#### Abstract

This paper is concerned with the dynamics of the following abstract retarded evolution equation: in a Hilbert space , where is a self-adjoint positive-definite operator with compact resolvent and is a locally Lipschitz continuous mapping. The dissipativity and pullback attractors are investigated, and the existence of locally almost periodic solutions is established.

#### 1. Introduction

This paper is concerned with the abstract retarded evolution equation in a Hilbert space , where is a self-adjoint positive-definite operator with compact resolvent, () is a locally Lipschitz continuous mapping with at most a linear growth rate, is a bounded function, and are constant time lags. The main purpose here is to study the dynamics of the equation by employing the theory of pullback attractors, Leray-Schauder fixed-point theorem, and some basic knowledge on the minimality and recurrence properties in topological dynamics. This consideration is motivated by an increasing interest on the dynamical behavior of retarded evolution equations in recent years.

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 [1, 2] and so forth. The situation in the infinite dimensional case seems to be more complicated. Although there have been a lot of works in this line for some types of retarded partial differential equations and the abstract equations as in (1) (see, e.g., [3–8]), due to some strict restrictions on the nonlinearities, we find that the known results in most of the existing works do not apply to many important PDE examples as the parabolic one given in Section 5 whose nonlinearity involves the gradient of the unknown function. In this present work, we will try to establish some new results in more regular spaces under weaker assumptions. In particular, instead of assuming that the corresponding nonlinearities belong to as in the literature, we will assume that is a mapping from to for some . It is worth noticing that this allows us to deal with retarded PDEs with more general nonlinearities (see Section 5). Since we are working in a space with more regularities than those in the literature, one has to overcome many extra difficulties in deriving the decay estimates and asymptotic compactness of the solutions.

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 *local almost periodicity*. We show that the local almost periodicity of a function is equivalent to the minimality of its hull under the Bebutov’s dynamical system (with respect to the compact-open topology). With this knowledge, we then prove that if is locally almost periodic, then (1) has at least one locally almost periodic solution.

This paper is organized as follows. In Section 2 we make some preliminary work. In Section 3 we discuss the dissipativity and establish the existence of pullback attractors. In Section 4 we prove the existence of locally almost periodic solutions. Section 5 consists of an example of retarded parabolic equation whose nonlinearity involves the gradient of the unknown function.

#### 2. Preliminaries

##### 2.1. Analytic Semigroups

Let be a Hilbert space with the inner product and norm . Let be a self-adjoint positive-definite operator with compact resolvent, and let be the eigenvalues of (counting with multiplicity) with the corresponding eigenvectors which form a canonical basis of .

For , define the powers as follows: Set Then, is a Hilbert space. The inner product and norm of are defined, respectively, as It is well known that, for any , the embedding is compact; moreover, it holds that

Denote by the analytic semigroup generated by in . The following proposition will play an important role in deriving the decay estimates and can be found in many text books (see, e.g., [9]).

Proposition 1. *The following estimates hold true.*(1)*Assume that*(2)*For any **, there exist constants ** and ** such that*(3)*For any **, there exists a constant ** such that *

##### 2.2. Solutions of the Equation and Its Cauchy Problem

We first give the definition of solutions to (1).

*Definition 2. *A function is said to be a solution of (1) on , if (1) for a.e. ,(2) and solves the equation at a.e. .

A solution on will be called an entire solution.

Denote by the space , which is endowed with the norm defined as

Let . For each , we will denote by the function in :
For convenience in statement, the function will be referred to as the *lifting* of in .

Given , consider the Cauchy problem:

*Definition 3. *A function is said to be a solution of the Cauchy problem (13), if is a solution of the equation on , and, moreover, it fulfills the initial-value condition .

*Remark 4. *A solution of the Cauchy problem (13) necessarily satisfies the integral equation:

Concerning the existence of solutions for the Cauchy problem, we have the following.

Theorem 5. *Suppose that is a locally Lipschitz continuous mapping, and .**Then, for any , the problem (13) has a unique solution on a maximal interval . *

*Proof. *The proof is quite standard and can be obtained by combining that of Theorem 3.1 in [10], Lemma 47.1 in [9], and Theorem 42.12 in [9]. Here, we give a sketch for completeness and the reader’s convenience.

We may assume that . Let . Define by
and consider the following initial-value problem:
By the assumptions on we know that is a continuous mapping which is locally Lipschitz in . It then follows by [9, Lemma 47.1] that the problem has a unique mild solution:
for some . Define
Then, is a mild solution of the delay system (13) on .

Further we can define a mapping by replacing in (15) with and consider the initial value problem:
Similar to the above one deduces that this problem has a unique mild solution . Set
One easily verifies that is a mild solution of the delay equation (13) on .

Repeating the above procedure, one can finally obtain a unique mild solution on some interval .

We can also establish a corresponding extension theorem. This can be done as follows. First, suppose that is bounded as . Then, we see that the function
is bounded in as . Therefore, by the basic knowledge on linear equations, it follows that is continuous in at . Further, by repeating some argument as above, one can obtain an extension of on some larger interval . With this fundamental result in hand, we immediately deduce that there exists a maximal interval such that the unique solution of the problem is defined.

This completes the proof of the theorem.

##### 2.3. Dynamical Systems

Now we recall some basic definitions and facts in the theory of nonautonomous dynamical systems on complete metric spaces.

Let be a complete metric space with the metric . Given any subsets , and of , define the *Hausdorff semidistance* of and as
where .

For any and , we will use to denote the ball in centered at with radius .

A dynamical system on is a continuous mapping from to fulfilling the following group properties: for all and . For notational simplicity we will rewrite as .

Let there be given a dynamical system on . A subset is said to be *invariant* (with respect to ) if

A compact invariant set is said to be *minimal* if it contains no proper compact invariant subset.

A point is said to be *almost recurrent* [11] if, for any , there exists an such that one can find on any segment of length a number such that .

Denote by the space which consists of all the bounded continuous functions from to . is usually equipped with the metric which yields the compact-open topology:
For convenience in statement, we will refer to as the *compact-open metric*.

Let be the translation operator on defined by
Then, is a dynamical system on , which is usually known as the *Bebutov’s dynamical system* [12].

For a function , the *hull * of in is defined as the closure of the set in ; namely,

##### 2.4. Pullback Attractors of Cocycles

*Definition 6. *Let and be two complete metric spaces, and let there be given a dynamical system on .

A continuous mapping is said to be a cocycle on with the base space and driving system if it satisfies the following conditions: (1) for any ;(2) for any and .

For the sake of simplicity, we will rewrite as .

*Definition 7. *A family of nonempty compact sets of is called the global pullback attractor of the cocycle if, for each , is the minimal compact set that enjoys the following pullback attracting property.

For any bounded subset of ,

The following existence result on pullback attractors is well known and can be found in [13–15] and so forth.

Theorem 8. *Let be a cocycle on with the base space and driving system . Suppose that there exists a nonempty compact subset of such that, for any bounded subset of ,
**
for sufficiently large.**Then, has a unique (global) pullback attractor :
*

#### 3. Dissipativity of (1)

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 is a locally Lipschitz continuous mapping from for some , unless otherwise stated.

Let be a bounded function. Denote

##### 3.1. Decay Estimate

The main result in this subsection is contained in the following lemma.

Lemma 9. *Suppose that satisfies the linear growth condition.*(H1)* There exist with and such that
**
Then, there exist positive constants , , and depending only on the parameters in (H1) and such that, for any solution of (13), one has
*

*Proof. *We fix a small enough so that
For simplicity, denote . Taking the inner product of (1) with in , one finds that
By using (H1) and the Young inequality, we obtain that
where . We infer from (7) that
Further by (36) one easily deduces that
where .

We rewrite (38) as
Multiplying (39) with and integrating from to , it yields
We observe that
Therefore, by (40), it holds that
where , from which one immediately concludes that
This completes the proof of the lemma.

##### 3.2. Existence of Pullback Attractor

Denote by the space which is equipped with the compact-open metric . Let be the translation operator on .

Set , where is the function in (1). For each , consider the Cauchy problem: By Theorem 5 we know that, under the hypotheses of Lemma 9, (44) has a unique global solution . Moreover, using some standard argument, it can be easily shown that is continuous in .

Define a continuous mapping as follows: where is the lifting of the solution of (44) in . Then, is a cocycle on with the base space and driving system .

Lemma 10. *Assume the hypotheses in Lemma 9. Then, there exists a bounded uniformly (with respect to ) absorbing set for the system .*

*Proof. *It is clear that the estimate in (33) holds true for solutions of the system (44).

Let , where is the positive number in (33). Given any bounded set in , by (33) we deduce that there exists a such that
for all and . It follows that is a uniformly absorbing set as we desired.

*Remark 11. *By using the smoothing property of the operator , it can be easily shown that the set
is contained in a compact subset of .

Now we state and prove the existence result on pullback attractors.

Theorem 12. * Assume the hypotheses in Lemma 9. Then has a unique global pullback attractor . *

*Proof. *By virtue of Theorem 8, it suffices to show that has a compact uniformly (with respect ) absorbing set.

Let
where is chosen such that (46) holds for . Then, we infer from Lemma 10 that
moreover, is a uniformly (with respect ) absorbing set in .

To complete the proof of the theorem, there remains to check that is relatively compact in . For this purpose, by Remark 11 and the classical Ascoli-Arzela theorem, one only needs to verify that consists of a family of equicontinuous functions.

For any , by definition it can be easily seen that there exist and such that
where is the lifting of the solution of (44). Note that if , then . Therefore, to prove the equicontinuity of , we need to check that there exist independent of and such that

For convenience, we rewrite as . Then, by (14) we have
where
Since , by Lemma 9 we see that
Hence one has
for all , where . We fix a . Then, by using (9) and (10) we deduce, recalling that ,
Now by (52) one concludes that
Note that both and are independent of and .

#### 4. Locally Almost Periodic Solutions

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 (1).

We first make a general discussion on locally almost periodic functions.

##### 4.1. Locally Almost Periodic Functions

Let be a complete metric space with metric , and let be the space which is always equipped with the compact-open metric . Denote by the translation operator on .

*Definition 13. *A function is said to be locally almost periodic if, for any , there exists an such that, for every , one can find on any segment of length a number such that

One easily verifies the validity of the following easy proposition, which actually gives another equivalent definition for locally almost periodic functions.

Proposition 14. * is locally almost periodic If, for every , there exists an such that, for every , one can find on any segment of length a number such that
*

Proposition 15. *If is a locally almost periodic function, then the following affirmations hold: *(1)* is uniformly continuous on **; *(2)*is compact in **.*

*Proof. *(1) Let be given arbitrarily. We need to prove that there exists a such that
for all with .

By local almost periodicity of , there exists an such that, for any , there exists a such that
Since is uniformly continuous on the interval , one can find a positive number such that
We show that satisfies (60).

Indeed, for any with , we can pick an such that . We may write and as
Then, by (61), there exists a such that
Now we have that

(2) To prove the compactness of , we employ the Kuratowski’s measure of noncompactness on which is defined as follows: for any ,
It is well known that is precompact if and only if .

In what follows we show that , thus proving what we desired.

Let be given arbitrarily. Then, by local almost periodicity of , there exists an such that, for any , one can find on any segment of length a number such that
In particular, we see that, for any , there exists a such that
It then follows that
where . Since is compact, one easily deduces by (69) that . Hence, . The proof is complete.

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.

Theorem 16. *A function is locally almost periodic if and only if is minimal under the Bebutov’s dynamical system. *

*Proof. * “” Assume that is minimal. Then, by the basic knowledge in the theory of dynamical systems (see, e.g., [11, 16]), we know that every is almost recurren; that is, for any , there exists an such that one can find on any segment of length a number such that
By a simple argument via contradiction, one can easily show that the length of the segments can be independent of (see also the work of Birkhoff [16]). Thanks to Proposition 14, one immediately concludes by (70) and the independence of on that is locally almost periodic.

“" Conversely, if is locally almost periodic, then by Proposition 15 is uniformly continuous on with being precompact in . Therefore, by the classical Ascoli-Arzela theorem, one deduces that, for any sequence , the sequence has a subsequence that converges uniformly on any compact interval to a function . Hence, we have
This implies that the hull is compact.

On the other hand, by the local almost periodicity of and Proposition 14, we see that is an almost recurrent point of the system . Since is compact, by adopting some argument as in [16], it can be easily shown that is minimal. We omit the details.

##### 4.2. Existence of Locally Almost Periodic Solutions

Now we consider the existence of locally almost periodic solutions for (1). The main result is contained in the following theorem.

Theorem 17. *Suppose that satisfies (H1). Then, if is locally almost periodic, (1) has at least one locally almost periodic solution.*

*Proof. *Denote by the space which is equipped with the compact-open metric . Denote by the translation operator on .

Let be the cocycle on the phase space generated by (44) (see (45) for the definition). Define the skew-product flow as follows:
Then, as in the proof of Theorem 12, one can easily show that has a global attractor in which is the maximal compact invariant set that attracts each bounded subset of . Further, by a recurrence theorem due to Birkhoff and Bebutov (see, e.g., [12]), contains a compact minimal invariant set of the system .

Fix a . Then, by the invariance property of , there exists a trajectory of which is defined on the whole line and contained in such that . Let . Define a function as
Then, one easily sees that is an entire solution of the equation in (44) with therein replaced by . We also infer from the proof of Theorem 12 that is uniformly continuous on with being contained in a compact subset of . Therefore by the classical Ascoli-Arzela theorem, the hull in is compact.

Let . Note that if , then it solves the following equation:

Observe that
where and are projection operators from to and , respectively. Since both and are compact, we deduce that is compact.

Now we show that is minimal. Suppose the contrary. Then, it would contain a proper compact invariant subset . Let
where is the restriction of on ,
It is trivial to check that is a proper compact invariant subset of , which contradicts to the minimality of .

By the first equation in (75), we see that there exists a such that . Further, by the minimality of , one finds that . Hence is minimal. It then follows immediately by Theorem 16 that is a locally almost periodic function. By (69), we see that is precisely an entire solution of (1).

#### 5. An Example

We now give an example to demonstrate how the abstract results in previous sections can be applied to nonautonomous parabolic equations with delays.

Let be a differential operator on a bounded domain : where and for all . We assume that there exists a constant such that, for a.e. , and that Hence, is uniformly elliptic on .

Let : be a locally Lipschitz continuous function. Consider the retarded parabolic equation on : associated with the homogeneous Dirichlet boundary condition: where denote time lags.

Let , . Define a symmetric bilinear form on as follows: It is clear that is bounded and coercive. Thanks to the Lax-Milgram theorem, generates a self-adjoint positive-definite operator on with compact resolvent. Note that

We assume that satisfies the following linear growth condition. (F1) There exist positive constants and , such that

Then, one easily sees that the mapping defined by makes sense and is locally Lipschitz. Setting , the problem (81)-(82) can be reformulated as an abstract equation in as follows: Now we are in a situation of (1).

Since satisfies (F1), then one can easily verify that the mapping satisfies (H1) in the previous sections, and hence the abstract results obtained therein apply. In particular, we have the following.

Theorem 18. *Assume that satisfies the linear growth condition (F1) with the positive constants ’s therein satisfying
**
where is the first eigenvalue of . Let be a locally almost periodic (resp., periodic) function.**Then, (87) has at least one locally almost periodic (resp., periodic) solution. *

#### 6. Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This paper is supported by the Grant of NSF of China (10771159, 11071185).