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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).