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.
Dissipative Equation; Global Periodic Attractor; One-Dimensional System1. Introduction
In this paper, we consider a class of infinite dimensional dissipative equations with the Dirichlet boundary condition
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
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
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)
4) is a compact operator whenever;
5), for;
6) There is a constant such that,;
7) If and then
for some;
8) If is continuous on, then the function is Holder continuous with any exponent.
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
Taking the inner scalar product of each side of (3.1) with in H, and we see that
For the third term on the left of (3.2), by (1.2), we have
From (3.1)-(3.3), we find
and if
we might as well assume
from the Gronwall’s inequality, we have that
Now considering ω-mapping
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
and
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]:
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.
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doi:10.1016/0022-247X(82)90211-6W. E. Fitzgibbon, “Semilinear Functional Equations in Banach Space,” Journal of Differential Equations, Vol. 29, No. 1, 1978, pp. 1-14.
doi:10.1016/0022-0396(78)90037-2C. Zhong, C. 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/S0252959905000312H. R. Henriquez, “Periodic Solutions of Quasi-Linear Partial Functional Differential Equations with Unbounded Delay,” Funkcialaj Ekvacioj, Vol. 37, No. 2, 1994, pp. 329-343.E. Hernandez and H. R. Henriquez, “Existence Results for Partial Neutral Functional Differential Equations with Unbounded Delay,” Journal of Mathematical Analysis and Applications, Vol. 221, No. 2, 1998, pp. 452-475.
doi:10.1006/jmaa.1997.5875E. Hernandez and H. R. Henriquez, “Existence of Periodic Solutions of Partial Neutral Functional Differential Equations with Unbounded Delay,” Journal of Mathematical Analysis and Applications, Vol. 221, No. 2, 1998, pp. 499-522. doi:10.1006/jmaa.1997.5899X. Fu and X. Liu, “Existence of Periodic Solutions for Abstract Neutral Non-Autonomous Equations with Infinite Delay,” Journal of Mathematical Analysis and Applications, Vol. 325, No. 1, 2007, pp. 249-267.
doi:10.1016/j.jmaa.2006.01.048
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
denotes the absolute value of the number in R.
, 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 [
.
, we discuss the existence of a global periodic attractor for abstract parabolic problems.
of linear operators, we impose on the following restrictions:
of 
is dense in Hilbert space H and independent of t, 
is a closed linear operator;
, the resolvent 
exists for all
, with 
and there exists 
so that
;
and 
such that
;
and some
, the resolvent set of
, the resolvent
, is a compact operator.
generates a unique linear evolution operator
, satisfying the following properties:
, the space of bounded linear transformations on H, whenever 
and for each
, the mapping 
is continuous;
for
;
is a compact operator whenever
;
, for
;
such that
,
;
and 
then
;
is continuous on
, then the function 
is Holder continuous with any exponent
.
is continuous in t in the uniform operator topology uniformly for s.
, there exists a unique continuous function 
such that 
and 
satisfies the integral equation
.
is called a mild solution of (1.1).
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
.
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.
be two solutions of problem (1.1) with initial values
, and
. Then by (1.1), we find
in H, and we see that
from the Gronwall’s inequality, we have that
is the solution of (1.1),
. From (3.4), 
, we obtain
.
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
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.
in (1.1)) studied in [2,3]:
(where 
is the first eigenvalue of operator 
that subjects to the homogeneous Dirichlet boundary condition).