functions in 𝐻⁴(Ω)∩𝐻²₀(Ω). Then one has _ΩΔ²𝑤𝑣𝑑𝑦=_Ω𝑎()+𝑤,𝑣𝑑𝑦_Γ1ℬ₂𝑤𝑣−ℬ₁𝑤𝜕𝑣𝜕𝜈𝑑Γ. (33)

Lemma 3 (see [18]). Suppose that 𝑓∈𝐿²(Ω), 𝑔∈𝐻^1/2(Γ₁), and ℎ∈𝐻^3/2(Γ₁); then, any solution of _Ω𝑎(𝑣,𝑤)𝑑𝑦=_Ω𝑓𝑤𝑑𝑦+_Γ1+𝑔𝑤𝑑Γ_Γ1ℎ𝜕𝑤𝜕𝜈𝑑Γ,∀𝑤∈𝐻²₀(Ω) (34) satisfies 𝑣∈𝐻⁴(Ω) and also Δ²𝑣=𝑓,𝑣=𝜕𝑣𝜕𝜈=0onΓ₀,ℬ₁𝑣=ℎ,ℬ₂𝑣=𝑔onΓ₁. (35)

To show the existence of solution, we will use the following hypotheses: 𝛾^′≤0,𝛾∈𝐿^∞(0,∞),inf_0≤𝑡<∞𝛾(𝑡)=𝛾₀𝛾>0,_′∈𝑊^2,∞(0,∞)∩𝑊^2,1(0,∞),0<max_0≤𝑡<∞𝛾^′(𝑡𝛾(𝑡)≤1)2𝑐₁𝑐₀⁻¹,𝑀𝑑 (36)(37)(38) where 𝑑=diam(Ω), 𝑀=meas(Ω), and 𝑐₀ is a positive imbedding constant such that ‖∇𝑣‖²≤𝑐₁‖Δ𝑣‖², for all 𝑣∈𝐻²₀(Ω).

3. Existence and Regularity

In this section we will study the existence and regularity of solutions for system (4)–(8).

The well posedness of system (17)–(21) is given by the following theorem.

Theorem 4. Let 𝑘_𝑖∈𝐶²(ℝ⁺) be such that 𝑘_𝑖,−𝑘^′_𝑖,𝑘_𝑖^′′≥0. (39) The function 𝛾 satisfies that 𝛾^′(𝑡)𝛾⁻¹(𝑡)𝑘<min1,−^′𝑖(𝑡)2. (40) If (𝑣₀,𝑣₁)∈(𝐻⁴(Ω)∩𝐻²₀(Ω))×𝐻²₀(Ω) satisfy the compatibility condition ℬ₂𝑣₀−𝜏₁𝛾²(0)𝑣₁=0,ℬ₁𝑣₀+𝜏₂𝛾²(0)𝜕𝑣₁𝜕𝜈𝑜𝑛Γ₁ (41) then there exists only one solution for system (17)–(21) satisfying 𝑣∈𝐿^∞0,𝑇;𝐻⁴(Ω)∩𝐻²₀(Ω,𝑣_′)∈𝐿^∞0,𝑇;𝐻²₀(Ω),𝑣^′′∈𝐿^∞0,𝑇;𝐿²(Ω). (42)

Proof. The main idea is to use the Galerkin method. To do this let us denote by 𝐵 the operator 𝐵𝑤=Δ²𝑤,𝐷(𝐵)=𝐻²₀(Ω)∩𝐻⁴(Ω). (43)
It is well known that 𝐵 is a positive self-adjoint operator in the Hilbert space 𝐿²(Ω) for which there exist sequences {𝑤_𝑛}_𝑛∈ℕ and {𝜆_𝑛}_𝑛∈ℕ of eigenfunctions and eigenvalues of 𝐵 such that the set of linear combinations of {𝑤_𝑛}_𝑛∈ℕ is dense in 𝐷(𝐵) and 𝜆₁<𝜆₂≤⋯≤𝜆_𝑛→∞ as 𝑛→∞. Let us define 𝑣_0𝑚=^𝑚_𝑗=1𝑣₀,𝑤_𝑗𝑤_𝑗,𝑣_1𝑚=^𝑚_𝑗=1𝑣₁,𝑤_𝑗𝑤_𝑗 (44) Note that for any (𝑣₀,𝑣₁)∈𝐷(𝐵)×𝐻²₀(Ω), we have 𝑣_0𝑚→𝑣₀ strong in 𝐷(𝐵) and 𝑣_1𝑚→𝑣₁ strong in 𝐻²₀(Ω).
Let us denote by 𝑉_𝑚 the space generated by 𝑤₁,𝑤₂,…,𝑤_𝑚. Standard results on ordinary differential equations guarantee that there exists only one local solution 𝑣_𝑚(𝑡)=^𝑚_𝑗=1𝑔_𝑗𝑚(𝑡)𝑤_𝑗, (45) of the approximate system _Ω𝑣_𝑚^′′𝑤_𝑗𝑑𝑦+𝛾⁻⁴_Ω𝑎𝑣_𝑚,𝑤_𝑗𝑑𝑦+_Ω𝐴(𝑡)𝑣_𝑚𝑤_𝑗+𝑑𝑦_Ω𝑐𝑦,𝑡⋅∇𝑣^′_𝑚𝑤_𝑗𝑑𝑦+_Ω𝑏𝑦,𝑡⋅∇𝑣_𝑚𝑤_𝑗𝑑𝑦=−𝜏₁𝛾⁻²_Γ1𝑣^′_𝑚+𝑘₁(0)𝑣_𝑚−𝑘₁(𝑡)𝑣_0𝑚+𝑘^′₁∗𝑣_𝑚𝑤_𝑗𝑑Γ−𝜏₂𝛾⁻²_Γ1𝜕𝑣^′_𝑚𝜕𝜈+𝑘₂(0)𝜕𝑣_𝑚𝜕𝜈−𝑘₂(𝑡)𝜕𝑣_0𝑚𝜕𝜈+𝑘^′₂∗𝜕𝑣_𝑚𝑤𝜕𝜈^𝑗,𝑣𝜕𝜈𝑑Γ𝑗=1,2,…,𝑚_𝑚(𝑥,0)=𝑣_0𝑚,𝑣^′_𝑚(𝑥,0)=𝑣_1𝑚. (46)(47) By standard methods for differential equations, we prove the existence of solutions to the approximate equation (46) on some interval [0,𝑡_𝑚). Then, this solution can be extended to the whole interval [0,𝑇], for all 𝑇>0, by using the following first estimate.
 
The First Estimate. Multiplying (46) by 𝑔^′_𝑗𝑚(𝑡), summing up the product result 𝑗=1,2,…,𝑚, and making some calculations using Lemma 1, we get 12𝑑𝑑𝑡_Ω𝑣^′_𝑚²𝑑𝑦+𝛾⁻⁴_Ω𝑎𝑣_𝑚,𝑣_𝑚𝑑𝑦+𝜏₁𝛾⁻²_Γ1𝑘₁(𝑡)𝑣_𝑚²−𝑘^′₁□𝑣_𝑚𝑑Γ+𝜏₂𝛾⁻²_Γ1𝑘₂(𝑡)𝜕𝑣_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣_𝑚𝜕𝜈𝑑Γ+2𝛾⁻⁵𝛾^′_Ω𝑎𝑣_𝑚,𝑣_𝑚𝑑𝑦+𝜏₁𝛾⁻³𝛾^′_Γ1𝑘₁(𝑡)𝑣_𝑚²−𝑘^′₁□𝑣_𝑚𝑑Γ+𝜏₂𝛾⁻³𝛾^′_Γ1𝑘₂(𝑡)𝜕𝑣_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣_𝑚𝜕𝜈𝑑Γ=−_Ω𝐴(𝑡)𝑣_𝑚𝑣^′_𝑚𝑑𝑦−_Ω𝑐𝑦,𝑡⋅∇𝑣^′_𝑚𝑣^′_𝑚−𝑑𝑦_Ω𝑏𝑦,𝑡⋅∇𝑣_𝑚𝑣^′_𝑚𝑑𝑦−𝜏₁𝛾⁻²_Γ1𝑣^′_𝑚²−𝑘₁(𝑡)𝑣_0𝑚𝑣^′_𝑚−12𝑘^′₁(𝑡)𝑣_𝑚²+12𝑘₁^′′□𝑣_𝑚𝑑Γ−𝜏₂𝛾⁻²_Γ1𝜕𝑣^′_𝑚𝜕𝜈²−𝑘₂(𝑡)𝜕𝑣_0𝑚𝜕𝜈𝜕𝑣^′_𝑚−1𝜕𝜈2𝑘^′₂(𝑡)𝜕𝑣_𝑚𝜕𝜈²+12𝑘₂^′′□𝜕𝑣_𝑚𝜕𝜈𝑑Γ. (48) Now we will estimate terms of the right-hand side of (48). From the hypotheses on 𝛾 and Green’s formula, we get −_Ω𝐴(𝑡)𝑣_𝑚𝑣^′_𝑚𝑑𝑦=−_Ω²_𝑖,𝑗=1𝜕_𝑦𝑖𝑎_𝑖𝑗𝜕_𝑦𝑗𝑣_𝑚𝑣^′_𝑚=𝑑𝑦_{Ω2𝑖,𝑗=1}𝑎_𝑖𝑗𝜕_𝑦𝑗𝑣_𝑚𝜕_𝑦𝑖𝑣^′_𝑚=𝑑𝑦_{Ω2𝑖,𝑗=1}𝛾^′𝛾⁻¹²𝑦_𝑖𝑦_𝑗𝜕_𝑦𝑗𝑣_𝑚𝜕_𝑦𝑖𝑣^′_𝑚=𝑑𝑑𝑦𝑑𝑡_Ω12𝛾^′𝛾⁻¹²∇𝑣_𝑚⋅𝑦²−𝛾𝑑𝑦_′𝛾⁻¹𝛾^′′𝛾⁻¹−𝛾^′𝛾⁻¹²∇𝑣_𝑚⋅𝑦²₂,_Ω𝑐𝑦,𝑡⋅∇𝑣^′_𝑚𝑣^′_𝑚𝑑𝑦=−_Ω2𝛾^′𝛾⁻¹𝑦⋅∇𝑣^′_𝑚𝑣^′_𝑚𝑑𝑦=−_Ω𝛾^′𝛾⁻¹𝑦⋅∇𝑣^′_𝑚²𝑑𝑦=2𝛾^′𝛾⁻¹𝑣^′_𝑚²₂,−_Ω𝑏𝑦,𝑡⋅∇𝑣_𝑚𝑣^′_𝑚=𝑑𝑦_Ω𝛾⁻²𝛾^′′𝛾+𝛾^′2𝑦⋅∇𝑣_𝑚𝑣^′_𝑚≤𝛾𝑑𝑦_′′𝛾⁻¹+𝛾^′𝛾⁻¹²2𝑦⋅∇𝑣_𝑚²₂+𝑣^′_𝑚²₂≤𝐶₁∇𝑣_𝑚²₂+𝑣^′_𝑚²₂. (49)
Young’s inequality yields _Γ1𝑘₁(𝑡)𝑣_0𝑚𝑣^′_𝑚1𝑑Γ≤2_Γ1𝑣^′_𝑚²𝑘𝑑Γ+²¹(𝑡)2_Γ1𝑣_0𝑚²𝑑Γ,_Γ1𝑘₂(𝑡)𝜕𝑣_0𝑚𝜕𝜈𝜕𝑣^′_𝑚≤1𝜕𝜈𝑑Γ2_Γ1𝜕𝑣^′_𝑚𝜕𝜈²𝑘𝑑Γ+²²(𝑡)2_Γ1𝜕𝑣_0𝑚𝜕𝜈²𝑑Γ. (50) Replacing the above calculations in (48) and using our assumptions 𝑘_𝑖,−𝑘^′_𝑖,𝑘_𝑖^′′≥0 and (30), we have 12𝑑𝑣𝑑𝑡^′𝑚22+𝛾⁻⁴_Ω𝑎𝑣_𝑚,𝑣_𝑚𝑑𝑦−𝛾^′𝛾⁻¹²∇𝑣_𝑚⋅𝑦²₂+𝜏₁𝛾⁻²_Γ1𝑘₁(𝑡)𝑣_𝑚²−𝑘^′₁□𝑣_𝑚𝑑Γ+𝜏₂𝛾⁻²_Γ1𝑘₂(𝑡)𝜕𝑣_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣_𝑚𝜕𝜈𝑑Γ≤𝐶₂𝑣^′_𝑚²₂+_Ω𝑎𝑣_𝑚,𝑣_𝑚𝑑𝑦+_Γ1𝑘₁(𝑡)𝑣_𝑚²−𝑘^′₁□𝑣_𝑚+𝑑Γ_Γ1𝑘₂(𝑡)𝜕𝑣_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣_𝑚+𝜏𝜕𝜈𝑑Γ₁𝛾⁻²2𝑘²₁(𝑡)_Γ1𝑣_0𝑚²𝜏𝑑Γ+²𝛾⁻²2𝑘²₂(𝑡)_Γ1𝜕𝑣_0𝑚𝜕𝜈²𝑑Γ. (51)
From our choice of 𝑣_0𝑚 and 𝑣_1𝑚 and integrating (51) over (0,𝑡) with 𝑡∈(0,𝑡_𝑚), we obtain 𝑣^′_𝑚²₂+𝛾⁻⁴_Ω𝑎𝑣_𝑚,𝑣_𝑚𝑑𝑦−𝛾^′𝛾⁻¹²∇𝑣_𝑚⋅𝑦²₂+𝜏₁𝛾⁻²_Γ1𝑘₁(𝑡)𝑣_𝑚²−𝑘^′₁□𝑣_𝑚𝑑Γ+𝜏₂𝛾⁻²_Γ1𝑘₂(𝑡)𝜕𝑣_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣_𝑚𝜕𝜈𝑑Γ≤𝐶₃^𝑡₀𝑣^′_𝑚(𝑠)²₂+_Ω𝑎𝑣_𝑚(𝑠),𝑣_𝑚(𝑠)+𝑑𝑦_Γ1𝑘₁(𝑠)𝑣_𝑚(𝑠)²−𝑘^′₁□𝑣_𝑚(𝑠)+𝑑Γ_Γ1𝑘₂(𝑠)𝜕𝑣_𝑚(𝑠)𝜕𝜈²−𝑘^′₂□𝜕𝑣_𝑚(𝑠)𝜕𝜈𝑑Γ𝑑𝑠+𝐶₄. (52) We observe that, from (30) and (38), 𝛾^′𝛾⁻¹²∇𝑣_𝑚⋅𝑦²₂≤𝛾^′𝛾⁻¹²𝑀𝑑²∇𝑣_𝑚²₂≤𝛾^′𝛾⁻¹²𝑐₁𝑐₀⁻¹𝑀𝑑²_Ω𝑎𝑣_𝑚,𝑣_𝑚≤𝛾𝑑𝑦₋₄2_Ω𝑎𝑣_𝑚,𝑣_𝑚𝑑𝑦, (53) for all 𝑡≥0. Hence, by Gronwall’s lemma we get 𝑣^′_𝑚²₂+_Ω𝑎𝑣_𝑚,𝑣_𝑚𝑑𝑦+_Γ1𝑘₁(𝑡)𝑣_𝑚²−𝑘^′₁□𝑣_𝑚+𝑑Γ_Γ1𝑘₂(𝑡)𝜕𝑣_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣_𝑚𝜕𝜈𝑑Γ≤𝐶₅, (54) where 𝐶₅ is a positive constant which is independent of 𝑚 and 𝑡.
 
The Second Estimate. First of all, we are going to estimate 𝑣_𝑚^′′(0) in 𝐿²(Ω)-norm. Letting 𝑡→0⁺ in (46), multiplying the result by 𝑔^′′_𝑗𝑚(0), and using the compatibility condition (41), we have 𝑣_𝑚^′′(0)²₂≤𝐶₆. (55) Now, differentiating (46) with respect to 𝑡, we obtain _Ω𝑣_𝑚^′′′𝑤_𝑗𝑑𝑦+𝛾⁻⁴_Ω𝑎𝑣^′_𝑚,𝑤_𝑗𝑑𝑦−4𝛾⁻⁵𝛾^′_Ω𝑎𝑣_𝑚,𝑤_𝑗𝑑𝑦=−_Ω𝑑𝐴(𝑡)𝑣𝑑𝑡^𝑚𝑤_𝑗𝑑𝑦−_Ω𝑑𝑐𝑑𝑡𝑦,𝑡⋅∇𝑣^′_𝑚𝑤_𝑗−𝑑𝑦_Ω𝑑𝑏𝑑𝑡𝑦,𝑡⋅∇𝑣_𝑚𝑤_𝑗𝑑𝑦−𝜏₁𝛾⁻²_Γ1𝑣_𝑚^′′+𝑘₁(0)𝑣^′_𝑚+𝑘^′₁∗𝑣^′_𝑚𝑤_𝑗𝑑Γ−𝜏₂𝛾⁻²_Γ1𝜕𝑣_𝑚^′′𝜕𝜈+𝑘₂(0)𝜕𝑣^′_𝑚𝜕𝜈+𝑘^′₂∗𝜕𝑣^′_𝑚𝑤𝜕𝜈𝜕𝜈^𝑗𝑑Γ+2𝜏₁𝛾⁻³𝛾^′_Γ1𝑣^′_𝑚+𝑘₁(0)𝑣_𝑚−𝑘₁(𝑡)𝑣_0𝑚+𝑘^′₁∗𝑣_𝑚𝑤_𝑗𝑑Γ+2𝜏₂𝛾⁻³𝛾^′_Γ1𝜕𝑣^′_𝑚𝜕𝜈+𝑘₂(0)𝜕𝑣_𝑚𝜕𝜈−𝑘₂(𝑡)𝜕𝑣_0𝑚𝜕𝜈+𝑘^′₂∗𝜕𝑣_𝑚𝑤𝜕𝜈𝜕𝜈^𝑗𝑑Γ. (56) Multiplying (56) by 𝑔^′′_𝑗𝑚(𝑡), summing up the product result in 𝑗, and using Lemma 1, we have 12𝑑𝑣𝑑𝑡^𝑚′′22+𝛾⁻⁴_Ω𝑎𝑣^′_𝑚,𝑣^′_𝑚𝑑𝑦−8𝛾⁻⁵𝛾^′_Ω𝑎𝑣_𝑚,𝑣^′_𝑚𝑑𝑦+𝜏₁𝛾⁻²_Γ1𝑘₁(𝑡)𝑣^′_𝑚²−𝑘^′₁□𝑣^′_𝑚𝑑Γ+𝜏₂𝛾⁻²_Γ1𝑘₂(𝑡)𝜕𝑣^′_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣^′_𝑚𝜕𝜈𝑑Γ+6𝛾⁻⁵𝛾^′_Ω𝑎𝑣^′_𝑚,𝑣^′_𝑚𝑑𝑦+4𝛾⁻⁴𝛾^′′𝛾⁻¹−5𝛾^′𝛾⁻¹²_Ω𝑎𝑣_𝑚,𝑣^′_𝑚𝑑𝑦+𝜏₁𝛾⁻³𝛾^′_Γ1𝑘₁(𝑡)𝑣^′_𝑚²−𝑘^′₁□𝑣^′_𝑚𝑑Γ+𝜏₂𝛾⁻³𝛾^′_Γ1𝑘₂(𝑡)𝜕𝑣^′_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣^′_𝑚𝜕𝜈𝑑Γ=−_Ω𝑑𝐴(𝑡)𝑣𝑑𝑡^𝑚𝑣_𝑚^′′𝑑𝑦−_Ω𝑑𝑐𝑑𝑡𝑦,𝑡⋅∇𝑣^′_𝑚𝑣_𝑚^′′−𝑑𝑦_Ω𝑑𝑏𝑑𝑡𝑦,𝑡⋅∇𝑣_𝑚𝑣_𝑚^′′𝑑𝑦−𝜏₁𝛾⁻²_Γ1𝑣_𝑚^′′2+12𝑘^''₁□𝑣^′_𝑚−12𝑘^′₁(𝑡)𝑣^′_𝑚²𝑑Γ−𝜏₂𝛾⁻²_Γ1𝜕𝑣_𝑚^′′𝜕𝜈²+12𝑘₂^′′□𝜕𝑣^′_𝑚−1𝜕𝜈2𝑘^′₂(𝑡)𝜕𝑣^′_𝑚𝜕𝜈²𝑑Γ+2𝜏₁𝛾⁻³𝛾^′_Γ1𝑣^′_𝑚+𝑘₁(0)𝑣_𝑚−𝑘₁(𝑡)𝑣_0𝑚+𝑘^′₁∗𝑣_𝑚𝑣_𝑚^′′𝑑Γ+2𝜏₂𝛾⁻³𝛾^′_Γ1𝜕𝑣^′_𝑚𝜕𝜈+𝑘₂(0)𝜕𝑣_𝑚𝜕𝜈−𝑘₂(𝑡)𝜕𝑣_0𝑚𝜕𝜈+𝑘^′₂∗𝜕𝑣_𝑚𝑣𝜕𝜈𝜕𝜈^𝑚′′𝑑Γ. (57) Now we will estimate terms of the right-hand side of (57). From the hypotheses on 𝛾 and Green’s formula, we get −_Ω𝑑𝐴(𝑡)𝑣𝑑𝑡^𝑚𝑣_𝑚^′′𝑑𝑦=−_Ω𝑑𝑑𝑡²_𝑖,𝑗=1𝜕_𝑦𝑖𝛾^′𝛾⁻¹²𝑦_𝑖𝑦_𝑗𝜕_𝑦𝑗𝑣_𝑚𝑣_𝑚^′′𝑑𝑦=−_Ω²_𝑖,𝑗=1𝜕_𝑦𝑖2𝛾^′𝛾⁻¹𝛾^′′𝛾⁻¹−𝛾^′𝛾⁻¹²𝑦_𝑖𝑦_𝑗𝜕_𝑦𝑗𝑣_𝑚+𝛾^′𝛾⁻¹²𝑦_𝑖𝑦_𝑗𝜕_𝑦𝑗𝑣^′_𝑚𝑣_𝑚^′′𝑑𝑦=−_Ω²_𝑖,𝑗=1𝜕_𝑦𝑖2𝛾^′𝛾⁻¹𝛾^′′𝛾⁻¹−𝛾^′𝛾⁻¹²×𝑦_𝑖𝑦_𝑗𝜕_𝑦𝑗𝑣_𝑚𝑣_𝑚^′′+𝑑𝑦_{Ω2𝑖,𝑗=1}𝛾^′𝛾⁻¹²𝑦_𝑖𝑦_𝑗𝜕_𝑦𝑗𝑣^′_𝑚𝜕_𝑦𝑖𝑣_𝑚^′′𝑑𝑦=−_Ω²_𝑖,𝑗=1𝜕_𝑦𝑖2𝛾^′𝛾⁻¹𝛾^′′𝛾⁻¹−𝛾^′𝛾⁻¹²𝑦_𝑖𝑦_𝑗𝜕_𝑦𝑗𝑣_𝑚𝑣_𝑚^′′+𝑑𝑑𝑦𝑑𝑡_Ω12𝛾^′𝛾⁻¹²∇𝑣^′_𝑚⋅𝑦²−𝛾𝑑𝑦_′𝛾⁻¹𝛾^′′𝛾⁻¹−𝛾^′𝛾⁻¹²∇𝑣^′_𝑚⋅𝑦²₂,−_Ω𝑑𝑐𝑑𝑡𝑦,𝑡⋅∇𝑣^′_𝑚𝑣_𝑚^′′=𝑑𝑦_Ω𝑑𝑑𝑡2𝛾^′𝛾⁻¹𝑦⋅∇𝑣^′_𝑚𝑣_𝑚^′′𝑑𝑦=_Ω2𝛾^′′𝛾⁻¹−𝛾^′𝛾⁻¹²𝑦⋅∇𝑣^′_𝑚+2𝛾^′𝛾⁻¹𝑦⋅∇𝑣_𝑚^′′𝑣_𝑚^′′=𝑑𝑦_Ω2𝛾^′′𝛾⁻¹−𝛾^′𝛾⁻¹²𝑦⋅∇𝑣^′_𝑚𝑣_𝑚^′′+𝑑𝑦_Ω𝛾^′𝛾⁻¹𝑦⋅∇|𝑣_𝑚^′′|²≤𝛾𝑑𝑦_′′𝛾⁻¹+𝛾^′𝛾⁻¹²𝑦⋅∇𝑣^′_𝑚²₂+𝑣_𝑚^′′2₂−2𝛾^′𝛾⁻¹𝑣_𝑚^′′2₂,−_Ω𝑑𝑏𝑑𝑡𝑦,𝑡⋅∇𝑣_𝑚𝑣_𝑚^′′=𝑑𝑦_Ω𝑑𝛾𝑑𝑡^′′𝛾⁻¹+𝛾^′𝛾⁻¹²𝑦⋅∇𝑣_𝑚𝑣_𝑚^′′=𝑑𝑦_Ω𝑑𝛾𝑑𝑡^′′𝛾⁻¹+𝛾^′𝛾⁻¹²𝑦⋅∇𝑣_𝑚𝑣_𝑚^′′𝑑𝑦+_Ω𝛾^′′𝛾⁻¹+𝛾^′𝛾⁻¹²𝑦⋅∇𝑣^′_𝑚𝑣_𝑚^′′𝑑𝑦≤𝐶₇𝑦⋅∇𝑣_𝑚²₂+𝑦⋅∇𝑣^′_𝑚²₂+𝑣_𝑚^′′2₂. (58)(59)(60) We know that 𝑘^′₁∗𝑣_𝑚(𝑡)=^𝑡₀𝑘^′₁(𝑡−𝑠)𝑣_𝑚(𝑠)−𝑣_𝑚(𝑡)𝑑𝑠+𝑘₁(𝑡)𝑣_𝑚(𝑡)−𝑘₁(0)𝑣_𝑚(𝑡). (61) By using Hölder’s inequality and our assumption 𝑘^′₁≤0, we note that ^𝑡₀𝑘^′₁(𝑡−𝑠)(𝑣(𝑡)−𝑣(𝑠))𝑑𝑠²_Γ1≤^𝑡₀𝑘^′₁(𝑠)𝑑s_Γ1^𝑡₀𝑘^′₁(𝑡−𝑠)(𝑣(𝑡)−𝑣(𝑠))²≤𝑑𝑠𝑑Γ_Γ1𝑘₁(0)𝑘^′₁□𝑣𝑑Γ (62) and, hence, by applying Young’s inequality, we obtain 2𝜏₁𝛾⁻³𝛾^′_Γ1𝑣^′_𝑚+𝑘₁(0)𝑣_𝑚−𝑘₁(𝑡)𝑣_0𝑚+𝑘^′₁∗𝑣_𝑚𝑣_𝑚^′′𝑑Γ≤𝜏₁𝛾⁻³𝛾^′_Γ1𝑣_𝑚^′′2𝑑Γ+𝜏₁𝛾⁻³𝛾^′×_Γ1𝑣^′_𝑚²+𝑘²₁(𝑡)𝑣_0𝑚²+𝑘₁(0)𝑘^′₁□𝑣_𝑚+𝑘²₁(𝑡)𝑣_𝑚²𝑑Γ. (63) By the same argument of (63), we can obtain the similar estimate 2𝜏₂𝛾⁻³𝛾^′_Γ1𝜕𝑣^′_𝑚𝜕𝜈+𝑘₂(0)𝜕𝑣_𝑚𝜕𝜈−𝑘₂(𝑡)𝜕𝑣_0𝑚𝜕𝜈+𝑘^′₂∗𝜕𝑣_𝑚𝑣𝜕𝜈𝜕𝜈^𝑚′′𝑑Γ≤𝜏₂𝛾⁻³𝛾^′_Γ1𝑣_𝑚^′′𝜕𝜈²𝑑Γ+𝜏₂𝛾⁻³𝛾^′×_Γ1𝑣^′_𝑚𝜕𝜈²+𝑘²₂(𝑡)𝜕𝑣_0𝑚𝜕𝜈²+𝑘₂(0)𝑘^′₂□𝜕𝑣_𝑚𝜕𝜈+𝑘²₂(𝑡)𝜕𝑣_𝑚𝜕𝜈²𝑑Γ. (64) Applying (58)–(64) to (57) and using the first estimate (54) and our assumptions 𝑘_𝑖,−𝑘^′_𝑖,𝑘_𝑖^′′≥0 and |𝛾^′|𝛾⁻¹<min{1,−(𝑘^′_𝑖/2)}, we have 12𝑑𝑣𝑑𝑡^𝑚′′22+𝛾⁻⁴_Ω𝑎𝑣^′_𝑚,𝑣^′_𝑚𝑑𝑦−𝛾^′𝛾⁻¹²∇𝑣^′_𝑚⋅𝑦²₂−8𝛾⁻⁵𝛾^′_Ω𝑎𝑣_𝑚,𝑣^′_𝑚𝑑𝑦+𝜏₁𝛾⁻²_Γ1𝑘₁(𝑡)𝑣^′_𝑚²−𝑘^′₁□𝑣^′_𝑚𝑑Γ+𝜏₂𝛾⁻²_Γ1𝑘₂(𝑡)𝜕𝑣^′_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣^′_𝑚𝜕𝜈𝑑Γ≤𝐶₈𝑣_𝑚^′′2₂+_Ω𝑎𝑣^′_𝑚,𝑣^′_𝑚𝑑𝑦+_Ω𝑎𝑣_𝑚,𝑣^′_𝑚+𝑑𝑦_Γ1𝑘₁(𝑡)𝑣^′_𝑚²−𝑘^′₁□𝑣^′_𝑚+𝑑Γ_Γ1𝑘₂(𝑡)𝜕𝑣^′_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣^′_𝑚𝜕𝜈𝑑Γ+𝜏₁𝛾⁻³𝛾^′_Γ1𝑘²₁(𝑡)𝑣_0𝑚²𝑑Γ+𝜏₂𝛾⁻³𝛾^′_Γ1𝑘²₂(𝑡)𝜕𝑣_0𝑚𝜕𝜈²𝑑Γ+𝐶₉. (65) From (55) and our choice of 𝑣_0𝑚 and 𝑣_1𝑚 and integrating (65) over (0,𝑡) with 𝑡∈(0,𝑡_𝑚), we obtain 𝑣_𝑚^′′2₂+𝛾⁻⁴_Ω𝑎𝑣^′_𝑚,𝑣^′_𝑚𝑑𝑦−𝛾^′𝛾⁻¹²∇𝑣^′_𝑚⋅𝑦²₂−8𝛾⁻⁵𝛾^′_Ω𝑎𝑣_𝑚,𝑣^′_𝑚𝑑𝑦+𝜏₁𝛾⁻²_Γ1𝑘₁(𝑡)𝑣^′_𝑚²−𝑘^′₁□𝑣^′_𝑚𝑑Γ+𝜏₂𝛾⁻²_Γ1𝑘₂(𝑡)𝜕𝑣^′_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣^′_𝑚𝜕𝜈𝑑Γ≤2𝐶₈^𝑡₀𝑣_𝑚^′′(𝑠)²₂+_Ω𝑎𝑣^′_𝑚(𝑠),𝑣^′_𝑚(𝑠)+𝑑𝑦_Ω𝑎𝑣_𝑚(𝑠),𝑣^′_𝑚(𝑠)𝑑𝑦+_Γ1𝑘₁(𝑠)𝑣^′_𝑚(𝑠)²−𝑘^′₁□𝑣^′_𝑚(𝑠)+𝑑Γ_Γ1𝑘₂(𝑠)𝜕𝑣^′_𝑚(𝑠)𝜕𝜈²−𝑘^′₂□𝜕𝑣^′_𝑚(𝑠𝜕𝜈)𝑑Γ𝑑𝑠+𝐶₁₀. (66) Using the same arguments as for (53), we get 𝛾^′𝛾⁻¹²∇𝑣^′_𝑚⋅𝑦²₂<𝛾⁻⁴2_Ω𝑎𝑣^′_𝑚,𝑣^′_𝑚𝑑𝑦, (67) for all 𝑡≥0. Therefore, by Gronwall’s lemma, we obtain 𝑣_𝑚^′′2₂+_Ω𝑎𝑣^′_𝑚,𝑣^′_𝑚𝑑𝑦+_Γ1𝑘₁(𝑡)𝑣^′_𝑚²−𝑘^′₁□𝑣^′_𝑚+𝑑Γ_Γ1𝑘₂(𝑡)𝜕𝑣^′_𝑚𝜕𝜈²−𝑘^′₂□𝜕𝑣^′_𝑚𝜕𝜈𝑑Γ≤𝐶₁₁, (68) where 𝐶₁₁ is a positive constant which is independent of 𝑚 and 𝑡.
According to (54) and (68), we get 𝑣_𝑚isboundedin𝐿^∞0,𝑇;𝐻²₀(Ω𝑣),_′𝑚isboundedin𝐿^∞0,𝑇;𝐻²₀(Ω𝑣),_𝑚′′isboundedin𝐿^∞0,𝑇;𝐿²(Ω). (69)(70)(71)
From (69) to (71), there exists a subsequence of {𝑣_𝑚}, which we still denote by {𝑣_𝑚}, such that 𝑣_𝑚⟶𝑣weakstarin𝐿^∞0,𝑇;𝐻²₀(Ω𝑣),_′𝑚⟶𝑣^′weakstarin𝐿^∞0,𝑇;𝐻²₀(Ω,𝑣_𝑚′′)⟶𝑣^′′weakstarin𝐿^∞0,𝑇;𝐿²(Ω). (72)(73)(74) Letting 𝑚→∞ in (46) and using (72)–(74), we obtain _Ω𝑎(𝑣,𝑤)𝑑𝑦=−𝛾⁴_Ω𝑣^′′𝑤𝑑𝑦−𝛾⁴_Ω𝐴(𝑡)𝑣𝑤𝑑𝑦−𝛾⁴_Ω𝑐𝑦,𝑡⋅∇𝑣^′𝑤𝑑𝑦−𝛾⁴_Ω𝑏𝑦,𝑡⋅∇𝑣𝑤𝑑𝑦−𝜏₁𝛾²_Γ1𝑣^′+𝑘₁(0)𝑣−𝑘₁(𝑡)𝑣₀+𝑘^′₁∗𝑣𝑤𝑑Γ−𝜏₂𝛾²_Γ1𝜕𝑣^′𝜕𝜈+𝑘₂(0)𝜕𝑣𝜕𝜈−𝑘₂(𝑡)𝜕𝑣₀𝜕𝜈+𝑘^′₂∗𝜕𝑣𝜕𝜈𝜕𝑤𝜕𝜈𝑑Γ (75) for any 𝑤∈𝐻²₀(Ω). From Lemma 3 we obtain that 𝑣∈𝐿^∞(0,𝑇;𝐻⁴(Ω)). The uniqueness of solutions follows by using standard arguments.

Theorem 5. Under the hypotheses of Theorem 4, let 𝑢₀∈𝐻²₀(Ω₀)∩𝐻⁴(Ω₀), 𝑢₁∈𝐻²₀(Ω₀). Then there exists a unique solution 𝑢 of the problem (4)–(8) satisfying 𝑢∈𝐿^∞0,∞;𝐻²₀Ω_𝑡∩𝐻⁴Ω_𝑡,𝑢^′∈𝐿^∞0,∞;𝐻²₀Ω_𝑡,𝑢^′′∈𝐿^∞0,∞;𝐿²Ω_𝑡. (76)

Proof. This idea was used in [11, 13, 14, 16, 17]. To show the existence in noncylindrical domains, we return to our original problem in the noncylindrical domains by using the change variable given in (14) by (𝑦,𝑡)=𝜏(𝑥,𝑡), (𝑥,𝑡)∈𝑄. Let 𝑣 be the solution obtained from Theorem 4 and 𝑢 defined by (16); then 𝑢 belongs to the class 𝑢∈𝐿^∞0,∞;𝐻²₀Ω_𝑡∩𝐻⁴Ω_𝑡,𝑢^′∈𝐿^∞0,∞;𝐻²₀Ω_𝑡,𝑢^′′∈𝐿^∞0,∞;𝐿²Ω_𝑡. (77) Denoting by 𝑢(𝑥,𝑡)=𝑣𝑦,𝑡=(𝑣∘𝜏)(𝑥,𝑡), (78) then from (15) it is easy to see that 𝑢 satisfies (4)–(8) in the sense of 𝐿^∞(0,∞;𝐿²(Ω_𝑡)). If 𝑢₁, 𝑢₂ are two solutions obtained through the diffeomorphism 𝜏 given by (14), then 𝑣₁=𝑣₂, so 𝑢₁=𝑢₂. Thus the proof of Theorem 5 is completed.

4. Exponential Decay

In this section, we show that the solution of system (4)–(8) decays exponentially. First of all, we introduce the useful lemma for a noncylindrical domain.

Lemma 6 (see [11, 12]). Let 𝐺(⋅,⋅) be the smooth function defined in Ω_𝑡×[0,∞[. Then 𝑑𝑑𝑡_Ω𝑡𝐺(𝑥,𝑡)𝑑𝑥=_Ω𝑡𝑑𝐺()𝑑𝑡𝑥,𝑡𝑑𝑥+𝛾^′𝛾⁻¹_Γ𝑡𝐺(𝑥,𝑡)(𝑥⋅𝜈)𝑑Γ, (79) where 𝜈 is the 𝑥-component of the unit normal exterior 𝜈.

By the same argument of (27) and (28), it can be written as ℬ₂𝑢=𝜏₁𝑢^′+𝑘₁(0)𝑢−𝑘₁(𝑡)𝑢₀+𝑘^′₁ℬ∗𝑢,₁𝑢=−𝜏₂𝜕𝑢^′𝜕𝜈+𝑘₂(0)𝜕𝑢𝜕𝜈−𝑘₂(𝑡)𝜕𝑢₀𝜕𝜈+𝑘^′₂∗𝜕𝑢.𝜕𝜈 (80)(81) We use (80) and (81) instead of the boundary conditions (6) and (7).

We will use the following lemma.

Lemma 7 (see [4]). For every 𝑢∈𝐻⁴(Ω) and for every 𝜇∈ℝ, one has _Ω𝑡(𝑚⋅∇𝑢)Δ²=𝑢𝑑𝑥_Ω𝑡𝑎()1𝑢,𝑢𝑑𝑥+2_Γ𝑡()𝑎()+𝑚⋅𝜈𝑢,𝑢𝑑Γ_Γ𝑡ℬ₂𝑢(𝑚⋅∇𝑢)−ℬ₁𝑢𝜕(𝜕𝜈𝑚⋅∇𝑢)𝑑Γ. (82)

Now, we define the energy of problem (4)–(8) by 𝐸(𝑡)=12𝑢^′2_2,𝑡+_Ω𝑡𝑎(𝑢,𝑢)𝑑𝑥+𝜏_1Γ1,𝑡𝑘₁(𝑡)|𝑢|²−𝑘^′₁□𝑢𝑑Γ+𝜏_2Γ1,𝑡𝑘₂(𝑡)𝜕𝑢𝜕𝜈²−𝑘^′₂□𝜕𝑢.𝜕𝜈𝑑Γ (83) We observe that 𝐸(𝑡) is a positive function. Using Lemmas 6 and 1, we have 𝐸^′(𝑡)≤𝛾^′𝛾⁻¹2_Γ1,𝑡𝑢^′2+𝑎(𝑢,𝑢)(𝑥⋅𝜈)−𝜏𝑑Γ₁2_Γ1,𝑡𝑢^′2𝜏𝑑Γ+¹2𝑘²₁(𝑡)_Γ1,𝑡𝑢₀²+𝜏𝑑Γ₁2𝑘^′₁(𝑡)_Γ1,𝑡|𝑢|²𝜏𝑑Γ−¹2_Γ1,𝑡𝑘₁^′′−𝜏□𝑢𝑑Γ₂2_Γ1,𝑡𝜕𝑢^′𝜕𝜈²𝜏𝑑Γ+²2𝑘²₂(𝑡)_Γ1,𝑡𝜕𝑢₀𝜕𝜈²+𝜏𝑑Γ₂2𝑘^′₂(𝑡)_Γ1,𝑡𝜕𝑢𝜕𝜈²𝜏𝑑Γ−²2_Γ1,𝑡𝑘₂^′′□𝜕𝑢𝜕𝜈𝑑Γ. (84) Let us consider the following functional: 𝜓(𝑡)=_Ω𝑡(𝑚⋅∇𝑢)𝑢^′𝑑𝑥. (85)

The following lemma plays an important role for the construction of the Lyapunov functional.

Lemma 8. Let one suppose that the initial data (𝑢₀,𝑢₁)∈(𝐻⁴(Ω₀)∩𝐻²₀(Ω₀))×𝐻²₀(Ω₀) and satisfies the compatibility condition (41). Then the solution of system (4)–(8) satisfies 𝜓^′(𝑡)≤12_Γ1,𝑡(𝑚⋅𝜈)𝑢^′2𝑑Γ−_Ω𝑡𝑢^′2−𝑑𝑥_Ω𝑡𝑎()1𝑢,𝑢𝑑𝑥−2_Γ1,𝑡()𝑎()−𝑚⋅𝜈𝑢,𝑢𝑑Γ_Γ1,𝑡ℬ₂𝑢(𝑚⋅∇𝑢)−ℬ₁𝑢𝜕(𝜕𝜈𝑚⋅∇𝑢)𝑑Γ+𝛾^′𝛾⁻¹_Γ1,𝑡(𝑚⋅∇𝑢)𝑢^′(𝑥⋅𝜈)𝑑Γ. (86)

Proof. Differentiating 𝜓 and using (4) and Lemmas 6 and 7, we get 𝜓^′(𝑡)=_Ω𝑡𝑚⋅∇𝑢^′𝑢^′𝑑𝑥+_Ω𝑡(𝑚⋅∇𝑢)𝑢^′′𝑑𝑥+𝛾^′𝛾⁻¹_Γ1,𝑡(𝑚⋅∇𝑢)𝑢^′(𝑥⋅𝜈)=1𝑑Γ2_Γ1,𝑡(𝑚⋅𝜈)𝑢^′2𝑑Γ−_Ω𝑡𝑢^′2−𝑑𝑥_Ω𝑡𝑎()1𝑢,𝑢𝑑𝑥−2_Γ𝑡()𝑎()−𝑚⋅𝜈𝑢,𝑢𝑑Γ_Γ𝑡ℬ₂𝑢(𝑚⋅∇𝑢)−ℬ₁𝑢𝜕(𝜕𝜈𝑚⋅∇𝑢)𝑑Γ+𝛾^′𝛾⁻¹_Γ𝑡(𝑚⋅∇𝑢)𝑢^′(𝑥⋅𝜈)𝑑Γ. (87) Let us next examine the integrals over Γ_0,𝑡 in (87). Since 𝑢=𝜕𝑢/𝜕𝜈=0 on Γ_0,𝑡, we have 𝐵₁𝑢=𝐵₂𝑢=∇𝑢=0onΓ_0,𝑡,𝑢_𝑥1=𝜕𝑢𝜈𝜕𝜈¹,𝑢_𝑥2=𝜕𝑢𝜈𝜕𝜈², (88) and hence _Γ0,𝑡ℬ₁𝑢𝜕()𝜕𝜈𝑚⋅∇𝑢𝑑Γ=_Γ0,𝑡()𝜕Δ𝑢𝑚⋅𝜈²𝑢𝜕𝜈²=𝑑Γ_Γ0,𝑡(𝑚⋅𝜈)|Δ𝑢|²𝑑Γ,_Γ0,𝑡(𝑚⋅𝜈)𝑎(𝑢,𝑢)𝑑Γ=_Γ0,𝑡(𝑚⋅𝜈)|Δ𝑢|²𝑑Γ. (89)(90) Therefore, from (87)–(90) we have 𝜓^′(𝑡)=12_Γ1,𝑡(𝑚⋅𝜈)𝑢^′2𝑑Γ−_Ω𝑡𝑢^′2𝑑𝑥−_Ω𝑡𝑎()+1𝑢,𝑢𝑑𝑥2_Γ0,𝑡(𝑚⋅𝜈)|Δ𝑢|²−1𝑑Γ2_Γ1,𝑡(𝑚⋅𝜈)𝑎(𝑢,𝑢)𝑑Γ−_Γ1,𝑡ℬ₂𝑢)+(𝑚⋅∇𝑢𝑑Γ_Γ1,𝑡ℬ₁𝑢𝜕()𝜕𝜈𝑚⋅∇𝑢𝑑Γ+𝛾^′𝛾⁻¹_Γ1,𝑡(𝑚⋅∇𝑢)𝑢^′(𝑥⋅𝜈)𝑑Γ. (91) Noting that 𝑚⋅𝜈≤0 on Γ_0,𝑡 follows from (91), we have the conclusion of the lemma.

Let us introduce the Lyapunov functional ℒ(𝑡)=𝑁𝐸(𝑡)+𝜓(𝑡), (92) with 𝑁>0. Using Young’s inequality and choosing 𝑁>0 sufficiently large, we see that 𝑞₀𝐸(𝑡)≤ℒ(𝑡)≤𝑞₁𝐸(𝑡) (93) for 𝑞₀ and 𝑞₁ are positive constants. We will show later that the functional ℒ satisfies the inequality of the following result.

Lemma 9 (see [7]). Let 𝑓 be a real positive function of class 𝐶¹. If there exist positive constants 𝑝₀, 𝑝₁, and 𝑝₂ such that 𝑓^′(𝑡)≤−𝑝₀𝑓(𝑡)+𝑝₁𝑒^−𝑝2𝑡 (94) then there exist positive constants 𝑝 and 𝑐 such that 𝑓(𝑡)≤𝑓(0)+𝑐𝑒^−𝑝𝑡. (95)

Finally, we will show the main result of this section.

Theorem 10. Assume that there exist positive constants 𝛽₁ and 𝛽₂ such that 𝑘_𝑖(0)>0,𝑘^′_𝑖(𝑡)≤−𝛽₁𝑘_𝑖(𝑡),𝑘_𝑖^′′(𝑡)≥−𝛽₂𝑘^′_𝑖(𝑡),𝑖=1,2. (96) If (𝑢₀,𝑢₁)∈𝐻²₀(Ω₀)×𝐿²(Ω₀) then there exist constants 𝜔,𝐶>0 such that 𝐸(𝑡)≤𝐶𝐸(0)𝑒^−𝜔𝑡,∀𝑡≥0. (97)

Proof. From (84) and Lemma 8 we have ℒ^′(𝑡)≤𝛾^′𝛾⁻¹𝑁2_Γ1,𝑡𝑢^′2+𝑎(𝑢,𝑢)(𝑥⋅𝜈)−𝜏𝑑Γ₁𝑁2_Γ1,𝑡𝑢^′2𝜏𝑑Γ+¹𝑁2𝑘²₁(𝑡)_Γ1,𝑡𝑢₀²+𝜏𝑑Γ₁𝑁2𝑘^′₁(𝑡)_Γ1,𝑡|𝑢|²𝜏𝑑Γ−¹𝑁2_Γ1,𝑡𝑘₁^′′−𝜏□𝑢𝑑Γ₂𝑁2_Γ1,𝑡𝜕𝑢^′𝜕𝜈²+𝜏𝑑Γ₂𝑁2𝑘²₂(𝑡)_Γ1,𝑡𝜕𝑢₀𝜕𝜈²+𝜏𝑑Γ₂𝑁2𝑘^′₂(𝑡)_Γ1,𝑡𝜕𝑢𝜕𝜈²−𝜏𝑑Γ₂𝑁2_Γ1,𝑡𝑘₂^′′□𝜕𝑢+1𝜕𝜈𝑑Γ2_Γ1,𝑡(𝑚⋅𝜈)𝑢^′2−𝑑Γ_Ω𝑡𝑢^′2𝑑𝑥−_Ω𝑡𝑎()−1𝑢,𝑢𝑑𝑥2_Γ1,𝑡()𝑎()−𝑚⋅𝜈𝑢,𝑢𝑑Γ_Γ1,𝑡ℬ₂𝑢(𝑚⋅∇𝑢)−ℬ₁𝑢𝜕(𝜕𝜈𝑚⋅∇𝑢)𝑑Γ+𝛾^′𝛾⁻¹_Γ1,𝑡(𝑚⋅∇𝑢)𝑢^′(𝑥⋅𝜈)𝑑Γ. (98) Since the boundary conditions (80) and (81) 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 (99) by using Young’s inequality we obtain −_Γ1,𝑡ℬ₂𝑢()≤𝜏𝑚⋅∇𝑢𝑑Γ¹2𝜖_Γ1,𝑡𝑢^′2+𝜏𝑑Γ₁𝑘2𝜖²¹(𝑡)_Γ1,𝑡|𝑢|²+𝜏𝑑Γ₁𝑘2𝜖²¹(𝑡)_Γ1,𝑡𝑢₀²+𝜏𝑑Γ₁2𝜖_Γ1,𝑡𝑘₁(0)𝑘^′₁+𝜖□𝑢𝑑Γ2_Γ1,𝑡|𝑚⋅∇𝑢|²𝑑Γ,_Γ1,𝑡ℬ₁𝑢𝜕()≤𝜏𝜕𝜈𝑚⋅∇𝑢𝑑Γ²2𝜖_Γ1,𝑡𝜕𝑢^′𝜕𝜈²+𝜏𝑑Γ₂𝑘2𝜖²²(𝑡)_Γ1,𝑡𝜕𝑢𝜕𝜈²+𝜏𝑑Γ₂𝑘2𝜖²²(𝑡)_Γ1,𝑡𝜕𝑢₀𝜕𝜈²+𝜏𝑑Γ₂2𝜖_Γ1,𝑡𝑘₂(0)𝑘^′₂□𝜕𝑢+𝜖𝜕𝜈𝑑Γ2_Γ1,𝑡𝜕(𝜕𝜈𝑚⋅∇𝑢)²𝑑Γ, (100)(101) where 𝜖 is a positive constant. Since the bilinear form 𝑎(𝑢,𝑢) is strictly coercive, using the trace theory and the fact 𝑚⋅𝜈≥𝛿₀ on Γ_1,𝑡, we get _Γ1,𝑡|𝑚⋅∇𝑢|²𝑑Γ+_Γ1,𝑡𝜕(𝜕𝜈𝑚⋅∇𝑢)²𝑑Γ≤𝜆_0Ω𝑡𝑎()𝜆𝑢,𝑢𝑑𝑥+⁰𝛿_0Γ1,𝑡(𝑚⋅𝜈)𝑎(𝑢,𝑢)𝑑Γ, (102) where 𝜆₀ is a constant depending on Ω and 𝜇. Substituting inequalities (100)–(102) into (98) we have ℒ^′(𝑡)≤𝛾^′𝛾⁻¹𝑁2_Γ1,𝑡𝑢^′2+𝑎(𝑢,𝑢)(𝑥⋅𝜈)−𝜏𝑑Γ₁𝑁2_Γ1,𝑡𝑢^′2𝜏𝑑Γ+¹𝑁2𝑘²₁(𝑡)_Γ1,𝑡𝑢₀²−𝜏𝑑Γ₁𝛽₁𝑁2𝑘₁(𝑡)_Γ1,𝑡|𝑢|²𝜏𝑑Γ+¹𝛽₂𝑁2_Γ1,𝑡𝑘^′₁−𝜏□𝑢𝑑Γ₂𝑁2_Γ1,𝑡𝜕𝑢^′𝜕𝜈²+𝜏𝑑Γ₂𝑁2𝑘²₂(𝑡)_Γ1,𝑡𝜕𝑢₀𝜕𝜈²−𝜏𝑑Γ₂𝛽₁𝑁2𝑘₂(𝑡)_Γ1,𝑡𝜕𝑢𝜕𝜈²+𝜏𝑑Γ₂𝛽₂𝑁2_Γ1,𝑡𝑘^′₂□𝜕𝑢−𝜕𝜈𝑑Γ_Ω𝑡𝑢^′2𝑑𝑥−1−𝜖𝜆₀2_Ω𝑡𝑎()−1𝑢,𝑢𝑑𝑥2−𝜖𝜆₀2𝛿_0Γ1,𝑡()𝑎()+1𝑚⋅𝜈𝑢,𝑢𝑑Γ2_Γ1,𝑡(𝑚⋅𝜈)𝑢^′2𝜏𝑑Γ+¹2𝜖_Γ1,𝑡𝑢^′2+𝜏𝑑Γ₁𝑘2𝜖²¹(𝑡)_Γ1,𝑡|𝑢|²𝜏𝑑Γ+¹𝑘2𝜖²¹(𝑡)_Γ1,𝑡𝑢₀²+𝜏𝑑Γ₁2𝜖_Γ1,𝑡𝑘₁(0)𝑘^′₁𝜏□𝑢𝑑Γ+²2𝜖_Γ1,𝑡𝜕𝑢^′𝜕𝜈²+𝜏𝑑Γ₂𝑘2𝜖²²(𝑡)_Γ1,𝑡𝜕𝑢𝜕𝜈²𝜏𝑑Γ+²𝑘2𝜖²²(𝑡)_Γ1,𝑡𝜕𝑢₀𝜕𝜈²+𝜏𝑑Γ₂2𝜖_Γ1,𝑡𝑘₂(0)𝑘^′₂□𝜕𝑢𝜕𝜈𝑑Γ+𝛾^′𝛾⁻¹_Γ1,𝑡(𝑚⋅∇𝑢)𝑢^′(𝑥⋅𝜈)𝑑Γ. (103) First, choose 𝜖>0 sufficiently small such that 1−𝜖𝜆₀21>0,2−𝜖𝜆₀2𝛿₀>0. (104) Then, choosing 𝑁 large enough, we have ℒ^′(𝑡)≤−𝑐₂𝐸(𝑡)+𝑐₃𝐾²(𝑡)𝐸(0), (105) where 𝑐₂,𝑐₃>0 and 𝐾(𝑡)=𝑘₁(𝑡)+𝑘₂(𝑡). From (93), (96), and (105), we obtain ℒ^′(𝑡)𝑐≤−²𝑞₁ℒ(𝑡)+𝑐₄𝑐₃𝐸(0)𝑒^{−2𝛽1𝑡}forsome𝑐₄>0. (106) By Lemma 9, there exist positive constants 𝑐₅ and 𝑐₆ such that ℒ(𝑡)≤ℒ(0)+𝑐₅𝐸(0)𝑒^−𝑐6𝑡,∀𝑡≥0. (107) Using (93), we conclude that 𝐸(𝑡)≤𝐶𝐸(0)𝑒^−𝜔𝑡,∀𝑡≥0 (108) for some positive constants 𝐶 and 𝜔.

Acknowledgment

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