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

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


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 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 .
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 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 ∈ C 1 0, ∞ and M λ ≥ m 0 > 0, for all λ ≥ 0. The existence, uniqueness, and boundary regularity of weak solutions were considered by Ramos 8 with K x ≥ k 0 > 0, x ∈ Ω. See also Pereira et al. 9 .The abstract model 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 ∈ C 1 0, ∞ with M λ ≥ m 0 m 1 λ, for all λ ≥ 0, where m 0 and m 1 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: where λ 1 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.

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 L 2 Ω endowed with the inner product We also consider the Sobolev space H 1 Ω endowed with the scalar product We define the subspace of H 1 Ω , denoted by H 1 0 Ω .This space endowed with the norm induced by the scalar product is a Hilbert space.

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: and there exists γ > 0 such that K i x, 0 ≥ γ > 0, 2.5

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 .
and let one suppose that assumptions 2.5 , 2.6 and 2.7 hold.Then, there exist unique functions u, v :

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 .

Perturbed System
Consider the perturbed system

3.4
Let w ν ν∈N be a basis of H 2 0 Ω formed by the eigenvectors of the operator −Δ, that is, . ., w m be the subspace generated by the first m vectors of w ν ν∈N .
For each fixed ε, we consider as solutions of the approximated perturbed system

3.9
The local existence of the approximated solutions u ε m , v ε m is guaranteed by the standard results of ordinary differential equations.The extension of the solutions u ε m , v ε m to the whole interval 0, T is a consequence of the first estimate below.

The First Estimate
Setting w u ε tm and z v ε tm 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

3.10
where From 2.7 and 2.8 , we have

3.12
Since β < λ 1 and so by 2.5 -2.7 and convergences 3.8 , 3.9 , and 3.12 , we obtain with 0 < δ < 1 and C 0 being a positive constant independent of ε, m, and t.Employing Gronwall's lemma in 3.13 , we obtain the first estimate

3.14
where C 1 is a positive constant independent of ε, m, and t.Then, we can conclude that

3.15
The Second Estimate Substituting w −Δu ε tm t and z −Δv ε tm t in 3.6 and 3.7 , respectively, it holds that

3.17
where C 2 is a positive constant independent of ε, m, and t.From the above estimate we conclude that

3.18
The Third Estimate Differentiating 3.6 and 3.7 with respect to t and setting w u ε ttm and v ε ttm , respectively, we arrive at

3.20
where C 5 is a positive constant independent of ε, m, and t.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 u ε m m∈N and v ε m m∈N such that and since M is continuous, it follows that

3.23
From the above estimate we can conclude that there exist subsequences of u ε m m∈N and v ε m m∈N , that we denote also by u ε m m∈N and v ε m m∈N such that as m → ∞ and ε → 0 we have

3.25
The convergences 3.24 are sufficient to pass to the limit in 3.25 in order to obtain and u, v satisfies 3.1 .

ISRN Mathematical Physics
The uniqueness and initial conditions follow by using the standard arguments as in Lions 6 .The proof is now complete.

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.
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 where K 0 max{K 4.9 Taking τ 1 t and τ 2 t 1 in 4.9 , we get