Global Periodic Attractors for a Class of Infinite Dimensional Dissipative Dynamical Systems ()
1. Introduction
In this paper, we consider a class of infinite dimensional dissipative equations with the Dirichlet boundary condition
(1.1)
where
is a real-valued function on
,
is an open bounded set of
with a smooth boundary
,
takes values in a Hilbert space H, the family
of unbounded linear operators generates a linear evolution operator. The external force term
is continuous and ω-periodic function in t, where ω is a positive constant. Let
. There exists a nonnegative constant
, such that
(1.2)
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 [1-7]. 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,
, especially,
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
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
.
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
, we discuss the existence of a global periodic attractor for abstract parabolic problems.
2. Preliminaries
For the family
of linear operators, we impose on the following restrictions:
1) The domain
of
is dense in Hilbert space H and independent of t,
is a closed linear operator;
2) For each
, the resolvent
exists for all
, with
and there exists
so that
;
3) There exists
and
such that
![](https://www.scirp.org/html/5-5300478\2321bbbf-d1eb-4145-b005-0b01190bc1ed.jpg)
for all
;
4) For each
and some
, the resolvent set of
, the resolvent
, is a compact operator.
Then the family
generates a unique linear evolution operator
, satisfying the following properties:
1)
, the space of bounded linear transformations on H, whenever
and for each
, the mapping
is continuous;
2)
for
;
3) ![](https://www.scirp.org/html/5-5300478\8ebe89f3-522a-4fbc-8888-2bd496cd8523.jpg)
4)
is a compact operator whenever
;
5)
, for
;
6) There is a constant
such that
,
;
7) If
and
then
![](https://www.scirp.org/html/5-5300478\963d7cb6-ca0c-4878-a920-db7e185a0b05.jpg)
for some
;
8) If
is continuous on
, then the function
is Holder continuous with any exponent
.![](https://www.scirp.org/html/5-5300478\1c355253-89be-4fe2-b9f1-7e2c7184f503.jpg)
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.
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
and
satisfies the integral equation
.
is called a mild solution of (1.1).
By Lemma 1, the (mild) solution
of (1.1) determines a map
from H into itself:
. Obviously,
is a discrete semidynamic system in H, since
is a ω-periodic function with respect to
.
3. Main Result
Theorem 1 Assume that (1.2), 1)-4) and
hold, then system (1.1) has a unique continuous ω-periodic solution which attracts any bounded set exponentially. The process
associated with (1.1) possesses a global periodic attractor.
Proof. Let
be two solutions of problem (1.1) with initial values
, and
. Then by (1.1), we find
(3.1)
Taking the inner scalar product of each side of (3.1) with
in H, and we see that
(3.2)
For the third term on the left of (3.2), by (1.2), we have
(3.3)
From (3.1)-(3.3), we find
![](https://www.scirp.org/html/5-5300478\b2f39670-ecca-4b1a-85af-fb8a6368c8d7.jpg)
and if
![](https://www.scirp.org/html/5-5300478\05c303a2-30c3-4dae-870c-af0c8fd2c558.jpg)
we might as well assume
from the Gronwall’s inequality, we have that
(3.4)
Now considering ω-mapping
![](https://www.scirp.org/html/5-5300478\c6e96abc-424e-46e9-b9d0-5253abe237db.jpg)
where
is the solution of (1.1),
. From (3.4),
, we obtain
.
Thus
is a contraction mapping. By Banach’s fixed point theorem, there exists a unique fixed point
for
in H such that
. At the same time, since
is a discrete semidynamic system in H, we can deduce
![](https://www.scirp.org/html/5-5300478\e629a18f-7224-464c-8de4-127d9a794b9f.jpg)
and
![](https://www.scirp.org/html/5-5300478\398aedee-9aa0-4fe4-bf01-bce0abc35f38.jpg)
where
is the solution passing
. Thus
is a ω-periodic solution of system (1.1). By (3.4),
attracts any bounded set exponentially, which is a global periodic attractor of System (1.1). The proof is completed.
4. Examples
In this section, as an illustration of the main result in Section 3, we consider one example of System (1.1) and get the corresponding results. We consider an evolution equation (i.e.,
in (1.1)) studied in [2,3]:
(4.1)
and 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
(where
is the first eigenvalue of operator
that subjects to the homogeneous Dirichlet boundary condition).
5. Acknowledgements
This work is supported by the National Natural Science Foundation of China under Grant 11101265 and 61075115.