Asymptotic Energy Estimates for Nonlinear Petrovsky Plate Model Subject to Viscoelastic Damping

Abstract

We consider the nonlinear Petrovsky plate model under the presence of long-time memory. Under suitable conditions, we show that the energy functional associated with the equation decays exponentially or polynomially to zero as time goes to infinity.

1. Introduction

Let Ω be a bounded domain of ℝ² with a sufficiently smooth boundary Γ=Γ₀∪Γ₁. We will assume that Γ₀∩Γ₁=∅ and Γ₀,Γ₁ have positive measures. The vector 𝝂=(𝜈₁,𝜈₂) is the unit exterior normal and 𝝉=(−𝜈₂,𝜈₁) represents the tangential direction to Γ. Here, the variable 𝑤 represents displacement of a plate occupying the domain Ω. The governing equation is given by 𝑤^′′(𝑡)−𝛾Δ𝑤^′′(𝑡)+Δ²𝑤(𝑡)+Δ^2∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)[𝒞(𝜎(𝑡(𝑡(),𝑑𝑠=div))∇𝑤)]inΩ×0,∞ (1.1) where the notations 𝑤^𝑡(𝑠)=𝑤(𝑡−𝑠),^′=𝜕_𝑡. The positive constant 𝛾 is the proportional to the thickness of the plate, ℎ(𝑡) is relaxation function, div stands for scalar divergence of a vector field, the stress resultant 𝜎(𝑡) is given by 𝜎(𝑡)∶=𝑓(∇𝑤(𝑡)), (1.2) and the nonlinear function 𝑓∶ℝ²→𝑀 is defined as 𝑓(𝑠)=(1/2)𝑠⊗𝑠 for all 𝑠∈ℝ².𝒞 is a linear operator defined on 𝑀 with value on 𝑀, the space of 2×2 symmetric matrices, and defined as 𝒞(𝜎)=𝐸𝑑1−𝜇²𝜇tr𝜎𝐼+1−𝜇𝜎 (1.3) for any 𝜎∈𝑀, where 𝐼is the identity matrix and tr𝜎 denotes the trace of 𝜎. Moreover, 𝑑>0 is the density of the shell, 𝐸>0 denotes the Young modulus, and 𝜇(0<𝜇<1/2) is the Poisson’s ratio.

With (1.1), we associate the boundary conditions on the portion of the boundary Γ₀, 𝑤=𝜕_𝜈𝑤=0onΓ₀×(0,∞), (1.4) and the boundary conditions on the remaining portion of the boundary Γ₁, ℬ₁𝑤(𝑡)+ℬ₁^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠=0onΓ₁×(),ℬ0,∞₂𝑤(𝑡)−𝛾𝜕_𝜈𝑤^′′(𝑡)+ℬ₂^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠−𝒞(𝜎(𝑡))𝜈⋅∇𝑤(𝑡)=0onΓ₁×(0,∞), (1.5) where ℬ₁𝑤=Δ𝑤+1−𝜇𝐵₁𝑤,ℬ₂𝑤=𝜕_𝜈Δ𝑤+1−𝜇𝜕_𝜏𝐵₂𝑤 (1.6) and the boundary operators 𝐵₁ and 𝐵₂ are defined by 𝐵₁𝑤=2𝜈₁𝜈₂𝑤_𝑥𝑦−𝜈²₁𝑤_𝑦𝑦−𝜈²₂𝑤_𝑥𝑥,𝐵₂𝑤=𝜈²₁−𝜈²₂𝑤_𝑥𝑦+𝜈₁𝜈₂𝑤_𝑦𝑦−𝑤_𝑥𝑥. (1.7)

With (1.1), we also associate the initial conditions 𝑤(0)=𝑤⁰,𝑤^′(0)=𝑤¹inΩ,𝑤(𝑠)=𝜒(𝑠),−∞<𝑠<0, (1.8) where 𝜒(𝑠),𝜒^′(𝑠)∈𝐿^∞_𝑐−∞,0;𝐻³×𝐻², (1.9) the symbol 𝐿^∞_𝑐 denotes the subspace of 𝐿^∞ such that there exists a constant 𝑇 such that the functions vanish as 𝑠<−𝑇.

This problem has its origin in the mathematical description of viscoelastic materials. It is well known that viscoelastic materials exhibit natural damping, which is due to the special property of these materials, to retain a memory of their past history. From the mathematical point of view, these damping effects are modeled by integro-differential operators. Therefore, the dynamics of viscoelastic materials are of great importance and interest as they have wide applications in natural sciences. From the physical point of view, the problem (1.1) describes the position 𝑤(𝑥,𝑦,𝑡) of the material particle (𝑥,𝑦) at time 𝑡, which is clamped in the portion Γ₀ of its boundary.

Models of Petrovsky type are of interest in applications in various areas in mathematical physics, as well as in geophysics and ocean acoustics [1, 2]. The Petrovsky type models without memory were discussed in [3, 4]. Messaoudi [3] considered the initial-boundary value problem 𝑢_𝑡𝑡+Δ²𝑢+𝑢_𝑡^𝑚−2𝑢_𝑡=|𝑢|^𝑝−2𝑢()𝑢,𝑥∈Ω,𝑡>0,𝑥,𝑡=𝜕_𝜈𝑢()𝑢()𝑥,𝑡=0,𝑥∈𝜕Ω,𝑡≥0,𝑥,0=𝑢₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥),𝑥∈Ω, (1.10) established an existence result for (1.10), and showed that the solution continues to exist globally if 𝑚≥𝑝, however if 𝑚<𝑝 and the initial energy is negative, the solution blows up in finite time. Chen and Zhou [4] proved that the solution of (1.10) blows up with positive initial energy. Moreover, she claimed that the solution blows up in finite time for vanishing initial energy under the condition 𝑚=2 by different method.

The Petrovsky type equations with memory arouse the attention of mathematicians to study them. Alabau-Boussouira et al. [5] discussed the initial-boundary value problem of linear Petrovsky equation related to a plate model with memory, 𝑢_𝑡𝑡+Δ²𝑢−^𝑡₀𝑔(𝑡−𝑠)Δ²𝑢(𝑡,𝑠)𝑑𝑠=0inΩ×(0,∞),𝑢=𝜕_𝜈𝑢=0on𝜕Ω×(0,∞),𝑢|_𝑡=0=𝑢₀,𝑢_𝑡|_𝑡=0=𝑢₁inΩ, (1.11) and showed that the solution decays exponentially or polynomially as 𝑡→+∞ if the initial data is sufficient small. Yang [6] considered the problem in 𝑁-dimensional space, 𝑢_𝑡𝑡+Δ²𝑢+𝜆𝑢_𝑡=^𝑁_𝑖=1𝜕𝜕𝑥_𝑖𝜎_𝑖𝑢_𝑥𝑖inΩ×(0,∞),𝑢|_𝜕Ω=0,𝜕𝑢𝜕𝑛_𝜕Ω[),𝑢()=0on0,∞𝑥,0=𝑢₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥),𝑥∈Ω, (1.12) and proved that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as 𝑡→∞ in the states of large initial data and small initial energy. In particular, in the case of space dimension 𝑁=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. Muñoz Rivera et al. [7] considered the initial-boundary problem for viscoelastic plate equation, 𝑢_𝑡𝑡−𝛾Δ𝑢_𝑡𝑡+Δ²𝑢(𝑡)−^𝑡₀𝑔(𝑡−𝜏)Δ²𝑢(𝜏)(),𝑢𝑑𝜏=0inΩ×0,∞𝑥,𝑦,0=𝑢₀𝑥,𝑦,𝑢_𝑡𝑥,𝑦,0=𝑢₁𝑥,𝑦inΩ,𝑢=𝜕_𝜈𝑢=0onΓ₀×(),ℬ0,∞₁𝑢(𝑡)+ℬ₁^∞₀𝑔(𝑠)𝑢^𝑡(𝑠)𝑑𝑠=0onΓ₁×(),ℬ0,∞₂𝑢(𝑡)−𝛾𝜕_𝜈𝑢^′′(𝑡)+ℬ₂^∞₀𝑔(𝑠)𝑢^𝑡(𝑠)𝑑𝑠=0onΓ₁×(0,∞), (1.13) and proved that the first and second order energies associated with its solution decay exponentially provided the kernel of the convolution also decays exponentially. When the kernel decays polynomially then the energy also decays polynomially. More precisely if the kernel 𝑔 satisfies 𝑔(𝑡)≤−𝑐₀𝑔^1+(1/𝑝)(𝑡),𝑔,𝑔^1+(1/𝑝)∈𝐿¹(𝑅)with𝑝>2, (1.14) then the energy decays as 1/(1+𝑡)^𝑝. On the recently related papers concerning the Petrovsky type models, the readers can see references [8–12].

In [13–15], Li et al. proved the existence uniqueness, uniform rates of decay, and limit behavior of the solution to nonlinear viscoelastic Marguerre-von Karman shallow shells system, respectively. To our best knowledge, we do not find the research report on the problem (1.1) which is considered in this paper.

Motivated by the above work, we obtain the energy functional associated with the equation decays exponentially or polynomially to zero as time goes to infinity. The main contribution of this paper are as follows. (a) The problem considered in this paper is nonlinear equation with integral dissipation, to our knowledge this model has not been considered; (b) the hypothesis on ℎ and initial data are weaker; (c) we naturally define the energy by simple computation and only define simple auxiliary functionals to prove our result by precise priori estimates.

The outline of this paper is the following. In Section 2, we present some material needed to be proved. Section 3 contains the statement and the proof of our results.

2. Notations and Preliminaries

In this section, we will prepare some material needed in the proof of our main results. We use standard Lebesgue space 𝐿^𝑝(Ω) and Sobolev space 𝐻^𝑚(Ω) and adopt the following notations (𝑢,𝑣)∶=(𝑢,𝑣)_𝐿²_(Ω),⟨𝑢,𝑣⟩∶=(𝑢,𝑣)_𝐿²_(Γ1),‖𝑢‖_𝑝∶=‖𝑢‖_𝐿^𝑝_(Ω),‖𝑢‖∶=‖𝑢‖_𝐿²_(Ω). (2.1) We denote |𝐀(𝑡)|²∶=_𝑖,𝑗𝑎²_𝑖𝑗(𝑡),𝑑𝐀(𝑡)𝑑𝑡∶=𝐀^′(𝑡)=𝑎^′_𝑖𝑗(𝑡),𝐀(𝑡)⋅𝐁(𝑡)∶=_𝑖,𝑗𝑎_𝑖𝑗(𝑡)𝑏_𝑖𝑗(𝑡),(𝐀(𝑡),𝐁(𝑡))∶=_𝑖,𝑗𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡) (2.2) for any pair of tensors 𝐀(𝑡)=(𝑎_𝑖𝑗(𝑡)) and 𝐁(𝑡)=(𝑏_𝑖𝑗(𝑡)). We introduce the following space 𝑊(Ω)=𝑤∈𝐻²(Ω),𝑤=𝜕_𝜈𝑤=0onΓ₀. (2.3) Define the bilinear form 𝑎(⋅,⋅) as follows 𝑎(𝑤,𝑣)=_Ω𝑤_𝑥𝑥𝑣_𝑥𝑥+𝑤_𝑦𝑦𝑣_𝑦𝑦+𝜇𝑤_𝑥𝑥𝑣_𝑦𝑦+𝑤_𝑦𝑦𝑣_𝑥𝑥+21−𝜇𝑤_𝑥𝑦𝑣_𝑥𝑦𝑑𝑥𝑑𝑦. (2.4) For simplicity, we denote 𝑎(𝑤,𝑤) by 𝑎(𝑤). For the relaxation function ℎ(𝑡), we assume that(𝐴₁)ℎ∶[0,+∞)→(0,+∞) is a 𝐶² function satisfying ℎ(𝑡)>0,ℎ^′(𝑡)<0,ℎ^′′(𝑡)≥0. (2.5)(𝐴₂)Both ℎ_∞∶=ℎ(∞) and ℎ^′_∞∶=ℎ^′(∞) exist, and ℎ_∞>0,ℎ^′_∞=0,ℎ(0)=1. (2.6)Hypotheses (𝐴₁) assure that the viscoelastic energy (defined below) is nonincreasing and the assumption (𝐴₂) means that the material behaves like an viscoelastic solid at 𝑡=+∞ (cf. [16]).

Our results are based on the following existence theorem.

Theorem 2.1. One assumes that ℎ(𝑡) satisfies conditions (𝐴₁)-(𝐴₂). For any initial data 𝑤⁰,𝑤¹∈𝐻³(Ω)×𝐻²(Ω) (2.7) subject to the compatibility conditions satisfied on the boundary Γ, ∇𝑤⁰=∇𝑤¹=𝟎,𝑤¹=𝑤⁰=0onΓ₀,ℬ₁𝑤⁰+^∞₀ℎ^′(𝑠)𝜒(−𝑠)𝑑𝑠=0onΓ₁. (2.8) Then for any 𝑇>0, there exists a unique global solution to (1.1)–(1.10) satisfying 𝑤,𝑤^′,𝑤^′′∈𝐿^∞0,𝑇;𝐻³(Ω)×𝐻²(Ω)×𝐻¹(Ω). (2.9)

Proof. We can use the method used in [13, 17] to show the existence and uniqueness as well as the regularity of the global solution.

At First, the energy of the system must be properly defined. The total energy can be expected to consist of two parts. One part involves the current kinetic and strain energies, and the other will involve the past history of strains. To obtain the appropriate expression of energy functional, using assumption (𝐴₂), we rewrite system (1.1) and (1.5) in the form 𝑤^′′(𝑡)−𝛾Δ𝑤^′′(𝑡)+ℎ_∞Δ²𝑤(𝑡)+Δ^2∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)(𝑡[𝒞(𝜎(𝑡(𝑡(),ℎ−𝑤)𝑑𝑠=div))∇𝑤)]inΩ×0,∞_∞ℬ₁𝑤(𝑡)+ℬ₁^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠=0onΓ₁×(),ℎ0,∞_∞ℬ₂𝑤(𝑡)−𝛾𝜕_𝜈𝑤^′′(𝑡)+ℬ₂^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)(𝑡(𝜎(𝑡(𝑡)−𝑤)𝑑𝑠+𝒞))𝜈⋅∇𝑤=0onΓ₁×(0,∞). (2.10)(2.11)

In order to define the energy functional, we give the following lemma.

Lemma 2.2. Let 𝑤 and 𝑣 be the functions in 𝐻⁴(Ω)∩𝑊. Then, one has _ΩΔ²𝑤𝑣𝑑𝑥𝑑𝑦=𝑎(𝑤,𝑣)+_Γ1ℬ₂𝑤𝑣−ℬ₁𝑤𝜕_𝜈𝑣𝑑Γ. (2.12)

Proof.. The definition of 𝑎(𝑤,𝑣) gives Δ𝑤Δ𝑣𝑑𝑥𝑑𝑦=𝑎(_Ω𝑤,𝑣)+_Ω1−𝜇𝑤_𝑥𝑥𝑣_𝑦𝑦+𝑤_𝑦𝑦𝑣_𝑥𝑥−21−𝜇𝑤_𝑥𝑦𝑣_𝑥𝑦𝑑𝑥𝑑𝑦. (2.13) Using Green’s formula, we see _ΩΔ²𝑤𝑣𝑑𝑥𝑑𝑦=_Γ1𝜕_𝜈Δ𝑤𝑣𝑑Γ−_Γ1Δ𝑤𝜕_𝜈𝑣𝑑Γ+_ΩΔ𝑤Δ𝑣𝑑𝑥𝑑𝑦=_Γ1𝜕_𝜈Δ𝑤𝑣𝑑Γ−_Γ1Δ𝑤𝜕_𝜈()+𝑣𝑑Γ+𝑎𝑤,𝑣_Ω1−𝜇𝑤_𝑥𝑥𝑣_𝑦𝑦+𝑤_𝑦𝑦𝑣_𝑥𝑥−21−𝜇𝑤_𝑥𝑦𝑣_𝑥𝑦𝑑𝑥𝑑𝑦, (2.14)_Ω𝑤_𝑥𝑥𝑣_𝑦𝑦+𝑤_𝑦𝑦𝑣_𝑥𝑥−2𝑤_𝑥𝑦𝑣_𝑥𝑦𝑑𝑥𝑑𝑦=_Γ1𝑤_𝑥𝑥𝑣_𝑦𝜈₂+𝑤_𝑦𝑦𝑣_𝑥𝜈₁𝑑Γ−_Ω𝑤_𝑥𝑥𝑦𝑣_𝑦+𝑤_𝑥𝑦𝑦𝑣_𝑥−𝑑𝑥𝑑𝑦_Γ1𝑤_𝑥𝑦𝑣_𝑦𝜈₁+𝑤_𝑥𝑦𝑣_𝑥𝜈₂+𝑑Γ_Ω𝑤_𝑥𝑥𝑦𝑣_𝑦+𝑤_𝑥𝑦𝑦𝑣_𝑥=𝑑𝑥𝑑𝑦_Γ1𝑤_𝑥𝑥𝑣_𝑦𝜈₂+𝑤_𝑦𝑦𝑣_𝑥𝜈₁−𝑑Γ_Γ1𝑤_𝑥𝑦𝑣_𝑦𝜈₁+𝑤_𝑥𝑦𝑣_𝑥𝜈₂𝑑Γ. (2.15) Using 𝜕_𝜈𝑣=𝑣_𝑥𝜈₁+𝑣_𝑦𝜈₂,𝜕_𝜏𝑣=−𝑣_𝑥𝜈₂+𝑣_𝑦𝜈₁, (2.16) we get 𝑣_𝑥=𝜕_𝜈𝑣𝜈₁−𝜕_𝜏𝑣𝜈₂,𝑣_𝑦=𝜕_𝜈𝑣𝜈₂+𝜕_𝜏𝑣𝜈₁. (2.17) Inserting (2.14) and (2.15) into (2.17) and noting _Γ1𝜕_𝜏(𝑤𝑣)𝑑Γ=_Γ𝜕_𝜏(𝑤𝑣)𝑑Γ=_Γ(𝑤𝑣)_𝑥𝑑𝑥+(𝑤𝑣)_𝑦𝑑𝑦=_Ω(𝑤𝑣)_𝑦𝑥−(𝑤𝑣)_𝑥𝑦𝑑𝑥𝑑𝑦=0, (2.18) we have _Γ1𝑤𝜕_𝜏𝑣𝑑Γ=−_Γ1𝑣𝜕_𝜏𝑤𝑑Γ, (2.19) we obtain the conclusion.

In order to define the energy function 𝐸(𝑡) of the problem (1.1)–(1.8), we give the following computations. Multiplying equation (2.10) by 𝑤^′, integrating the result over Ω and adding Green’s formula, we get from Lemma 2.2 and (1.4) that _Ω𝑤^′′(𝑡)𝑤^′(𝑡)𝑑𝑥𝑑𝑦+𝛾_Ω∇𝑤^′′(𝑡)⋅∇𝑤^′(𝑡)𝑑𝑥𝑑𝑦+ℎ_∞𝑎𝑤(𝑡),𝑤^′(𝑡)+_Γ1𝑤^′(𝑡)ℎ_∞ℬ₂𝑤(𝑡)−𝛾𝜕_𝜈𝑤^′′(𝑡)−ℎ_∞𝜕_𝜈𝑤^′(𝑡)ℬ₁𝑤(𝑡+)𝑑Γ_∞0ℎ^′(𝑠)Δ²𝑤^𝑡(𝑠)−Δ²𝑤(𝑡),𝑤^′(𝑡)𝑑𝑠=_Γ1𝒞(𝜎(𝑡))∇𝑤(𝑡)⋅𝜈𝑑Γ−_Ω𝒞(𝜎(𝑡))∇𝑤(𝑡)⋅∇𝑤^′(𝑡)=𝑑𝑥𝑑𝑦_Γ1𝒞(𝜎(𝑡))∇𝑤(𝑡)⋅𝜈𝑑Γ−_Ω𝒞(𝜎(𝑡))⋅∇𝑤^′(𝑡)⊗∇𝑤(𝑡)𝑑𝑥𝑑𝑦. (2.20) Clearly Δ²𝑤^𝑡(𝑠)−Δ²𝑤(𝑡),𝑤^′(𝑡)=−Δ²𝑤^𝑡(𝑠)−Δ²𝑤(𝑡),𝑤^′𝑡(𝑠)−𝑤^′(𝑡)+Δ²𝑤^′𝑡(𝑠)−Δ²𝑤(𝑡),𝑤^′𝑡(𝑠). (2.21) By applying Lemma 2.2, the first term on the right-hand side of (2.21) equals −Δ²𝑤^𝑡(𝑠)−Δ²𝑤(𝑡),𝑤^′𝑡(𝑠)−𝑤^′(𝑡1)=−2𝜕_𝑡𝑎𝑤^𝑡(𝑠)(𝑡−−𝑤)_Γ1𝑤^′𝑡(𝑠)−𝑤^′(𝑡)ℬ₂𝑤^𝑡(𝑠)−𝑤(𝑡)−𝜕_𝜈𝑤^′𝑡(𝑠)−𝑤^′(𝑡)ℬ₁𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑Γ. (2.22) In the same way, the second term on the right-hand side of (2.21) equals Δ²𝑤^′𝑡(𝑠)−Δ²𝑤(𝑡),𝑤^′𝑡(𝑠1)=−2𝜕_𝑠𝑎𝑤^𝑡(𝑠)(𝑡+−𝑤)_Γ1𝑤^′𝑡(𝑠)ℬ₂𝑤^𝑡(𝑠)−𝑤(𝑡)−𝜕_𝜈𝑤^′𝑡(𝑠)ℬ₁𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑Γ. (2.23) Therefore, from (2.21)–(2.23), we have Δ²𝑤^𝑡(𝑠)−Δ²𝑤(𝑡),𝑤^′(𝑡1)=−2𝜕_𝑡𝑎𝑤^𝑡(𝑠)(𝑡−1−𝑤)2𝜕_𝑠𝑎𝑤^𝑡(𝑠)(𝑡+−𝑤)_Γ1𝑤^′(𝑡)ℬ₂𝑤^𝑡(𝑠)−𝑤(𝑡)−𝜕_𝜈𝑤^′(𝑡)ℬ₁𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑Γ. (2.24) Note −12^∞₀ℎ^′(𝑠)𝜕_𝑡𝑎𝑤^𝑡(𝑠)(𝑡)1−𝑤𝑑𝑠=−2𝑑𝑑𝑡^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (2.25) and (by ℎ^′_∞=0) −12^∞₀ℎ^′(𝑠)𝜕_𝑠𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠=−ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)^∞₀+12^∞₀ℎ^′′(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)=1−𝑤𝑑𝑠2^∞₀ℎ^′′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (2.26) Consequently, we conclude form (2.24)–(2.26) that ^∞₀ℎ^′(𝑠)Δ²𝑤^𝑡(𝑠)−Δ²𝑤(𝑡),𝑤^′(𝑡)1𝑑𝑠=−2𝑑𝑑𝑡^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)+1−𝑤𝑑𝑠2^∞₀ℎ^′′(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)+−𝑤𝑑𝑠_∞0ℎ^′(𝑠)_Γ1𝑤^′(𝑡)ℬ₂𝑤^𝑡(𝑠)−𝑤(𝑡)−𝜕_𝜈𝑤^′(𝑡)ℬ₁𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑Γ𝑑𝑠. (2.27) Summing (2.20) and (2.27), using the boundary conditions (2.11) together with the symmetric of 𝜎(𝑡) yields 12𝑑𝑑𝑡‖𝑤^′(𝑡)‖²+𝛾‖∇𝑤^′(𝑡)‖²+ℎ_∞𝑎(𝑤(𝑡))−^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)+−𝑤𝑑𝑠_Ω𝒞(𝜎(𝑡))⋅∇𝑤^′(𝑡)(𝑡)1⊗∇𝑤𝑑𝑥𝑑𝑦+2^∞₀ℎ^′′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠=0. (2.28) Note the relation _Ω𝒞(𝜎(𝑡))⋅∇𝑤^′(𝑡)(𝑡)1⊗∇𝑤𝑑𝑥𝑑𝑦=4𝑑𝑑𝑡_Ω𝒞(𝜎(𝑡))⋅𝜎(𝑡)𝑑𝑥𝑑𝑦. (2.29) In fact _Ω𝒞(𝜎(𝑡))⋅∇𝑤^′(𝑡)⊗∇𝑤(𝑡)𝑑𝑥𝑑𝑦=𝛼_Ω𝑤_𝑥𝑤_𝑥+𝜇𝑤_𝑦𝑤_𝑦1−𝜇𝑤_𝑥𝑤_𝑦1−𝜇𝑤_𝑦𝑤_𝑥𝑤_𝑦𝑤_𝑦+𝜇𝑤_𝑥𝑤_𝑥⋅𝑤^′_𝑥𝑤_𝑥𝑤^′_𝑥𝑤_𝑦𝑤^′_𝑦𝑤_𝑥𝑤^′_𝑦𝑤_𝑦𝑑𝑥𝑑𝑦=𝛼_Ω𝑤_𝑥³𝑤^′_𝑥+𝑤_𝑦³𝑤^′_𝑦+𝑤_𝑦²𝑤_𝑥𝑤^′_𝑥+𝑤_𝑥²𝑤_𝑦𝑤^′_𝑦𝑑𝑥𝑑𝑦 (2.30) with 𝛼=𝐸/𝑑(1−𝜇²). Denote 𝜎∶=^𝜎11𝜎12_𝜎21𝜎22. Hence 𝒞(𝜎)𝜎=𝛼¹¹+𝜇𝜎₂₂1−𝜇𝜎₁₂1−𝜇𝜎₂₁𝜎₂₂+𝜇𝜎₁₁,[𝒞(𝜎(𝑡))]^′⋅𝜎(𝑡)=𝛼𝜎^′₁₁𝜎₁₁+𝜎^′₂₂𝜎₂₂+𝜇𝜎^′₁₁𝜎₂₂+𝜎^′₂₂𝜎₁₁+1−𝜇𝜎^′₁₂𝜎₁₂+𝜎^′₂₁𝜎₂₁=𝒞(𝜎(𝑡))⋅[𝜎(𝑡)]^′, (2.31) which implies that 𝑑𝑑𝑡_Ω𝒞(𝜎(𝑡(𝑡)=))⋅𝜎𝑑𝑥𝑑𝑦_Ω[𝒞(𝜎(𝑡))]^′⋅𝜎(𝑡)𝑑𝑥𝑑𝑦+_Ω𝒞(𝜎(𝑡))⋅𝜎^′(𝑡)𝑑𝑥𝑑𝑦=2_Ω[𝒞(𝜎(𝑡))]^′⋅𝜎(𝑡)𝑑𝑥𝑑𝑦=4𝛼_Ω𝑤_𝑥³𝑤^′_𝑥+𝑤_𝑦³𝑤^′_𝑦+𝑤_𝑦²𝑤_𝑥𝑤^′_𝑥+𝑤_𝑥²𝑤_𝑦𝑤^′_𝑦𝑑𝑥𝑑𝑦. (2.32) From (2.30) and (2.32), we conclude that (2.29) holds. Combining (2.28) and (2.29), we conclude that 12𝑑𝑤𝑑𝑡^′(𝑡)²+𝛾∇𝑤^′(𝑡)²+ℎ_∞𝑎(𝑤(𝑡−))_∞0ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)1−𝑤𝑑𝑠+2_Ω𝒞(𝜎(𝑡(𝑡)1))⋅𝜎𝑑𝑥𝑑𝑦=−2^∞₀ℎ^′′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (2.33) The relation (2.33) inspires us to define energy functional as the following 𝐸(𝑡)=12𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²+12ℎ_∞𝑎(𝑤(𝑡))−^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)1−𝑤𝑑𝑠+2_Ω𝒞(𝜎(𝑡))⋅𝜎(𝑡)𝑑𝑥𝑑𝑦. (2.34)

From (2.33) and (2.34), we obtain the following lemma.

Lemma 2.3. Under the above notations and assumptions (𝐴₁) and (𝐴₂), one has that 𝑑𝐸(𝑡)1𝑑𝑡=−2^∞₀ℎ^′′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠≤0. (2.35)

3. Main Results

In this section, 𝑐,𝑐,𝑐̃, and 𝑐_𝑖 denote some positive constants. Here, we need to point out that 𝛿 denotes the small enough different positive constant and 𝐶_𝛿 denotes the different positive constant depending on 𝛿 in a different place, respectively. The main goal of this paper is to show, respectively, that the solution decays exponentially or polynomially to zero as the time goes to infinity under suitable conditions. Our main results are formulated below.

Theorem 3.1. Let 𝑤 be the global solution of the problem (1.1)–(1.8) with the conditions ℎ^′(𝑡)≤𝑐ℎ^′′′(𝑡),−𝑐₃ℎ^′(𝑡)≤ℎ^′′(𝑡)≤−𝑐₄ℎ^′(𝑡). (3.1) Then the energy functional 𝐸(𝑡) decays exponentially to zero as the time goes to infinity.

To prove our main result, we will give some important preliminaries.

Lemma 3.2. Assume 𝑤∈𝑊(Ω), functional 𝑎(𝑤) and tensor 𝜎(𝑡) are defined as above. Then(1)𝑐₁‖∇²𝑤‖²≤𝑎(𝑤)≤𝑐₂‖∇²𝑤‖²(2)‖∇𝑤‖²_𝐿^𝑞≤𝑐₃𝑎(𝑤),𝑞≥2(3)‖𝑤‖²≤𝑐₄𝑎(𝑤)(4)𝑐₅|𝜎(𝑡)|²≤𝒞(𝜎(𝑡))⋅𝜎(𝑡)≤𝑐₆|𝜎(𝑡)|².

Proof. (1) In fact 𝑎(𝑤)=_Ω𝑤_𝑥𝑥²+𝑤_𝑦𝑦²+2𝜇𝑤_𝑥𝑥𝑤_𝑦𝑦+21−𝜇𝑤_𝑥𝑦²≤𝑑𝑥𝑑𝑦_Ω𝑤_𝑥𝑥²+𝑤_𝑦𝑦²+𝜇𝑤_𝑥𝑥²+𝑤_𝑦𝑦²+21−𝜇𝑤_𝑥𝑦²𝑑𝑥𝑑𝑦=_Ω1+𝜇𝑤_𝑥𝑥²+𝑤_𝑦𝑦²+21−𝜇𝑤_𝑥𝑦²𝑎(𝑤)=𝑑𝑥𝑑𝑦,_Ω𝑤_𝑥𝑥²+𝑤_𝑦𝑦²+2𝜇𝑤_𝑥𝑥𝑤_𝑦𝑦+21−𝜇𝑤_𝑥𝑦²≥𝑑𝑥𝑑𝑦_Ω𝑤_𝑥𝑥²+𝑤_𝑦𝑦²−𝜇𝑤_𝑥𝑥²+𝑤_𝑦𝑦²+21−𝜇𝑤_𝑥𝑦²𝑑𝑥𝑑𝑦=_Ω1−𝜇𝑤_𝑥𝑥²+𝑤_𝑦𝑦²+21−𝜇𝑤_𝑥𝑦²∇𝑑𝑥𝑑𝑦,₂𝑤²=∇²𝑤,∇²𝑤=_Ω𝑤_𝑥𝑥²+𝑤_𝑦𝑦²+2𝑤_𝑥𝑦²𝑑𝑥𝑑𝑦. (3.2)(3.3)(3.4) Combining (3.2)–(3.4) and noting 0<𝜇<1/2, we finish the desire conclusion (1).
(2) By 𝐻¹(Ω)↪𝐿^𝑞(Ω)(𝑞≥2), the definition of 𝑊(Ω), and Poincaré inquality, we know ‖∇𝑤(𝑡)‖²_𝐿^𝑞_(Ω)≤𝑐‖𝑤(𝑡)‖²_𝐻1(Ω)≤𝑐‖𝑤(𝑡)‖²_𝑊(Ω)≤𝑐₃𝑎(𝑤). (3.5)
(3) Using Poincaré inequality, we get ‖𝑤(𝑡)‖≤𝐶_∗‖∇𝑤(𝑡)‖, (3.6) which with conclusion (2) shows the desire result.
(4) Denote 𝜎∶=^𝜎11𝜎12_𝜎21𝜎22. Hence 𝒞(𝜎)𝜎=𝛼¹¹+𝜇𝜎₂₂1−𝜇𝜎₁₂1−𝜇𝜎₂₁𝜎₂₂+𝜇𝜎₁₁, (3.7) with 𝛼=𝐸/𝑑(1−𝜇²)>0, and 𝒞(𝜎)⋅𝜎=𝛼𝜎²₁₁+𝜎²₂₂+2𝜇𝜎₁₁𝜎₂₂+1−𝜇𝜎²₁₂+𝜎²₂₁, (3.8) which imply 𝛼1−𝜇𝜎²₁₁+𝜎²₂₂+1−𝜇𝜎²₁₂+𝜎²₂₁≤𝒞(𝜎)⋅𝜎≤𝛼1+𝜇𝜎²₁₁+𝜎²₂₂+1−𝜇𝜎²₁₂+𝜎²₂₁. (3.9) From (3.9) and noting 0<𝜇<1/2, we obtain the desire conclusion (4).
The proof is completed.

Lemma 3.3. Suppose 𝑤,𝑣∈𝑊 and 𝑎(𝑤,𝑣) is defined as above, then 𝑎()()+9𝑤,𝑣≤𝛿𝑎𝑤,𝑤𝑎().4𝛿𝑣,𝑣 (3.10)

Proof. In fact, for 𝜂>0 small enough, |𝑎(𝑤,𝑣)|≤_Ω𝑤_𝑥𝑥𝑣_𝑥𝑥+𝑤_𝑦𝑦𝑣_𝑦𝑦+𝜇𝑤_𝑥𝑥𝑣_𝑦𝑦+𝑤_𝑦𝑦𝑣_𝑥𝑥+21−𝜇𝑤_𝑥𝑦𝑣_𝑥𝑦𝑑𝑥𝑑𝑦≤𝜂_Ω1+𝜇𝑤_𝑥𝑥²+𝑤_𝑦𝑦²+21−𝜇𝑤_𝑥𝑦²+1𝑑𝑥𝑑𝑦4𝜂_Ω1+𝜇𝑣_𝑥𝑥²+𝑣_𝑦𝑦²+21−𝜇𝑣_𝑥𝑦²≤𝑑𝑥𝑑𝑦1+𝜇𝜂1−𝜇_Ω1−𝜇𝑤_𝑥𝑥²+𝑤_𝑦𝑦²+21−𝜇𝑤_𝑥𝑦²+𝑑𝑥𝑑𝑦1+𝜇11−𝜇4𝜂_Ω1−𝜇𝑣_𝑥𝑥²+𝑣_𝑦𝑦²+21−𝜇𝑣_𝑥𝑦²𝑑𝑥𝑑𝑦. (3.11) Denote 𝛿=((1+𝜇)/(1−𝜇))𝜂, then ((1+𝜇)/(1−𝜇))(1/4𝜂)=((1+𝜇)/(1−𝜇))²(1/4𝛿). Owing to 1<(1+𝜇)/(1−𝜇)<3, from (3.3) and (3.11), we obtain the conclusion.

Lemma 3.4. Define the functional 𝜑(𝑡)1∶=2_Ω𝑤(𝑡)𝑤^′(𝑡)+𝛾∇𝑤^′(𝑡)⋅∇𝑤(𝑡)𝑑𝑥𝑑𝑦, (3.12) then 𝑑𝜑(𝑡)≤1𝑑𝑡2𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²−121−𝛿+^∞₀ℎ^′(𝑠)𝑎(𝑤)−𝑑𝑠_Ω𝒞(𝜎(𝑡(𝑡)9))⋅𝜎𝑑𝑥𝑑𝑦+8𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.13)

Proof. Multiplying (1.1) with 𝑤 and integrating the result over Ω, we obtain _Ω𝑤^′′(𝑡)−𝛾Δ𝑤^′′(𝑡)+Δ²𝑤+Δ^2∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠−div[𝒞(𝜎(𝑡))∇𝑤(𝑡)]𝑤𝑑𝑥𝑑𝑦=0, (3.14) which with Lemma 2.2 and (1.5) implies 12𝑑𝑑𝑡_Ω𝑤(𝑡)𝑤^′(𝑡)+𝛾∇𝑤^′(𝑡)(𝑡=1⋅∇𝑤)𝑑𝑥𝑑𝑦2𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²+12_Ω𝑤(𝑡)𝑤^′′(𝑡)+𝛾∇𝑤(𝑡)⋅∇𝑤^′′(𝑡=1)𝑑𝑥𝑑𝑦2𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²+12_Ω𝑤(𝑡)𝑤^′′(𝑡)−𝛾Δ𝑤^′′(𝑡)1𝑑𝑥𝑑𝑦+2_Γ1𝑤𝜕_𝜈𝑤^′′(𝑡)=1𝑑Γ2𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²−12_Ω𝑤Δ²𝑤+Δ^2∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)[𝒞(𝜎(𝑡(𝑡1𝑑𝑠−div))∇𝑤)]𝑑𝑥𝑑𝑦+2_Γ1𝑤𝜕_𝜈𝑤^′′(𝑡)=1𝑑Γ2𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²−12_Ω𝒞(𝜎(𝑡(𝑡)(𝑡)1))⋅∇𝑤⊗∇𝑤𝑑𝑥𝑑𝑦−2𝑎(𝑤)−12𝑎^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)−1𝑑𝑠,𝑤2_Γ1𝑤(𝑡)ℬ₂𝑤+^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠+1)𝑑Γ2_Γ1𝜕_𝜈𝑤(𝑡)ℬ₁𝑤+^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)1𝑑Γ+2_Γ1[𝒞(𝜎(𝑡]+1))𝜈⋅∇𝑤𝑤𝑑Γ2_Γ1𝑤𝜕_𝜈𝑤^′′(𝑡)1𝑑Γ=2𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²−12_Ω𝒞(𝜎(𝑡(𝑡)(𝑡)1))⋅∇𝑤⊗∇𝑤𝑑𝑥𝑑𝑦−2𝑎(𝑤)−12𝑎^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)=1𝑑𝑠,𝑤2𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²−_Ω𝒞(𝜎(𝑡(𝑡)−1))⋅𝜎𝑑𝑥𝑑𝑦21+^∞₀ℎ^′(𝑠)𝑎(𝑤)−1𝑑𝑠2𝑎𝑤,^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.15) By Lemma 3.3, we know 12𝑎𝑤(𝑡),^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)(𝑡)≤1−𝑤𝑑𝑠2(𝑤(𝑡+9𝛿𝑎))8𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.16) Using (3.15)-(3.16), we get the conclusion.

Lemma 3.5. Suppose 𝑤 is the solution of (1.1)–(1.8) and (𝐴₁), (𝐴₂), and (3.1) hold, then(1)(∫^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠)^′=ℎ^′(0)𝑤(𝑡)+∫^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠=∫^∞₀ℎ^′′(𝑠)[𝑤^𝑡(𝑠)−𝑤(𝑡)]𝑑𝑠; (2)ℎ_∞^′′=0; (3)(∫^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠)^′=ℎ^′′(0)𝑤(𝑡)+∫^∞₀ℎ^′′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠=∫^∞₀ℎ^′′′(𝑠)[𝑤^𝑡(𝑠)−𝑤(𝑡)]𝑑𝑠.

Proof. (1) By the formula of integral by part and (𝐴₂), we deduce ^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠^′=^∞₀ℎ^′(𝑠)𝜕_𝑡𝑤^𝑡(𝑠)𝑑𝑠=−^∞₀ℎ^′(𝑠)𝜕_𝑠𝑤^𝑡(𝑠)𝑑𝑠=−ℎ^′(𝑠)𝑤^𝑡(𝑠)^∞₀+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠=ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)=𝑑𝑠_∞0ℎ^′′(𝑠)𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.17)
(2) Using (𝐴₂) and (3.1), we infer the conclusion.
(3) By ℎ_∞^′′=0 and the formula of integral by part, we get the conclusion (3).
The proof is completed.

Lemma 3.6. Define the functional 𝜓(𝑡)=𝛾_Γ1𝜕_𝜈𝑤^′(𝑡)ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑Γ+_Ω𝑤^′(𝑡)−𝛾Δ𝑤^′(𝑡)ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)+1𝑑𝑠𝑑𝑥𝑑𝑦2𝑎^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠. (3.18) then for some 𝛿(𝛿>0) small enough, there exist 𝐶_𝛿>0 and 𝑇(𝑇>0) as 𝑡>𝑇 the following inequality holds 𝑑𝜓(𝑡)≤ℎ𝑑𝑡^′(0)2𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²+𝛿_Ω𝒞(𝜎(𝑡))⋅𝜎(𝑡)𝑑𝑥𝑑𝑦+𝐶_𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.19)

Proof. Multiplying (1.1) with ℎ^′(0)𝑤(𝑡)+∫^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠 and integrating the result over Ω, we have _Ω𝑤^′′(𝑡)−𝛾Δ𝑤^′′(𝑡)+Δ²𝑤+Δ^2∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)[𝒞(𝜎(𝑡(𝑡⋅ℎ𝑑𝑠−div))∇𝑤)]_′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑𝑥𝑑𝑦=0, (3.20) which with Lemma 3.5 gives 𝑑𝑑𝑡_Ω𝑤^′(𝑡)−𝛾Δ𝑤^′(𝑡)ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)=𝑑𝑠𝑑𝑥𝑑𝑦_Ω𝑤^′′(𝑡)−𝛾Δ𝑤^′′(𝑡)ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)+𝑑𝑠𝑑𝑥𝑑𝑦_Ω𝑤^′(𝑡)−𝛾Δ𝑤^′(𝑡)ℎ^′(0)𝑤^′(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠^′𝑑𝑥𝑑𝑦=−_ΩΔ²𝑤+^∞₀ℎ^′(𝑠)Δ²𝑤^𝑡(𝑠)[𝒞(𝜎(𝑡(𝑡×ℎ𝑑𝑠−div))∇𝑤)]_′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)+𝑑𝑠𝑑𝑥𝑑𝑦_Ω𝑤^′(𝑡)−𝛾Δ𝑤^′(𝑡)ℎ^′(0)𝑤^′(𝑡)+ℎ^′′(0)𝑤(𝑡)+^∞₀ℎ^′′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑𝑥𝑑𝑦. (3.21) By Green formula, we have _Ω𝑤^′(𝑡)−𝛾Δ𝑤^′(𝑡)ℎ^′(0)𝑤^′(𝑡)+ℎ^′′(0)𝑤(𝑡)+^∞₀ℎ^′′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑𝑥𝑑𝑦=ℎ^′(0)𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²−𝛾_Γ1𝜕_𝜈𝑤^′(𝑡)×ℎ^′(0)𝑤^′(𝑡)+ℎ^′′(0)𝑤(𝑡)+^∞₀ℎ^′′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑Γ+𝛾_Ω∇𝑤^′(𝑡)⋅ℎ^′′(0)∇𝑤(𝑡)+^∞₀ℎ^′′′(𝑠)∇𝑤^𝑡(𝑠)𝑑𝑠𝑑𝑥𝑑𝑦+_Ω𝑤^′(𝑡)ℎ^′′(0)𝑤+^∞₀ℎ^′′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑𝑥𝑑𝑦. (3.22) Clearly 𝛾_Γ1𝜕_𝜈𝑤^′(𝑡)ℎ^′(0)𝑤^′(𝑡)+ℎ^′′(0)𝑤(𝑡)+^∞₀ℎ^′′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑑𝑠𝑑Γ=𝛾𝑑𝑡_Γ1𝜕_𝜈𝑤^′(𝑡)ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑Γ−𝛾_Γ1𝜕_𝜈𝑤^′′(𝑡)ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑Γ. (3.23) Inserting (3.22) and (3.23) into (3.21) yields 𝑑𝑑𝑡_Ω𝑤^′(𝑡)−𝛾Δ𝑤^′(𝑡)ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑑𝑠𝑑𝑥𝑑𝑦+𝛾𝑑𝑡_Γ1𝜕_𝜈𝑤^′(𝑡)ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑Γ=−_ΩΔ²𝑤+^∞₀ℎ^′(𝑠)Δ²𝑤^𝑡(𝑠)[𝒞(𝜎(𝑡(𝑡×ℎ𝑑𝑠−div))∇𝑤)]_′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑𝑥𝑑𝑦+ℎ^′(0)𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²+𝛾_Γ1𝜕_𝜈𝑤^′′(𝑡)^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑Γ+𝛾_Ω∇𝑤^′(𝑡)⋅ℎ^′′(0)∇𝑤(𝑡)+^∞₀ℎ^′′′(𝑠)∇𝑤^𝑡(𝑠)+𝑑𝑠𝑑𝑥𝑑𝑦_Ω𝑤^′(𝑡)ℎ^′′(0)𝑤+^∞₀ℎ^′′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑𝑥𝑑𝑦. (3.24) Applying Lemma 2.2, Lemma 3.5 and noting 𝑎(𝑤,𝑣)=𝑎(𝑣,𝑤), we have −_ΩΔ²𝑤+Δ^2∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)[𝒞(𝜎(𝑡(𝑡×ℎ𝑑𝑠−div))∇𝑤)]_′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝑑𝑥𝑑𝑦=−𝑎𝑤,^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)(𝑡−1−𝑤)𝑑𝑠2𝑑𝑎𝑑𝑡^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠,^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)−𝑑𝑠_Ω𝒞(𝜎(𝑡))∇𝑤(𝑡)⋅^∞₀ℎ^′′(𝑠)∇𝑤^𝑡(𝑠)(𝑡−−∇𝑤)𝑑𝑠𝑑𝑥𝑑𝑦_Γ1ℬ₂𝑤^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)(𝑡−ℬ−𝑤)𝑑𝑠₁𝑤𝜕_𝜈^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)(𝑡−−𝑤)𝑑𝑠𝑑Γ_Γ1ℬ₂^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠−(𝐶(𝜎(𝑡))𝜈⋅∇𝑤(𝑡))^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠𝑑Γ−_Γ1ℬ₁^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠𝜕_𝜈^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠𝑑Γ. (3.25) Inserting (3.25) into (3.24) and owing to (1.5) yield 𝑑𝜓(𝑡)=𝑑𝑑𝑡𝑑𝑡_Ω𝑤^′(𝑡)−𝛾Δ𝑤^′(𝑡)ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)+1𝑑𝑠𝑑𝑥𝑑𝑦2𝑑𝑎𝑑𝑡^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑑𝑠+𝛾𝑑𝑡_Γ1𝜕_𝜈𝑤^′(𝑡)ℎ^′(0)𝑤(𝑡)+^∞₀ℎ^′′𝑤^𝑡(𝑠)𝑑𝑠𝑑Γ=ℎ^′(0)𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²−_Ω𝒞(𝜎(𝑡))∇𝑤(𝑡)⋅^∞₀ℎ^′′(𝑠)∇𝑤^𝑡(𝑠)−∇𝑤(𝑡)𝑑𝑠𝑑𝑥𝑑𝑦−𝑎𝑤,^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠+𝛾_Ω∇𝑤^′(𝑡)⋅^∞₀ℎ^′′′(𝑠)∇𝑤^𝑡(𝑠)(𝑡+−∇𝑤)𝑑𝑠𝑑𝑥𝑑𝑦_Ω𝑤^′(𝑡)^∞₀ℎ^′′′(𝑠)∇𝑤^𝑡(𝑠)−∇𝑤(𝑡)𝑑𝑠𝑑𝑥𝑑𝑦. (3.26) Now, we estimate some terms on the right-hand side of (3.26). The Lemma 3.3 and (3.1) imply 𝑎𝑤,^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠≤𝛿𝑎(𝑤,𝑤)+𝐶_𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.27) Using Cauchy inequality, Hölder inequality, and Lemma 3.2 and noting (3.1), we obtain 𝛾_Ω∇𝑤^′(𝑡)⋅^∞₀ℎ^′′′(𝑠)∇𝑤^𝑡(𝑠)−∇𝑤(𝑡)𝑑𝑠𝑑𝑥𝑑𝑦≤𝛿𝛾∇𝑤^′(𝑡)²+𝐶_𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠,_Ω𝑤^′(𝑡)^∞₀ℎ^′′′(𝑠)∇𝑤^𝑡(𝑠)(𝑡𝑤−∇𝑤)𝑑𝑠𝑑𝑥𝑑𝑦≤𝛿^′(𝑡)²+𝐶_𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.28) Applying Hölder inequality, Lemma 3.2, and Lemma 2.3, we conclude _Ω𝒞(𝜎(𝑡))∇𝑤(𝑡)⋅^∞₀ℎ^′′(𝑠)∇𝑤^𝑡(𝑠)(𝑡=−∇𝑤)𝑑𝑠𝑑𝑥𝑑𝑦_Ω𝒞(𝜎(𝑡))⋅^∞₀ℎ^′′(𝑠)∇𝑤(𝑡)⊗∇𝑤^𝑡(𝑠)−∇𝑤(𝑡)𝑑𝑠𝑑𝑥𝑑𝑦≤𝛿_Ω𝒞(𝜎(𝑡))⋅𝜎(𝑡)𝑑𝑥𝑑𝑦+𝐶_𝛿^∞₀ℎ^′(𝑠)‖∇𝑤(𝑡)‖²∇𝑤^𝑡(𝑠)−∇𝑤(𝑡)²𝑑𝑠≤𝛿_Ω𝒞(𝜎(𝑡))⋅𝜎(𝑡)𝑑𝑥𝑑𝑦+𝑎(𝑤(𝑡))𝐶_𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠≤𝛿_Ω𝒞(𝜎(𝑡(𝑡)1))⋅𝜎𝑑𝑥𝑑𝑦+ℎ_∞𝐸(0)𝐶_𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠=𝛿_Ω𝒞(𝜎(𝑡))⋅𝜎(𝑡)𝑑𝑥𝑑𝑦+𝐶_𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.29) Combining (3.26)–(3.29), we conclude that 𝑑𝜓(𝑡)≤ℎ𝑑𝑡^′(0)2𝑤^′(𝑡)‖²+𝛾∇𝑤^′(𝑡)‖²+𝛿_Ω𝒞(𝜎(𝑡))⋅𝜎(𝑡)𝑑𝑥𝑑𝑦+𝐶_𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.30)
The proof is completed.

proof of Theorem 3.1. Let 𝐹(𝑡)=𝑃𝐸(𝑡)+𝑄𝜑(𝑡)+𝑅𝜓(𝑡). (3.31) By Lemma 2.3, Lemma 3.4, and Lemma 3.6 and noting (3.1), we have 𝐹^′(𝑡)1≤−𝑃2^∞₀ℎ^′′(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)1−𝑤𝑑𝑠+𝑄2‖𝑤^′(𝑡)‖²+𝛾‖∇𝑤^′(𝑡)‖²1−𝑄21−𝛿+^∞₀ℎ^′(𝑠)𝑑𝑠𝑎(𝑤(𝑡))−𝑄_Ω𝒞(𝜎(𝑡(𝑡)9))⋅𝜎𝑑𝑥𝑑𝑦+𝑄8𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)ℎ−𝑤𝑑𝑠+𝑅^′(0)2𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²+𝑅𝛿_Ω𝒞(𝜎(𝑡))⋅𝜎(𝑡)𝑑𝑥𝑑𝑦+𝑅𝐶_𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠≤𝐾₁𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²+𝐾₂𝑎(𝑤(𝑡))+𝐾_3Ω𝒞(𝜎(𝑡))×𝜎(𝑡)𝑑𝑥𝑑𝑦+𝐾₄^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠, (3.32) with 𝐾₁=12𝑅ℎ^′(0)+12𝑄,𝐾₂1=−2𝑄1−𝛿+^∞₀ℎ^′(𝑠)1𝑑𝑠=−2𝑄ℎ_∞−𝛿,𝐾₃𝐾=−𝑄+𝑅𝛿,₄1=−2𝑃𝑐₃+9𝑄8𝛿+𝑅𝐶_𝛿. (3.33) Fixing 𝑅=1,𝛿=(1/2)min{ℎ_∞,−ℎ^′(0)}>0,𝑄=(1/2)(𝛿−ℎ^′(0))>0,𝑃>9𝑄/4𝛿𝑐₃+2(𝐶_𝛿/𝑐₃)>0, we obtain 𝐾_𝑖<0,𝑖=1,2,3,4. Under the conditions, we obtain from (2.34) and (3.32) that there exists a positive 𝛼>0 such that 𝑑𝐹(𝑡)(𝑡).𝑑𝑡≤−𝛼𝐸 (3.34) Fixing a large enough positive constant 𝑃, we prove 𝐹(𝑡) and 𝐸(𝑡) are equivalent, that is, 𝑀₁𝐸(𝑡)≤𝐹(𝑡)≤𝑀₂𝐸(𝑡) (3.35) with 𝑀₁,𝑀₂>0. Indeed, using Cauchy inequality, Lemma 3.2, and (2.34), we have (𝑡=𝑄𝑅𝜑)2_Ω𝑤(𝑡)𝑤^′(𝑡)+𝛾∇𝑤^′(𝑡)(𝑡)≤𝑄⋅∇𝑤𝑑𝑥𝑑𝑦4‖𝑤(𝑡)‖²+𝛾‖∇𝑤(𝑡)‖²+𝑤^′(𝑡)²+𝛾∇𝑤^′(𝑡)²≤𝑐𝑎(𝑤(𝑡))+𝑐𝐸̃(𝑡)≤𝑐𝐸(𝑡). (3.36) Using the definition of 𝜓(𝑡) and Green’s formula, we infer 𝜓(𝑡)=_Ω𝑤^′(𝑡)^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)(𝑡+−𝑤)𝑑𝑠𝑑𝑥𝑑𝑦_Ω𝛾∇𝑤^′(𝑡)⋅^∞₀ℎ^′′(𝑠)∇𝑤^𝑡(𝑠)(𝑡1−∇𝑤)𝑑𝑠𝑑𝑥𝑑𝑦+2𝑎^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠. (3.37) Applying Cauchy inequality, Lemma 3.2 and noting (3.1), we conclude that _Ω𝑤^′(𝑡)^∞₀ℎ^′′(𝑠)𝑤^𝑡(𝑠)(𝑡+−𝑤)𝑑𝑠𝑑𝑥𝑑𝑦_Ω𝛾∇𝑤^′(𝑡)⋅^∞₀ℎ^′′(𝑠)∇𝑤^𝑡(𝑠)(𝑡𝑤−∇𝑤)𝑑𝑠𝑑𝑥𝑑𝑦≤𝛿^′(𝑡)²+𝛾𝛿∇𝑤^′(𝑡)²+𝐶_𝛿^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)1−𝑤𝑑𝑠,2𝑎^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)𝑑𝑠≤𝐶∇²ℎ(0)−ℎ_∞𝑤(𝑡)+𝑑𝑠_∞0ℎ^′(𝑠)𝑤^𝑡(𝑠)−𝑤(𝑡)²≤𝑐𝑎(𝑤(𝑡))+𝑐^∞₀ℎ^′(𝑠)𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.38)(3.39) Relations (3.37)-(3.39) imply that 𝜓(𝑡)≤𝑐𝐸(𝑡). (3.40) Fixing a large enough positive constant 𝑃, inequalities (3.36) and (3.40) show (3.35) hold. Relations (3.34) and (3.35) imply that there exists a positive constant 𝛽>0 such that 𝑑𝐹(𝑡)(𝑡),𝑑𝑡≤−𝛽𝐹 (3.41) which with Gronwall’s inequality gives 𝐹(𝑡)≤𝐹(0)𝑒^−𝛽𝑡. (3.42) Combining (3.35) and (3.42), we finish the proof.

Theorem 3.7. Assume that 𝑤 is the global solution of (1.1)–(1.8) with condition 𝐶|ℎ^′(𝑡)|^1+(1/𝑝)≤ℎ^′′𝑡),𝑝>2. then for some 𝑇>0 there exists a positive constant 𝐶, such that 𝐸(𝑡)≤𝐶(1+𝑡)^𝑝/2,∀𝑡≥𝑇. (3.43)

In order to prove Theorem 3.7, we quote the following technical lemma.

Lemma 3.8. Suppose that 𝑤∈𝐿^∞(0,∞;𝐻²(Ω)) and 𝑔 is a continuous function and there exists 0<𝜃<1, such that ^∞₀𝑔^1−𝜃(𝑠)𝑑𝑠<∞. (3.44) Then, there exists 𝐶>0 such that ^∞₀𝑔(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠≤𝐶^∞₀𝑔^1−𝜃(𝑠)𝑑𝑠‖𝑣‖²_{𝐿∞(0,𝑇;𝐻2(Ω))}^{1/(𝜃𝑝+1)}×^∞₀𝑔^1+(1/𝑝)(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑥𝑑𝑦^{𝜃𝑝/(𝜃𝑝+1)}. (3.45)

Proof. Applying Hölder inequality, we obtain ^∞₀𝑔(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)≤−𝑤𝑑𝑠_∞0𝑔(𝑠)^1−𝜃𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠^{1/(𝜃𝑝+1)}×^𝑡₀𝑔(𝑠)^1+(1/𝑝)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠^{𝜃𝑝/(𝜃𝑝+1)}≤𝐶^∞₀𝑔^1−𝜃(𝑠)𝑑𝑠‖𝑤‖²_{𝐿∞(0,𝑇;𝐻2(Ω))}^{1/(𝜃𝑝+1)}×^∞₀𝑔^1+(1/𝑝)(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠^{𝜃𝑝/(𝜃𝑝+1)}. (3.46)
This completes the proof of Lemma 3.8.

Proof of Theorem 3.7. From the hypothesis 𝑐|ℎ^′(𝑡)|^1+(1/𝑝)≤ℎ^′′(𝑡), we have [|ℎ^′|^−1/𝑝(𝑡)]^′≥𝑐/𝑝. Integrating the above inequality over [0,𝑡], we get ℎ^′(𝑡)≤𝑐₇(1+𝑡)^−𝑝,𝑐₇>0. (3.47) Fixing 𝜃=1/2 in Lemma 3.8, then (1−𝜃)𝑝=𝑝/2>1, hence we obtain ^∞₀ℎ^′1−𝜃(𝑠)𝑑𝑠≤𝑐₈^∞₀1(1+𝑠)^𝑝/2𝑑𝑠<∞. (3.48) Using this estimate into (3.46), noting (2.34) and Lemma 3.2, we get ^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠≤𝑐₉𝐸^2/(𝑝+2)(0)^∞₀ℎ^{′1+(1/𝑝)}𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠^{𝑝/(𝑝+2)}. (3.49) Denote 𝐺(𝑡)=𝑄𝜑(𝑡)+𝜓(𝑡). Applying (3.32) and (3.49), we arrive at 𝑑𝐺(𝑡)𝑑𝑡≤−𝛼₁𝐸(𝑡)+𝛼₂^∞₀ℎ^′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠≤−𝛼₁𝐸(𝑡)+𝛼₂𝑐₉𝐸^2/(𝑝+2)(0)^∞₀ℎ^{′1+(1/𝑝)}𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠^{𝑝/(𝑝+2)}. (3.50) Since |𝐺(𝑡)|≤𝑐𝐸(𝑡), applying Young inequality and from Lemma 3.2, (3.50), we deduce 𝑑𝐸𝑑𝑡^2/𝑝(𝑡)𝐺(𝑡=2)𝑝𝐺(𝑡)𝐸^(2/𝑝)−1(𝑡)𝑑𝐸(𝑡)𝑑𝑡+𝐸^2/𝑝(𝑡)𝑑𝐺(𝑡)2𝑑𝑡≤−𝑝𝑐𝐸^2/𝑝(𝑡)𝑑𝐸(𝑡)𝑑𝑡+𝐸^2/𝑝(𝑡)𝑑𝐺(𝑡)𝑑𝑡≤−𝑐₁₀𝑑𝐸𝑑𝑡^1+(2/𝑝)(𝑡)−𝛼₁𝐸^1+(2/𝑝)(𝑡)+𝛼₂𝐶₂𝐸^2/(𝑝+2)(0)𝐸^2/𝑝(𝑡)^∞₀ℎ^{′1+(1/𝑝)}𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠^{𝑝/(𝑝+2)}≤−𝑐₁₀𝑑𝐸𝑑𝑡^1+(2/𝑝)(𝑡)−𝛼₁𝐸^1+(2/𝑝)(𝑡)+𝛼₂𝐶₂𝐸^2/(𝑝+2)(0)𝜀𝐸^1+(2/𝑝)(𝑡)+𝛼₂𝐶₂𝐸^2/(𝑝+2)(0)𝐶_𝜀^∞₀ℎ^{′1+(1/𝑝)}𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.51) Since 𝑐|ℎ^′|^1+(1/𝑝)(𝑡)≤ℎ^′′(𝑡), then ^∞₀ℎ^{′1+(1/𝑝)}𝑎𝑤^𝑡(𝑠)(𝑡)1−𝑤𝑑𝑠≤𝑐^∞₀ℎ^′′(𝑠)𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠. (3.52) By Lemma 3.2, we have ^∞₀ℎ^′′(𝑠)𝑎𝑤^𝑡(𝑠)(𝑡)𝑑−𝑤𝑑𝑠=−2𝐸(𝑡).𝑑𝑡 (3.53) Hence, we get ^∞₀ℎ^{′1+(1/𝑝)}𝑎𝑤^𝑡(𝑠)−𝑤(𝑡)𝑑𝑠≤−𝑐₁₁𝑑𝐸(𝑡).𝑑𝑡 (3.54) Taking 𝜀 small enough, from (3.51) and (3.54), we get 𝑑𝐸𝑑𝑡^2/𝑝(𝑡)𝐺(𝑡)≤−𝑐₁₀𝑑𝐸𝑑𝑡^1+(2/𝑝)(𝑡)−𝛼₁2𝐸^1+(2/𝑝)(𝑡)−𝑐₁₂𝑑𝐸(𝑡).𝑑𝑡 (3.55) Thus, we have 𝑑𝐸𝑑𝑡^2/𝑝(𝑡)𝐺(𝑡)+𝑐₁₀𝐸(𝑡)≤−𝑐₁₂𝑑𝐸(𝑡)−𝛼𝑑𝑡2¹𝐸^1+(2/𝑝)(𝑡). (3.56)
Let 𝜆>0 be a positive constant and define 𝐻(𝑡)=𝜆𝐸(𝑡)+𝐸^2/𝑝(𝑡)𝐺(𝑡)+𝑐₁₀𝐸(𝑡). (3.57) Since |𝐺(𝑡)|≤𝑐𝐸(𝑡),(𝑑/𝑑𝑡)𝐸(𝑡)≤0, for 𝜆 large enough, we get 𝜆2𝐸(𝑡)≤𝐻(𝑡)≤2𝜆𝐸(𝑡),∀𝑡≥0. (3.58) From Lemma 3.2 and (3.56), taking 𝜆 large enough, we obtain 𝑑𝐻(𝑡)𝑑𝑑𝑡=𝜆𝐸(𝑡)+𝑑𝑑𝑡𝐸𝑑𝑡^2/𝑝(𝑡)𝐺(𝑡)+𝐶₀𝐸(𝑡)𝑑≤𝜆𝐸(𝑡)𝑑𝑡−𝑐₁₂𝑑𝐸(𝑡)−𝛼𝑑𝑡2¹𝐸^1+(2/𝑝)(𝑡)=𝜆−𝑐₁₂𝑑𝐸(𝑡)−𝛼𝑑𝑡2¹𝐸^1+(2/𝑝)(𝑡)𝛼≤−¹2𝐸^1+(2/𝑝)(𝑡). (3.59) From (3.58) and (3.59), we have 𝑑𝐻(𝑡)𝑑𝑡≤−𝑐₁₃𝐻^1+(2/𝑝)(𝑡). (3.60) Applying Gronwall inequality, we get 𝐻(𝑡)≤𝑐₁₄(1+𝑡)^𝑝/2. (3.61) Hence, we obtain 𝐸(𝑡)≤𝑐₁₅(1+𝑡)^𝑝/2. (3.62)
This completes the proof of Theorem 3.7.

4. Comment

In this paper, the key point is to construct suitable energy functional 𝐸(𝑡), find the appropriate auxiliary functionals 𝜑(𝑡) and 𝜓(𝑡) by multiplier method, and prove that 𝐹(𝑡) and 𝐸(𝑡) are equivalent and 𝐹(𝑡) satisfies 𝐹^′(𝑡)≤−𝑐𝐹(𝑡) by precise apriori estimates. Thanks to Lemma 3.8 and auxiliary functional 𝐻(𝑡), we show the polynomial decay result under suitable conditions on relaxation ℎ.

Acknowledgment

The authors would like to thank the referee for his/her careful reading and kind suggestions. The paper was supported by the National Natural Science Foundation of China (11201258), the Natural Science Foundation of Shandong Province of China (ZR2011AM008, ZR2011AQ006, and ZR2012AM010), the STPF of University in Shandong Province of China (J09LA04), and the Science Foundation of Qufu Normal University of China (XJ201114).