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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 659289, 14 pages
http://dx.doi.org/10.5402/2012/659289
Research Article

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

1Programa de Pós-Graduação, em Matemática Eestatística, Faculdade de Matemática, Universidade Federal do Pará, Campus Universitário do Guamá, Rua Augusto Corrêa 01, 66075-110 Belém, PA, Brazil
2Departamento de Matemática, Estatística e Informática, Universidade do Estado do Pará, Rua do Una 156, Telégrafo, 66113-200 Belém, PA, Brazil

Received 10 April 2012; Accepted 3 July 2012

Academic Editors: D. Dürr and W.-H. Steeb

Copyright © 2012 R. F. C. Lobato et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝐾1(𝑥,𝑡)𝑢𝑡𝑡+Δ2𝑢𝑀𝑢2+𝑣2Δ𝑢+𝑢𝑡𝐾=0inΩ×(0,𝑇),(1.1)2(𝑥,𝑡)𝑣𝑡𝑡+Δ2𝑣𝑀𝑢2+𝑣2Δ𝑣+𝑣𝑡=0inΩ×(0,𝑇),(1.2)𝑢=𝑣=𝜕𝑢=𝜕𝜂𝜕𝑣𝑢𝜕𝜂=0onΣ,(1.3)(𝑢(𝑥,0),𝑣(𝑥,0))=0,𝑣0𝑢inΩ,(1.4)𝑡(𝑥,0),𝑣𝑡=𝑢(𝑥,0)1(𝑥),𝑣1(𝑥)inΩ,(1.5) where Ω is a bounded domain of 𝑛, 𝑛1, 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 𝐾𝑖𝐶1([0,𝑇];𝐻10(Ω)𝐿(Ω)),   𝑖=1,2 and 𝑀𝐶1([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)𝑢𝑡𝑡+Δ2𝑢𝑀𝑢2Δ𝑢=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, 𝑥𝑛, and 𝑀𝐶1[0,[  and 𝑀(𝜆)𝑚0>0, for all 𝜆0. The existence, uniqueness, and boundary regularity of weak solutions were considered by Ramos [8] with 𝐾(𝑥)𝑘0>0,  𝑥Ω. See also Pereira et al. [9]. The abstract model 𝑢𝑡𝑡+𝐀2||𝐀𝑢+𝑀1/2||2𝐀𝑢=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 𝑀𝐶1[0,[  with 𝑀(𝜆)𝑚0+𝑚1𝜆, for all 𝜆0,  where 𝑚0 and 𝑚1 are positive constants. Pereira [11] considered the abstract model (1.7) with dissipative term 𝑢𝑡. He proved the existence, uniqueness, and exponential decay of the solutions with the following assumptions about 𝑀: 𝑀𝐶0([[0,)with𝑀(𝜆)𝛽,𝜆0,0<𝛽<𝜆1,(1.8) where 𝜆1 is the first eigenvalue of 𝐀2𝑢𝜆𝐀𝑢=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 𝐾𝑖, 𝑖=1,2, and 𝑀

To obtain the weak solution of the system (1.1)–(1.5) we consider the following hypothesis: 𝐾𝑖𝐶1[]0,𝑇;𝐻10(Ω)𝐿(Ω),𝑖=1,2,with𝐾𝑖(𝑥,𝑡)0,(𝑥,𝑡)Ω×(0,𝑇),andthereexists𝛾>0suchthat𝐾𝑖||||(𝑥,0)𝛾>0,(2.5)𝜕𝐾𝑖||||𝜕𝑡𝛿+𝐶(𝛿)𝐾𝑖,𝑖=1,2,𝛿>0,(2.6)𝑀𝐶1([[)0,with𝑀(𝜆)𝛽,𝜆0,0<𝛽<𝜆1,𝜆1Δisthersteingenvalueofthestationaryproblem,2𝑢𝜆(Δ𝑢)=0.(2.7)

Remark 2.1. Let 𝜆1 be the first eingevalue of Δ2𝑢𝜆(Δ𝑢)=0; then (see Miklin [13]) 𝜆1=inf𝑤𝐻20(Ω)||||Δ𝑤2||||𝑤2>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 𝐿(𝑢,𝑣)loc(0,)𝐻20(Ω)𝐻4(Ω)2,𝑢𝑡,𝑣𝑡𝐿loc(0,)𝐻20(Ω)2,𝑢𝑡𝑡,𝑣𝑡𝑡𝐿loc(0,)𝐿2(Ω)2(3.1) satisfying 𝐾1(𝑥,𝑡)𝑢𝑡𝑡+Δ2𝑢𝑀𝑢2+𝑣2Δ𝑢+𝑢𝑡=0in𝐿2loc0,;𝐿2,𝐾(Ω)2(𝑥,𝑡)𝑣𝑡𝑡+Δ2𝑣𝑀𝑢2+𝑣2Δ𝑣+𝑣𝑡=0in𝐿2loc0,;𝐿2(,𝑢Ω)(3.2)(𝑢(𝑥,0),𝑣(𝑥,0))=0(𝑥),𝑣0𝑢(𝑥)inΩ,𝑡(𝑥,0),𝑣𝑡(=𝑢𝑥,0)1(𝑥),𝑣1(𝑥)inΩ.(3.3)

Proof. Since 𝐾𝑖0, 𝑖=1,2, we first perturb the system (1.1)–(1.5) with the terms 𝜀𝑢𝑡𝑡,𝜀𝑣𝑡𝑡, with 0<𝜀<1, and we apply the Faedo-Galerkin method to the perturbed system. After we pass to the limit with 𝜀0 in the perturbed system and we obtain the solution for the system (1.1)–(1.5).
(1) Perturbed System
Consider the perturbed system 𝐾1𝑢+𝜀𝜀𝑡𝑡+Δ𝑢𝜀+𝑀𝑢𝜀2+𝑣𝜀2(Δ𝑢𝜀)+𝑢𝜀𝑡𝐾=0inΩ×(0,𝑇),2𝑣+𝜀𝜀𝑡𝑡+Δ𝑣𝜀+𝑀𝑢𝜀2+𝑣𝜀2(Δ𝑣𝜀)+𝑣𝜀𝑡𝑢=0inΩ×(0,𝑇),𝜀=𝑣𝜀=𝜕𝑢𝜀=𝜕𝜂𝜕𝑣𝜀𝜕𝜂=0onΣ,(𝑢𝜀(𝑥,0),𝑣𝜀𝑢(𝑥,0))=0(𝑥),𝑣0𝑢(𝑥)inΩ,𝜀𝑡(𝑥,0),𝑣𝜀𝑡(=𝑢𝑥,0)1(𝑥),𝑣1(𝑥)inΩ.(3.4) Let (𝑤𝜈)𝜈 be a basis of 𝐻20(Ω) formed by the eigenvectors of the operator Δ, that is, Δ𝑤𝜈=𝜆𝜈𝑤𝜈, with 𝜆𝜈 when 𝜈. Let 𝑉𝑚=[𝑤1,𝑤2,,𝑤𝑚] be the subspace generated by the first 𝑚 vectors of (𝑤𝜈)𝜈.
For each fixed 𝜀, we consider 𝑢𝜀𝑚(𝑡)=𝑚𝑗=1𝑔𝑗𝜀𝑚(𝑡)𝑤𝑗𝑉𝑚,𝑣𝜀𝑚(𝑡)=𝑚𝑗=1𝑗𝜀𝑚(𝑡)𝑤𝑗𝑉𝑚(3.5) as solutions of the approximated perturbed system 𝐾1𝑢+𝜀𝜀𝑡𝑡𝑚+(𝑡),𝑤Δ𝑢𝜀𝑚𝑢(𝑡),Δ𝑤+𝑀𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2Δ𝑢𝜀𝑚+𝑢(𝑡),𝑤𝜀𝑡𝑚(𝑡),𝑤=0,𝑤𝑉𝑚,𝐾(3.6)2𝑣+𝜀𝜀𝑡𝑡𝑚+(𝑡),𝑧Δ𝑣𝜀𝑚𝑢(𝑡),Δ𝑧+𝑀𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2Δ𝑣𝜀𝑚+𝑣(𝑡),𝑧𝜀𝑡𝑚(𝑡),𝑧=0,𝑧𝑉𝑚,𝑢(3.7)𝜀𝑚(0),𝑣𝜀𝑚=𝑢(0)0𝑚,𝑣0𝑚𝑢0,𝑣0𝐻in20(Ω)𝐻4(Ω)2,𝑢(3.8)𝜀𝑡𝑚(0),𝑣𝜀𝑡𝑚=𝑢(0)1𝑚,𝑣1𝑚𝑢1,𝑣1𝐻in20(Ω)2.(3.9) The local existence of the approximated solutions (𝑢𝜀𝑚,𝑣𝜀𝑚) is guaranteed by the standard results of ordinary differential equations. The extension of the solutions (𝑢𝜀𝑚,𝑣𝜀𝑚) to the whole interval [0,𝑇] is a consequence of the first estimate below.
The First Estimate
Setting 𝑤=𝑢𝜀𝑡𝑚 and 𝑧=𝑣𝜀𝑡𝑚 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,𝑢𝜀𝑡𝑚2||𝑢(𝑡)+𝜀𝜀𝑡𝑚||(𝑡)2+||Δ𝑢𝜀𝑚||(𝑡)2+𝐾2,𝑣𝜀𝑡𝑚2||𝑣(𝑡)+𝜀𝜀𝑡𝑚||2+||Δ𝑣𝜀𝑚||(𝑡)2+𝑀𝑢𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2+2𝑡0||𝑢𝜀𝑡𝑚||(𝑠)2+||𝑣𝜀𝑡𝑚||(𝑠)2𝑑𝑠𝑡0||||𝜕𝐾1,𝑢𝜕𝑡𝜀𝑡𝑚2(||||𝑠)+||||𝜕𝐾2,𝑣𝜕𝑡𝜀𝑡𝑚2(||||𝑠)𝐾𝑑𝑠+1(0),𝑢21𝑚||𝑢+𝜀1𝑚||2+||Δ𝑢0𝑚||2+𝐾2(0),𝑣21𝑚||𝑣+𝜀1𝑚||2+||Δ𝑣0𝑚||2+𝑀𝑢0𝑚2+𝑣0𝑚2,(3.10) where 𝑀(𝑠)=𝑠0𝑀(𝜏)𝑑𝜏.(3.11) From (2.7) and (2.8), we have 𝑀𝑢𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2𝛽𝜆1||Δ𝑢𝜀𝑚||(𝑡)2+||Δ𝑣𝜀𝑚||(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,𝑢𝜀𝑡𝑚2+𝐾(𝑡)2,𝑢𝜀𝑡𝑚2||𝑢(𝑡)+𝜀𝜀𝑡𝑚||(𝑡)2+||𝑣𝜀𝑡𝑚||2+𝛽(𝑡)1𝜆1||Δ𝑢𝜀𝑚||(𝑡)2+||Δ𝑣𝜀𝑚||(𝑡)2+(2𝛿)𝑡0||𝑢𝜀𝑡𝑚||(𝑠)2+||𝑢𝜀𝑡𝑚||(𝑠)2𝑑𝑠𝐶0+𝐶(𝛿)𝑡0𝐾1,𝑢𝜀𝑡𝑚2+𝐾(𝑠)2,𝑣𝜀𝑡𝑚2(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,𝑢𝜀𝑡𝑚2+𝐾(𝑡)2,𝑢𝜀𝑡𝑚2||𝑢(𝑡)+𝜀𝜀𝑡𝑚||(𝑡)2+||𝑣𝜀𝑡𝑚||2+𝛽(𝑡)1𝜆1||Δ𝑢𝜀𝑚||(𝑡)2+||Δ𝑣𝜀𝑚||(𝑡)2+(2𝛿)𝑡0||𝑢𝜀𝑡𝑚||(𝑠)2+||𝑢𝜀𝑡𝑚||(𝑠)2𝑑𝑠𝐶1,(3.14) where 𝐶1 is a positive constant independent of 𝜀, 𝑚, and 𝑡. Then, we can conclude that 𝐾11/2𝑢𝜀𝑡𝑚,𝐾21/2𝑣𝜀𝑡𝑚areboundedin𝐿0,𝑇;𝐿2,(Ω)𝜀𝑢𝜀𝑡𝑚,𝜀𝑣𝜀𝑡𝑚areboundedin𝐿0,𝑇;𝐿2,𝑢(Ω)𝜀𝑚,𝑣𝜀𝑚areboundedin𝐿0,𝑇;𝐻20,𝑢(Ω)𝜀𝑡𝑚,𝑣𝜀𝑡𝑚areboundedin𝐿20,𝑇;𝐿2.(Ω)(3.15)
The Second Estimate
Substituting 𝑤=Δ𝑢𝜀𝑡𝑚(𝑡) and 𝑧=Δ𝑣𝜀𝑡𝑚(𝑡) in (3.6) and (3.7), respectively, it holds that 𝑑𝐾𝑑𝑡1,𝑢𝜀𝑡𝑚2+𝐾(𝑡)2,𝑣𝜀𝑡𝑚2𝑢(𝑡)+𝜀𝜀𝑡𝑚(𝑡)2+𝑣𝜀𝑡𝑚(𝑡)2+Δ𝑢𝜀𝑚(𝑡)2+Δ𝑣𝜀𝑚(𝑡)2𝑢+2𝜀𝑡𝑚(𝑡)2+𝑣𝜀𝑡𝑚(𝑡)2=𝜕𝐾1,𝑢𝜕𝑡𝜀𝑡𝑚2+(𝑡)𝜕𝐾2,𝑣𝜕𝑡𝜀𝑡𝑚2𝑢(𝑡)2𝑀𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2Δ𝑢𝜀𝑚(𝑡),𝑢𝜀𝑡𝑚(+𝑡)Δ𝑣𝜀𝑚(𝑡),𝑣𝜀𝑡𝑚(.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,𝑢𝜀𝑡𝑚2+𝐾(𝑡)2,𝑣𝜀𝑡𝑚2𝑢(𝑡)+𝜀𝜀𝑡𝑚(𝑡)2+𝑣𝜀𝑡𝑚(𝑡)2+Δ𝑢𝜀𝑚(𝑡)2+Δ𝑣𝜀𝑚(𝑡)2𝑢+2𝜀𝑡𝑚(𝑡)2+𝑣𝜀𝑡𝑚(𝑡)2+(22𝛿)𝑡0𝑢𝜀𝑡𝑚(𝑡)2+𝑣𝜀𝑡𝑚(𝑡)2𝑑𝑠𝐶2,(3.17) where 𝐶2 is a positive constant independent of 𝜀, 𝑚, and 𝑡. From the above estimate we conclude that 𝐾11/2𝑢𝜀𝑡𝑚,𝐾21/2𝑣𝜀𝑡𝑚areboundedin𝐿0,𝑇;𝐻10,(Ω)𝜀𝑢𝜀𝑡𝑚,𝜀𝑣𝜀𝑡𝑚areboundedin𝐿0,𝑇;𝐻10,𝑢(Ω)𝜀𝑚,𝑣𝜀𝑚areboundedin𝐿0,𝑇;𝐻20(Ω)𝐻4,𝑢(Ω)𝜀𝑡𝑚,𝑣𝜀𝑡𝑚areboundedin𝐿20,𝑇;𝐻10.(Ω)(3.18)The Third Estimate
Differentiating (3.6) and (3.7) with respect to 𝑡 and setting 𝑤=𝑢𝜀𝑡𝑡𝑚 and 𝑣𝜀𝑡𝑡𝑚, respectively, we arrive at 𝑑𝐾𝑑𝑡1,𝑢𝜀𝑡𝑡𝑚(𝑡)2+𝐾2,𝑣𝜀𝑡𝑡𝑚(𝑡)2||𝑢+𝜀𝜀𝑡𝑡𝑚||(𝑡)2+||𝑣𝜀𝑡𝑡𝑚||2+||Δ𝑢𝜀𝑡𝑚||(𝑡)2+||Δ𝑣𝜀𝑡𝑚||(𝑡)2||𝑢+2𝜀𝑡𝑚||(𝑠)2+||𝑣𝜀𝑡𝑚||(𝑠)2𝑢=2𝑀𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2Δ𝑢𝜀𝑡𝑚(𝑡),𝑢𝜀𝑡𝑡𝑚+(𝑡)Δ𝑣𝜀𝑡𝑚(𝑡),𝑣𝜀𝑡𝑡𝑚𝑢(𝑡)4𝜀𝑚(t),Δ𝑢𝜀𝑡𝑚+𝑣(𝑡)𝜀𝑚(𝑡),Δ𝑣𝜀𝑡𝑚(𝑡)𝑀𝑢𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2Δ𝑢𝜀𝑡𝑚(𝑡),𝑢𝜀𝑡𝑡𝑚(+𝑡)Δ𝑣𝜀𝑡𝑚(𝑡),𝑣𝜀𝑡𝑡𝑚(+𝑡)𝜕𝐾1,𝑢𝜕𝑡𝜀𝑡𝑡𝑚2(+𝑡)𝜕𝐾2,𝑣𝜕𝑡𝜀𝑡𝑡𝑚2(.𝑡)(3.19) Integrating (3.19) over (0,𝑡), and using (2.5), (3.8), (3.9), and the norms |𝑢𝜀𝑡𝑡𝑚(0)|2𝐶3 and |𝑣𝜀𝑡𝑡𝑚(0)|2𝐶4 after employing Gronwall’s lemma, we obtain the third estimate 𝐾1,𝑢𝜀𝑡𝑡𝑚2+𝐾(𝑡)2,𝑣𝜀𝑡𝑡𝑚2||𝑢(𝑡)+𝜀𝜀𝑡𝑡𝑚||(𝑡)2+||𝑣𝜀𝑡𝑡𝑚||(𝑡)2+||Δ𝑢𝜀𝑡𝑚||(𝑡)2+||Δ𝑣𝜀𝑡𝑚||(𝑡)2+(22𝛿)𝑡0||𝑢𝜀𝑡𝑡𝑚||(𝑡)2+||𝑣𝜀𝑡𝑡𝑚||(𝑡)2𝑑𝑠𝐶5,(3.20) where 𝐶5 is a positive constant independent of 𝜀, 𝑚, and 𝑡. From the above estimate we conclude that 𝐾11/2𝑢𝜀𝑡𝑡𝑚,𝐾21/2𝑣𝜀𝑡𝑡𝑚areboundedin𝐿0,𝑇;𝐿2,(Ω)𝜀𝑢𝜀𝑡𝑡𝑚,𝜀𝑣𝜀𝑡𝑡𝑚areboundedin𝐿0,𝑇;𝐿2,𝑢(Ω)𝜀𝑡𝑚,𝑣𝜀𝑡𝑚areboundedin𝐿0,𝑇;𝐻20,𝑢(Ω)𝜀𝑡𝑡𝑚,𝑣𝜀𝑡𝑡𝑚areboundedin𝐿20,𝑇;𝐿2.(Ω)(3.21)(2) Limits of Approximated Solutions
From the Aubin-Lions theorem (see [6]) we deduce that there exist subsequences of (𝑢𝜀𝑚)𝑚 and (𝑣𝜀𝑚)𝑚 such that 𝑢𝜀𝑚𝑢𝜀stronglyin𝐿20,𝑇;𝐻10,𝑣(Ω)𝜀𝑚𝑣𝜀stronglyin𝐿20,𝑇;𝐻10,(Ω)(3.22) and since 𝑀 is continuous, it follows that 𝑀𝑢𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2𝑢𝑀𝜀(𝑡)2+𝑣𝜀(𝑡)2.(3.23) From the above estimate we can conclude that there exist subsequences of (𝑢𝜀𝑚)𝑚 and (𝑣𝜀𝑚)𝑚, that we denote also by (𝑢𝜀𝑚)𝑚 and (𝑣𝜀𝑚)𝑚 such that as 𝑚 and 𝜀0 we have 𝑢𝜀𝑚𝑢,𝑣𝜀𝑚𝑣weakstarin𝐿0,𝑇;𝐻20(Ω)𝐻4,𝑢(Ω)𝜀𝑡𝑚𝑢𝑡,𝑣𝜀𝑡𝑚𝑣𝑡weakstar𝐿0,𝑇;𝐻20,𝑢(Ω)𝜀𝑡𝑡𝑚𝑢𝑡𝑡,𝑣𝜀𝑡𝑡𝑚𝑣𝑡𝑡weakstar𝐿20,𝑇;𝐿2,(Ω)Δ𝑢𝜀𝑚Δ𝑢,Δ𝑣𝜀𝑚Δ𝑣weakstarin𝐿0,𝑇;𝐿2(,𝐾Ω)1𝑢𝜀𝑡𝑡𝑚𝐾1𝑢𝑡𝑡,𝐾2𝑣𝜀𝑡𝑡𝑚𝐾2𝑣𝑡𝑡weakstarin𝐿0,𝑇;𝐿2,Δ(Ω)2𝑢𝜀𝑚Δ2𝑢,Δ2𝑣𝜀𝑚Δ2𝑣weakstarin𝐿20,𝑇;𝐿2(,Ω)𝜀𝑢𝜀𝑡𝑡𝑚𝜀𝑢𝑡𝑡,𝜀𝑣𝜀𝑡𝑡𝑚𝜀𝑣𝑡𝑡weakstarin𝐿0,𝑇;𝐿2(,𝑀𝑢Ω)𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2Δ𝑢𝜀𝑚Δ𝑣𝜀𝑚(𝑀𝑢𝑡)2+𝑣(𝑡)2(Δ𝑢Δ𝑣)weakstarin𝐿0,𝑇;𝐿2(.Ω)(3.24) Now, multiplying (3.6), (3.7) by 𝜃𝒟(0,𝑇) and integrating over (0,𝑇), we arrive at 𝑇0𝐾1𝑢+𝜀𝜀𝑡𝑡𝑚(𝑡),𝑤𝜃(𝑡)𝑑𝑡+𝑇0Δ𝑢𝜀𝑚+(𝑡),Δ𝑤𝜃(𝑡)𝑑𝑡𝑇0𝑀𝑢𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2Δ𝑢𝜀𝑚+(𝑡),𝑤𝜃𝑑𝑡𝑇0𝑢𝜀𝑡𝑚,𝑤𝜃𝑑𝑡=0,𝑤𝑉𝑚,𝜃𝒟(0,𝑇),𝑇0𝐾2𝑣+𝜀𝜀𝑡𝑡𝑚(𝑡),𝑧𝜃(𝑡)𝑑𝑡+𝑇0Δ𝑣𝜀𝑚(+𝑡),Δ𝑧𝜃(𝑡)𝑑𝑡𝑇0𝑀𝑢𝜀𝑚(𝑡)2+𝑣𝜀𝑚(𝑡)2Δ𝑣𝜀𝑚(+𝑡),𝑧𝜃𝑑𝑡𝑇0𝑣𝜀𝑡𝑚,𝑧𝜃𝑑𝑡=0,𝑧𝑉𝑚,𝜃𝒟(0,𝑇).(3.25) The convergences (3.24) are sufficient to pass to the limit in (3.25) in order to obtain 𝐾1𝑢𝑡𝑡+Δ2𝑢+𝑀𝑢(𝑡)2+𝑣(𝑡)2(Δ𝑢)+𝑢𝑡=0in𝐿loc0,;𝐿2,𝐾(Ω)2𝑣𝑡𝑡+Δ2(𝑣+𝑀𝑢𝑡)2+𝑣(𝑡)2(Δ𝑣)+𝑣𝑡=0in𝐿loc0,;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 ||𝐾11/2𝑢𝑡||(𝑡)2+||𝐾21/2𝑣𝑡||(𝑡)2+||||Δ𝑢(𝑡)2+||||Δ𝑣(𝑡)2+𝑡𝑡+1||𝑢𝑡||(𝑠)2+||𝑣𝑡||(𝑠)2𝑑𝑠𝛼1𝑒𝛼2𝑡,(4.1) for all 𝑡1, where 𝛼1 and 𝛼2 are positive constants.

Proof. Multiplying (3.2) by 𝑢𝑡(𝑡) and 𝑣𝑡(𝑡), respectively, and integrating over Ω, we obtain 12𝑑||𝐾𝑑𝑡11/2𝑢𝑡||(𝑡)2+||𝐾21/2𝑣𝑡||(𝑡)2+||||Δ𝑢(𝑡)2+||||Δ𝑣(𝑡)2+𝑀𝑢(𝑡)2+𝑣(𝑡)2+||𝑢𝑡||(𝑡)2+||𝑣𝑡||(𝑡)2=𝜕𝐾1𝜕𝑡,𝑢2𝑡+(𝑡)𝜕𝐾2𝜕𝑡,𝑣2𝑡,(𝑡)(4.2) where 𝑀(𝑠)=𝑠0𝑀(𝜏)𝑑𝜏.(4.3) Using (2.6) and considering 𝛿>0 sufficiently small, we get 12𝑑||𝐾𝑑𝑡11/2𝑢𝑡||(𝑡)2+||𝐾21/2𝑣𝑡||(𝑡)2+||||Δ𝑢(𝑡)2+||||Δ𝑣(𝑡)2+𝑀𝑢(𝑡)2+𝑣(𝑡)2+1𝛿𝐾0||𝑢𝐶(𝛿)𝑡||(𝑡)2+||𝑣𝑡||(𝑡)20,(4.4) where 𝐾0=max{𝐾1,𝐾2} with 𝐾𝑖=max[]𝑠𝑡,𝑡+1esssup𝐾𝑖(𝑥,𝑠),𝑖=1,2.(4.5) Integrating (4.4) from 0to𝑡, we have 𝐸(𝑡)+1𝛿𝐾0𝐶(𝛿)𝑡0||𝑢𝑡(||𝑠)2+||𝑣𝑡(||𝑠)2𝑑𝑠𝐸(0),(4.6) where 1𝐸(𝑡)=2||𝐾11/2𝑢𝑡||(𝑡)2+||𝐾21/2𝑣𝑡||(𝑡)2+||||Δ𝑣(𝑡)2+||||Δ𝑣(𝑡)2+𝑀𝑢(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)0𝑡(0,),(4.8) that is, 𝐸(𝑡) is bounded and increasing in (0,).
Integrating (4.4) from 𝜏1to𝜏2, 0<𝜏1<𝜏2<, we arrive at 𝐸𝜏2+1𝛿𝐾0𝐶(𝛿)𝜏2𝜏1||𝑢𝑡||(𝑠)2+||𝑣𝑡||(𝑠)2𝜏𝑑𝑠𝐸1.(4.9) Taking 𝜏1=𝑡 and 𝜏2=𝑡+1 in (4.9), we get 𝑡𝑡+1||𝑢𝑡||(𝑠)2+||𝑣𝑡||(𝑠)21𝑑𝑠1𝛿𝐾0[]𝐶(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 ||𝑢𝑡𝑡𝑖||+||𝑣𝑡𝑡𝑖||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 𝑑𝐾𝑑𝑡1𝑢𝑡+𝑑(𝑡),𝑢(𝑡)𝐾𝑑𝑡2𝑣𝑡|||(𝑡),𝑣(𝑡)𝐾1𝑢𝑡|||(𝑡)2|||𝐾2𝑣𝑡|||(𝑡)2+||||Δ𝑢(𝑡)2+||||Δ𝑣(𝑡)2+𝑀𝑢(𝑡)2+𝑣(𝑡)2𝑢(𝑡)2+𝑣(𝑡)2+𝑢𝑡+𝑣(𝑡),𝑢(𝑡)𝑡=(𝑡),𝑣(𝑡)𝜕𝐾1𝑢𝜕𝑡𝑡+(𝑡),𝑢(𝑡)𝜕𝐾2𝑣𝜕𝑡𝑡.(𝑡),𝑣(𝑡)(4.12) Integrating from 𝑡1 to 𝑡2 and using (2.6), and (2.7) we have 𝛽1𝜆1𝑡2𝑡1||||Δ𝑢(𝑠)2+||||Δ𝑣(𝑠)2𝐾𝑑𝑠1𝑢𝑡𝑡1𝑡,𝑢1𝐾1𝑢𝑡𝑡2𝑡,𝑢2+𝐾2𝑣𝑡𝑡1𝑡,𝑣1𝐾2𝑣𝑡𝑡2𝑡,𝑣2+1+𝛿+𝐾0𝐶(𝛿)𝑡2𝑡1||𝑢𝑡||||||+||𝑣(𝑠)𝑢(𝑠)𝑡||||||(𝑠)𝑣(𝑠)𝑑𝑠+𝐾0𝑡2𝑡1||𝑢𝑡||(𝑠)2+||𝑣𝑡||(𝑠)2𝑑𝑠.(4.13) Let us consider 𝐶>0 such that ||||||||,||||||||𝑢(𝑠)𝐶Δ𝑢(𝑠)𝑣(𝑠)𝐶Δ𝑣(𝑠)(4.14) and we take 𝑑>0 sufficiently small Then we have. 1+𝛿+𝐾0𝐶||𝑢(𝛿)𝑡||||𝑢||+||𝑣(𝑠)(𝑠)𝑡||||𝑣||(𝑠)(𝑠)1+𝛿+𝐾0𝐶(𝛿)2𝑑||𝑢𝑡||(𝑠)2+||𝑣𝑡||(𝑠)2||||+𝑑Δ𝑢(𝑠)2+||||Δ𝑣(𝑠)2,||𝐾1𝑢𝑡𝑡1𝑡,𝑢1+𝐾2𝑣𝑡𝑡1𝑡,𝑣1𝐾1𝑢𝑡𝑡2𝑡,𝑢2𝐾2𝑣𝑡𝑡2𝑡,𝑣2||𝐶𝐾0esssup[]𝑠𝑡,𝑡+1||||||𝑢Δ𝑢(𝑠)𝑡𝑡1||+||𝑢𝑡𝑡2||+𝐶𝐾0esssup[]𝑠𝑡,𝑡+1||||||𝑣Δ𝑣(𝑠)𝑡𝑡1||+||𝑣𝑡𝑡2||.(4.15) Thus, substituting (4.15) into (4.13), we arrive at 𝛽1𝜆1𝑡2𝑡1||||Δ𝑢(𝑠)2+||||Δ𝑣(𝑠)2𝑑𝑠𝐾0𝑡2𝑡1||𝑢𝑡||(𝑠)2+||𝑣𝑡||(𝑠)2𝑑𝑠+𝑑𝑡2𝑡1||||Δ𝑢(𝑠)2+||||Δ𝑣(𝑠)2𝑑𝑠+𝐶𝐾0esssup[]𝑠𝑡,𝑡+1||||||𝑢Δ𝑢(𝑠)𝑡𝑡1||+||𝑢𝑡𝑡2||+𝐶𝐾0esssup[]𝑠𝑡,𝑡+1||||||𝑣Δ𝑣(𝑠)𝑡𝑡1||+||𝑣𝑡𝑡2||.(4.16) Applying (4.10) and (4.11) in (4.16), we have 𝑡2𝑡1||||Δ𝑢(𝑠)2+||||Δ𝑣(𝑠)2𝑑𝑠𝐶1𝐹2(𝑡)+esssup[]𝑠𝑡,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 𝑡2𝑡1||𝑢𝑡(||𝑠)2+||𝑣𝑡(||𝑠)2+||||Δ𝑢(𝑠)2+||||Δ𝑣(𝑠)2𝑑𝑠𝐹2(𝑡)+𝐺2(𝑡).(4.18) Hence, there exists 𝑡[𝑡1,𝑡2] such that ||𝑢𝑡𝑡||2+||𝑣𝑡𝑡||2+||𝑡Δ𝑢||2+||𝑡Δ𝑣||2𝐹22(𝑡)+𝐺2.(𝑡)(4.19) Consequently, 𝑀𝑢(𝑡)2+𝑣(𝑡)2𝐶2𝐹2(𝑡)+𝐺2,(𝑡)(4.20) where 𝐶2=2𝑚0𝐶,𝑚0=max0𝑠𝑢(𝑡)2+𝑣(𝑡)2<𝑀(𝑠)(4.21) and 𝐶 is a positive constant such that 𝑢(𝑡)2𝐶|Δ𝑢(𝑡)|2.
From (4.19) and (4.20), we have 𝐸𝑡𝐶3𝐹2(𝑡)+𝐺2.(𝑡)(4.22) Since 𝐸(𝑡) is increasing, we have esssup[]𝑠𝑡,𝑡+1𝑡𝐸(𝑠)𝐸+1+𝛿+𝐾0𝐶(𝛿)𝑡2𝑡1||𝑢𝑡(||𝑠)2+||𝑣𝑡(||𝑠)2𝑑𝑠.(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)𝑏1𝑒𝛼2𝑡,𝑡1,(4.25) where 𝑏1 and 𝛼2 are positive constants, that is, |||𝐾1𝑢𝑡|||(𝑡)2+|||𝐾2𝑣𝑡|||(𝑡)2+||||Δ𝑢(𝑡)2+||||Δ𝑣(𝑡)2𝑀𝑢(𝑡)2+𝑣(𝑡)22𝑏1𝑒𝛼2𝑡.(4.26) Using (2.7) we obtain |||𝐾1𝑢𝑡|||(𝑡)2+|||𝐾2𝑣𝑡|||(𝑡)2+||||Δ𝑢(𝑡)2+||||Δ𝑣(𝑡)22𝑏1𝑚1𝑒𝛼2𝑡,(4.27) where 𝑚1𝛽=1𝜆1>0.(4.28) From (4.10) we have 𝑡𝑡+1||𝑢𝑡||(𝑠)2+||𝑣𝑡||(𝑠)21𝑑𝑠1𝛿𝐾0[]𝐶(𝛿)𝐸(𝑡)𝐸(𝑡+1)𝐸(𝑡)𝑏1𝑒𝛼2𝑡.(4.29) Therefore, from (4.27) and (4.29) we conclude that |||𝐾1𝑢𝑡|||(𝑡)2+|||𝐾2𝑣𝑡|||(𝑡)2+||||Δ𝑢(𝑡)2+||||Δ𝑣(𝑡)2+𝑡𝑡+1||𝑢𝑡||(𝑠)2+||𝑣𝑡||(𝑠)2𝑑𝑠𝛼1𝑒𝛼2𝑡,𝑡1,(4.30) where 𝛼1 and 𝛼2 are positive constants. Now, the proof is complete.

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