Strong Attractor of Beam Equation with Structural Damping and Nonlinear Damping

This paper is mainly concerned with the existence of a global strong attractor for the nonlinear extensible beam equation with structural damping and nonlinear external damping. This kind of problem arises from the model of an extensible vibration beam. By the asymptotic compactness of the related continuous semigroup, we prove the existence of a strong global attractor which is connected with phase spaceD(Δ) × H1 0 (Ω) ∩ H 2 (Ω).


Introduction
Global attractor is a basic concept in the study of long-time behavior of nonlinear dissipative evolution equations with various dissipation.There have been many methods to prove the existence of the global attractor.It can be proved by the theory of -contractions of the solution semigroup (), such as [1][2][3] and the reference therein.It can also be proved by the decomposition of the solution semigroup () (see Hale [4], Temam [5], etc.).
In this paper, we use the method of the asymptotically compact property of the solution semigroup () which is different from the method of [1][2][3][4][5] to prove the existence of a strong global attractor for the Kirchhoff type equations with structural damping and nonlinear external damping which arises from the model of the nonlinear vibration beam (, 0) =  0 () , where , , and  are all positive constants, Ω is a bounded domain of   with smooth boundary Γ = Ω, (), (), (), and () are nonlinear functions specified later, and ℎ ∈  2 (Ω) is an external force term.() represents the vertical deflection of the beam, and  = (, ) is a real-valued function on Ω × [0, +∞).
On the other hand, the existence of the attractor for a related problem, with the boundary conditions  = Δ = 0 of (2) replaced with  = ∇ = 0, was considered by Ma and Narciso [7], Eden and Milani [8] with a linear damping   or nonlinear damping (  ) without structural damping, respectively.Chueshov and Lasiecka [10] considered a kind of boundary condition which is  = Δ = 0 but without structural damping.
By virtue of Galerkin method, we may prove Theorem 1 combined with the priori estimates of Section 3.
According by Theorem 1, for any  > 0, we may introduce the mapping It maps  into itself, and it enjoys the usual semigroup properties as follows: And it is obvious that the map {(),  > 0}, for all  ∈ , is continuous in space .In the following, we will introduce the existence of bounded absorbing set and global attractor in space  for map {(),  ≥ 0}.

The Existence of Bounded Absorbing Set in Space 𝐸
In this section, we will show boundedness of the solutions for systems (1)-( 3).
Theorem 2. Assume that these assumptions of Theorem 1 hold then for the dynamic system determined by problems (1)-( 3), there exists the boundary absorbing set in space .

The Existence of Global Attractor in Space 𝐸
The general theory [11] indicates that the continuous semigroup () defined on a Banach space  has a global attractor which is connected when the following conditions are satisfied.
Theorem 3.Under the assumptions of Theorem 1, the continuous semigroup () has a global attractor which is connected to .