# On the Dynamics of Abstract Retarded Evolution Equations

Authors: Desheng Li; Jinying Wei; Jintao Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with the dynamics of the following abstract
retarded evolution equation:d/dtut+Aut=Fut-r1,…,ut-rn+gt in a Hilbert space H,
where A:DA⊂H→H is a self-adjoint positive-definite operator with compact resolvent and
F:DAαn→Hα∈0,1/2 is a locally Lipschitz continuous mapping.
The dissipativity and pullback attractors are investigated, and the existence of locally almost periodic solutions is established.


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## Body

## 1. Introduction

This paper is concerned with the abstract retarded evolution equation(1)ddtu(t)+Au(t)=F(u(t-r1),…,u(t-rn))+g(t)
in a Hilbert space H, where A:D(A)⊂H→H is a self-adjoint positive-definite operator with compact resolvent, F:D(Aα)n→H (α∈[0,1/2]) is a locally Lipschitz continuous mapping with at most a linear growth rate, g∈C(ℝ;H) is a bounded function, and r1,…,rn≥0 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 ∇u 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 C(Hn;H) as in the literature, we will assume that F is a mapping from D(Aα)n to H for some α∈[0,1/2]. 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 thelocal almost periodicity. We show that the local almost periodicity of a function h is equivalent to the minimality of its hull ℋ(h) under the Bebutov’s dynamical system (with respect to the compact-open topology). With this knowledge, we then prove that if g is locally almost periodic, then (1) has at least one locally almost periodic solution.This paper is organized as follows. In Section2 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

LetH be a Hilbert space with the inner product (·,·) and norm |·|. Let
(2)A:D(A)⊂H⟶H
be a self-adjoint positive-definite operator with compact resolvent, and let
(3)0<λ1≤λ2≤⋯≤λk⟶∞(k⟶∞)
be the eigenvalues of A (counting with multiplicity) with the corresponding eigenvectors {ωi}i=1∞ which form a canonical basis of H.Forα∈ℝ, define the powers Aα as follows:
(4)Aαu=∑i=1∞λiαciωi,∀u:=∑i=1∞ciωi∈H.
Set
(5)Hα:=D(Aα):={u∈H∣∑i=1∞λi2αci2<∞}.
Then, Hα is a Hilbert space. The inner product (·,·)α and norm |·|α of Hα are defined, respectively, as
(6)(u,v)α=(Aαu,Aαv),|u|α=(u,u)α1/2,(u,v)α5.(Aαu,Aαv),.|u|α∀u,v∈Hα.
It is well known that, for any -∞<α<β<∞, the embedding Hβ⊂Hα is compact; moreover, it holds that
(7)|v|α2≤λ12(α-β)|v|β2,∀v∈Hβ.Denote byE(t) the analytic semigroup generated by A in H. 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(8)∥E(t)∥≤e-λ1t,∀t≥0.(2)
For any0<α≤1, there exist constantsMα>0anda>0such that(9)∥AαE(t)∥≤Mαt-αe-at,∀t>0.(3)
For any0<α≤1, there exists a constantKα>0such that(10)|(E(t)-I)u|≤Kαtα|u|α,∀t≥0,u∈Hα.


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

We first give the definition of solutions to (1).Definition 2.
A functionu∈C((a-r,b);Hα) is said to be a solution of (1) on (a,b), if (1)
u(t)∈H1=D(A) for a.e. t∈(a,b),(2)
u∈C1(J;H) and solves the equation at a.e. t∈(a,b).
A solution onℝ will be called an entire solution.Denote by𝒞 the space C([-r,0];Hα), which is endowed with the norm ∥·∥𝒞 defined as
(11)∥φ∥𝒞=maxs∈[-r,0]|φ(s)|α,∀φ∈𝒞.Letu∈C([-r,T);Hα). For each t∈[0,T), we will denote by ut the function in 𝒞:
(12)ut(s)=u(t+s),∀s∈[-r,0].
For convenience in statement, the function ut will be referred to as the  lifting of u in 𝒞.Givenφ∈𝒞, consider the Cauchy problem:
(13)ddtu(t)+Au(t)=F(u(t-r1),…,u(t-rn))F(u(t-r1),…,5.)+g(t),t>0,u0=φ.Definition 3.
A functionu∈C([-r,T);Hα) is said to be a solution of the Cauchy problem (13), if u is a solution of the equation on (0,T), and, moreover, it fulfills the initial-value condition u0=φ.Remark 4.
A solutionu of the Cauchy problem (13) necessarily satisfies the integral equation:
(14)u(t)=E(t)φ(0)+∫0t‍E(t-s)(F(u(s-r1),…,u(s-rn))+g(s))ds.Concerning the existence of solutions for the Cauchy problem, we have the following.Theorem 5.
Suppose thatF:Hαn→H(0≤α<1) is a locally Lipschitz continuous mapping, and g∈C(ℝ;H).
Then, for anyφ∈𝒞, the problem (13) has a unique solution u(t)=u(t;φ) on a maximal interval [-r,Tφ).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 thatr1=0. Let σ=min2≤k≤nrk(>0). Define G0:[0,σ]×Hα→H by
(15)G0(t,v)=F(v,φ(t-r2),…,φ(t-rn))+g(t),
and consider the following initial-value problem:
(16)ddtv(t)+Av(t)=G0(t,v(t)),t∈[0,σ],v(0)=φ(0).
By the assumptions on F we know that G0:[0,σ]×Hα→H is a continuous mapping which is locally Lipschitz in v. It then follows by [9, Lemma 47.1] that the problem has a unique mild solution:
(17)v1∈C([0,σ1];Hα)∩Lloc2(0,σ1;H1)
for some σ1≤σ. Define
(18)u1(t)={v1(t),t∈[0,σ1],φ(t),t∈[-r,0].
Then, u=u1 is a mild solution of the delay system (13) on [-r,σ1].
Further we can define a mappingG1:[σ1,σ1+σ]×Hα→H by replacing φ in (15) with u1 and consider the initial value problem:
(19)ddtv(t)+Av(t)=G1(t,v(t)),t∈[σ1,σ1+σ],v(σ1)=u1(σ1).
Similar to the above one deduces that this problem has a unique mild solution v2∈C([σ1,σ2];Hα)∩Lloc2(σ1,σ2;H1). Set
(20)u2(t)={v2(t),t∈[σ1,σ2],u1(t),t∈[-r,σ1].
One easily verifies that u2 is a mild solution of the delay equation (13) on [-r,σ2].
Repeating the above procedure, one can finally obtain a unique mild solutionu∈C([-r,T);Hα)∩Lloc2(0,T;H1) on some interval [-r,T).
We can also establish a corresponding extension theorem. This can be done as follows. First, suppose that|u(t)|α is bounded as t→T<∞. Then, we see that the function
(21)t⟼h(t):=F(u(t-r1),…,u(t-rn))+g(t)
is bounded in H as t→T. Therefore, by the basic knowledge on linear equations, it follows that u(t) is continuous in Hα at t=T. Further, by repeating some argument as above, one can obtain an extension of u on some larger interval [-r,T+δ). With this fundamental result in hand, we immediately deduce that there exists a maximal interval [-r,Tφ) such that the unique solution u 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.LetX be a complete metric space with the metric d. Given any subsets A, and B of X, define the  Hausdorff semidistance  dH(A,B) of A and B as
(22)dH(A,B)=supx∈Ad(x,B),
where d(x,B)=miny∈Bd(x,y).For anyx∈X and R>0, we will use BX(x,R) to denote the ball in X centered at x with radius R.A dynamical systemS on X is a continuous mapping from ℝ×X to X fulfilling the following group properties:
(23)S(0,x)=x,S(t+s,x)=S(s,S(t,x))
for all t,s∈ℝ and x∈X. For notational simplicity we will rewrite S(t,x) as Stx.Let there be given a dynamical systemS on X. A subset M⊂X is said to be  invariant (with respect to S) if
(24)StM=M,∀t∈ℝ.A compact invariant setM is said to be minimal if it contains no proper compact invariant subset.A pointx∈X is said to be  almost recurrent [11] if, for any ε>0, there exists an l>0 such that one can find on any segment of length l a number τ such that d(Sτx,x)<ε.Denote by𝒳 the space Cb(ℝ;X) which consists of all the bounded continuous functions from ℝ to X. 𝒳 is usually equipped with the metric ϱ which yields the compact-open topology:
(25)ϱ(u,v)=∑n=1∞12nmax|t|≤nd(u(t),v(t))1+max|t|≤nd(u(t),v(t)),max|t|≤nd(u(t),v(t))1+max|t|≤nd{u(t),.}∀u,v∈𝒳.
For convenience in statement, we will refer to ϱ as the  compact-open metric.Letθ=θt be the translation operator on 𝒳 defined by
(26)θtu=u(t+·),u∈𝒳.
Then, θ is a dynamical system on 𝒳, which is usually known as the Bebutov’s dynamical system [12].For a functionu∈𝒳, the hullℋ𝒳(u) of u in 𝒳 is defined as the closure of the set {θtu∣t∈ℝ} in 𝒳; namely,
(27)ℋ𝒳(u)={θtu∣t∈ℝ}¯.


### 2.4. Pullback Attractors of Cocycles

Definition 6.
LetX and Σ be two complete metric spaces, and let there be given a dynamical system θ on Σ.
A continuous mappingΦ:ℝ+×Σ×X→X is said to be a  cocycle on X with the base space Σ and driving system θ if it satisfies the following conditions:  (1)
Φ(0,h,x)=x for any (h,x)∈Σ×X;(2)
Φ(t+s,h,x)=Φ(t,θsh,Φ(s,h,x)) for any t,s≥0 and (h,x)∈Σ×X.For the sake of simplicity, we will rewriteΦ(t,h,x) as Φ(t,h)x.Definition 7.
A familyA={A(h)}h∈Σ of nonempty compact sets of X is called the global pullback attractor of the cocycle Φ if, for each h, A(h) is the minimal compact set that enjoys the following pullback attracting property.
For any bounded subsetB of X,
(28)limt→+∞dH(Φ(t,θ-th)B,A(h))=0.The following existence result on pullback attractors is well known and can be found in [13–15] and so forth.Theorem 8.
LetΦ be a cocycle on X with the base space Σ and driving system θ. Suppose that there exists a nonempty compact subset K of X such that, for any bounded subset B of X,
(29)Φ(t,h)B⊂K
for t>0 sufficiently large.
Then,Φ has a unique (global) pullback attractor A={A(h)}h∈Σ:
(30)A(h)=⋂τ≥0‍⋃t>τ‍Φ(t,θ-th)K¯.


## 2.1. Analytic Semigroups

LetH be a Hilbert space with the inner product (·,·) and norm |·|. Let
(2)A:D(A)⊂H⟶H
be a self-adjoint positive-definite operator with compact resolvent, and let
(3)0<λ1≤λ2≤⋯≤λk⟶∞(k⟶∞)
be the eigenvalues of A (counting with multiplicity) with the corresponding eigenvectors {ωi}i=1∞ which form a canonical basis of H.Forα∈ℝ, define the powers Aα as follows:
(4)Aαu=∑i=1∞λiαciωi,∀u:=∑i=1∞ciωi∈H.
Set
(5)Hα:=D(Aα):={u∈H∣∑i=1∞λi2αci2<∞}.
Then, Hα is a Hilbert space. The inner product (·,·)α and norm |·|α of Hα are defined, respectively, as
(6)(u,v)α=(Aαu,Aαv),|u|α=(u,u)α1/2,(u,v)α5.(Aαu,Aαv),.|u|α∀u,v∈Hα.
It is well known that, for any -∞<α<β<∞, the embedding Hβ⊂Hα is compact; moreover, it holds that
(7)|v|α2≤λ12(α-β)|v|β2,∀v∈Hβ.Denote byE(t) the analytic semigroup generated by A in H. 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(8)∥E(t)∥≤e-λ1t,∀t≥0.(2)
For any0<α≤1, there exist constantsMα>0anda>0such that(9)∥AαE(t)∥≤Mαt-αe-at,∀t>0.(3)
For any0<α≤1, there exists a constantKα>0such that(10)|(E(t)-I)u|≤Kαtα|u|α,∀t≥0,u∈Hα.


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

We first give the definition of solutions to (1).Definition 2.
A functionu∈C((a-r,b);Hα) is said to be a solution of (1) on (a,b), if (1)
u(t)∈H1=D(A) for a.e. t∈(a,b),(2)
u∈C1(J;H) and solves the equation at a.e. t∈(a,b).
A solution onℝ will be called an entire solution.Denote by𝒞 the space C([-r,0];Hα), which is endowed with the norm ∥·∥𝒞 defined as
(11)∥φ∥𝒞=maxs∈[-r,0]|φ(s)|α,∀φ∈𝒞.Letu∈C([-r,T);Hα). For each t∈[0,T), we will denote by ut the function in 𝒞:
(12)ut(s)=u(t+s),∀s∈[-r,0].
For convenience in statement, the function ut will be referred to as the  lifting of u in 𝒞.Givenφ∈𝒞, consider the Cauchy problem:
(13)ddtu(t)+Au(t)=F(u(t-r1),…,u(t-rn))F(u(t-r1),…,5.)+g(t),t>0,u0=φ.Definition 3.
A functionu∈C([-r,T);Hα) is said to be a solution of the Cauchy problem (13), if u is a solution of the equation on (0,T), and, moreover, it fulfills the initial-value condition u0=φ.Remark 4.
A solutionu of the Cauchy problem (13) necessarily satisfies the integral equation:
(14)u(t)=E(t)φ(0)+∫0t‍E(t-s)(F(u(s-r1),…,u(s-rn))+g(s))ds.Concerning the existence of solutions for the Cauchy problem, we have the following.Theorem 5.
Suppose thatF:Hαn→H(0≤α<1) is a locally Lipschitz continuous mapping, and g∈C(ℝ;H).
Then, for anyφ∈𝒞, the problem (13) has a unique solution u(t)=u(t;φ) on a maximal interval [-r,Tφ).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 thatr1=0. Let σ=min2≤k≤nrk(>0). Define G0:[0,σ]×Hα→H by
(15)G0(t,v)=F(v,φ(t-r2),…,φ(t-rn))+g(t),
and consider the following initial-value problem:
(16)ddtv(t)+Av(t)=G0(t,v(t)),t∈[0,σ],v(0)=φ(0).
By the assumptions on F we know that G0:[0,σ]×Hα→H is a continuous mapping which is locally Lipschitz in v. It then follows by [9, Lemma 47.1] that the problem has a unique mild solution:
(17)v1∈C([0,σ1];Hα)∩Lloc2(0,σ1;H1)
for some σ1≤σ. Define
(18)u1(t)={v1(t),t∈[0,σ1],φ(t),t∈[-r,0].
Then, u=u1 is a mild solution of the delay system (13) on [-r,σ1].
Further we can define a mappingG1:[σ1,σ1+σ]×Hα→H by replacing φ in (15) with u1 and consider the initial value problem:
(19)ddtv(t)+Av(t)=G1(t,v(t)),t∈[σ1,σ1+σ],v(σ1)=u1(σ1).
Similar to the above one deduces that this problem has a unique mild solution v2∈C([σ1,σ2];Hα)∩Lloc2(σ1,σ2;H1). Set
(20)u2(t)={v2(t),t∈[σ1,σ2],u1(t),t∈[-r,σ1].
One easily verifies that u2 is a mild solution of the delay equation (13) on [-r,σ2].
Repeating the above procedure, one can finally obtain a unique mild solutionu∈C([-r,T);Hα)∩Lloc2(0,T;H1) on some interval [-r,T).
We can also establish a corresponding extension theorem. This can be done as follows. First, suppose that|u(t)|α is bounded as t→T<∞. Then, we see that the function
(21)t⟼h(t):=F(u(t-r1),…,u(t-rn))+g(t)
is bounded in H as t→T. Therefore, by the basic knowledge on linear equations, it follows that u(t) is continuous in Hα at t=T. Further, by repeating some argument as above, one can obtain an extension of u on some larger interval [-r,T+δ). With this fundamental result in hand, we immediately deduce that there exists a maximal interval [-r,Tφ) such that the unique solution u 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.LetX be a complete metric space with the metric d. Given any subsets A, and B of X, define the  Hausdorff semidistance  dH(A,B) of A and B as
(22)dH(A,B)=supx∈Ad(x,B),
where d(x,B)=miny∈Bd(x,y).For anyx∈X and R>0, we will use BX(x,R) to denote the ball in X centered at x with radius R.A dynamical systemS on X is a continuous mapping from ℝ×X to X fulfilling the following group properties:
(23)S(0,x)=x,S(t+s,x)=S(s,S(t,x))
for all t,s∈ℝ and x∈X. For notational simplicity we will rewrite S(t,x) as Stx.Let there be given a dynamical systemS on X. A subset M⊂X is said to be  invariant (with respect to S) if
(24)StM=M,∀t∈ℝ.A compact invariant setM is said to be minimal if it contains no proper compact invariant subset.A pointx∈X is said to be  almost recurrent [11] if, for any ε>0, there exists an l>0 such that one can find on any segment of length l a number τ such that d(Sτx,x)<ε.Denote by𝒳 the space Cb(ℝ;X) which consists of all the bounded continuous functions from ℝ to X. 𝒳 is usually equipped with the metric ϱ which yields the compact-open topology:
(25)ϱ(u,v)=∑n=1∞12nmax|t|≤nd(u(t),v(t))1+max|t|≤nd(u(t),v(t)),max|t|≤nd(u(t),v(t))1+max|t|≤nd{u(t),.}∀u,v∈𝒳.
For convenience in statement, we will refer to ϱ as the  compact-open metric.Letθ=θt be the translation operator on 𝒳 defined by
(26)θtu=u(t+·),u∈𝒳.
Then, θ is a dynamical system on 𝒳, which is usually known as the Bebutov’s dynamical system [12].For a functionu∈𝒳, the hullℋ𝒳(u) of u in 𝒳 is defined as the closure of the set {θtu∣t∈ℝ} in 𝒳; namely,
(27)ℋ𝒳(u)={θtu∣t∈ℝ}¯.


## 2.4. Pullback Attractors of Cocycles

Definition 6.
LetX and Σ be two complete metric spaces, and let there be given a dynamical system θ on Σ.
A continuous mappingΦ:ℝ+×Σ×X→X is said to be a  cocycle on X with the base space Σ and driving system θ if it satisfies the following conditions:  (1)
Φ(0,h,x)=x for any (h,x)∈Σ×X;(2)
Φ(t+s,h,x)=Φ(t,θsh,Φ(s,h,x)) for any t,s≥0 and (h,x)∈Σ×X.For the sake of simplicity, we will rewriteΦ(t,h,x) as Φ(t,h)x.Definition 7.
A familyA={A(h)}h∈Σ of nonempty compact sets of X is called the global pullback attractor of the cocycle Φ if, for each h, A(h) is the minimal compact set that enjoys the following pullback attracting property.
For any bounded subsetB of X,
(28)limt→+∞dH(Φ(t,θ-th)B,A(h))=0.The following existence result on pullback attractors is well known and can be found in [13–15] and so forth.Theorem 8.
LetΦ be a cocycle on X with the base space Σ and driving system θ. Suppose that there exists a nonempty compact subset K of X such that, for any bounded subset B of X,
(29)Φ(t,h)B⊂K
for t>0 sufficiently large.
Then,Φ has a unique (global) pullback attractor A={A(h)}h∈Σ:
(30)A(h)=⋂τ≥0‍⋃t>τ‍Φ(t,θ-th)K¯.


## 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 thatF is a locally Lipschitz continuous mapping from D(Aα)n→H for some 0≤α≤1/2, unless otherwise stated.Letg∈Cb(ℝ;H) be a bounded function. Denote
(31)g0=supt∈ℝ|g(t)|.


### 3.1. Decay Estimate

The main result in this subsection is contained in the following lemma.Lemma 9.
Suppose thatF satisfies the linear growth condition.(H1)
There existβi>0(1≤i≤n) with ∑i=1n‍βi<λ11-α and k1>0 such that
(32)|F(v1,…,vn)|≤∑i=1n‍βi|vi|α+k1,|F(v1,…,)|1∀(v1,…,vn)∈Hαn.
Then, there exist positive constantsδ, C1, and C2 depending only on the parameters in (H1) and g0 such that, for any solution u(t)=u(t;φ) of (13), one has
(33)|u(t)|α2≤C1∥φ∥𝒞2e-δt+C2,∀t≥0.Proof.
We fix aδ>0 small enough so that
(34)c0:=λ11-α-∑i=1n‍eδri/2βi-λ1α+λ1-α2δ>0.
For simplicity, denote li=λ1-αeδri/2(i=1,…,n). Taking the inner product of (1) with A2αu in H, one finds that
(35)12ddt|u(t)|α2+|u(t)|(1/2)+α2=(F(u(t-r1),…,u(t-rn))+g(t),A2αu(t))≤|F(u(t-r1),…,u(t-rn))+g(t)|·|u(t)|2α.
By using (H1) and the Young inequality, we obtain that
(36)ddt|u(t)|α2+2|u(t)|(1/2)+α2≤∑i=1n‍βili|u(t-ri)|α2+(∑i=1n‍liβi)|u(t)|2α2+δ|u|2α2+k2δ,
where k=k1+g0. We infer from (7) that
(37)|u(t)|2α2≤λ12α-1|u(t)|(1/2)+α2,|u(t)|α2≤λ1-1|u(t)|(1/2)+α2.
Further by (36) one easily deduces that
(38)ddt|u(t)|α2+c1|u(t)|α2≤∑i=1n‍βili|u(t-ri)|α2+k2δ,
where c1=2λ1-∑i=1n‍βiliλ12α-δλ12α.
We rewrite (38) as
(39)ddt|u(t)|α2+δ|u(t)|α2≤-(c1-δ)|u(t)|α2+∑i=1n‍βili|u(t-ri)|α2+k2δ.
Multiplying (39) with eδt and integrating from 0 to t, it yields
(40)|u(t)|α2≤e-δt|φ(0)|α2-(c1-δ)∫0t‍|u(s)|α2e-δ(t-s)ds+∑i=1n‍βili∫0t‍|u(s-ri)|α2e-δ(t-s)ds+k2δ2.
We observe that
(41)∫0t‍|u(s-ri)|α2e-δ(t-s)ds=∫-rit-ri‍|u(s)|α2e-δ(t-s-ri)ds≤eδri∫0t‍|u(s)|α2e-δ(t-s)ds+eδri∫-ri0‍|u(s)|α2e-δ(t-s)ds.
Therefore, by (40), it holds that
(42)|u(t)|α2≤e-δt|φ(0)|α2-c2∫0t‍|u(s)|α2e-δ(t-s)ds+∑i=1n‍βilieδri∫-ri0‍|u(s)|α2e-δ(t-s)ds+k2δ2≤e-δt|φ(0)|α2+∑i=1n‍βilieδri∫-ri0‍|u(s)|α2e-δ(t-s)ds+k2δ2,
where c2=2λ1αc0, from which one immediately concludes that
(43)|u(t)|α2≤(1+∑i=1n‍1δβiλ1αeδri/2)∥φ∥𝒞2e-δt+K2δ2.
This completes the proof of the lemma.


### 3.2. Existence of Pullback Attractor

Denote by𝒳α the space Cb(ℝ;Hα) which is equipped with the compact-open metric ϱ. Let θ=θt be the translation operator on 𝒳α.SetΣ=ℋ𝒳0(g), where g∈𝒳0:=Cb(ℝ;H) is the function in (1). For each (h,φ)∈Σ×𝒞, consider the Cauchy problem:
(44)ddtu(t)+Au(t)=F(u(t-r1),…,u(t-rn))+h(t),ddtu(t)+Au(t)=F(u(t-r1),…5,u(t-rn))t>0,u0=φ.
By Theorem 5 we know that, under the hypotheses of Lemma 9, (44) has a unique global solution u(t)=u(t;h,φ). Moreover, using some standard argument, it can be easily shown that u(t;h,φ) is continuous in (t,h,φ).Define a continuous mappingℝ+×Σ×𝒞→𝒞 as follows:
(45)Φ(t,h)φ=ut,∀(t,h,φ)∈ℝ+×Σ×𝒞,
where ut is the lifting of the solution u(t)=u(t;h,φ) of (44) in 𝒞. Then, Φ is a cocycle on 𝒞 with the base space Σ and driving system θ.Lemma 10.
Assume the hypotheses in Lemma9. Then, there exists a bounded uniformly (with respect to h∈Σ) absorbing set 𝒰⊂𝒞 for the system Φ.Proof.
It is clear that the estimate in (33) holds true for solutions of the system (44).
Letρα2=1+C2, where C2 is the positive number in (33). Given any bounded set D in 𝒞, by (33) we deduce that there exists a t0=t0(D)>r>0 such that
(46)|u(t;h,φ)|α≤ρα,∀t>t0
for all φ∈D and h∈Σ. It follows that 𝒰=B𝒞(0,ρα) is a uniformly absorbing set as we desired.Remark 11.
By using the smoothing property of the operatorA, it can be easily shown that the set
(47){u(t;h,φ)∣t>t0,φ∈D,h∈Σ}
is contained in a compact subset of Hα.Now we state and prove the existence result on pullback attractors.Theorem 12.
Assume the hypotheses in Lemma9. Then Φ has a unique global pullback attractor A={A(h)}h∈Σ.Proof.
By virtue of Theorem8, it suffices to show that Φ has a compact uniformly (with respect h∈Σ) absorbing set.
Let(48)M=⋃h∈Σ‍(⋃t≥t0(𝒰)+r‍Φ(t,h)𝒰),N=⋃h∈Σ‍(⋃t≥t0(𝒰)+2r+1‍Φ(t,h)𝒰),
where t0:=t0(𝒰) is chosen such that (46) holds for 𝒰. Then, we infer from Lemma 10 that
(49)N⊂M⊂𝒰:=B𝒞(0,ρα);
moreover, N is a uniformly (with respect h∈Σ) absorbing set in 𝒞.
To complete the proof of the theorem, there remains to check thatN is relatively compact in 𝒞. For this purpose, by Remark 11 and the classical Ascoli-Arzela theorem, one only needs to verify that N consists of a family of equicontinuous functions.
For anyψ∈N, by definition it can be easily seen that there exist h∈Σ and φ∈M such that
(50)ψ=Φ(r+1,h)φ=ur+1,
where ut is the lifting of the solution u(t)=u(t;h,φ) of (44). Note that if s∈[-r,0], then r+1+s∈[1,r+1]. Therefore, to prove the equicontinuity of N, we need to check that there exist L,γ>0 independent of h∈Σ and φ∈M such that
(51)|u(t′)-u(t)|α≤L|t′-t|γ,1≤t≤t′≤r+1.
For convenience, we rewritet′ as t+δ. Then, by (14) we have
(52)|u(t+δ)-u(t)|α≤J1+J2+J3,
where
(53)J1=|(E(δ)-I)(E(t)φ(0))|α,J2=∫0t‍|(E(δ)-I)E(t-s)G(s)|αds,J3=∫tt+δ‍|E(t+δ-s)G(s)|αds,G(s)=F(u(s-r1),…,u(s-rn))+h(s).
Since ∥φ∥𝒞≤ρα, by Lemma 9 we see that
(54)|u(s)|α2≤(C1+1)ρα2,∀s≥-r.
Hence one has
(55)|G(s)|≤∑i=1n‍βi|u(s-ri)|α+k1+|h(s)|≤(∑i=1n‍βi)C1+1ρα+K1+g0∶=Cα
for all s≥0, where g0=sups∈ℝ|g(s)|. We fix a 0<γ<1/2. Then, by using (9) and (10) we deduce, recalling that t≥1,
(56)J1≤|(E(δ)-I)(AαE(t)φ(0))|≤Kγδγ|Aγ+αE(t)φ(0)|≤KγδγMγt-γe-at|φ(0)|α≤KγδγMγt-γe-atρα≤C3δγ,J2≤∫0t‍|(E(δ)-I)AαE(t-s)G(s)|ds≤∫0t‍Kγδγ|Aγ+αE(t-s)G(s)|ds≤KγδγMα+γCα∫0t‍(t-s)-(γ+α)e-a(t-s)ds≤C4δγ,J3≤∫tt+δ‍|AαE(t+δ-s)G(s)|ds≤Cα∫tt+δ‍Mα(t+δ-s)-αe-a(t+δ-s)ds≤CαMα∫0δ‍(δ-s)-αe-a(δ-s)ds≤C5′δ1-α≤C5δγ.
Now by (52) one concludes that
(57)|u(t′)-u(t)|α=|u(t+δ)-u(t)|α≤(C3+C4+C5)δγ=L|t′-t|γ.
Note that both L and γ are independent of h∈Σ and φ∈M.


## 3.1. Decay Estimate

The main result in this subsection is contained in the following lemma.Lemma 9.
Suppose thatF satisfies the linear growth condition.(H1)
There existβi>0(1≤i≤n) with ∑i=1n‍βi<λ11-α and k1>0 such that
(32)|F(v1,…,vn)|≤∑i=1n‍βi|vi|α+k1,|F(v1,…,)|1∀(v1,…,vn)∈Hαn.
Then, there exist positive constantsδ, C1, and C2 depending only on the parameters in (H1) and g0 such that, for any solution u(t)=u(t;φ) of (13), one has
(33)|u(t)|α2≤C1∥φ∥𝒞2e-δt+C2,∀t≥0.Proof.
We fix aδ>0 small enough so that
(34)c0:=λ11-α-∑i=1n‍eδri/2βi-λ1α+λ1-α2δ>0.
For simplicity, denote li=λ1-αeδri/2(i=1,…,n). Taking the inner product of (1) with A2αu in H, one finds that
(35)12ddt|u(t)|α2+|u(t)|(1/2)+α2=(F(u(t-r1),…,u(t-rn))+g(t),A2αu(t))≤|F(u(t-r1),…,u(t-rn))+g(t)|·|u(t)|2α.
By using (H1) and the Young inequality, we obtain that
(36)ddt|u(t)|α2+2|u(t)|(1/2)+α2≤∑i=1n‍βili|u(t-ri)|α2+(∑i=1n‍liβi)|u(t)|2α2+δ|u|2α2+k2δ,
where k=k1+g0. We infer from (7) that
(37)|u(t)|2α2≤λ12α-1|u(t)|(1/2)+α2,|u(t)|α2≤λ1-1|u(t)|(1/2)+α2.
Further by (36) one easily deduces that
(38)ddt|u(t)|α2+c1|u(t)|α2≤∑i=1n‍βili|u(t-ri)|α2+k2δ,
where c1=2λ1-∑i=1n‍βiliλ12α-δλ12α.
We rewrite (38) as
(39)ddt|u(t)|α2+δ|u(t)|α2≤-(c1-δ)|u(t)|α2+∑i=1n‍βili|u(t-ri)|α2+k2δ.
Multiplying (39) with eδt and integrating from 0 to t, it yields
(40)|u(t)|α2≤e-δt|φ(0)|α2-(c1-δ)∫0t‍|u(s)|α2e-δ(t-s)ds+∑i=1n‍βili∫0t‍|u(s-ri)|α2e-δ(t-s)ds+k2δ2.
We observe that
(41)∫0t‍|u(s-ri)|α2e-δ(t-s)ds=∫-rit-ri‍|u(s)|α2e-δ(t-s-ri)ds≤eδri∫0t‍|u(s)|α2e-δ(t-s)ds+eδri∫-ri0‍|u(s)|α2e-δ(t-s)ds.
Therefore, by (40), it holds that
(42)|u(t)|α2≤e-δt|φ(0)|α2-c2∫0t‍|u(s)|α2e-δ(t-s)ds+∑i=1n‍βilieδri∫-ri0‍|u(s)|α2e-δ(t-s)ds+k2δ2≤e-δt|φ(0)|α2+∑i=1n‍βilieδri∫-ri0‍|u(s)|α2e-δ(t-s)ds+k2δ2,
where c2=2λ1αc0, from which one immediately concludes that
(43)|u(t)|α2≤(1+∑i=1n‍1δβiλ1αeδri/2)∥φ∥𝒞2e-δt+K2δ2.
This completes the proof of the lemma.


## 3.2. Existence of Pullback Attractor

Denote by𝒳α the space Cb(ℝ;Hα) which is equipped with the compact-open metric ϱ. Let θ=θt be the translation operator on 𝒳α.SetΣ=ℋ𝒳0(g), where g∈𝒳0:=Cb(ℝ;H) is the function in (1). For each (h,φ)∈Σ×𝒞, consider the Cauchy problem:
(44)ddtu(t)+Au(t)=F(u(t-r1),…,u(t-rn))+h(t),ddtu(t)+Au(t)=F(u(t-r1),…5,u(t-rn))t>0,u0=φ.
By Theorem 5 we know that, under the hypotheses of Lemma 9, (44) has a unique global solution u(t)=u(t;h,φ). Moreover, using some standard argument, it can be easily shown that u(t;h,φ) is continuous in (t,h,φ).Define a continuous mappingℝ+×Σ×𝒞→𝒞 as follows:
(45)Φ(t,h)φ=ut,∀(t,h,φ)∈ℝ+×Σ×𝒞,
where ut is the lifting of the solution u(t)=u(t;h,φ) of (44) in 𝒞. Then, Φ is a cocycle on 𝒞 with the base space Σ and driving system θ.Lemma 10.
Assume the hypotheses in Lemma9. Then, there exists a bounded uniformly (with respect to h∈Σ) absorbing set 𝒰⊂𝒞 for the system Φ.Proof.
It is clear that the estimate in (33) holds true for solutions of the system (44).
Letρα2=1+C2, where C2 is the positive number in (33). Given any bounded set D in 𝒞, by (33) we deduce that there exists a t0=t0(D)>r>0 such that
(46)|u(t;h,φ)|α≤ρα,∀t>t0
for all φ∈D and h∈Σ. It follows that 𝒰=B𝒞(0,ρα) is a uniformly absorbing set as we desired.Remark 11.
By using the smoothing property of the operatorA, it can be easily shown that the set
(47){u(t;h,φ)∣t>t0,φ∈D,h∈Σ}
is contained in a compact subset of Hα.Now we state and prove the existence result on pullback attractors.Theorem 12.
Assume the hypotheses in Lemma9. Then Φ has a unique global pullback attractor A={A(h)}h∈Σ.Proof.
By virtue of Theorem8, it suffices to show that Φ has a compact uniformly (with respect h∈Σ) absorbing set.
Let(48)M=⋃h∈Σ‍(⋃t≥t0(𝒰)+r‍Φ(t,h)𝒰),N=⋃h∈Σ‍(⋃t≥t0(𝒰)+2r+1‍Φ(t,h)𝒰),
where t0:=t0(𝒰) is chosen such that (46) holds for 𝒰. Then, we infer from Lemma 10 that
(49)N⊂M⊂𝒰:=B𝒞(0,ρα);
moreover, N is a uniformly (with respect h∈Σ) absorbing set in 𝒞.
To complete the proof of the theorem, there remains to check thatN is relatively compact in 𝒞. For this purpose, by Remark 11 and the classical Ascoli-Arzela theorem, one only needs to verify that N consists of a family of equicontinuous functions.
For anyψ∈N, by definition it can be easily seen that there exist h∈Σ and φ∈M such that
(50)ψ=Φ(r+1,h)φ=ur+1,
where ut is the lifting of the solution u(t)=u(t;h,φ) of (44). Note that if s∈[-r,0], then r+1+s∈[1,r+1]. Therefore, to prove the equicontinuity of N, we need to check that there exist L,γ>0 independent of h∈Σ and φ∈M such that
(51)|u(t′)-u(t)|α≤L|t′-t|γ,1≤t≤t′≤r+1.
For convenience, we rewritet′ as t+δ. Then, by (14) we have
(52)|u(t+δ)-u(t)|α≤J1+J2+J3,
where
(53)J1=|(E(δ)-I)(E(t)φ(0))|α,J2=∫0t‍|(E(δ)-I)E(t-s)G(s)|αds,J3=∫tt+δ‍|E(t+δ-s)G(s)|αds,G(s)=F(u(s-r1),…,u(s-rn))+h(s).
Since ∥φ∥𝒞≤ρα, by Lemma 9 we see that
(54)|u(s)|α2≤(C1+1)ρα2,∀s≥-r.
Hence one has
(55)|G(s)|≤∑i=1n‍βi|u(s-ri)|α+k1+|h(s)|≤(∑i=1n‍βi)C1+1ρα+K1+g0∶=Cα
for all s≥0, where g0=sups∈ℝ|g(s)|. We fix a 0<γ<1/2. Then, by using (9) and (10) we deduce, recalling that t≥1,
(56)J1≤|(E(δ)-I)(AαE(t)φ(0))|≤Kγδγ|Aγ+αE(t)φ(0)|≤KγδγMγt-γe-at|φ(0)|α≤KγδγMγt-γe-atρα≤C3δγ,J2≤∫0t‍|(E(δ)-I)AαE(t-s)G(s)|ds≤∫0t‍Kγδγ|Aγ+αE(t-s)G(s)|ds≤KγδγMα+γCα∫0t‍(t-s)-(γ+α)e-a(t-s)ds≤C4δγ,J3≤∫tt+δ‍|AαE(t+δ-s)G(s)|ds≤Cα∫tt+δ‍Mα(t+δ-s)-αe-a(t+δ-s)ds≤CαMα∫0δ‍(δ-s)-αe-a(δ-s)ds≤C5′δ1-α≤C5δγ.
Now by (52) one concludes that
(57)|u(t′)-u(t)|α=|u(t+δ)-u(t)|α≤(C3+C4+C5)δγ=L|t′-t|γ.
Note that both L and γ are independent of h∈Σ and φ∈M.


## 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

LetX be a complete metric space with metric d, and let 𝒳 be the space Cb(ℝ;X) which is always equipped with the compact-open metric ϱ. Denote by θ the translation operator on 𝒳.Definition 13.
A functionu∈𝒳 is said to be locally almost periodic if, for any ε,T>0, there exists an l>0 such that, for every s∈ℝ, one can find on any segment of length l a number τ=τ(s) such that
(58)max|t|≤Td(θsu(t),θτu(t))<ε.One easily verifies the validity of the following easy proposition, which actually gives another equivalent definition for locally almost periodic functions.Proposition 14.
u
∈
𝒳 is locally almost periodic If, for every ε>0, there exists an l>0 such that, for every s∈ℝ, one can find on any segment of length l a number τ=τ(s) such that
(59)ϱ(θsu,θτu)<ε.Proposition 15.
Ifu is a locally almost periodic function, then the following affirmations hold: (1)
uis uniformly continuous onℝ;(2)
U:={u(t)∣t∈ℝ}¯is compact inX.Proof.
(1) Letε>0 be given arbitrarily. We need to prove that there exists a 0<δ<1 such that
(60)d(u(s1),u(s2))<ε
for all s1,s2∈ℝ with |s1-s2|<δ.
By local almost periodicity ofu, there exists an l>0 such that, for any s∈ℝ, there exists a τ=τ(s)∈[0,l] such that
(61)d(u(s+t),u(τ+t))<ε3,∀t∈[-1,1].
Since u is uniformly continuous on the interval [-1,1], one can find a positive number δ<1 such that
(62)d(u(s),u(t))<ε3,∀s,t∈[-1,l+1],|s-t|<δ.
We show that δ satisfies (60).
Indeed, for anys1,s2∈ℝ with |s1-s2|<δ, we can pick an s∈ℝ such that s1,s2∈[s-1,s+1]. We may write s1 and s2 as
(63)s1=s+t1,s2=s+t2.
Then, by (61), there exists a τ∈[0,l] such that
(64)d(u(s+ti),u(τ+ti))<ε3,i=1,2.
Now we have that
(65)d(u(s1),u(s2))=d(u(s+t1),u(s+t2))≤d(u(s+t1),u(τ+t1))+d(u(s+t2),u(τ+t2))+d(u(τ+t1),u(τ+t2))≤ε3+ε3+ε3=ε.
(2) To prove the compactness ofU, we employ the Kuratowski’s measure β of noncompactness on X which is defined as follows: for any K⊂X,
(66)β(K)={⋃‍1≤i≤ninfδ>0∣thereexistafinitenumberofinfδ>055.ballsB1,…,BninXwithradiusinfδ>055.δsuchthatK⊂⋃‍1≤i≤nBi}.
It is well known that K is precompact if and only if β(K)=0.
In what follows we show thatβ(U)=0, thus proving what we desired.
Letε>0 be given arbitrarily. Then, by local almost periodicity of u, there exists an l>0 such that, for any s∈ℝ, one can find on any segment of length l a number τ=τ(s) such that
(67)max|t|≤1d(θsu(t),θτu(t))<ε.
In particular, we see that, for any s∈ℝ, there exists a τ∈I:=[0,l] such that
(68)d(u(s),u(τ))<ε.
It then follows that
(69)U⊂{x∈X∣d(x,u(I))<ε},
where u(I)={u(t)∣t∈I}. Since u(I) is compact, one easily deduces by (69) that β(U)≤2ε. Hence, β(U)=0. 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 functionu∈𝒳 is locally almost periodic if and only if  ℋ𝒳(u) is minimal under the Bebutov’s dynamical system.Proof.
“⇐”  Assume that ℋ𝒳(u) is minimal. Then, by the basic knowledge in the theory of dynamical systems (see, e.g., [11, 16]), we know that every v∈ℋ𝒳(u) is almost recurren; that is, for any ε>0, there exists an l>0 such that one can find on any segment of length l a number τ such that
(70)ϱ(v,θτv)<ε.
By a simple argument via contradiction, one can easily show that the length l of the segments can be independent of v (see also the work of Birkhoff [16]). Thanks to Proposition 14, one immediately concludes by (70) and the independence of l on v that u is locally almost periodic.
“⇒"  Conversely, if u is locally almost periodic, then by Proposition 15  u is uniformly continuous on ℝ with {u(t)∣t∈ℝ} being precompact in X. Therefore, by the classical Ascoli-Arzela theorem, one deduces that, for any sequence τn∈ℝ, the sequence θτnu has a subsequence θτnku that converges uniformly on any compact interval I to a function v∈𝒳. Hence, we have
(71)limk→∞ϱ(θτnku,v)=0.
This implies that the hull ℋ𝒳(u) is compact.
On the other hand, by the local almost periodicity ofu and Proposition 14, we see that u is an almost recurrent point of the system θ. Since ℋ𝒳(u) is compact, by adopting some argument as in [16], it can be easily shown that ℋ𝒳(u) 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 thatF satisfies (H1). Then, if g is locally almost periodic, (1) has at least one locally almost periodic solution.Proof.
Denote by𝒳α the space Cb(ℝ;Hα) which is equipped with the compact-open metric ϱ. Denote by θ the translation operator on 𝒳α.
LetΦ be the cocycle on the phase space 𝒞:=C([-r,0];Hα) generated by (44) (see (45) for the definition). Define the skew-product flow π:R+×Σ×𝒞→Σ×𝒞 as follows:
(72)π(t,h,φ):=(θth,Φ(t,h)φ),∀(h,φ)∈Σ×𝒞,t≥0.
Then, as in the proof of Theorem 12, one can easily show that π has a global attractor 𝒜 in Σ×𝒞 which is the maximal compact invariant set that attracts each bounded subset of Σ×𝒞. Further, by a recurrence theorem due to Birkhoff and Bebutov (see, e.g., [12]), 𝒜 contains a compact minimal invariant set ℳ of the system π.
Fix a(h0,φ0)∈ℳ. Then, by the invariance property of ℳ, there exists a trajectory γ of π which is defined on the whole line ℝ and contained in ℳ such that γ(0)=(h0,φ0). Let γ(t)=(θth0,vt). Define a function v as
(73)v(t)=vt(0),t∈ℝ.
Then, one easily sees that v is an entire solution of the equation in (44) with h therein replaced by h0. We also infer from the proof of Theorem 12 that v is uniformly continuous on ℝ with {v(t)∣t∈ℝ} being contained in a compact subset of Hα. Therefore by the classical Ascoli-Arzela theorem, the hull ℋ𝒳α(v) in 𝒳α is compact.
Letℑ:=ℋ𝒳0×𝒳a(h0,v). Note that if (h,w)∈ℑ, then it solves the following equation:
(74)ddtw(t)+Aw(t)=F(w(t-r1),…,w(t-rn))+h(t),t∈R.
Observe that(75)Pℑ=ℋ𝒳0(h0)=Σ,Qℑ=ℋ𝒳α(v),
where P and Q are projection operators from 𝒳0×𝒳a to 𝒳0 and 𝒳α, respectively. Since both Σ and ℋ𝒳α(v) are compact, we deduce that ℑ is compact.
Now we show thatℑ is minimal. Suppose the contrary. Then, it would contain a proper compact invariant subset K. Let
(76)𝒦={h,w0∣(h,w)∈K},
where w0∈𝒞 is the restriction of w on [-r,0],
(77)w0(s)=w(s),s∈[-r,0].
It is trivial to check that 𝒦 is a proper compact invariant subset of ℳ, which contradicts to the minimality of ℳ.
By the first equation in (75), we see that there exists a u∈𝒳α such that (g,u)∈ℑ. Further, by the minimality of ℑ, one finds that ℋ𝒳0×𝒳a(g,u)=ℑ. Hence ℋ𝒳0×𝒳a(g,u) is minimal. It then follows immediately by Theorem 16 that u is a locally almost periodic function. By (69), we see that u is precisely an entire solution of (1).


## 4.1. Locally Almost Periodic Functions

LetX be a complete metric space with metric d, and let 𝒳 be the space Cb(ℝ;X) which is always equipped with the compact-open metric ϱ. Denote by θ the translation operator on 𝒳.Definition 13.
A functionu∈𝒳 is said to be locally almost periodic if, for any ε,T>0, there exists an l>0 such that, for every s∈ℝ, one can find on any segment of length l a number τ=τ(s) such that
(58)max|t|≤Td(θsu(t),θτu(t))<ε.One easily verifies the validity of the following easy proposition, which actually gives another equivalent definition for locally almost periodic functions.Proposition 14.
u
∈
𝒳 is locally almost periodic If, for every ε>0, there exists an l>0 such that, for every s∈ℝ, one can find on any segment of length l a number τ=τ(s) such that
(59)ϱ(θsu,θτu)<ε.Proposition 15.
Ifu is a locally almost periodic function, then the following affirmations hold: (1)
uis uniformly continuous onℝ;(2)
U:={u(t)∣t∈ℝ}¯is compact inX.Proof.
(1) Letε>0 be given arbitrarily. We need to prove that there exists a 0<δ<1 such that
(60)d(u(s1),u(s2))<ε
for all s1,s2∈ℝ with |s1-s2|<δ.
By local almost periodicity ofu, there exists an l>0 such that, for any s∈ℝ, there exists a τ=τ(s)∈[0,l] such that
(61)d(u(s+t),u(τ+t))<ε3,∀t∈[-1,1].
Since u is uniformly continuous on the interval [-1,1], one can find a positive number δ<1 such that
(62)d(u(s),u(t))<ε3,∀s,t∈[-1,l+1],|s-t|<δ.
We show that δ satisfies (60).
Indeed, for anys1,s2∈ℝ with |s1-s2|<δ, we can pick an s∈ℝ such that s1,s2∈[s-1,s+1]. We may write s1 and s2 as
(63)s1=s+t1,s2=s+t2.
Then, by (61), there exists a τ∈[0,l] such that
(64)d(u(s+ti),u(τ+ti))<ε3,i=1,2.
Now we have that
(65)d(u(s1),u(s2))=d(u(s+t1),u(s+t2))≤d(u(s+t1),u(τ+t1))+d(u(s+t2),u(τ+t2))+d(u(τ+t1),u(τ+t2))≤ε3+ε3+ε3=ε.
(2) To prove the compactness ofU, we employ the Kuratowski’s measure β of noncompactness on X which is defined as follows: for any K⊂X,
(66)β(K)={⋃‍1≤i≤ninfδ>0∣thereexistafinitenumberofinfδ>055.ballsB1,…,BninXwithradiusinfδ>055.δsuchthatK⊂⋃‍1≤i≤nBi}.
It is well known that K is precompact if and only if β(K)=0.
In what follows we show thatβ(U)=0, thus proving what we desired.
Letε>0 be given arbitrarily. Then, by local almost periodicity of u, there exists an l>0 such that, for any s∈ℝ, one can find on any segment of length l a number τ=τ(s) such that
(67)max|t|≤1d(θsu(t),θτu(t))<ε.
In particular, we see that, for any s∈ℝ, there exists a τ∈I:=[0,l] such that
(68)d(u(s),u(τ))<ε.
It then follows that
(69)U⊂{x∈X∣d(x,u(I))<ε},
where u(I)={u(t)∣t∈I}. Since u(I) is compact, one easily deduces by (69) that β(U)≤2ε. Hence, β(U)=0. 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 functionu∈𝒳 is locally almost periodic if and only if  ℋ𝒳(u) is minimal under the Bebutov’s dynamical system.Proof.
“⇐”  Assume that ℋ𝒳(u) is minimal. Then, by the basic knowledge in the theory of dynamical systems (see, e.g., [11, 16]), we know that every v∈ℋ𝒳(u) is almost recurren; that is, for any ε>0, there exists an l>0 such that one can find on any segment of length l a number τ such that
(70)ϱ(v,θτv)<ε.
By a simple argument via contradiction, one can easily show that the length l of the segments can be independent of v (see also the work of Birkhoff [16]). Thanks to Proposition 14, one immediately concludes by (70) and the independence of l on v that u is locally almost periodic.
“⇒"  Conversely, if u is locally almost periodic, then by Proposition 15  u is uniformly continuous on ℝ with {u(t)∣t∈ℝ} being precompact in X. Therefore, by the classical Ascoli-Arzela theorem, one deduces that, for any sequence τn∈ℝ, the sequence θτnu has a subsequence θτnku that converges uniformly on any compact interval I to a function v∈𝒳. Hence, we have
(71)limk→∞ϱ(θτnku,v)=0.
This implies that the hull ℋ𝒳(u) is compact.
On the other hand, by the local almost periodicity ofu and Proposition 14, we see that u is an almost recurrent point of the system θ. Since ℋ𝒳(u) is compact, by adopting some argument as in [16], it can be easily shown that ℋ𝒳(u) 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 thatF satisfies (H1). Then, if g is locally almost periodic, (1) has at least one locally almost periodic solution.Proof.
Denote by𝒳α the space Cb(ℝ;Hα) which is equipped with the compact-open metric ϱ. Denote by θ the translation operator on 𝒳α.
LetΦ be the cocycle on the phase space 𝒞:=C([-r,0];Hα) generated by (44) (see (45) for the definition). Define the skew-product flow π:R+×Σ×𝒞→Σ×𝒞 as follows:
(72)π(t,h,φ):=(θth,Φ(t,h)φ),∀(h,φ)∈Σ×𝒞,t≥0.
Then, as in the proof of Theorem 12, one can easily show that π has a global attractor 𝒜 in Σ×𝒞 which is the maximal compact invariant set that attracts each bounded subset of Σ×𝒞. Further, by a recurrence theorem due to Birkhoff and Bebutov (see, e.g., [12]), 𝒜 contains a compact minimal invariant set ℳ of the system π.
Fix a(h0,φ0)∈ℳ. Then, by the invariance property of ℳ, there exists a trajectory γ of π which is defined on the whole line ℝ and contained in ℳ such that γ(0)=(h0,φ0). Let γ(t)=(θth0,vt). Define a function v as
(73)v(t)=vt(0),t∈ℝ.
Then, one easily sees that v is an entire solution of the equation in (44) with h therein replaced by h0. We also infer from the proof of Theorem 12 that v is uniformly continuous on ℝ with {v(t)∣t∈ℝ} being contained in a compact subset of Hα. Therefore by the classical Ascoli-Arzela theorem, the hull ℋ𝒳α(v) in 𝒳α is compact.
Letℑ:=ℋ𝒳0×𝒳a(h0,v). Note that if (h,w)∈ℑ, then it solves the following equation:
(74)ddtw(t)+Aw(t)=F(w(t-r1),…,w(t-rn))+h(t),t∈R.
Observe that(75)Pℑ=ℋ𝒳0(h0)=Σ,Qℑ=ℋ𝒳α(v),
where P and Q are projection operators from 𝒳0×𝒳a to 𝒳0 and 𝒳α, respectively. Since both Σ and ℋ𝒳α(v) are compact, we deduce that ℑ is compact.
Now we show thatℑ is minimal. Suppose the contrary. Then, it would contain a proper compact invariant subset K. Let
(76)𝒦={h,w0∣(h,w)∈K},
where w0∈𝒞 is the restriction of w on [-r,0],
(77)w0(s)=w(s),s∈[-r,0].
It is trivial to check that 𝒦 is a proper compact invariant subset of ℳ, which contradicts to the minimality of ℳ.
By the first equation in (75), we see that there exists a u∈𝒳α such that (g,u)∈ℑ. Further, by the minimality of ℑ, one finds that ℋ𝒳0×𝒳a(g,u)=ℑ. Hence ℋ𝒳0×𝒳a(g,u) is minimal. It then follows immediately by Theorem 16 that u is a locally almost periodic function. By (69), we see that u 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.LetL be a differential operator on a bounded domain Ω⊂ℝN:
(78)Lu∶=-∑i,j=1N‍∂∂xi(aij(x)∂u∂xj)+a0(x)u,
where aij,a0∈L∞(Ω) and aij=aji for all 1≤i,j≤N. We assume that there exists a constant ν>0 such that, for a.e. x∈Ω,
(79)∑i,j=1N‍aij(x)ξiξj≥ν|ξ|2,∀ξ=(ξ1,…,ξN)∈ℝN
and that
(80)a0(x)≥0,a.e.x∈Ω¯.
Hence, L is uniformly elliptic on Ω.Letf: ℝN+1→ℝ be a locally Lipschitz continuous function. Consider the retarded parabolic equation on Ω:
(81)∂u∂t+Lu=f(u(x,t-r1),∇u(x,t-r2))+g(x,t)
associated with the homogeneous Dirichlet boundary condition:
(82)u|∂Ω=0,
where r1,r2>0 denote time lags.LetH=L2(Ω), V=H01(Ω). Define a symmetric bilinear form B(u,v) on V as follows:
(83)B(u,v)=∫Ω‍(aij(x)∂u∂xj∂v∂xj+a0(x)uv)dx,u,v∈V.
It is clear that B(u,v) is bounded and coercive. Thanks to the Lax-Milgram theorem, B(u,v) generates a self-adjoint positive-definite operator A on H with compact resolvent. Note that
(84)V=D(A1/2):=H1/2.We assume thatf satisfies the following linear growth condition. (F1)
There exist positive constantsb0,…,bN and k, such that
(85)|f(z)|≤∑i=0N‍bi|zi|+k,∀z=(z0,…,zN)∈ℝN+1.Then, one easily sees that the mappingF:V2→H defined by
(86)F(u,v)=f(u,∇v),u,v∈V,
makes sense and is locally Lipschitz. Setting g(t)=g(·,t), the problem (81)-(82) can be reformulated as an abstract equation in H as follows:
(87)dudt+Au=F(u(t-r1),u(t-r2))+g(t).
Now we are in a situation of (1).Sincef satisfies (F1), then one can easily verify  that the mapping F satisfies (H1) in the previous sections, and hence the abstract results obtained therein apply. In particular, we have the following.Theorem 18.
Assume thatf satisfies the linear growth condition (F1) with the positive constants bi’s therein satisfying
(88)b0λ1-1/2+∑i=1N‍bi<λ11/2,
where λ1 is the first eigenvalue of A. Let g∈Cb(ℝ;H) 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.



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