This paper is concerned with the dynamics of the following abstract
retarded evolution equation:
This paper is concerned with the abstract retarded evolution equation
First, we make a discussion on the dissipativity and existence of pullback attractors of the equation. In the finite dimensional case, Caraballo et al. made a systematic study on such problems for retarded differential equations with and without uniqueness in [
Then, we are interested in the existence of locally almost periodic solutions. It is well known that an evolution equation with almost periodic external force may fail to have almost periodic solutions (in the Bohr’s sense), even in the case where the equation is of a dissipative type. In this paper, we consider a local version of the concept of almost periodicity which will be referred to as the
This paper is organized as follows. In Section
Let
For
Denote by
The following estimates hold true.
We first give the definition of solutions to (
A function
A solution on
Denote by
Let
Given
A function
A solution
Concerning the existence of solutions for the Cauchy problem, we have the following.
Suppose that
Then, for any
The proof is quite standard and can be obtained by combining that of Theorem 3.1 in [
We may assume that
Further we can define a mapping
Repeating the above procedure, one can finally obtain a unique mild solution
We can also establish a corresponding extension theorem. This can be done as follows. First, suppose that
This completes the proof of the theorem.
Now we recall some basic definitions and facts in the theory of nonautonomous dynamical systems on complete metric spaces.
Let
For any
A dynamical system
Let there be given a dynamical system
A compact invariant set
A point
Denote by
Let
For a function
Let
A continuous mapping
For the sake of simplicity, we will rewrite
A family
For any bounded subset
The following existence result on pullback attractors is well known and can be found in [
Let
Then,
In this section, we give a decay estimate and prove the existence of pullback attractors for the Cauchy problem of the equation under appropriate conditions.
From now on we will always assume that
Let
The main result in this subsection is contained in the following lemma.
Suppose that There exist
Then, there exist positive constants
We fix a
We rewrite (
Denote by
Set
Define a continuous mapping
Assume the hypotheses in Lemma
It is clear that the estimate in (
Let
By using the smoothing property of the operator
Now we state and prove the existence result on pullback attractors.
Assume the hypotheses in Lemma
By virtue of Theorem
Let
To complete the proof of the theorem, there remains to check that
For any
For convenience, we rewrite
In general we know that a system with almost periodic forcing term may have no almost periodic solutions even if the system is dissipative. Here, we consider a local version of the concept of almost periodicity in the sense of Bohr. Namely, we introduce a concept of local almost periodicity and prove the existence of locally almost periodic solutions for (
We first make a general discussion on locally almost periodic functions.
Let
A function
One easily verifies the validity of the following easy proposition, which actually gives another equivalent definition for locally almost periodic functions.
If
(1) Let
By local almost periodicity of
Indeed, for any
(2) To prove the compactness of
In what follows we show that
Let
The following result shows that the local almost periodicity of a function is actually equivalent to the minimality of the hull of the function under the Bebutov’s dynamical system and is of crucial importance in proving the existence of locally almost periodic solutions.
A function
“
“
On the other hand, by the local almost periodicity of
Now we consider the existence of locally almost periodic solutions for (
Suppose that
Denote by
Let
Fix a
Let
Observe that
Now we show that
By the first equation in (
We now give an example to demonstrate how the abstract results in previous sections can be applied to nonautonomous parabolic equations with delays.
Let
Let
Let
We assume that There exist positive constants
Then, one easily sees that the mapping
Since
Assume that
Then, (
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper is supported by the Grant of NSF of China (10771159, 11071185).