Exponential Decay to the Degenerate Nonlinear Coupled Beams System with Weak Damping

Abstract

We consider a nonlinear degenerate coupled beams system with weak damping. We show using the Nakao method that the solution of this system decays exponentially when the time tends to infinity.

1. Introduction

For the last several decades, various types of equations have been employed as some mathematical models describing physical, chemical, biological, and engineering systems. Among them, the mathematical models of vibrating, flexible structures have been considerably stimulated in recent years by an increasing number of questions of practical concern. Research on stabilization of distributed parameter systems has largely focused on the stabilization of dynamic models of individual structural members such as strings, membranes, and beams.

This paper is devoted to the study of the existence, uniqueness, and uniform decay rates of the energy of solution for the nonlinear degenerate coupled beams system with weak damping given by 𝐾𝐾₁(𝑥,𝑡)𝑢_𝑡𝑡+Δ²𝑢−𝑀‖𝑢‖²+‖𝑣‖²Δ𝑢+𝑢_𝑡=0inΩ×(0,𝑇),(1.1)₂(𝑥,𝑡)𝑣_𝑡𝑡+Δ²𝑣−𝑢𝑀=𝑣‖=𝑢‖𝜕²𝑢+‖𝑣‖²Δ𝑣+𝑣_𝑡=0inΩ×(0,𝑇),(1.2)(𝑢(𝑥,0),𝑣(𝜕𝑥𝜂,0=))𝜕𝜕=𝑣𝜂=𝑢₀0onΣ,(1.3)𝑢,𝑣₀inΩ,(1.4)_𝑡(𝑥,0),𝑣_𝑡(𝑥,0)=𝑢₁(𝑥),𝑣₁(𝑥)inΩ,(1.5) where Ω is a bounded domain of ℝ^𝑛, 𝑛≥1, with smooth boundary Γ,   𝑇>0 is a real arbitrary number, and 𝜂 is the unit normal at Σ=Γ×(0,𝑇) direct towards the exterior of Ω×(0,𝑇). Here 𝐾_𝑖∈𝐶¹([0,𝑇];𝐻¹⁰(Ω)∩𝐿^∞(Ω)),   𝑖=1,2 and 𝑀∈𝐶¹([0,∞[), see Section 2 for more details.

Problems related to the system (1.1)–(1.5) are interesting not only from the point of view of PDE general theory, but also due to its applications in mechanics. For instance, when we consider only one equation without the dissipative term, that is, 𝐾(𝑥,𝑡)𝑢_𝑡𝑡+Δ²𝑢−𝑀‖𝑢‖²Δ𝑢=0inΩ×(0,𝑇)(1.6) and with 𝐾(𝑥,𝑡)=1, it is a generalization of one-dimensional model proposed by Woinowsky-Krieger [1] as a model for the transverse deflection 𝑢(𝑥,𝑡) of an extensible beam of natural length whose ends are held a fixed distance apart. The nonlinear term represents the change in the tension of the beam due to its extensibility. The model has also been discussed by Eisley [2], while related experimental results have been given by Burgreen [3]. Dickey [4] considered the initial-boundary value problem for one-dimensional case of (1.6) with 𝐾(𝑥,𝑡)=1 in the case when the ends of the beam are hinged. He showed how the model affords a description of the phenomenon of “dynamic buckling.” The one-dimensional case has also been studied by Ball [5]. He extended the work of Dickey [4] in several directions. In both cases he used the techniques of Lions [6] to prove that the initial boundary value problem is weakly well-posed. Menzala [7] studied the existence and uniqueness of solutions of (1.6) with 𝐾(𝑥,𝑡)=1, 𝑥∈ℝ^𝑛, and 𝑀∈𝐶¹[0,∞[  and 𝑀(𝜆)≥𝑚₀>0, for all 𝜆≥0. The existence, uniqueness, and boundary regularity of weak solutions were considered by Ramos [8] with 𝐾(𝑥)≥𝑘₀>0,  𝑥∈Ω. See also Pereira et al. [9]. The abstract model 𝑢_𝑡𝑡+𝐀²𝑢+𝑀||𝐀^1/2||²𝐀𝑢=0(1.7) of (1.6), where 𝐀 is a nonbounded self-adjoint operator in a conveniently Hilbert space has been studied by Medeiros [10]. He proved that the abstract model is well-posed in the weak sense, since 𝑀∈𝐶¹[0,∞[  with 𝑀(𝜆)≥𝑚₀+𝑚₁𝜆, for all 𝜆≥0,  where 𝑚₀ and 𝑚₁ are positive constants. Pereira [11] considered the abstract model (1.7) with dissipative term 𝑢_𝑡. He proved the existence, uniqueness, and exponential decay of the solutions with the following assumptions about 𝑀: 𝑀∈𝐶⁰([0,∞[)with𝑀(𝜆)≥−𝛽,∀𝜆≥0,0<𝛽<𝜆₁,(1.8) where 𝜆₁ is the first eigenvalue of 𝐀²𝑢−𝜆𝐀𝑢=0.(1.9) Our main goal here is to extend the previous results for a nonlinear degenerate coupled beams system of type (1.1)–(1.5). We show the existence, uniqueness, and uniform exponential decay rates.

Our paper is organized as follows. In Section 2 we give some notations and state our main result. In Section 3 we obtain the existence and uniqueness for global weak solutions. To obtain the global weak solution we use the Faedo-Galerkin method. Finally, in Section 4 we use the Nakao method (see Nakao [12]) to derive the exponential decay of the energy.

2. Assumptions and Main Result

In what follows we are going to use the standard notations established in Lions [6].

Let us consider the Hilbert space 𝐿²(Ω) endowed with the inner product (𝑢,𝑣)=_Ω𝑢(𝑥)𝑣(𝑥)𝑑𝑥(2.1) and norm |𝑢|=√(𝑢,𝑣).(2.2) We also consider the Sobolev space 𝐻¹(Ω) endowed with the scalar product (𝑢,𝑣)_𝐻¹_(Ω)=(𝑢,𝑣)+(∇𝑢,∇𝑣).(2.3) We define the subspace of 𝐻¹(Ω), denoted by 𝐻¹⁰(Ω). This space endowed with the norm induced by the scalar product ((𝑢,𝑣))_𝐻¹⁰_(Ω)=(∇𝑢,∇𝑣)(2.4) is a Hilbert space.

2.1. Assumptions on the Functions 𝐾_𝑖, 𝑖=1,2, and 𝑀

To obtain the weak solution of the system (1.1)–(1.5) we consider the following hypothesis: w𝐾i_𝑖th∈𝐾𝐶¹_𝑖[0,𝑇];𝐻¹⁰(Ω)∩𝐿^∞(Ω),𝑖=1,2,andther(e𝑥e,x𝑡)is≥ts0𝛾,>∀0(𝑥su,c𝑡)h∈thΩat×𝐾_𝑖(0,𝑇),||||𝜕𝐾_𝑖(𝑥,0)≥𝛾>0,(2.5)𝜕𝑡||||_ℝ𝑀∈𝐶₁≤𝛿+𝐶(𝛿)𝐾_𝑖,𝑖=1,2,∀𝛿>0,(2.6)0<𝛽<𝜆₁([0,∞[)with𝑀(𝜆)≥−𝛽,∀𝜆≥0,,𝜆₁isthefiΔrsteingenvalueofthestationaryproblem,₂𝑢−𝜆(−Δ𝑢)=0.(2.7)

Remark 2.1. Let 𝜆₁ be the first eingevalue of Δ²𝑢−𝜆(−Δ𝑢)=0; then (see Miklin [13]) 𝜆₁=inf_{𝑤∈𝐻20(Ω)}||Δ𝑤||²||∇𝑤||²>0.(2.8)

3. Existence and Uniqueness Results

Now, we are in a position to state our result about the existence of weak solution to the system (1.1)–(1.5).

Theorem 3.1. Let one take (𝑢₀,𝑣₀)∈(𝐻¹⁰(Ω)∩𝐻⁴(Ω))²  and   (𝑢₁,𝑣₁)∈(𝐻²⁰(Ω))², and let one suppose that assumptions (2.5), (2.6) and (2.7) hold. Then, there exist unique functions 𝑢,𝑣∶[0,𝑇]→𝐿²(Ω) in the class (𝑢,𝑣)𝑢∈_𝑡𝐿^∞loc(0,∞)∶𝐻²⁰(Ω)∩𝐻⁴(Ω)²,𝑢_𝑡𝑡,𝑣_𝑡∈𝐿^∞loc(0,∞)∶𝐻²⁰(Ω)²,,𝑣_𝑡𝑡∈𝐿^∞loc(0,∞)∶𝐿²(Ω)²(3.1) satisfying 𝐾𝐾₁(𝑥,𝑡)𝑢_𝑡𝑡+Δ²𝑢−𝑀‖𝑢‖²+‖𝑣‖²Δ𝑢+𝑢_𝑡=0in𝐿^2loc0,∞;𝐿²(Ω)₂,(𝑥,𝑡)𝑣_𝑡𝑡+Δ²𝑣(𝑢−(𝑥𝑀,0)‖,𝑣𝑢‖(𝑥²,+0)‖)𝑣=‖²𝑢₀Δ𝑣+𝑣_𝑡=0in𝐿^2loc0,∞;𝐿²(Ω),(3.2)𝑢(𝑥),𝑣₀(𝑥)inΩ,_𝑡(𝑥,0),𝑣_𝑡(𝑥,0)=𝑢₁(𝑥),𝑣₁(𝑥)inΩ.(3.3)

Proof. Since 𝐾_𝑖≥0, 𝑖=1,2, we first perturb the system (1.1)–(1.5) with the terms 𝜀𝑢_𝑡𝑡,𝜀𝑣_𝑡𝑡, with 0<𝜀<1, and we apply the Faedo-Galerkin method to the perturbed system. After we pass to the limit with 𝜀→0 in the perturbed system and we obtain the solution for the system (1.1)–(1.5).
(1) Perturbed System
Consider the perturbed system 𝐾𝐾₁+𝜀𝑢^𝜀𝑡𝑡+Δ𝑢^𝜀+𝑀‖𝑢^𝜀‖²+‖𝑣^𝜀‖²(−Δ𝑢^𝜀)+𝑢^𝜀𝑡=0inΩ×(0,𝑇),₂+𝜀𝑣^𝜀𝑡𝑡+Δ𝑣^𝜀+𝑢𝑀‖𝑢^𝜀‖²+‖𝑣^𝜀‖²(−Δ𝑣^𝜀)+𝑣^𝜀𝑡=0inΩ×(0,𝑇),_𝜀=𝑣^𝜀=𝜕𝑢^𝜀(𝑢_𝜀𝜕𝜂=𝜕𝜕𝑣𝜂^𝜀=0onΣ,𝑢(𝑥,0),𝑣^𝜀(𝑥,0))=𝑢₀(𝑥),𝑣₀(𝑥)inΩ,_𝜀𝑡(𝑥,0),𝑣^𝜀𝑡(𝑥,0)=𝑢₁(𝑥),𝑣₁(𝑥)inΩ.(3.4) Let (𝑤_𝜈)_𝜈∈ℕ be a basis of 𝐻²⁰(Ω) formed by the eigenvectors of the operator −Δ, that is, −Δ𝑤_𝜈=𝜆_𝜈𝑤_𝜈, with 𝜆_𝜈→∞ when 𝜈→∞. Let 𝑉_𝑚=[𝑤₁,𝑤₂,…,𝑤_𝑚] be the subspace generated by the first 𝑚 vectors of (𝑤_𝜈)_𝜈∈ℕ.
For each fixed 𝜀, we consider 𝑢^𝜀𝑚(𝑡)=^𝑚_𝑗=1𝑣_𝜀𝑚𝑔_𝑗𝜀𝑚(𝑡)𝑤_𝑗∈𝑉_𝑚,(𝑡)=^𝑚_𝑗=1ℎ_𝑗𝜀𝑚(𝑡)𝑤_𝑗∈𝑉_𝑚(3.5) as solutions of the approximated perturbed system 𝐾₁++𝜀𝑀𝑢‖‖^{𝜀𝑡𝑡𝑚}𝑢(_𝜀𝑚𝑡),𝑤+−Δ𝑢^𝜀𝑚(𝑡),−Δ𝑤𝐾₂(𝑡)‖‖²+‖‖𝑣^𝜀𝑚(𝑡)‖‖²−Δ𝑢^𝜀𝑚(𝑡),𝑤+𝑢^𝜀𝑡𝑚(𝑡),𝑤=0,∀𝑤∈𝑉_𝑚,(3.6)++𝜀𝑀𝑣‖‖^{𝜀𝑡𝑡𝑚}𝑢(_𝜀𝑚𝑡),𝑧+−Δ𝑣^𝜀𝑚(𝑡),−Δ𝑧𝑢(𝑡)‖‖²+‖‖𝑣^𝜀𝑚(𝑡)‖‖²−Δ𝑣^𝜀𝑚(𝑡),𝑧+𝑣^𝜀𝑡𝑚(𝑡),𝑧=0,∀𝑧∈𝑉_𝑚,(3.7)_𝜀𝑚(0𝑢),𝑣^𝜀𝑚(0)=𝑢_0𝑚,𝑣_0𝑚⟶𝑢₀,𝑣₀in𝐻²⁰(Ω)∩𝐻⁴(Ω)²,(3.8)_𝜀𝑡𝑚(0),𝑣^𝜀𝑡𝑚(0)=𝑢_1𝑚,𝑣_1𝑚⟶𝑢₁,𝑣₁in𝐻²⁰(Ω)².(3.9) The local existence of the approximated solutions (𝑢^𝜀𝑚,𝑣^𝜀𝑚) is guaranteed by the standard results of ordinary differential equations. The extension of the solutions (𝑢^𝜀𝑚,𝑣^𝜀𝑚) to the whole interval [0,𝑇] is a consequence of the first estimate below.
The First Estimate
Setting 𝑤=𝑢^𝜀𝑡𝑚 and 𝑧=𝑣^𝜀𝑡𝑚 in (3.6) and (3.7), respectively, integrating over (0,𝑡), and taking the convergences (3.8) and (3.9) in consideration, we arrive at 𝐾¹,𝑢^𝜀𝑡𝑚²(𝑡)+𝜀||𝑢^𝜀𝑡𝑚(𝑡)||²+||Δ𝑢^𝜀𝑚(𝑡)||²+𝐾₂,𝑣^𝜀𝑡𝑚²(𝑡)+𝜀||𝑣^𝜀𝑡𝑚||²≤+||Δ𝑣^𝜀𝑚(𝑡)||²+𝑀‖‖𝑢^𝜀𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑚(𝑡)‖‖²+2^𝑡0||𝑢^𝜀𝑡𝑚(𝑠)||²+||𝑣^𝜀𝑡𝑚(𝑠)||²𝑑𝑠_𝑡0||||𝜕𝜕𝐾𝑡¹,𝑢^𝜀𝑡𝑚²(𝑠)||||_ℝ+||||𝜕𝜕𝐾𝑡²,𝑣^𝜀𝑡𝑚²(𝑠)||||_ℝ𝑑𝑠+𝐾¹(0),𝑢^21𝑚+𝜀||𝑢_1𝑚||²+||Δ𝑢_0𝑚||²+𝐾₂(0),𝑣^21𝑚+𝜀||𝑣_1𝑚||²+||Δ𝑣_0𝑚||²+𝑀‖‖𝑢_0𝑚‖‖²+‖‖𝑣_0𝑚‖‖²,(3.10) where 𝑀(𝑠)=^𝑠0𝑀(𝜏)𝑑𝜏.(3.11) From (2.7) and (2.8), we have 𝑀‖‖𝑢^𝜀𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑚(𝑡)‖‖²≥−𝜆𝛽₁||Δ𝑢^𝜀𝑚(𝑡)||²+||Δ𝑣^𝜀𝑚(𝑡)||².(3.12) Since 𝛽<𝜆₁ and so by (2.5)–(2.7) and convergences (3.8), (3.9), and (3.12), we obtain 𝐾¹,𝑢^𝜀𝑡𝑚²(𝑡)+𝐾₂,𝑢^𝜀𝑡𝑚²(𝑡)+𝜀||𝑢^𝜀𝑡𝑚(𝑡)||²+||𝑣^𝜀𝑡𝑚+1−𝛽||²(𝑡)𝜆₁||Δ𝑢^𝜀𝑚≤𝐶₀(𝑡)||²+||Δ𝑣^𝜀𝑚(𝑡)||²+(2−𝛿)^𝑡0||𝑢^𝜀𝑡𝑚(𝑠)||²+||𝑢^𝜀𝑡𝑚(𝑠)||²𝑑𝑠+𝐶(𝛿)^𝑡0𝐾¹,𝑢^𝜀𝑡𝑚²(𝑠)+𝐾₂,𝑣^𝜀𝑡𝑚²(𝑠)𝑑𝑠(3.13) with 0<𝛿<1 and 𝐶₀ being a positive constant independent of 𝜀, 𝑚, and 𝑡.
Employing Gronwall’s lemma in (3.13), we obtain the first estimate 𝐾¹,𝑢^𝜀𝑡𝑚²(𝑡)+𝐾₂,𝑢^𝜀𝑡𝑚²(𝑡)+𝜀||𝑢^𝜀𝑡𝑚(𝑡)||²+||𝑣^𝜀𝑡𝑚+1−𝛽||²(𝑡)𝜆₁||Δ𝑢^𝜀𝑚(𝑡)||²+||Δ𝑣^𝜀𝑚(𝑡)||²+(2−𝛿)^𝑡0||𝑢^𝜀𝑡𝑚(𝑠)||²+||𝑢^𝜀𝑡𝑚(𝑠)||²𝑑𝑠≤𝐶¹,(3.14) where 𝐶₁ is a positive constant independent of 𝜀, 𝑚, and 𝑡. Then, we can conclude that 𝐾√^11/2𝑢^𝜀𝑡𝑚,𝐾^21/2𝑣^𝜀𝑡𝑚areboundedin𝐿^∞0,𝑇;𝐿²(Ω),𝜀𝑢𝑢^𝜀𝑡𝑚,√𝜀𝑣^𝜀𝑡𝑚areboundedin𝐿^∞0,𝑇;𝐿²(Ω)_𝜀𝑚,𝑢,𝑣^𝜀𝑚areboundedin𝐿^∞0,𝑇;𝐻²⁰(Ω)_𝜀𝑡𝑚,,𝑣^𝜀𝑡𝑚areboundedin𝐿²0,𝑇;𝐿²(Ω).(3.15)
The Second Estimate
Substituting 𝑤=−Δ𝑢^𝜀𝑡𝑚(𝑡) and 𝑧=−Δ𝑣^𝜀𝑡𝑚(𝑡) in (3.6) and (3.7), respectively, it holds that 𝑑𝑑𝑡𝐾¹+‖‖Δ𝑢,_𝜀𝑚𝑢^𝜀𝑡𝑚²(𝑡)+𝐾₂,𝑣^𝜀𝑡𝑚²(𝑡)+𝜀‖‖𝑢^𝜀𝑡𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑡𝑚(𝑡)‖‖²=𝜕𝐾(𝑡)₁‖‖²+‖‖Δ𝑣^𝜀𝑚(𝑡)‖‖²+2‖‖𝑢^𝜀𝑡𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑡𝑚(𝑡)‖‖²𝜕𝑡,𝑢^𝜀𝑡𝑚²(𝑡)+𝜕𝐾²𝜕𝑡,𝑣^𝜀𝑡𝑚²−2𝑀‖‖𝑢_𝜀𝑚(𝑡)(𝑡)‖‖²+‖‖𝑣^𝜀𝑚(𝑡)‖‖²⋅−Δ𝑢^𝜀𝑚(𝑡),𝑢^𝜀𝑡𝑚(𝑡)+−Δ𝑣^𝜀𝑚(𝑡),𝑣^𝜀𝑡𝑚(𝑡).(3.16)
Integrating (3.16) over (0,𝑡), 0<𝑡<𝑇, and taking (2.5)–(2.7) and (3.8), (3.9), and first estimate into account, we infer 𝐾¹,+𝑢‖‖Δ^𝜀𝑡𝑚𝑢²_𝜀𝑚(𝑡)+𝐾₂,𝑣^𝜀𝑡𝑚²(𝑡)+𝜀‖‖𝑢^𝜀𝑡𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑡𝑚(𝑡)‖‖²+(2−2(𝛿𝑡))‖‖²_𝑡0+‖‖Δ𝑣^𝜀𝑚(𝑡)‖‖²+2‖‖𝑢^𝜀𝑡𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑡𝑚(𝑡)‖‖²‖‖𝑢^𝜀𝑡𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑡𝑚(𝑡)‖‖²𝑑𝑠≤𝐶²,(3.17) where 𝐶₂ is a positive constant independent of 𝜀, 𝑚, and 𝑡. From the above estimate we conclude that 𝐾√^11/2𝑢^𝜀𝑡𝑚,𝐾^21/2𝑣^𝜀𝑡𝑚areboundedin𝐿^∞0,𝑇;𝐻¹⁰(Ω),𝑢𝜀𝑢^𝜀𝑡𝑚,√𝜀𝑣^𝜀𝑡𝑚areboundedin𝐿^∞0,𝑇;𝐻¹⁰(Ω)_𝜀𝑚,,𝑢𝑣^𝜀𝑚areboundedin𝐿^∞0,𝑇;𝐻²⁰(Ω)∩𝐻⁴(Ω)_𝜀𝑡𝑚,,𝑣^𝜀𝑡𝑚areboundedin𝐿²0,𝑇;𝐻¹⁰(Ω).(3.18)The Third Estimate
Differentiating (3.6) and (3.7) with respect to 𝑡 and setting 𝑤=𝑢^{𝜀𝑡𝑡𝑚} and 𝑣^{𝜀𝑡𝑡𝑚}, respectively, we arrive at 𝑑𝑑𝑡𝐾¹+,2𝑢^{𝜀𝑡𝑡𝑚}||𝑢(𝑡_𝜀𝑡𝑚)²+𝐾₂,𝑣^{𝜀𝑡𝑡𝑚}(𝑡)²+𝜀||𝑢^{𝜀𝑡𝑡𝑚}(𝑡)||²+||𝑣^{𝜀𝑡𝑡𝑚}||²+||Δ𝑢^𝜀𝑡𝑚(𝑡)||²+||Δ𝑣^𝜀𝑡𝑚(𝑡)||²=−2𝑀‖‖(𝑢𝑠_𝜀𝑚)||²+||𝑣^𝜀𝑡𝑚(𝑠)||²−4𝑢_𝜀𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑚(𝑡)‖‖²⋅−Δ𝑢^𝜀𝑡𝑚(𝑡),𝑢^{𝜀𝑡𝑡𝑚}(𝑡)+−Δ𝑣^𝜀𝑡𝑚(𝑡),𝑣^{𝜀𝑡𝑡𝑚}(𝑡)⋅−Δ𝑢_𝜀𝑡𝑚(t),−Δ𝑢^𝜀𝑡𝑚(𝑡)+𝑣^𝜀𝑚(𝑡),−Δ𝑣^𝜀𝑡𝑚(𝑡)⋅𝑀^‖‖𝑢^𝜀𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑚(𝑡)‖‖²(𝑡),𝑢^{𝜀𝑡𝑡𝑚}(𝑡)+−Δ𝑣^𝜀𝑡𝑚(𝑡),𝑣^{𝜀𝑡𝑡𝑚}(𝑡)+𝜕𝐾¹𝜕𝑡,𝑢^{𝜀𝑡𝑡𝑚}²(𝑡)+𝜕𝐾²𝜕𝑡,𝑣^{𝜀𝑡𝑡𝑚}²(𝑡).(3.19) Integrating (3.19) over (0,𝑡), and using (2.5), (3.8), (3.9), and the norms |𝑢^{𝜀𝑡𝑡𝑚}(0)|²≤𝐶₃ and |𝑣^{𝜀𝑡𝑡𝑚}(0)|²≤𝐶₄ after employing Gronwall’s lemma, we obtain the third estimate 𝐾¹,+𝑢||Δ^{𝜀𝑡𝑡𝑚}𝑢²_𝜀𝑡𝑚(𝑡)+𝐾₂,𝑣^{𝜀𝑡𝑡𝑚}²(𝑡)+𝜀||𝑢^{𝜀𝑡𝑡𝑚}(𝑡)||²+||𝑣^{𝜀𝑡𝑡𝑚}(𝑡)||²(𝑡)||²+||Δ𝑣^𝜀𝑡𝑚(𝑡)||²+(2−2𝛿)^𝑡0||𝑢^{𝜀𝑡𝑡𝑚}(𝑡)||²+||𝑣^{𝜀𝑡𝑡𝑚}(𝑡)||²𝑑𝑠≤𝐶⁵,(3.20) where 𝐶₅ is a positive constant independent of 𝜀, 𝑚, and 𝑡. From the above estimate we conclude that 𝐾√^11/2𝑢^{𝜀𝑡𝑡𝑚},𝐾^21/2𝑣^{𝜀𝑡𝑡𝑚}areboundedin𝐿^∞0,𝑇;𝐿²(Ω),𝜀𝑢𝑢^{𝜀𝑡𝑡𝑚},√𝜀𝑣^{𝜀𝑡𝑡𝑚}areboundedin𝐿^∞0,𝑇;𝐿²(Ω)_𝜀𝑡𝑚,𝑢,𝑣^𝜀𝑡𝑚areboundedin𝐿^∞0,𝑇;𝐻²⁰(Ω)_{𝜀𝑡𝑡𝑚},,𝑣^{𝜀𝑡𝑡𝑚}areboundedin𝐿²0,𝑇;𝐿²(Ω).(3.21)(2) Limits of Approximated Solutions
From the Aubin-Lions theorem (see [6]) we deduce that there exist subsequences of (𝑢^𝜀𝑚)_𝑚∈ℕ and (𝑣^𝜀𝑚)_𝑚∈ℕ such that 𝑣𝑢^𝜀𝑚⟶𝑢^𝜀stronglyin𝐿²0,𝑇;𝐻¹⁰(Ω)_𝜀𝑚,⟶𝑣^𝜀stronglyin𝐿²0,𝑇;𝐻¹⁰(Ω),(3.22) and since 𝑀 is continuous, it follows that 𝑀‖‖𝑢^𝜀𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑚(𝑡)‖‖²⟶𝑀‖𝑢^𝜀(𝑡)‖²+‖𝑣^𝜀(𝑡)‖².(3.23) From the above estimate we can conclude that there exist subsequences of (𝑢^𝜀𝑚)_𝑚∈ℕ and (𝑣^𝜀𝑚)_𝑚∈ℕ, that we denote also by (𝑢^𝜀𝑚)_𝑚∈ℕ and (𝑣^𝜀𝑚)_𝑚∈ℕ such that as 𝑚→∞ and 𝜀→0 we have 𝑢𝑢^𝜀𝑚⟶𝑢,𝑣^𝜀𝑚⟶𝑣weakstarin𝐿^∞0,𝑇;𝐻²⁰(Ω)∩𝐻⁴(Ω)_𝜀𝑡𝑚,𝑢⟶𝑢_𝑡,𝑣^𝜀𝑡𝑚⟶𝑣_𝑡weakstar𝐿^∞0,𝑇;𝐻²⁰(Ω)_{𝜀𝑡𝑡𝑚},Δ𝑢_𝜀𝑚⟶𝑢_𝑡𝑡,𝑣^{𝜀𝑡𝑡𝑚}⟶𝑣_𝑡𝑡weakstar𝐿²0,𝑇;𝐿²(Ω),𝐾⟶Δ𝑢,Δ𝑣^𝜀𝑚⟶Δ𝑣weakstarin𝐿^∞0,𝑇;𝐿²(Ω)₁,Δ𝑢^{𝜀𝑡𝑡𝑚}⟶𝐾₁𝑢_𝑡𝑡,𝐾₂𝑣^{𝜀𝑡𝑡𝑚}⟶𝐾₂𝑣_𝑡𝑡weakstarin𝐿^∞0,𝑇;𝐿²(Ω)₂,√𝑢^𝜀𝑚⟶Δ²𝑢,Δ²𝑣^𝜀𝑚⟶Δ²𝑣weakstarin𝐿²0,𝑇;𝐿²(Ω),𝑀𝜀𝑢‖‖^{𝜀𝑡𝑡𝑚}𝑢⟶√𝜀𝑢_𝑡𝑡,√𝜀𝑣^{𝜀𝑡𝑡𝑚}⟶√𝜀𝑣_𝑡𝑡weakstarin𝐿^∞0,𝑇;𝐿²(Ω)_𝜀𝑚,⟶(𝑀𝑡)‖‖²‖+𝑢(‖‖𝑡𝑣)‖^𝜀𝑚₂(𝑡)‖‖²−Δ𝑢^𝜀𝑚−Δ𝑣^𝜀𝑚+‖𝑣(𝑡)‖²(−Δ𝑢−Δ𝑣)weakstarin𝐿^∞0,𝑇;𝐿²(Ω).(3.24) Now, multiplying (3.6), (3.7) by 𝜃∈𝒟(0,𝑇) and integrating over (0,𝑇), we arrive at ^𝑇0𝐾₁++𝜀𝑢^{𝜀𝑡𝑡𝑚}(𝑡),𝑤𝜃(𝑡)𝑑𝑡+^𝑇0−Δ𝑢^𝜀𝑚(𝑡),−Δ𝑤𝜃(𝑡)𝑑𝑡_𝑇0+𝑀‖‖𝑢^𝜀𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑚(𝑡)‖‖²−Δ𝑢^𝜀𝑚(𝑡),𝑤𝜃𝑑𝑡_𝑇0𝑢^𝜀𝑡𝑚,𝑤𝜃𝑑𝑡=0,∀𝑤∈𝑉_𝑚,∀𝜃∈𝒟(0,𝑇),_𝑇0𝐾+₂+𝜀𝑣^{𝜀𝑡𝑡𝑚}(𝑡),𝑧𝜃(𝑡)𝑑𝑡+^𝑇0−Δ𝑣^𝜀𝑚(𝑡),−Δ𝑧𝜃(𝑡)𝑑𝑡_𝑇0+𝑀‖‖𝑢^𝜀𝑚(𝑡)‖‖²+‖‖𝑣^𝜀𝑚(𝑡)‖‖²−Δ𝑣^𝜀𝑚(𝑡),𝑧𝜃𝑑𝑡_𝑇0𝑣^𝜀𝑡𝑚,𝑧𝜃𝑑𝑡=0,∀𝑧∈𝑉_𝑚,∀𝜃∈𝒟(0,𝑇).(3.25) The convergences (3.24) are sufficient to pass to the limit in (3.25) in order to obtain 𝐾𝐾₁𝑢_𝑡𝑡+Δ²𝑢+𝑀‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²(−Δ𝑢)+𝑢_𝑡=0in𝐿^∞loc0,∞;𝐿²(Ω)₂,𝑣_𝑡𝑡+Δ²𝑣+𝑀‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²(−Δ𝑣)+𝑣_𝑡=0in𝐿^∞loc0,∞;𝐿²(Ω),(3.26) and (𝑢,𝑣) satisfies (3.1).
The uniqueness and initial conditions follow by using the standard arguments as in Lions [6]. The proof is now complete.

4. Asymptotic Behavior

In this section we study the asymptotic behavior of solutions to the system (1.1)–(1.5). We show using the Nakao method that the system (1.1)–(1.5) is exponentially stable. The main result of this paper is given by the following theorem.

Theorem 4.1. Let one take (𝑢₀,𝑣₀)∈(𝐻¹⁰(Ω)∩𝐻⁴(Ω))², and   (𝑢₁,𝑣₁)∈(𝐻²⁰(Ω))² and let one suppose that assumptions (2.5), (2.6), and (2.7) hold. Then, the solution (𝑢,𝑣) of system (1.1)–(1.5) satisfies ||𝐾^11/2𝑢_𝑡(𝑡)||²+||𝐾^21/2𝑣_𝑡(𝑡)||²+||Δ𝑢(𝑡)||²+||Δ𝑣(𝑡)||²+^𝑡𝑡+1||𝑢^𝑡(𝑠)||²+||𝑣_𝑡(𝑠)||²𝑑𝑠≤𝛼¹𝑒^−𝛼2𝑡,(4.1) for all 𝑡≥1, where 𝛼₁ and 𝛼₂ are positive constants.

Proof. Multiplying (3.2) by 𝑢_𝑡(𝑡) and 𝑣_𝑡(𝑡), respectively, and integrating over Ω, we obtain 12𝑑𝑑𝑡||𝐾^11/2+||𝑢_𝑡𝑢_𝑡(𝑡)||²+||𝐾^21/2𝑣_𝑡(𝑡)||²+||Δ𝑢(𝑡)||²+||Δ𝑣(𝑡)||²+𝑀‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²(𝑡)||²+||𝑣_𝑡(𝑡)||²=𝜕𝐾¹𝜕𝑡,𝑢^2𝑡(𝑡)+𝜕𝐾²𝜕𝑡,𝑣^2𝑡(𝑡),(4.2) where 𝑀(𝑠)=^𝑠0𝑀(𝜏)𝑑𝜏.(4.3) Using (2.6) and considering 𝛿>0 sufficiently small, we get 12𝑑𝑑𝑡||𝐾^11/2+1𝑢−_𝑡(𝑡𝛿)||−²+𝐾||₀𝐾^21/2𝑣_𝑡(𝑡)||²+||Δ𝑢(𝑡)||²+||Δ𝑣(𝑡)||²+𝑀‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²𝐶(𝛿)||𝑢_𝑡(𝑡)||²+||𝑣_𝑡(𝑡)||²≤0,(4.4) where 𝐾₀=max{𝐾₁,𝐾₂} with 𝐾_𝑖=max_{[]𝑠∈𝑡,𝑡+1}esssup𝐾_𝑖(𝑥,𝑠),𝑖=1,2.(4.5) Integrating (4.4) from 0to𝑡, we have 𝐸(𝑡)+1−𝛿−𝐾₀𝐶(𝛿)^𝑡0||𝑢^𝑡(𝑠)||²+||𝑣_𝑡(𝑠)||²𝑑𝑠≤𝐸(0),(4.6) where 𝐸(𝑡)=12||𝐾^11/2𝑢_𝑡(𝑡)||²+||𝐾^21/2𝑣_𝑡(𝑡)||²+||Δ𝑣(𝑡)||²+||Δ𝑣(𝑡)||²+𝑀‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²(4.7) is the energy associated with the system (1.1)–(1.5). From (4.4) we conclude that 𝑑𝑑𝑡𝐸(𝑡)≤0∀𝑡∈(0,∞),(4.8) that is, 𝐸(𝑡) is bounded and increasing in (0,∞).
Integrating (4.4) from 𝜏₁to𝜏₂, 0<𝜏₁<𝜏₂<∞, we arrive at 𝐸𝜏₂+1−𝛿−𝐾₀𝐶(𝛿)^𝜏2_𝜏1||𝑢^𝑡(𝑠)||²+||𝑣_𝑡(𝑠)||²𝑑𝑠≤𝐸𝜏¹.(4.9) Taking 𝜏₁=𝑡 and 𝜏₂=𝑡+1 in (4.9), we get ^𝑡𝑡+1||𝑢^𝑡(𝑠)||²+||𝑣_𝑡(𝑠)||²𝑑𝑠≤11−𝛿−𝐾₀𝐶(𝛿)[𝐸(𝑡)−𝐸(𝑡+1)]=𝐹²(𝑡).(4.10) Therefore, there exist two points 𝑡₁∈[𝑡,𝑡+1/4] and 𝑡₂∈[𝑡+3/4,𝑡+1], such that ||𝑢_𝑡𝑡_𝑖||+||𝑣_𝑡𝑡_𝑖||≤4𝐹(𝑡),𝑖=1,2.(4.11) Making the inner product in 𝐿²(Ω) of (1.1) and (1.2) by 𝑢(𝑡) and 𝑣(𝑡), respectively, and summing up the result we obtain 𝑑𝑑𝑡𝐾¹𝑢_𝑡(𝑡),𝑢(𝑡)+𝑑𝑑𝑡𝐾²𝑣_𝑡(𝑡),𝑣(𝑡)−|||√𝐾₁𝑢_𝑡(𝑡)|||²−|||√𝐾₂𝑣_𝑡(𝑡)|||²++||Δ𝑣𝑢(𝑡)||²+||Δ𝑣(𝑡)||²+𝑀‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²+𝑢_𝑡(𝑡),𝑢(𝑡)_𝑡(𝑡),𝑣(𝑡)=𝜕𝐾¹𝜕𝑡𝑢^𝑡(𝑡),𝑢(𝑡)+𝜕𝐾²𝜕𝑡𝑣^𝑡(𝑡),𝑣(𝑡).(4.12) Integrating from 𝑡₁ to 𝑡₂ and using (2.6), and (2.7) we have 1−𝜆𝛽₁^𝑡2_𝑡1||Δ𝑢(𝑠)||²≤𝐾+||Δ𝑣(𝑠)||²𝑑𝑠₁+1𝑢+_𝑡𝛿𝑡₁+𝐾,𝑢₀𝑡₁−𝐾₁𝑢_𝑡𝑡₂,𝑢𝑡₂+𝐾₂𝑣_𝑡𝑡₁,𝑣𝑡₁−𝐾₂𝑣_𝑡𝑡₂,𝑣𝑡₂+𝐾₀𝐶(𝛿)^𝑡2_𝑡1||𝑢_𝑡(𝑠)||||𝑢(𝑠)||+||𝑣_𝑡(𝑠)||||𝑣(𝑠)||𝑑𝑠^𝑡2_𝑡1||𝑢^𝑡(𝑠)||²+||𝑣_𝑡(𝑠)||²𝑑𝑠.(4.13) Let us consider 𝐶>0 such that ||𝑢(𝑠)||≤𝐶||Δ𝑢(𝑠)||,||𝑣(𝑠)||≤𝐶||Δ𝑣(𝑠)||(4.14) and we take 𝑑>0 sufficiently small Then we have. 1+𝛿≤+𝐾1₀+𝐶𝛿(𝛿+)𝐾₀||𝑢_𝑡(𝑠)||||𝑢(𝑠)||+||𝑣_𝑡(𝑠)||||𝑣(𝑠)||𝐶(𝛿)²𝑑||𝑢^𝑡||𝐾₁(𝑠)||²+||𝑣_𝑡(𝑠)||²+𝑑||Δ𝑢(𝑠)||²+||Δ𝑣(𝑠)||²,𝑢_𝑡≤𝑡𝐶₁𝐾,₀𝑢𝑡₁+𝐾₂𝑣_𝑡𝑡₁,𝑣𝑡₁−𝐾₁𝑢_𝑡𝑡₂,𝑢𝑡₂−𝐾₂𝑣_𝑡𝑡₂,𝑣𝑡₂||esssup_{[]𝑠∈𝑡,𝑡+1}||Δ𝑢(𝑠)||||𝑢_𝑡𝑡₁||+||𝑢_𝑡𝑡₂||+𝐶𝐾₀esssup_{[]𝑠∈𝑡,𝑡+1}||Δ𝑣(𝑠)||||𝑣_𝑡𝑡₁||+||𝑣_𝑡𝑡₂||.(4.15) Thus, substituting (4.15) into (4.13), we arrive at 1−𝜆𝛽₁^𝑡2_𝑡1||Δ𝑢(𝑠)||²≤𝐾₀+||Δ𝑣(𝑠)||²𝑑𝑠^𝑡2_𝑡1||𝑢^𝑡+𝐶𝐾₀(𝑠)||²+||𝑣_𝑡(𝑠)||²𝑑𝑠+𝑑^𝑡2_𝑡1||Δ𝑢(𝑠)||²+||Δ𝑣(𝑠)||²𝑑𝑠esssup_{[]𝑠∈𝑡,𝑡+1}||Δ𝑢(𝑠)||||𝑢_𝑡𝑡₁||+||𝑢_𝑡𝑡₂||+𝐶𝐾₀esssup_{[]𝑠∈𝑡,𝑡+1}||Δ𝑣(𝑠)||||𝑣_𝑡𝑡₁||+||𝑣_𝑡𝑡₂||.(4.16) Applying (4.10) and (4.11) in (4.16), we have ^𝑡2_𝑡1||Δ𝑢(𝑠)||²+||Δ𝑣(𝑠)||²𝑑𝑠≤𝐶¹𝐹²(𝑡)+esssup_{[]𝑠∈𝑡,𝑡+1}||Δ𝑢(𝑠)||+||Δ𝑣(𝑠)||𝐹(𝑡)=𝐺²(𝑡),(4.17) where 𝐶₁ is a positive constant independent of 𝑡. Therefore, from (4.10) and (4.17) we obtain ^𝑡2_𝑡1||𝑢^𝑡(𝑠)||²+||𝑣_𝑡(𝑠)||²+||Δ𝑢(𝑠)||²+||Δ𝑣(𝑠)||²𝑑𝑠≤𝐹²(𝑡)+𝐺²(𝑡).(4.18) Hence, there exists 𝑡^∗∈[𝑡₁,𝑡₂] such that ||𝑢_𝑡𝑡^∗||²+||𝑣_𝑡𝑡^∗||²+||Δ𝑢𝑡^∗||²+||Δ𝑣𝑡^∗||²≤2𝐹²(𝑡)+𝐺²(𝑡).(4.19) Consequently, 𝑀‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²≤𝐶₂𝐹²(𝑡)+𝐺²(𝑡),(4.20) where 𝐶₂=2𝑚₀𝐶,𝑚₀=max_{0≤𝑠≤‖𝑢(𝑡∗)‖2+‖𝑣(𝑡∗)‖2<∞}𝑀(𝑠)(4.21) and 𝐶 is a positive constant such that ‖𝑢(𝑡^∗)‖²≤𝐶|Δ𝑢(𝑡^∗)|².
From (4.19) and (4.20), we have 𝐸𝑡^∗≤𝐶₃𝐹²(𝑡)+𝐺²(𝑡).(4.22) Since 𝐸(𝑡) is increasing, we have esssup_{[]𝑠∈𝑡,𝑡+1}𝐸(𝑠)≤𝐸𝑡^∗+1+𝛿+𝐾₀𝐶(𝛿)^𝑡2_𝑡1||𝑢^𝑡(𝑠)||²+||𝑣_𝑡(𝑠)||²𝑑𝑠.(4.23) Now, by (4.10), (4.22), and (4.23) we get 𝐸(𝑡)≤𝐶₄[𝐸(𝑡)−𝐸(𝑡+1)],(4.24) where 𝐶₄ is a positive constant. Then, by the Nakao lemma (see [12]) we conclude that 𝐸(𝑡)≤𝑏₁𝑒^−𝛼2𝑡,∀𝑡≥1,(4.25) where 𝑏₁ and 𝛼₂ are positive constants, that is, |||√𝐾₁𝑢_𝑡(𝑡)|||²+|||√𝐾₂𝑣_𝑡(𝑡)|||²+||Δ𝑢(𝑡)||²+||Δ𝑣(𝑡)||²𝑀‖𝑢(𝑡)‖²+‖𝑣(𝑡)‖²≤2𝑏₁𝑒^−𝛼2𝑡.(4.26) Using (2.7) we obtain |||√𝐾₁𝑢_𝑡(𝑡)|||²+|||√𝐾₂𝑣_𝑡(𝑡)|||²+||Δ𝑢(𝑡)||²+||Δ𝑣(𝑡)||²≤2𝑏¹𝑚₁𝑒^−𝛼2𝑡,(4.27) where 𝑚₁=1−𝛽𝜆₁>0.(4.28) From (4.10) we have ^𝑡𝑡+1||𝑢^𝑡(𝑠)||²+||𝑣_𝑡(𝑠)||²𝑑𝑠≤11−𝛿−𝐾₀𝐶(𝛿)[𝐸(𝑡)−𝐸(𝑡+1)]≤𝐸(𝑡)≤𝑏¹𝑒^−𝛼2𝑡.(4.29) Therefore, from (4.27) and (4.29) we conclude that |||√𝐾₁𝑢_𝑡(𝑡)|||²+|||√𝐾₂𝑣_𝑡(𝑡)|||²+||Δ𝑢(𝑡)||²+||Δ𝑣(𝑡)||²+^𝑡𝑡+1||𝑢^𝑡(𝑠)||²+||𝑣_𝑡(𝑠)||²𝑑𝑠≤𝛼¹𝑒^−𝛼2𝑡,∀𝑡≥1,(4.30) where 𝛼₁ and 𝛼₂ are positive constants. Now, the proof is complete.