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 š¾1(š‘„,š‘”)š‘¢š‘”š‘”+Ī”2ī€·š‘¢āˆ’š‘€ā€–š‘¢ā€–2+ā€–š‘£ā€–2ī€øĪ”š‘¢+š‘¢š‘”š¾=0inĪ©Ć—(0,š‘‡),(1.1)2(š‘„,š‘”)š‘£š‘”š‘”+Ī”2ī€·š‘£āˆ’š‘€ā€–š‘¢ā€–2+ā€–š‘£ā€–2ī€øĪ”š‘£+š‘£š‘”=0inĪ©Ć—(0,š‘‡),(1.2)š‘¢=š‘£=šœ•š‘¢=šœ•šœ‚šœ•š‘£ī€·š‘¢šœ•šœ‚=0onĪ£,(1.3)(š‘¢(š‘„,0),š‘£(š‘„,0))=0,š‘£0ī€øī€·š‘¢inĪ©,(1.4)š‘”(š‘„,0),š‘£š‘”ī€ø=ī€·š‘¢(š‘„,0)1(š‘„),š‘£1ī€ø(š‘„)inĪ©,(1.5) where Ī© is a bounded domain of ā„š‘›, š‘›ā‰„1, with smooth boundary Ī“,ā€‰ā€‰ā€‰š‘‡>0 is a real arbitrary number, and šœ‚ is the unit normal at Ī£=Ī“Ć—(0,š‘‡) direct towards the exterior of Ī©Ć—(0,š‘‡). Here š¾š‘–āˆˆš¶1([0,š‘‡];š»10(Ī©)āˆ©šæāˆž(Ī©)), ā€‰ā€‰š‘–=1,2 and š‘€āˆˆš¶1([0,āˆž[), see Section 2 for more details.

Problems related to the system (1.1)ā€“(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, š¾(š‘„,š‘”)š‘¢š‘”š‘”+Ī”2ī€·š‘¢āˆ’š‘€ā€–š‘¢ā€–2ī€øĪ”š‘¢=0inĪ©Ć—(0,š‘‡)(1.6) and with š¾(š‘„,š‘”)=1, it is a generalization of one-dimensional model proposed by Woinowsky-Krieger [1] as a model for the transverse deflection š‘¢(š‘„,š‘”) 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 š¾(š‘„,š‘”)=1 in the case when the ends of the beam are hinged. He showed how the model affords a description of the phenomenon of ā€œdynamic buckling.ā€ 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 š¾(š‘„,š‘”)=1, š‘„āˆˆā„š‘›, and š‘€āˆˆš¶1[0,āˆž[ā€‰ā€‰and š‘€(šœ†)ā‰„š‘š0>0, for all šœ†ā‰„0. The existence, uniqueness, and boundary regularity of weak solutions were considered by Ramos [8] with š¾(š‘„)ā‰„š‘˜0>0,ā€‰ā€‰š‘„āˆˆĪ©. See also Pereira et al. [9]. The abstract model š‘¢š‘”š‘”+š€2ī‚€||š€š‘¢+š‘€1/2||2ī‚š€š‘¢=0(1.7) of (1.6), where š€ 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 š‘€āˆˆš¶1[0,āˆž[ā€‰ā€‰with š‘€(šœ†)ā‰„š‘š0+š‘š1šœ†, for all šœ†ā‰„0,ā€‰ā€‰where š‘š0 and š‘š1 are positive constants. Pereira [11] considered the abstract model (1.7) with dissipative term š‘¢š‘”. He proved the existence, uniqueness, and exponential decay of the solutions with the following assumptions about š‘€: š‘€āˆˆš¶0([[0,āˆž)withš‘€(šœ†)ā‰„āˆ’š›½,āˆ€šœ†ā‰„0,0<š›½<šœ†1,(1.8) where šœ†1 is the first eigenvalue of š€2š‘¢āˆ’šœ†š€š‘¢=0.(1.9) Our main goal here is to extend the previous results for a nonlinear degenerate coupled beams system of type (1.1)ā€“(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 šæ2(Ī©) endowed with the inner product (ī€œš‘¢,š‘£)=Ī©š‘¢(š‘„)š‘£(š‘„)š‘‘š‘„(2.1) and norm āˆš|š‘¢|=(š‘¢,š‘£).(2.2) We also consider the Sobolev space š»1(Ī©) endowed with the scalar product (š‘¢,š‘£)š»1(Ī©)=(š‘¢,š‘£)+(āˆ‡š‘¢,āˆ‡š‘£).(2.3) We define the subspace of š»1(Ī©), denoted by š»10(Ī©). This space endowed with the norm induced by the scalar product ((š‘¢,š‘£))š»10(Ī©)=(āˆ‡š‘¢,āˆ‡š‘£)(2.4) is a Hilbert space.

2.1. Assumptions on the Functions š¾š‘–, š‘–=1,2, and š‘€

To obtain the weak solution of the system (1.1)ā€“(1.5) we consider the following hypothesis: š¾š‘–āˆˆš¶1ī€·[]0,š‘‡;š»10(Ī©)āˆ©šæāˆžī€ø(Ī©),š‘–=1,2,withš¾š‘–(š‘„,š‘”)ā‰„0,āˆ€(š‘„,š‘”)āˆˆĪ©Ć—(0,š‘‡),andthereexistsš›¾>0suchthatš¾š‘–||||(š‘„,0)ā‰„š›¾>0,(2.5)šœ•š¾š‘–||||šœ•š‘”ā„ā‰¤š›æ+š¶(š›æ)š¾š‘–,š‘–=1,2,āˆ€š›æ>0,(2.6)š‘€āˆˆš¶1([[)0,āˆžwithš‘€(šœ†)ā‰„āˆ’š›½,āˆ€šœ†ā‰„0,0<š›½<šœ†1,šœ†1Ī”istheļ¬rsteingenvalueofthestationaryproblem,2š‘¢āˆ’šœ†(āˆ’Ī”š‘¢)=0.(2.7)

Remark 2.1. Let šœ†1 be the first eingevalue of Ī”2š‘¢āˆ’šœ†(āˆ’Ī”š‘¢)=0; then (see Miklin [13]) šœ†1=infš‘¤āˆˆš»20(Ī©)||||Ī”š‘¤2||||āˆ‡š‘¤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)ā€“(1.5).

Theorem 3.1. Let one take (š‘¢0,š‘£0)āˆˆ(š»10(Ī©)āˆ©š»4(Ī©))2ā€‰ā€‰and ā€‰ā€‰(š‘¢1,š‘£1)āˆˆ(š»20(Ī©))2, and let one suppose that assumptions (2.5), (2.6) and (2.7) hold. Then, there exist unique functions š‘¢,š‘£āˆ¶[0,š‘‡]ā†’šæ2(Ī©) in the class šæ(š‘¢,š‘£)āˆˆī€·ī€·āˆžloc(0,āˆž)āˆ¶š»20(Ī©)āˆ©š»4(Ī©)ī€øī€ø2,ī€·š‘¢š‘”,š‘£š‘”ī€øāˆˆšæī€·ī€·āˆžloc(0,āˆž)āˆ¶š»20(Ī©)ī€øī€ø2,ī€·š‘¢š‘”š‘”,š‘£š‘”š‘”ī€øāˆˆšæī€·ī€·āˆžloc(0,āˆž)āˆ¶šæ2(Ī©)ī€øī€ø2(3.1) satisfying š¾1(š‘„,š‘”)š‘¢š‘”š‘”+Ī”2ī€·š‘¢āˆ’š‘€ā€–š‘¢ā€–2+ā€–š‘£ā€–2ī€øĪ”š‘¢+š‘¢š‘”=0inšæ2locī€·0,āˆž;šæ2ī€ø,š¾(Ī©)2(š‘„,š‘”)š‘£š‘”š‘”+Ī”2ī€·š‘£āˆ’š‘€ā€–š‘¢ā€–2+ā€–š‘£ā€–2ī€øĪ”š‘£+š‘£š‘”=0inšæ2locī€·0,āˆž;šæ2(ī€ø,ī€·š‘¢Ī©)(3.2)(š‘¢(š‘„,0),š‘£(š‘„,0))=0(š‘„),š‘£0ī€øī€·š‘¢(š‘„)inĪ©,š‘”(š‘„,0),š‘£š‘”(ī€ø=ī€·š‘¢š‘„,0)1(š‘„),š‘£1(ī€øš‘„)inĪ©.(3.3)

Proof. Since š¾š‘–ā‰„0, š‘–=1,2, we first perturb the system (1.1)ā€“(1.5) with the terms šœ€š‘¢š‘”š‘”,šœ€š‘£š‘”š‘”, with 0<šœ€<1, and we apply the Faedo-Galerkin method to the perturbed system. After we pass to the limit with šœ€ā†’0 in the perturbed system and we obtain the solution for the system (1.1)ā€“(1.5).
(1) Perturbed System
Consider the perturbed system ī€·š¾1ī€øš‘¢+šœ€šœ€š‘”š‘”+Ī”š‘¢šœ€ī€·+š‘€ā€–š‘¢šœ€ā€–2+ā€–š‘£šœ€ā€–2ī€ø(āˆ’Ī”š‘¢šœ€)+š‘¢šœ€š‘”ī€·š¾=0inĪ©Ć—(0,š‘‡),2ī€øš‘£+šœ€šœ€š‘”š‘”+Ī”š‘£šœ€ī€·+š‘€ā€–š‘¢šœ€ā€–2+ā€–š‘£šœ€ā€–2ī€ø(āˆ’Ī”š‘£šœ€)+š‘£šœ€š‘”š‘¢=0inĪ©Ć—(0,š‘‡),šœ€=š‘£šœ€=šœ•š‘¢šœ€=šœ•šœ‚šœ•š‘£šœ€šœ•šœ‚=0onĪ£,(š‘¢šœ€(š‘„,0),š‘£šœ€ī€·š‘¢(š‘„,0))=0(š‘„),š‘£0ī€øī€·š‘¢(š‘„)inĪ©,šœ€š‘”(š‘„,0),š‘£šœ€š‘”(ī€ø=ī€·š‘¢š‘„,0)1(š‘„),š‘£1(ī€øš‘„)inĪ©.(3.4) Let (š‘¤šœˆ)šœˆāˆˆā„• be a basis of š»20(Ī©) formed by the eigenvectors of the operator āˆ’Ī”, that is, āˆ’Ī”š‘¤šœˆ=šœ†šœˆš‘¤šœˆ, with šœ†šœˆā†’āˆž when šœˆā†’āˆž. Let š‘‰š‘š=[š‘¤1,š‘¤2,ā€¦,š‘¤š‘š] be the subspace generated by the first š‘š vectors of (š‘¤šœˆ)šœˆāˆˆā„•.
For each fixed šœ€, we consider š‘¢šœ€š‘š(š‘”)=š‘šī“š‘—=1š‘”š‘—šœ€š‘š(š‘”)š‘¤š‘—āˆˆš‘‰š‘š,š‘£šœ€š‘š(š‘”)=š‘šī“š‘—=1ā„Žš‘—šœ€š‘š(š‘”)š‘¤š‘—āˆˆš‘‰š‘š(3.5) as solutions of the approximated perturbed system š¾ī€·ī€·1ī€øš‘¢+šœ€šœ€š‘”š‘”š‘šī€ø+ī€·(š‘”),š‘¤āˆ’Ī”š‘¢šœ€š‘šī€øī‚€ā€–ā€–š‘¢(š‘”),āˆ’Ī”š‘¤+š‘€šœ€š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘šā€–ā€–(š‘”)2ī‚ī€·āˆ’Ī”š‘¢šœ€š‘šī€ø+ī€·š‘¢(š‘”),š‘¤šœ€š‘”š‘šī€ø(š‘”),š‘¤=0,āˆ€š‘¤āˆˆš‘‰š‘š,š¾(3.6)ī€·ī€·2ī€øš‘£+šœ€šœ€š‘”š‘”š‘šī€ø+ī€·(š‘”),š‘§āˆ’Ī”š‘£šœ€š‘šī€øī‚€ā€–ā€–š‘¢(š‘”),āˆ’Ī”š‘§+š‘€šœ€š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘šā€–ā€–(š‘”)2ī‚ī€·āˆ’Ī”š‘£šœ€š‘šī€ø+ī€·š‘£(š‘”),š‘§šœ€š‘”š‘šī€ø(š‘”),š‘§=0,āˆ€š‘§āˆˆš‘‰š‘š,ī€·š‘¢(3.7)šœ€š‘š(0),š‘£šœ€š‘šī€ø=ī€·š‘¢(0)0š‘š,š‘£0š‘šī€øāŸ¶ī€·š‘¢0,š‘£0ī€øī€·š»in20(Ī©)āˆ©š»4ī€ø(Ī©)2,ī€·š‘¢(3.8)šœ€š‘”š‘š(0),š‘£šœ€š‘”š‘šī€ø=ī€·š‘¢(0)1š‘š,š‘£1š‘šī€øāŸ¶ī€·š‘¢1,š‘£1ī€øī€·š»in20ī€ø(Ī©)2.(3.9) The local existence of the approximated solutions (š‘¢šœ€š‘š,š‘£šœ€š‘š) is guaranteed by the standard results of ordinary differential equations. The extension of the solutions (š‘¢šœ€š‘š,š‘£šœ€š‘š) to the whole interval [0,š‘‡] is a consequence of the first estimate below.
The First Estimate
Setting š‘¤=š‘¢šœ€š‘”š‘š and š‘§=š‘£šœ€š‘”š‘š in (3.6) and (3.7), respectively, integrating over (0,š‘”), and taking the convergences (3.8) and (3.9) in consideration, we arrive at ī‚€š¾1,ī€·š‘¢šœ€š‘”š‘šī€ø2ī‚||š‘¢(š‘”)+šœ€šœ€š‘”š‘š||(š‘”)2+||Ī”š‘¢šœ€š‘š||(š‘”)2+ī‚€š¾2,ī€·š‘£šœ€š‘”š‘šī€ø2ī‚||š‘£(š‘”)+šœ€šœ€š‘”š‘š||2+||Ī”š‘£šœ€š‘š||(š‘”)2+ī‚Šš‘€ī‚€ā€–ā€–š‘¢šœ€š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘šā€–ā€–(š‘”)2ī‚ī€œ+2š‘”0ī‚ƒ||š‘¢šœ€š‘”š‘š||(š‘ )2+||š‘£šœ€š‘”š‘š||(š‘ )2ī‚„ā‰¤ī€œš‘‘š‘ š‘”0īƒ¬||||ī‚µšœ•š¾1,ī€·š‘¢šœ•š‘”šœ€š‘”š‘šī€ø2(ī‚¶||||š‘ )ā„+||||ī‚µšœ•š¾2,ī€·š‘£šœ•š‘”šœ€š‘”š‘šī€ø2(ī‚¶||||š‘ )ā„īƒ­ī€·š¾š‘‘š‘ +1(0),š‘¢21š‘šī€ø||š‘¢+šœ€1š‘š||2+||Ī”š‘¢0š‘š||2+ī€·š¾2(0),š‘£21š‘šī€ø||š‘£+šœ€1š‘š||2+||Ī”š‘£0š‘š||2+ī‚Šš‘€ī‚€ā€–ā€–š‘¢0š‘šā€–ā€–2+ā€–ā€–š‘£0š‘šā€–ā€–2ī‚,(3.10) where ī‚Šī€œš‘€(š‘ )=š‘ 0š‘€(šœ)š‘‘šœ.(3.11) From (2.7) and (2.8), we have ī‚Šš‘€ī‚€ā€–ā€–š‘¢šœ€š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘šā€–ā€–(š‘”)2ī‚š›½ā‰„āˆ’šœ†1ī‚€||Ī”š‘¢šœ€š‘š||(š‘”)2+||Ī”š‘£šœ€š‘š||(š‘”)2ī‚.(3.12) Since š›½<šœ†1 and so by (2.5)ā€“(2.7) and convergences (3.8), (3.9), and (3.12), we obtain ī‚€š¾1,ī€·š‘¢šœ€š‘”š‘šī€ø2ī‚+ī‚€š¾(š‘”)2,ī€·š‘¢šœ€š‘”š‘šī€ø2ī‚ī‚€||š‘¢(š‘”)+šœ€šœ€š‘”š‘š||(š‘”)2+||š‘£šœ€š‘”š‘š||2ī‚+ī‚µš›½(š‘”)1āˆ’šœ†1ī‚¶ī‚€||Ī”š‘¢šœ€š‘š||(š‘”)2+||Ī”š‘£šœ€š‘š||(š‘”)2ī‚+ī€œ(2āˆ’š›æ)š‘”0ī‚ƒ||š‘¢šœ€š‘”š‘š||(š‘ )2+||š‘¢šœ€š‘”š‘š||(š‘ )2ī‚„š‘‘š‘ ā‰¤š¶0ī€œ+š¶(š›æ)š‘”0š¾ī‚ƒī‚€1,ī€·š‘¢šœ€š‘”š‘šī€ø2ī‚+ī‚€š¾(š‘ )2,ī€·š‘£šœ€š‘”š‘šī€ø2(š‘ )ī‚ī‚„š‘‘š‘ (3.13) with 0<š›æ<1 and š¶0 being a positive constant independent of šœ€, š‘š, and š‘”.
Employing Gronwallā€™s lemma in (3.13), we obtain the first estimate ī‚€š¾1,ī€·š‘¢šœ€š‘”š‘šī€ø2ī‚+ī‚€š¾(š‘”)2,ī€·š‘¢šœ€š‘”š‘šī€ø2ī‚ī‚€||š‘¢(š‘”)+šœ€šœ€š‘”š‘š||(š‘”)2+||š‘£šœ€š‘”š‘š||2ī‚+ī‚µš›½(š‘”)1āˆ’šœ†1ī‚¶ī‚€||Ī”š‘¢šœ€š‘š||(š‘”)2+||Ī”š‘£šœ€š‘š||(š‘”)2ī‚+ī€œ(2āˆ’š›æ)š‘”0ī‚ƒ||š‘¢šœ€š‘”š‘š||(š‘ )2+||š‘¢šœ€š‘”š‘š||(š‘ )2ī‚„š‘‘š‘ ā‰¤š¶1,(3.14) where š¶1 is a positive constant independent of šœ€, š‘š, and š‘”. Then, we can conclude that ī€·š¾11/2š‘¢šœ€š‘”š‘šī€ø,ī€·š¾21/2š‘£šœ€š‘”š‘šī€øareboundedinšæāˆžī€·0,š‘‡;šæ2ī€ø,ī‚€āˆš(Ī©)šœ€š‘¢šœ€š‘”š‘šī‚,ī‚€āˆššœ€š‘£šœ€š‘”š‘šī‚areboundedinšæāˆžī€·0,š‘‡;šæ2ī€ø,ī€·š‘¢(Ī©)šœ€š‘šī€ø,ī€·š‘£šœ€š‘šī€øareboundedinšæāˆžī€·0,š‘‡;š»20ī€ø,ī€·š‘¢(Ī©)šœ€š‘”š‘šī€ø,ī€·š‘£šœ€š‘”š‘šī€øareboundedinšæ2ī€·0,š‘‡;šæ2ī€ø.(Ī©)(3.15)
The Second Estimate
Substituting š‘¤=āˆ’Ī”š‘¢šœ€š‘”š‘š(š‘”) and š‘§=āˆ’Ī”š‘£šœ€š‘”š‘š(š‘”) in (3.6) and (3.7), respectively, it holds that š‘‘š¾š‘‘š‘”ī‚ƒī‚€ī‚€1,ī€·š‘¢šœ€š‘”š‘šī€ø2+š¾(š‘”)ī‚ī‚ī‚€ī‚€2,ī€·š‘£šœ€š‘”š‘šī€ø2ī‚€ā€–ā€–š‘¢(š‘”)ī‚ī‚+šœ€šœ€š‘”š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘”š‘šā€–ā€–(š‘”)2ī‚+ā€–ā€–Ī”š‘¢šœ€š‘šā€–ā€–(š‘”)2+ā€–ā€–Ī”š‘£šœ€š‘šā€–ā€–(š‘”)2ī‚„ī‚€ā€–ā€–š‘¢+2šœ€š‘”š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘”š‘šā€–ā€–(š‘”)2ī‚=ī‚µī‚µšœ•š¾1,ī€·š‘¢šœ•š‘”šœ€š‘”š‘šī€ø2+(š‘”)ī‚¶ī‚¶ī‚µī‚µšœ•š¾2,ī€·š‘£šœ•š‘”šœ€š‘”š‘šī€ø2ī‚€ā€–ā€–š‘¢(š‘”)ī‚¶ī‚¶āˆ’2š‘€šœ€š‘š(ā€–ā€–š‘”)2+ā€–ā€–š‘£šœ€š‘š(ā€–ā€–š‘”)2ī‚ā‹…ī€ŗī€·ī€·āˆ’Ī”š‘¢šœ€š‘š(š‘”),š‘¢šœ€š‘”š‘š(+š‘”)ī€øī€øī€·ī€·āˆ’Ī”š‘£šœ€š‘š(š‘”),š‘£šœ€š‘”š‘š(.š‘”)ī€øī€øī€»(3.16)
Integrating (3.16) over (0,š‘”), 0<š‘”<š‘‡, and taking (2.5)ā€“(2.7) and (3.8), (3.9), and first estimate into account, we infer š¾ī‚€ī‚€1,ī€·š‘¢šœ€š‘”š‘šī€ø2+š¾(š‘”)ī‚ī‚ī‚€ī‚€2,ī€·š‘£šœ€š‘”š‘šī€ø2ī‚€ā€–ā€–š‘¢(š‘”)ī‚ī‚+šœ€šœ€š‘”š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘”š‘šā€–ā€–(š‘”)2ī‚+ā€–ā€–Ī”š‘¢šœ€š‘šā€–ā€–(š‘”)2+ā€–ā€–Ī”š‘£šœ€š‘šā€–ā€–(š‘”)2ī‚€ā€–ā€–š‘¢+2šœ€š‘”š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘”š‘šā€–ā€–(š‘”)2ī‚ī€œ+(2āˆ’2š›æ)š‘”0ī‚€ā€–ā€–š‘¢šœ€š‘”š‘š(ā€–ā€–š‘”)2+ā€–ā€–š‘£šœ€š‘”š‘š(ā€–ā€–š‘”)2ī‚š‘‘š‘ ā‰¤š¶2,(3.17) where š¶2 is a positive constant independent of šœ€, š‘š, and š‘”. From the above estimate we conclude that ī€·š¾11/2š‘¢šœ€š‘”š‘šī€ø,ī€·š¾21/2š‘£šœ€š‘”š‘šī€øareboundedinšæāˆžī€·0,š‘‡;š»10ī€ø,ī‚€āˆš(Ī©)šœ€š‘¢šœ€š‘”š‘šī‚,ī‚€āˆššœ€š‘£šœ€š‘”š‘šī‚areboundedinšæāˆžī€·0,š‘‡;š»10ī€ø,ī€·š‘¢(Ī©)šœ€š‘šī€ø,ī€·š‘£šœ€š‘šī€øareboundedinšæāˆžī€·0,š‘‡;š»20(Ī©)āˆ©š»4ī€ø,ī€·š‘¢(Ī©)šœ€š‘”š‘šī€ø,ī€·š‘£šœ€š‘”š‘šī€øareboundedinšæ2ī€·0,š‘‡;š»10ī€ø.(Ī©)(3.18)The Third Estimate
Differentiating (3.6) and (3.7) with respect to š‘” and setting š‘¤=š‘¢šœ€š‘”š‘”š‘š and š‘£šœ€š‘”š‘”š‘š, respectively, we arrive at š‘‘ī‚ƒš¾š‘‘š‘”1,ī€·š‘¢šœ€š‘”š‘”š‘šī€ø(š‘”)2+ī‚€š¾2,ī€·š‘£šœ€š‘”š‘”š‘šī€ø(š‘”)2ī‚ī‚€||š‘¢+šœ€šœ€š‘”š‘”š‘š||(š‘”)2+||š‘£šœ€š‘”š‘”š‘š||2ī‚+||Ī”š‘¢šœ€š‘”š‘š||(š‘”)2+||Ī”š‘£šœ€š‘”š‘š||(š‘”)2ī‚„ī‚€||š‘¢+2šœ€š‘”š‘š||(š‘ )2+||š‘£šœ€š‘”š‘š||(š‘ )2ī‚ī‚€ā€–ā€–š‘¢=āˆ’2š‘€šœ€š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘šā€–ā€–(š‘”)2ī‚ā‹…ī€ŗī€·āˆ’Ī”š‘¢šœ€š‘”š‘š(š‘”),š‘¢šœ€š‘”š‘”š‘šī€ø+ī€·(š‘”)āˆ’Ī”š‘£šœ€š‘”š‘š(š‘”),š‘£šœ€š‘”š‘”š‘šš‘¢(š‘”)ī€øī€»āˆ’4ī€ŗī€·šœ€š‘š(t),āˆ’Ī”š‘¢šœ€š‘”š‘šī€ø+ī€·š‘£(š‘”)šœ€š‘š(š‘”),āˆ’Ī”š‘£šœ€š‘”š‘š(š‘”)ī€øī€»ā‹…š‘€ī…žī‚€ā€–ā€–š‘¢šœ€š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘šā€–ā€–(š‘”)2ī‚ā‹…ī€ŗī€·āˆ’Ī”š‘¢šœ€š‘”š‘š(š‘”),š‘¢šœ€š‘”š‘”š‘š(ī€ø+ī€·š‘”)āˆ’Ī”š‘£šœ€š‘”š‘š(š‘”),š‘£šœ€š‘”š‘”š‘š(+ī‚µš‘”)ī€øī€»šœ•š¾1,ī€·š‘¢šœ•š‘”šœ€š‘”š‘”š‘šī€ø2(ī‚¶+ī‚µš‘”)šœ•š¾2,ī€·š‘£šœ•š‘”šœ€š‘”š‘”š‘šī€ø2(ī‚¶.š‘”)(3.19) Integrating (3.19) over (0,š‘”), and using (2.5), (3.8), (3.9), and the norms |š‘¢šœ€š‘”š‘”š‘š(0)|2ā‰¤š¶3 and |š‘£šœ€š‘”š‘”š‘š(0)|2ā‰¤š¶4 after employing Gronwallā€™s lemma, we obtain the third estimate ī‚€š¾1,ī€·š‘¢šœ€š‘”š‘”š‘šī€ø2ī‚+ī‚€š¾(š‘”)2,ī€·š‘£šœ€š‘”š‘”š‘šī€ø2ī‚ī‚€||š‘¢(š‘”)+šœ€šœ€š‘”š‘”š‘š||(š‘”)2+||š‘£šœ€š‘”š‘”š‘š||(š‘”)2ī‚+||Ī”š‘¢šœ€š‘”š‘š||(š‘”)2+||Ī”š‘£šœ€š‘”š‘š||(š‘”)2ī‚„+ī€œ(2āˆ’2š›æ)š‘”0ī‚€||š‘¢šœ€š‘”š‘”š‘š||(š‘”)2+||š‘£šœ€š‘”š‘”š‘š||(š‘”)2ī‚š‘‘š‘ ā‰¤š¶5,(3.20) where š¶5 is a positive constant independent of šœ€, š‘š, and š‘”. From the above estimate we conclude that ī€·š¾11/2š‘¢šœ€š‘”š‘”š‘šī€ø,ī€·š¾21/2š‘£šœ€š‘”š‘”š‘šī€øareboundedinšæāˆžī€·0,š‘‡;šæ2ī€ø,ī‚€āˆš(Ī©)šœ€š‘¢šœ€š‘”š‘”š‘šī‚,ī‚€āˆššœ€š‘£šœ€š‘”š‘”š‘šī‚areboundedinšæāˆžī€·0,š‘‡;šæ2ī€ø,ī€·š‘¢(Ī©)šœ€š‘”š‘šī€ø,ī€·š‘£šœ€š‘”š‘šī€øareboundedinšæāˆžī€·0,š‘‡;š»20ī€ø,ī€·š‘¢(Ī©)šœ€š‘”š‘”š‘šī€ø,ī€·š‘£šœ€š‘”š‘”š‘šī€øareboundedinšæ2ī€·0,š‘‡;šæ2ī€ø.(Ī©)(3.21)(2) Limits of Approximated Solutions
From the Aubin-Lions theorem (see [6]) we deduce that there exist subsequences of (š‘¢šœ€š‘š)š‘šāˆˆā„• and (š‘£šœ€š‘š)š‘šāˆˆā„• such that š‘¢šœ€š‘šāŸ¶š‘¢šœ€stronglyinšæ2ī€·0,š‘‡;š»10ī€ø,š‘£(Ī©)šœ€š‘šāŸ¶š‘£šœ€stronglyinšæ2ī€·0,š‘‡;š»10ī€ø,(Ī©)(3.22) and since š‘€ is continuous, it follows that š‘€ī‚€ā€–ā€–š‘¢šœ€š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘šā€–ā€–(š‘”)2ī‚ī€·ā€–š‘¢āŸ¶š‘€šœ€ā€–(š‘”)2+ā€–š‘£šœ€ā€–(š‘”)2ī€ø.(3.23) From the above estimate we can conclude that there exist subsequences of (š‘¢šœ€š‘š)š‘šāˆˆā„• and (š‘£šœ€š‘š)š‘šāˆˆā„•, that we denote also by (š‘¢šœ€š‘š)š‘šāˆˆā„• and (š‘£šœ€š‘š)š‘šāˆˆā„• such that as š‘šā†’āˆž and šœ€ā†’0 we have š‘¢šœ€š‘šāŸ¶š‘¢,š‘£šœ€š‘šāŸ¶š‘£weakstarinšæāˆžī€·0,š‘‡;š»20(Ī©)āˆ©š»4ī€ø,š‘¢(Ī©)šœ€š‘”š‘šāŸ¶š‘¢š‘”,š‘£šœ€š‘”š‘šāŸ¶š‘£š‘”weakstaršæāˆžī€·0,š‘‡;š»20ī€ø,š‘¢(Ī©)šœ€š‘”š‘”š‘šāŸ¶š‘¢š‘”š‘”,š‘£šœ€š‘”š‘”š‘šāŸ¶š‘£š‘”š‘”weakstaršæ2ī€·0,š‘‡;šæ2ī€ø,(Ī©)Ī”š‘¢šœ€š‘šāŸ¶Ī”š‘¢,Ī”š‘£šœ€š‘šāŸ¶Ī”š‘£weakstarinšæāˆžī€·0,š‘‡;šæ2(ī€ø,š¾Ī©)1š‘¢šœ€š‘”š‘”š‘šāŸ¶š¾1š‘¢š‘”š‘”,š¾2š‘£šœ€š‘”š‘”š‘šāŸ¶š¾2š‘£š‘”š‘”weakstarinšæāˆžī€·0,š‘‡;šæ2ī€ø,Ī”(Ī©)2š‘¢šœ€š‘šāŸ¶Ī”2š‘¢,Ī”2š‘£šœ€š‘šāŸ¶Ī”2š‘£weakstarinšæ2ī€·0,š‘‡;šæ2(ī€ø,āˆšĪ©)šœ€š‘¢šœ€š‘”š‘”š‘šāŸ¶āˆššœ€š‘¢š‘”š‘”,āˆššœ€š‘£šœ€š‘”š‘”š‘šāŸ¶āˆššœ€š‘£š‘”š‘”weakstarinšæāˆžī€·0,š‘‡;šæ2(ī€ø,š‘€ī‚€ā€–ā€–š‘¢Ī©)šœ€š‘š(ā€–ā€–š‘”)2+ā€–ā€–š‘£šœ€š‘š(ā€–ā€–š‘”)2ī‚ī€·āˆ’Ī”š‘¢šœ€š‘šāˆ’Ī”š‘£šœ€š‘šī€øī€·(āŸ¶š‘€ā€–š‘¢š‘”)ā€–2+ā€–š‘£(š‘”)ā€–2ī€ø(āˆ’Ī”š‘¢āˆ’Ī”š‘£)weakstarinšæāˆžī€·0,š‘‡;šæ2(ī€ø.Ī©)(3.24) Now, multiplying (3.6), (3.7) by šœƒāˆˆš’Ÿ(0,š‘‡) and integrating over (0,š‘‡), we arrive at ī€œš‘‡0š¾ī€·ī€·1ī€øš‘¢+šœ€šœ€š‘”š‘”š‘šī€øī€œ(š‘”),š‘¤šœƒ(š‘”)š‘‘š‘”+š‘‡0ī€·āˆ’Ī”š‘¢šœ€š‘šī€ø+ī€œ(š‘”),āˆ’Ī”š‘¤šœƒ(š‘”)š‘‘š‘”š‘‡0š‘€ī‚€ā€–ā€–š‘¢šœ€š‘šā€–ā€–(š‘”)2+ā€–ā€–š‘£šœ€š‘šā€–ā€–(š‘”)2ī‚ī€·āˆ’Ī”š‘¢šœ€š‘šī€ø+ī€œ(š‘”),š‘¤šœƒš‘‘š‘”š‘‡0ī€·š‘¢šœ€š‘”š‘šī€ø,š‘¤šœƒš‘‘š‘”=0,āˆ€š‘¤āˆˆš‘‰š‘šī€œ,āˆ€šœƒāˆˆš’Ÿ(0,š‘‡),š‘‡0š¾ī€·ī€·2ī€øš‘£+šœ€šœ€š‘”š‘”š‘š(ī€øī€œš‘”),š‘§šœƒ(š‘”)š‘‘š‘”+š‘‡0ī€·āˆ’Ī”š‘£šœ€š‘š(ī€ø+ī€œš‘”),āˆ’Ī”š‘§šœƒ(š‘”)š‘‘š‘”š‘‡0š‘€ī‚€ā€–ā€–š‘¢šœ€š‘š(ā€–ā€–š‘”)2+ā€–ā€–š‘£šœ€š‘š(ā€–ā€–š‘”)2ī‚ī€·āˆ’Ī”š‘£šœ€š‘š(ī€ø+ī€œš‘”),š‘§šœƒš‘‘š‘”š‘‡0ī€·š‘£šœ€š‘”š‘šī€ø,š‘§šœƒš‘‘š‘”=0,āˆ€š‘§āˆˆš‘‰š‘š,āˆ€šœƒāˆˆš’Ÿ(0,š‘‡).(3.25) The convergences (3.24) are sufficient to pass to the limit in (3.25) in order to obtain š¾1š‘¢š‘”š‘”+Ī”2ī€·ā€–š‘¢+š‘€ā€–š‘¢(š‘”)2ā€–+ā€–š‘£(š‘”)2ī€ø(āˆ’Ī”š‘¢)+š‘¢š‘”=0inšæāˆžlocī€·0,āˆž;šæ2ī€ø,š¾(Ī©)2š‘£š‘”š‘”+Ī”2ī€·(š‘£+š‘€ā€–š‘¢š‘”)ā€–2+ā€–š‘£(š‘”)ā€–2ī€ø(āˆ’Ī”š‘£)+š‘£š‘”=0inšæāˆžlocī€·0,āˆž;šæ2(ī€ø,Ī©)(3.26) and (š‘¢,š‘£) 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)ā€“(1.5). We show using the Nakao method that the system (1.1)ā€“(1.5) is exponentially stable. The main result of this paper is given by the following theorem.

Theorem 4.1. Let one take (š‘¢0,š‘£0)āˆˆ(š»10(Ī©)āˆ©š»4(Ī©))2, and ā€‰ā€‰(š‘¢1,š‘£1)āˆˆ(š»20(Ī©))2 and let one suppose that assumptions (2.5), (2.6), and (2.7) hold. Then, the solution (š‘¢,š‘£) of system (1.1)ā€“(1.5) satisfies ||š¾11/2š‘¢š‘”||(š‘”)2+||š¾21/2š‘£š‘”||(š‘”)2+||||Ī”š‘¢(š‘”)2+||||Ī”š‘£(š‘”)2+ī€œš‘”š‘”+1ī‚€||š‘¢š‘”||(š‘ )2+||š‘£š‘”||(š‘ )2ī‚š‘‘š‘ ā‰¤š›¼1š‘’āˆ’š›¼2š‘”,(4.1) for all š‘”ā‰„1, where š›¼1 and š›¼2 are positive constants.

Proof. Multiplying (3.2) by š‘¢š‘”(š‘”) and š‘£š‘”(š‘”), respectively, and integrating over Ī©, we obtain 12š‘‘ī‚ƒ||š¾š‘‘š‘”11/2š‘¢š‘”||(š‘”)2+||š¾21/2š‘£š‘”||(š‘”)2+||||Ī”š‘¢(š‘”)2+||||Ī”š‘£(š‘”)2+ī‚Šš‘€ī€·ā€–š‘¢(š‘”)ā€–2+ā€–š‘£(š‘”)ā€–2ī€øī‚„+||š‘¢š‘”||(š‘”)2+||š‘£š‘”||(š‘”)2=ī‚µšœ•š¾1šœ•š‘”,š‘¢2š‘”ī‚¶+ī‚µ(š‘”)šœ•š¾2šœ•š‘”,š‘£2š‘”ī‚¶,(š‘”)(4.2) where ī‚Šī€œš‘€(š‘ )=š‘ 0š‘€(šœ)š‘‘šœ.(4.3) Using (2.6) and considering š›æ>0 sufficiently small, we get 12š‘‘ī‚ƒ||š¾š‘‘š‘”11/2š‘¢š‘”||(š‘”)2+||š¾21/2š‘£š‘”||(š‘”)2+||||Ī”š‘¢(š‘”)2+||||Ī”š‘£(š‘”)2+ī‚Šš‘€ī€·ā€–š‘¢(š‘”)ā€–2+ā€–š‘£(š‘”)ā€–2ī€øī‚„+ī€·1āˆ’š›æāˆ’š¾0ī€øī‚€||š‘¢š¶(š›æ)š‘”||(š‘”)2+||š‘£š‘”||(š‘”)2ī‚ā‰¤0,(4.4) where š¾0=max{š¾1,š¾2} with š¾š‘–=max[]š‘ āˆˆš‘”,š‘”+1ī€½esssupš¾š‘–(ī€¾š‘„,š‘ ),š‘–=1,2.(4.5) Integrating (4.4) from 0toš‘”, we have ī€·šø(š‘”)+1āˆ’š›æāˆ’š¾0ī€øī€œš¶(š›æ)š‘”0ī‚€||š‘¢š‘”(||š‘ )2+||š‘£š‘”(||š‘ )2ī‚š‘‘š‘ ā‰¤šø(0),(4.6) where 1šø(š‘”)=2ī‚ƒ||š¾11/2š‘¢š‘”||(š‘”)2+||š¾21/2š‘£š‘”||(š‘”)2+||||Ī”š‘£(š‘”)2+||||Ī”š‘£(š‘”)2+ī‚Šš‘€ī€·ā€–š‘¢(š‘”)ā€–2+ā€–š‘£(š‘”)ā€–2ī€øī‚„(4.7) is the energy associated with the system (1.1)ā€“(1.5). From (4.4) we conclude that š‘‘š‘‘š‘”šø(š‘”)ā‰¤0āˆ€š‘”āˆˆ(0,āˆž),(4.8) that is, šø(š‘”) is bounded and increasing in (0,āˆž).
Integrating (4.4) from šœ1tošœ2, 0<šœ1<šœ2<āˆž, we arrive at šøī€·šœ2ī€ø+ī€·1āˆ’š›æāˆ’š¾0ī€øī€œš¶(š›æ)šœ2šœ1ī‚€||š‘¢š‘”||(š‘ )2+||š‘£š‘”||(š‘ )2ī‚ī€·šœš‘‘š‘ ā‰¤šø1ī€ø.(4.9) Taking šœ1=š‘” and šœ2=š‘”+1 in (4.9), we get ī€œš‘”š‘”+1ī‚€||š‘¢š‘”||(š‘ )2+||š‘£š‘”||(š‘ )2ī‚1š‘‘š‘ ā‰¤1āˆ’š›æāˆ’š¾0[]š¶(š›æ)šø(š‘”)āˆ’šø(š‘”+1)=š¹2(š‘”).(4.10) Therefore, there exist two points š‘”1āˆˆ[š‘”,š‘”+1/4] and š‘”2āˆˆ[š‘”+3/4,š‘”+1], such that ||š‘¢š‘”ī€·š‘”š‘–ī€ø||+||š‘£š‘”ī€·š‘”š‘–ī€ø||ā‰¤4š¹(š‘”),š‘–=1,2.(4.11) Making the inner product in šæ2(Ī©) of (1.1) and (1.2) by š‘¢(š‘”) and š‘£(š‘”), respectively, and summing up the result we obtain š‘‘ī€·š¾š‘‘š‘”1š‘¢š‘”ī€ø+š‘‘(š‘”),š‘¢(š‘”)ī€·š¾š‘‘š‘”2š‘£š‘”ī€øāˆ’|||āˆš(š‘”),š‘£(š‘”)š¾1š‘¢š‘”|||(š‘”)2āˆ’|||āˆšš¾2š‘£š‘”|||(š‘”)2+||||Ī”š‘¢(š‘”)2+||||Ī”š‘£(š‘”)2ī€·+š‘€ā€–š‘¢(š‘”)ā€–2+ā€–š‘£(š‘”)ā€–2ī€øī€·ā€–š‘¢(š‘”)ā€–2+ā€–š‘£(š‘”)ā€–2ī€ø+ī€·š‘¢š‘”ī€ø+ī€·š‘£(š‘”),š‘¢(š‘”)š‘”ī€ø=ī‚µ(š‘”),š‘£(š‘”)šœ•š¾1š‘¢šœ•š‘”š‘”ī‚¶+ī‚µ(š‘”),š‘¢(š‘”)šœ•š¾2š‘£šœ•š‘”š‘”ī‚¶.(š‘”),š‘£(š‘”)(4.12) Integrating from š‘”1 to š‘”2 and using (2.6), and (2.7) we have ī‚µš›½1āˆ’šœ†1ī‚¶ī€œš‘”2š‘”1ī‚€||||Ī”š‘¢(š‘ )2+||||Ī”š‘£(š‘ )2ī‚ā‰¤ī€·š¾š‘‘š‘ 1š‘¢š‘”ī€·š‘”1ī€øī€·š‘”,š‘¢1āˆ’ī€·š¾ī€øī€ø1š‘¢š‘”ī€·š‘”2ī€øī€·š‘”,š‘¢2+ī€·š¾ī€øī€ø2š‘£š‘”ī€·š‘”1ī€øī€·š‘”,š‘£1āˆ’ī€·š¾ī€øī€ø2š‘£š‘”ī€·š‘”2ī€øī€·š‘”,š‘£2+ī€·ī€øī€ø1+š›æ+š¾0ī€øī€œš¶(š›æ)š‘”2š‘”1ī€·||š‘¢š‘”||||||+||š‘£(š‘ )š‘¢(š‘ )š‘”||||||ī€ø(š‘ )š‘£(š‘ )š‘‘š‘ +š¾0ī€œš‘”2š‘”1ī‚€||š‘¢š‘”||(š‘ )2+||š‘£š‘”||(š‘ )2ī‚š‘‘š‘ .(4.13) Let us consider š¶>0 such that ||||||||,||||||||š‘¢(š‘ )ā‰¤š¶Ī”š‘¢(š‘ )š‘£(š‘ )ā‰¤š¶Ī”š‘£(š‘ )(4.14) and we take š‘‘>0 sufficiently small Then we have. ī€·1+š›æ+š¾0š¶||š‘¢(š›æ)ī€øī€·š‘”||||š‘¢||+||š‘£(š‘ )(š‘ )š‘”||||š‘£||ī€øā‰¤ī€·(š‘ )(š‘ )1+š›æ+š¾0ī€øš¶(š›æ)2š‘‘ī‚€||š‘¢š‘”||(š‘ )2+||š‘£š‘”||(š‘ )2ī‚ī‚€||||+š‘‘Ī”š‘¢(š‘ )2+||||Ī”š‘£(š‘ )2ī‚,||ī€·š¾1š‘¢š‘”ī€·š‘”1ī€øī€·š‘”,š‘¢1+ī€·š¾ī€øī€ø2š‘£š‘”ī€·š‘”1ī€øī€·š‘”,š‘£1āˆ’ī€·š¾ī€øī€ø1š‘¢š‘”ī€·š‘”2ī€øī€·š‘”,š‘¢2āˆ’ī€·š¾ī€øī€ø2š‘£š‘”ī€·š‘”2ī€øī€·š‘”,š‘£2||ī€øī€øā‰¤š¶š¾0esssup[]š‘ āˆˆš‘”,š‘”+1||||ī€·||š‘¢Ī”š‘¢(š‘ )š‘”ī€·š‘”1ī€ø||+||š‘¢š‘”ī€·š‘”2ī€ø||ī€ø+š¶š¾0esssup[]š‘ āˆˆš‘”,š‘”+1||||ī€·||š‘£Ī”š‘£(š‘ )š‘”ī€·š‘”1ī€ø||+||š‘£š‘”ī€·š‘”2ī€ø||ī€ø.(4.15) Thus, substituting (4.15) into (4.13), we arrive at ī‚µš›½1āˆ’šœ†1ī‚¶ī€œš‘”2š‘”1ī‚€||||Ī”š‘¢(š‘ )2+||||Ī”š‘£(š‘ )2ī‚š‘‘š‘ ā‰¤š¾0ī€œš‘”2š‘”1ī‚€||š‘¢š‘”||(š‘ )2+||š‘£š‘”||(š‘ )2ī‚ī€œš‘‘š‘ +š‘‘š‘”2š‘”1ī‚€||||Ī”š‘¢(š‘ )2+||||Ī”š‘£(š‘ )2ī‚š‘‘š‘ +š¶š¾0esssup[]š‘ āˆˆš‘”,š‘”+1||||ī€·||š‘¢Ī”š‘¢(š‘ )š‘”ī€·š‘”1ī€ø||+||š‘¢š‘”ī€·š‘”2ī€ø||ī€ø+š¶š¾0esssup[]š‘ āˆˆš‘”,š‘”+1||||ī€·||š‘£Ī”š‘£(š‘ )š‘”ī€·š‘”1ī€ø||+||š‘£š‘”ī€·š‘”2ī€ø||ī€ø.(4.16) Applying (4.10) and (4.11) in (4.16), we have ī€œš‘”2š‘”1ī‚€||||Ī”š‘¢(š‘ )2+||||Ī”š‘£(š‘ )2ī‚š‘‘š‘ ā‰¤š¶1īƒ¬š¹2(š‘”)+esssup[]š‘ āˆˆš‘”,š‘”+1ī€·||||+||||ī€øīƒ­Ī”š‘¢(š‘ )Ī”š‘£(š‘ )š¹(š‘”)=šŗ2(š‘”),(4.17) where š¶1 is a positive constant independent of š‘”. Therefore, from (4.10) and (4.17) we obtain ī€œš‘”2š‘”1ī‚€||š‘¢š‘”(||š‘ )2+||š‘£š‘”(||š‘ )2+||||Ī”š‘¢(š‘ )2+||||Ī”š‘£(š‘ )2ī‚š‘‘š‘ ā‰¤š¹2(š‘”)+šŗ2(š‘”).(4.18) Hence, there exists š‘”āˆ—āˆˆ[š‘”1,š‘”2] such that ||š‘¢š‘”ī€·š‘”āˆ—ī€ø||2+||š‘£š‘”ī€·š‘”āˆ—ī€ø||2+||ī€·š‘”Ī”š‘¢āˆ—ī€ø||2+||ī€·š‘”Ī”š‘£āˆ—ī€ø||2ī€ŗš¹ā‰¤22(š‘”)+šŗ2ī€».(š‘”)(4.19) Consequently, ī‚Šš‘€ī€·ā€–š‘¢(š‘”)ā€–2+ā€–š‘£(š‘”)ā€–2ī€øā‰¤š¶2ī€ŗš¹2(š‘”)+šŗ2ī€»,(š‘”)(4.20) where š¶2=2š‘š0ī‚š¶,š‘š0=maxī€·0ā‰¤š‘ ā‰¤ā€–š‘¢(š‘”āˆ—)ā€–2+ā€–š‘£(š‘”āˆ—)ā€–2ī€ø<āˆžš‘€(š‘ )(4.21) and ī‚š¶ is a positive constant such that ā€–š‘¢(š‘”āˆ—)ā€–2ā‰¤ī‚š¶|Ī”š‘¢(š‘”āˆ—)|2.
From (4.19) and (4.20), we have šøī€·š‘”āˆ—ī€øā‰¤š¶3ī€ŗš¹2(š‘”)+šŗ2ī€».(š‘”)(4.22) Since šø(š‘”) is increasing, we have esssup[]š‘ āˆˆš‘”,š‘”+1ī€·š‘”šø(š‘ )ā‰¤šøāˆ—ī€ø+ī€·1+š›æ+š¾0ī€øī€œš¶(š›æ)š‘”2š‘”1ī‚€||š‘¢š‘”(||š‘ )2+||š‘£š‘”(||š‘ )2ī‚š‘‘š‘ .(4.23) Now, by (4.10), (4.22), and (4.23) we get šø(š‘”)ā‰¤š¶4[],šø(š‘”)āˆ’šø(š‘”+1)(4.24) where š¶4 is a positive constant. Then, by the Nakao lemma (see [12]) we conclude that šø(š‘”)ā‰¤š‘1š‘’āˆ’š›¼2š‘”,āˆ€š‘”ā‰„1,(4.25) where š‘1 and š›¼2 are positive constants, that is, |||āˆšš¾1š‘¢š‘”|||(š‘”)2+|||āˆšš¾2š‘£š‘”|||(š‘”)2+||||Ī”š‘¢(š‘”)2+||||Ī”š‘£(š‘”)2ī‚Šš‘€ī€·ā€–š‘¢(š‘”)ā€–2+ā€–š‘£(š‘”)ā€–2ī€øā‰¤2š‘1š‘’āˆ’š›¼2š‘”.(4.26) Using (2.7) we obtain |||āˆšš¾1š‘¢š‘”|||(š‘”)2+|||āˆšš¾2š‘£š‘”|||(š‘”)2+||||Ī”š‘¢(š‘”)2+||||Ī”š‘£(š‘”)2ā‰¤2š‘1š‘š1š‘’āˆ’š›¼2š‘”,(4.27) where š‘š1š›½=1āˆ’šœ†1>0.(4.28) From (4.10) we have ī€œš‘”š‘”+1ī‚€||š‘¢š‘”||(š‘ )2+||š‘£š‘”||(š‘ )2ī‚1š‘‘š‘ ā‰¤1āˆ’š›æāˆ’š¾0[]š¶(š›æ)šø(š‘”)āˆ’šø(š‘”+1)ā‰¤šø(š‘”)ā‰¤š‘1š‘’āˆ’š›¼2š‘”.(4.29) Therefore, from (4.27) and (4.29) we conclude that |||āˆšš¾1š‘¢š‘”|||(š‘”)2+|||āˆšš¾2š‘£š‘”|||(š‘”)2+||||Ī”š‘¢(š‘”)2+||||Ī”š‘£(š‘”)2+ī€œš‘”š‘”+1ī‚€||š‘¢š‘”||(š‘ )2+||š‘£š‘”||(š‘ )2ī‚š‘‘š‘ ā‰¤š›¼1š‘’āˆ’š›¼2š‘”,āˆ€š‘”ā‰„1,(4.30) where š›¼1 and š›¼2 are positive constants. Now, the proof is complete.