Global Periodic Attractors for a Class of Infinite Dimensional Dissipative Dynamical Systems

In this paper we consider the existence of a global periodic attractor for a class of infinite dimensional dissipative equations under homogeneous Dirichlet boundary conditions. It is proved that in a certain parameter, for an arbitrary timeperiodic driving force, the system has a unique periodic solution attracting any bounded set exponentially in the phase space, which implies that the system behaves exactly as a one-dimensional system. We mention, in particular, that the obtained result can be used to prove the existence of the global periodic attractor for abstract parabolic problems.


Introduction
In this paper, we consider a class of infinite dimensional dissipative equations with the Dirichlet boundary condition where .denotes the absolute value of the number in R.
There has been an increasing interest in the study of the evolution equations of form (1.1), such as existence and asymptotic behavior of solutions (mild solutions, strong solutions and classical solutions), and existence of global attractors, etc. Especially in physics and mechanics, many important results associated with this problem have been obtained in [1][2][3][4][5][6][7].In [9] and [10], Hernandez and Henriquez have extended the problem studied in [8] to neutral equations and established the corresponding existence results of solutions and periodic solutions.In their work,   is a negative Laplacian operator, and A generates an analytic semigroup so that the theory of the fractional power has been used effectively there.However, their results clearly cannot apply to Equation (1.1) with A t is non-autonomous which is a more general and maybe more important case [11].So we will use the appropriate assumptions to overcome the difficulty for the non-autonomous operator   A t .We arrange this paper as follows.Firstly we present the existence and uniqueness of solutions.Then we obtain a nonstandard estimation under which system (1.1) possesses a global periodic attractor.Finally, for the special case

 
A t   , we discuss the existence of a global periodic attractor for abstract parabolic problems.

Preliminaries
For the family       : 0, A t t   of linear operators, we impose on the following restrictions: 1) The domain exists for all  , with Re 0   and there exists 0 3) There exists 0 1 for all ;   , , 0, Then the family      : 0, A t t    generates a unique linear evolution operator   , ,0 U t s s t     , satisfying the following properties: 1) , the space of bounded linear transformations on H, whenever and for each , the mapping is continuous; for some ; Condition 4) ensures the generated evolution operator satisfies 4) (see [6], Proposition 2.1).
Proposition 1 (see [11]) The family of operators is continuous in t in the uniform operator topology uniformly for s.

   , :
U t s t s    0 Lemma 1 (see [11]) Consider the initial value problem (1.1) in E. If 1)-4) hold, then, for any , there exists a unique continuous function such that   u t is called a mild solution of (1.1).By Lemma 1, the (mild) solution of (1.1) determines a map from H into itself: , is a discrete semidynamic system in H, since f x t t R  is a ω-periodic function with respect to .

Main Result
Taking the inner scalar product of each side of (3.1) with   u t in H, and we see that For the third term on the left of (3.2), by (1.2), we have , 2


where   0 , u t u is the solution of (1.1),   Thus   S  is a contraction mapping.By Banach's fixed point theorem, there exists a unique fixed point u  is a ω-periodic solution of system (1.1).By (3.4) , , Sun and M. Niu, "On the Existence of Global Attractor for a Class of Infinite Dimensional Dissipative Nonlinear Dynamical Systems," Chinese Annals of Mathematics, Vol. 26, No. 3, 2005, pp.393-400.doi:10.1142/S0252959905000312and if the function f is continuous ω-periodic in t, we have the following theorem.Theorem 2 System (4.1) possesses a global ω-periodic attractor which attracts any bounded set exponentially, if [8] H. R. Henriquez, "Periodic Solutions of Quasi-Linear Partial Functional Differential Equations with Unbounded Delay," Funkcialaj Ekvacioj, Vol.37, No. 2, 1994, pp.329-343. 