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 where is a bounded domain of , , with smooth boundary ,    is a real arbitrary number, and is the unit normal at direct towards the exterior of . Here ,    and , 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, and with , it is a generalization of one-dimensional model proposed by Woinowsky-Krieger [1] as a model for the transverse deflection 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 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 , , and   and , for all . The existence, uniqueness, and boundary regularity of weak solutions were considered by Ramos [8] with ,  . See also Pereira et al. [9]. The abstract model of (1.6), where 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   with , for all ,  where and are positive constants. Pereira [11] considered the abstract model (1.7) with dissipative term . He proved the existence, uniqueness, and exponential decay of the solutions with the following assumptions about : where is the first eigenvalue of 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 endowed with the inner product and norm We also consider the Sobolev space endowed with the scalar product We define the subspace of , denoted by . This space endowed with the norm induced by the scalar product is a Hilbert space.

2.1. Assumptions on the Functions , , and

To obtain the weak solution of the system (1.1)–(1.5) we consider the following hypothesis:

Remark 2.1. Let be the first eingevalue of ; then (see Miklin [13])

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   and   , and let one suppose that assumptions (2.5), (2.6) and (2.7) hold. Then, there exist unique functions in the class satisfying

Proof. Since , , we first perturb the system (1.1)–(1.5) with the terms , with , and we apply the Faedo-Galerkin method to the perturbed system. After we pass to the limit with in the perturbed system and we obtain the solution for the system (1.1)–(1.5).
(1) Perturbed System
Consider the perturbed system Let be a basis of formed by the eigenvectors of the operator , that is, , with when . Let be the subspace generated by the first vectors of .
For each fixed , we consider as solutions of the approximated perturbed system The local existence of the approximated solutions is guaranteed by the standard results of ordinary differential equations. The extension of the solutions to the whole interval is a consequence of the first estimate below.
The First Estimate
Setting and in (3.6) and (3.7), respectively, integrating over , and taking the convergences (3.8) and (3.9) in consideration, we arrive at where From (2.7) and (2.8), we have Since and so by (2.5)–(2.7) and convergences (3.8), (3.9), and (3.12), we obtain with and being a positive constant independent of , , and .
Employing Gronwall’s lemma in (3.13), we obtain the first estimate where is a positive constant independent of , , and . Then, we can conclude that
The Second Estimate
Substituting and in (3.6) and (3.7), respectively, it holds that
Integrating (3.16) over , , and taking (2.5)–(2.7) and (3.8), (3.9), and first estimate into account, we infer where is a positive constant independent of , , and . From the above estimate we conclude that The Third Estimate
Differentiating (3.6) and (3.7) with respect to and setting and , respectively, we arrive at Integrating (3.19) over , and using (2.5), (3.8), (3.9), and the norms and after employing Gronwall’s lemma, we obtain the third estimate where is a positive constant independent of , , and . From the above estimate we conclude that (2) Limits of Approximated Solutions
From the Aubin-Lions theorem (see [6]) we deduce that there exist subsequences of and such that and since is continuous, it follows that From the above estimate we can conclude that there exist subsequences of and , that we denote also by and such that as and we have Now, multiplying (3.6), (3.7) by and integrating over , we arrive at The convergences (3.24) are sufficient to pass to the limit in (3.25) in order to obtain and 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 , and    and let one suppose that assumptions (2.5), (2.6), and (2.7) hold. Then, the solution of system (1.1)–(1.5) satisfies for all , where and are positive constants.

Proof. Multiplying (3.2) by and , respectively, and integrating over , we obtain where Using (2.6) and considering sufficiently small, we get where with Integrating (4.4) from , we have where is the energy associated with the system (1.1)–(1.5). From (4.4) we conclude that that is, is bounded and increasing in .
Integrating (4.4) from , , we arrive at Taking and in (4.9), we get Therefore, there exist two points and , such that Making the inner product in of (1.1) and (1.2) by and , respectively, and summing up the result we obtain Integrating from to and using (2.6), and (2.7) we have Let us consider such that and we take sufficiently small Then we have. Thus, substituting (4.15) into (4.13), we arrive at Applying (4.10) and (4.11) in (4.16), we have where is a positive constant independent of . Therefore, from (4.10) and (4.17) we obtain Hence, there exists such that Consequently, where and is a positive constant such that .
From (4.19) and (4.20), we have Since is increasing, we have Now, by (4.10), (4.22), and (4.23) we get where is a positive constant. Then, by the Nakao lemma (see [12]) we conclude that where and are positive constants, that is, Using (2.7) we obtain where From (4.10) we have Therefore, from (4.27) and (4.29) we conclude that where and are positive constants. Now, the proof is complete.