global attractor for some partial functional diferential equations with finite delay

Contents lists available atScienceDirect Nonlinear Analysis journal homepage:www.elsevier.com/locate/na Global attractor for some partial functional differential equations with finite de

Contents lists available atScienceDirect Nonlinear Analysis

journal homepage:www.elsevier.com/locate/na

Global attractor for some partial functional differential equations with finite delayI

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China

a r t i c l e i n f o

Article history:

Received 27 June 2009

Accepted 10 December 2009

MSC:

35B41

34K30

35B40

Keywords:

Global attractor

Hille–Yosida

Finite delay

Point dissipative

a b s t r a c t

In this paper, we study a class of partial functional differential equations with finite delay, whose linear part is not necessarily densely defined but satisfies the Hille–Yosida condi-tion Using the classical theory about global attractors in infinite dimensional dynamical systems, we establish some sufficient conditions for guaranteeing the existence of a global attractor under small delays

© 2009 Elsevier Ltd All rights reserved

1 Introduction

The purpose of this paper is to investigate the existence of a global attractor for the following partial differential equations with finite delay



x0(t) =Ax(t) +F(x t), t ≥0,

Here C := C([−r,0] ,E), r > 0, is the space of continuous functions from[−r,0]to the Banach space(E, | · |), equipped with the uniform normk φk =sup−r≤θ≤ 0| φ(θ)|; the linear operator A:D(A) ⊂E→E satisfies the following condition

(H1) there exist two constants M≥1 andω ∈R such that(ω, +∞) ⊂ ρ(A)and

(λ − ω)n, λ > ω,

whereρ(A)is the resolvent set of A,k · kLdenotes the operator norm;

F :C→E is globally Lipschitz continuous, i.e., there exists a constant L>0 such that

(H2) |F(φ1) −F(φ2)| ≤Lk φ1− φ2kfor anyφ1,φ2∈C

I Supported by National Natural Science Foundation of China and RFDP.

∗Corresponding author.

E-mail addresses:hlyou@mail.bnu.edu.cn (H You), ryuan@bnu.edu.cn (R Yuan).

0362-546X/$ – see front matter © 2009 Elsevier Ltd All rights reserved.

For every t≥0, the history function x t ∈C is defined by

x t(θ) =x(t+ θ), forθ ∈ [−r,0]

It is well known that retarded functional differential equations can describe some realistic situations in biological models such as the non-instant transmission due to physical reasons, pregnant period of species or latency time of disease Recently, with the development of theories on retarded functional differential equations and the operator theory, partial functional differential equations like(1.1)have been extensively considered; see for instance [1–12] and the references listed therein for more information However, it is worth pointing out that all of the papers mentioned above are mainly devoted to the existence and stability of equilibria or steady states

In the present paper, we are going to investigate the global dynamic behavior of Eq.(1.1) Explicitly, we consider the existence of a global attractor of Eq.(1.1) It is known that the global attractor is a very useful tool, which is valid for more general situations than those for stability, to investigate the asymptotical behavior As far as we know, only a few works dealt with this subject; see [13–16] For the case A=0 in Eq.(1.1), i.e.,

with finite delay, the authors in [14] showed some results on the existence of a global attractor of Eq.(1.2)without uniqueness, which generalize the results on uniqueness in [16] The authors in [14] also studied the non-autonomous case, i.e.,

by using the theory of pullback attractors which has been developed for stochastic and non-autonomous systems When the delay is infinite, some results of the existence of attractors for Eq.(1.2)and pullback attractors for Eq.(1.3)are established

in [15]

The theory of pullback attractors are also employed to deal with the case A6=0, i.e.,

In [13], the authors studied a class of integro-differential equations, which can be transformed into(1.4)with infinite delay

and A generating a C0contraction semigroup which is exponentially stable By the way, the recent paper [17] dealt with the existence of attractors for multi-valued non-autonomous and random dynamical systems and the results are then applied

to a random evolution equation with infinite delay

However, there are very few works about global attractors of Eq.(1.1)with A6=0 To our knowledge, only the paper [18] dealt with this problem, in which the authors obtained the existence of a global attractor of Eq.(1.1)with A not necessarily

densely defined and infinite delay

Motivated by the above results, in this paper we investigate the existence of a global attractor of Eq.(1.1)with finite delay

and A being a Hille–Yosida operator but not necessarily densely defined In fact, operators with non-dense domain occur in

many situations owing to restrictions on the space where the equations are considered For example, periodic continuous functions and Holder continuous functions are not dense in the space of continuous functions; see more examples in [19] Besides, the boundary conditions may also give rise to operators with non-dense domains, e.g., the domain of the Laplacian operator with Dirichlet boundary condition is not dense in the space of continuous functions Therefore, it is of great importance to study the existence of a global attractor of Eq.(1.1)with A non-densely defined.

To obtain the existence of a global attractor of Eq.(1.1), the part of difficulty is to prove the point dissipativeness of the

solution semigroup U(t)corresponding to Eq.(1.1) Fortunately, this can be done by proving a so-called dissipative estimate

of the following form

where the monotonic function Q and the positive constantsβand C∗are independent ofφ ∈C ; see [20] To get this estimate

The content of this paper is organized as follows In Section2, we present some basic definitions and results about dissipative dynamics and integrated semigroup theory Section3is devoted to establishing the main result of this paper Concretely, firstly, using a Gronwall inequality, we give a dissipative estimate of the integral solution, which leads to the point dissipativeness of the semigroup; secondly, we show the compactness of the strongly continuous semigroup generated

by the integral solution of Eq.(1.1); finally, the existence of a global attractor is a direct consequence of some fundamental theory in [21] In the last section we give an example to illustrate our result

2 Preliminaries

Firstly we recall some definitions and results from the integrated semigroup theory A is called a Hille–Yosida operator

on Banach space(E, | · |), if it satisfies the Hille–Yosida condition (H1)

Definition 2.1 ([ 22 ]) Let T >0 A continuous function x: [−r,T] →E is called an integral solution of Eq.(1.1)if

(i) R0t x(s)ds∈D(A)for t≥0;

(ii) x(t) = φ(0) +A(Rt

0x(s)ds) + Rt

0F(x s)ds;

(iii) x0= φ

Remark 2.1 From (i) we know that if x is an integral solution of(1.1), then x t(0) =x(t) ∈D(A)for t∈ [0,T] In particular,

φ(0) ∈D(A), which is a necessary condition for the existence of an integral solution

Definition 2.2 ([ 23 ]) An integrated semigroup is a family S(t), t≥0, of bounded linear operators on E with the following

properties:

(i) S(0) =0;

(ii) t7→S(t)is strongly continuous;

(iii) S(s)S(t) = Rs

0(S(t+r) −S(r))dr, for all t,s≥0

Definition 2.3 ([ 24 ]) An integrated semigroup S(t), t≥0, is called locally Lipschitz continuous, if for allτ >0 there exists

a constant l(τ) >0 such that

kS(t) −S(s)kL≤l(τ)|t−s| , for all t,s∈ [0, τ].

Definition 2.4 ([ 24 ]) An operator A is called a generator of an integrated semigroup if there existsω ∈ R such that

such that S(0) =0 and

R(λ,A) := (λI−A)− 1= λ

0

e−λt S(t)dt

exists for allλ > ω

Lemma 2.1 ([ 24 ]) The following assertions are equivalent:

(i) A is the generator of a locally Lipschitz continuous integrated semigroup;

(ii) A is a Hille–Yosida operator.

Now we introduce the part A0of A in D(A):

A0=A on D(A0) = {x∈D(A);Ax∈D(A)}.

Proposition 2.1 ([ 11 ]) The part A0of A in D(A)generates a strongly continuous semigroup on D(A).

Remark 2.2 From (H1), Lemma 2.1 and Proposition 2.1, we see that the operator A in(1.1) generates an integrated

semigroup S(t), t ≥ 0; A0 generates a C0-semigroup T0(t), t ≥ 0 Moreover, the author in [11] gives the relationship

between S(t)and T0(t):

S(t)x= lim

λ→+∞

Z t

0

T0(s)λ(λI−A)− 1xds, for x∈E,t≥0. (2.1) Based on the above abstract results, we give some concrete results for Eq.(1.1); see [3,22]

Definition 2.5 Let T >0 For any givenφ ∈C withφ(0) ∈D(A), the function x(·) :=x(·, φ) : [−r,T] →E is said to be an

integral solution of Eq.(1.1)with initial functionφat time t=0, if

x(t) =







T0(t)φ(0) + d

dt

Z t

0

S(t− τ)λ(λI−A)− 1F(xτ)dτ, 0≤t≤T,

Remark 2.3 From(2.1), we can rewrite(2.2)as following

x(t) =







T0(t)φ(0) + lim

λ→+∞

Z t

0

T0(t− τ)λ(λI−A)− 1F(xτ)dτ, 0≤t ≤T,

(2.3)

Lemma 2.2 ([ 22 , Proposition 2.3]) Under the assumption(H1)and(H2), ifφ ∈C withφ(0) ∈D(A), then Eq.(1.1)possesses a unique global integral solution x(·, φ) : [−r, +∞) →E with initial functionφat time t=0, which can be expressed by(2.3).

According toRemark 2.1, let us denoteC0 = { φ ∈C : φ(0) ∈D(A)} Then, fromLemma 2.2, for anyφ ∈C0, we define

the following operator U(t)onC0by

where x(·, φ)is the unique global integral solution of Eq.(1.1)inLemma 2.2 Moreover, U(t), t≥0, is a strongly continuous semigroup onC0; see [22]

At the end of this section we state some definitions about dissipative dynamics and a functional theory for the existence

of a global attractor, which is necessary for the proof of our main result; see [21,16]

Definition 2.6 ([ 16 ]) An invariant setAis said to be a global attractor ifAis a maximal compact invariant set which attracts

each bounded set B⊂X

Definition 2.7 ([ 16 ]) A semigroup U(t) :X→X , t ≥0, is said to be point dissipative if there is a bounded set B⊆X that

attracts each point of E under U(t)

Lemma 2.3 ([ 21 ]) If

(i) there is a t0≥0 such that U(t)is compact for t>t0,

(ii) U(t)is point dissipative in X ,

then there exists a nonempty global attractorAin X

3 The global attractor

In this section, we applyLemma 2.3to the strongly continuous semigroup U(t), t ≥ 0, which is defined by(2.4), to obtain the existence of a global attractor of Eq.(1.1) Firstly, in order to prove the point dissipativeness of U(t), which can

be obtained from a dissipative estimate as(1.5), we need an estimate on the integral solution of Eq.(1.1) For this purpose,

we make the following assumption on the C0-semigroup T0(t), t ≥0

(H3)kT0(t)kL≤e−αtfor some constantα >0

Furthermore, the following generalized Gronwall inequality, which can be found in [25, page 10], is crucial to obtain the estimate

Lemma 3.1 ([ 25 ]) If

x(t) ≤h(t) +

t0

k(s)x(s)ds, t∈ [t0,T),

where all the functions involved are continuous on[t0,T), T ≤ +∞, and k(t) ≥0, then x(t)satisfies

x(t) ≤h(t) +

t0

h(s)k(s)e

Rt

k(u)du ds, t∈ [t0,T).

Proposition 3.1 Suppose that(H1)–(H3)hold Then, for anyφ ∈ C0, there exists a constant γ > αsuch that the integral solution x(·, φ)of Eq.(1.1)satisfies the following inequality

kx tk ≤ c1eγr

α −Leγr +eγr



α −Leγr



e(Leγ r− α)t, t≥0, (3.1)

where c1= |F(0)|,α 6=Leγr

Proof By (H2), for anyφ ∈C0, we have

|F(φ)| = |F(φ) −F(0) +F(0)|

≤ |F(0)| + |F(φ) −F(0)|

≤ c1+Lk φk.

Instead of considering the normkx tkdirectly, let us firstly estimatekeγ ·x

tkfor some constantγ > α Case 1 For 0≤t≤r, from the expression of the integral solution in(2.3), we have

sup

−r≤ θ≤ 0

|eγ θx



sup

−r≤ θ≤−t

|eγ θφ(t+ θ)|, sup

−t≤ θ≤ 0

|eγ θx

t(θ)|





e−γtk φk, sup

−t≤ θ≤ 0

eγ θe− α(t+ θ)| φ(0)|

θ≤ e

γ θ lim λ→+∞

Z t+ θ

e−α(t+ θ−τ)k λ(λI−A)− 1kL(c1+Lkxτk )dτ



In the following, for simplicity, we take M=1 in (H1), i.e.,

k (λI−A)− 1k ≤ 1

In fact, this can be done if we employ the renorming lemma in [26, Page 17] to introduce an equivalent norm in E Therefore,

sup

−r≤ θ≤ 0

|eγ θx



e−γtk φk,e−αt| φ(0)| + sup

−t≤ θ≤ 0

c1e−α(t+ θ)eγ θ lim

λ→+∞

Z t+ θ

0

λ

λ − ωeατdτ

−t≤ θ≤ 0

Le−α(t+ θ)eγ θ lim

λ→+∞

Z t+ θ

0

λ

λ − ωeατkxτkdτ





e−γtk φk,e−αt| φ(0)| + sup

−t≤ θ≤ 0

c1e−α(t+ θ)eγ θZ t+θ

0

eατdτ

−t≤ θ≤ 0

Le−α(t+ θ)eγ θZ t+θ

0

eατkx

τkdτ



≤ e−αtk φk +c1e−αt

0

eατdτ +Le−αt

Z t

0

eατkx

τkdτ

= e−αtk φk +c1

α (1−e

− αt) +Le−αt

Z t

0

eατkx

τkdτ.

Case 2 For t ≥r, the integral solution does not include the initial part Thus, the estimate ofkeγ ·x

tkis as following sup

−r≤ θ≤ 0

|eγ θx

0 ≤t+ θ≤t

|eγ θx(t+ θ)|

0 ≤t+ θ≤t

eγ θe− α(t+ θ)| φ(0)|

0 ≤t+ θ≤t

eγ θ lim λ→+∞

Z t+ θ

0

e−α(t+ θ−τ)k λ(λI−A)− 1kL(c1+Lkxτk )dτ

≤ e−αt| φ(0)| + sup

0 ≤t+ θ≤t

c1e−α(t+ θ)eγ θZ t+θ

0

eατdτ + sup

0 ≤t+ θ≤t

Le−α(t+ θ)eγ θZ t+θ

0

eατkx

τkdτ

≤ e−αtk φk +c1e−αt

0

eατdτ +Le−αt

Z t

0

eατkx

τkdτ

= e−αtk φk +c1

α (1−e

− αt) +Le−αt

Z t

0

eατkx

τkdτ.

Therefore, for t ≥0, we obtain

sup

−r≤ θ≤ 0

|eγ θx

t(θ)| ≤e−αtk φk +c1

α (1−e

− αt) +Le−αt

0

eατkx

On the other hand, we have

sup

−r≤ θ≤ 0

|eγ θx

−r≤ θ≤ 0

eγ θ|x

t(θ)|

≥ e−γr sup

−r≤ θ≤ 0

|x t(θ)|

= e−γrkx tk ,

which combines with(3.2)yields that

e−γrkx tk ≤e−αtk φk +c1

α (1−e

− αt) +Le−αt

Z t

0

eατkx

τkdτ.

So we get

eαtkx tk ≤eγrh k φk +c1

α (eαt−1)

i +Leγr

Z t

0

eατkx

τkdτ.

By the generalized Gronwall inequality inLemma 3.1, we obtain that

eαtkx tk ≤eγrk φk +c1

αeγr(eαt−1) +Leγr

Z t

0

h

eγrk φk +c1

αeγr(eαs−1)

i

e

Rt Leγ rd τds

=eγrk φk +c1

αeγ

r(eαt−1) +Leγreγrk φkeLeγ r t

Z t

0

e−Leγ r s ds

+c1L

α eγreγreLeγ

r t

0

e(α−Leγ r)s ds−

Z t

0

e−Leγ r s ds



=eγrk φk +c1

αeγr(eαt−1) +eγrk φk(eLeγ

r t−1) + c1Leγreγr

α(α −Leγr)



eαt−eLeγ r t



− c1Leγreγr

αLeγr



eLeγ r t−1



=



c1eγr

c1Leγreγr

α(α −Leγr)



eαt+



eγrk φk − c1Leγreγr

α(α −Leγr) −

c1eγr

α



eLeγ r t

= c1eγr

α −Leγreαt+eγr



α −Leγr



eLeγ r t.

Consequently, we get

kx tk ≤ c1eγr

α −Leγr +eγr



α −Leγr



e(Leγ r− α)t, t≥0. 

Remark 3.1 If F(0) = 0, then 0 is an equilibrium of Eq.(1.1) In this case, from3.1we know that, for anyφ ∈ C0, the

integral solution x(·, φ)satisfies

kx tk ≤eγrk φke(Leγ r− α)t, t ≥0.

Furthermore, if we require thatα >Leγr, then 0 is globally exponentially asymptotically stable

Lemma 3.2 Under the hypotheses in Proposition 3.1 , further more, assume α > Leγr , where γ is the constant given in

Proposition 3.1 Then U(t), t≥0, is point dissipative.

Proof FromProposition 3.1, we know that, for anyφ ∈C0, sinceα >Leγr >0, there exists a t0=t0(φ) >0 such that for

t≥t0,

kx tk ≤ c1eγr

α −Leγr +1 (independent ofφ)

Therefore, B X0



0, c1 eγr

α−Leγ r +1



∩X0attracts each point of X0, where B X0



0, c1 eγr

α−Leγ r +1



denotes the open ball inC0with center 0 and radius c1 eγr

α−Leγ r +1 

Now we show the compactness of the operator U(t) To this end, we make the next assumption about the C0-semigroup

T0(t), t ≥0

(H4) T0(t)is compact for t>0

The following lemma is similar with [22, Theorem 2.7] But here, for the reader convenience, we give the details of its proof

Lemma 3.3 Under the assumptions(H1)–(H4), U(t)is compact for t >r.

Proof Let t > r and{ φn}be any bounded sequence ofC0 In the following we use Ascoli–Arzelà theorem to show that

{U(t)φn:n∈N}is pre-compact inC0 We achieve this by two steps: first, for any fixedθ ∈ [−r,0], the set

Z(θ) = {(U(t)φn) (θ) :n∈N}

is pre-compact; second,{U(t)φn:n∈N}is equicontinuous in[−r,0]

For t>r andθ ∈ [−r,0], byLemma 2.2and(2.3), we have

(U(t)φn) (θ) =T0(t+ θ)φn(0) + lim

λ→+∞

Z t+ θ

0

where x n(·)is the integral solution of Eq.(1.1)with initial functionφn

By (H4) and the boundedness of{ φn}, we know that

is pre-compact With regard to the second term in(3.3), for sufficiently small >0, we have

lim

λ→+∞

Z t+ θ

0

T0(t+ θ − τ)λ(λI−A)− 1F(x nτ)dτ

=T0() lim

λ→+∞

Z t+ θ−

0

T0(t+ θ − τ − )λ(λI−A)− 1F(x nτ)dτ

λ→+∞

Z t+ θ

t+ θ−

T0(t+ θ − τ)λ(λI−A)− 1F(x nτ)dτ.

Note that as{ φn}is bounded inC0, byProposition 3.1, we have

sup

n∈ N

By (H2), we get

Therefore, combining(3.5)and(3.6), (H3) and (H4), there exist some constants M1, M2>0, such that

λ→+∞

Z t+ θ

t+ θ−

T0(t+ θ − τ)λ(λI−A)− 1F(x nτ)dτ

and

λ→+∞

Z t+ θ−

0

T0(t+ θ − τ − )λ(λI−A)− 1F(x nτ)dτ

≤M2,

which implies that

T0()



lim

λ→+∞

Z t+ θ−

0

T0(t+ θ − τ − )λ(λI−A)− 1F(x nτ)dτ :n∈N



where Kis a compact set Consequently,(3.4),(3.7)and(3.8)show the pre-compactness of Z(θ)

To establish the equicontinuity of{U(t)φn:n∈N}in[−r,0], let−r≤ θ1< θ2≤0, we have

(U(t)φn)(θ2) − (U(t)φn)(θ1) =T0(t+ θ2)φn(0) −T0(t+ θ1)φn(0)

λ→+∞

Z t+ θ 2

0

T0(t+ θ2− τ)λ(λI−A)− 1F(x nτ)dτ

λ→+∞

Z t+ θ 1

0

T0(t+ θ1− τ)λ(λI−A)− 1

F(x nτ)dτ

=T0(t+ θ1) (T0(θ2− θ1) −I) φn(0)

λ→+∞

Z t+ θ 2

t+ θ 1

T0(t+ θ2− τ)λ(λI−A)− 1F(x nτ)dτ

λ→+∞

Z t+ θ 1

0

(T0(t+ θ2− τ) −T0(t+ θ1− τ)) λ(λI−A)− 1F(x nτ)dτ,

which leads to

| (U(t)φn)(θ2) − (U(t)φn)(θ1)| ≤ kT0(t+ θ1) (T0(θ2− θ1) −I)kL| φn(0)|

λ→+∞

Z t+ θ 2

t+ θ 1

T0(t+ θ2− τ)λ(λI−A)− 1F(x nτ) dτ

+ (T0(θ2− θ1) −I) lim

λ→+∞

Z t+ θ 1

0

T0(t+ θ1− τ)λ(λI−A)− 1F(x nτ) dτ.

Since T0(t)is compact for t >0, we know that the mapping t→T0(t)is norm-continuous for t>0 Putting

T0(t+ θ1) (T0(θ2− θ1) −I) =T0(t+ θ1− δ)(T0(θ2− θ1+ δ) −T0(δ))

for someδ ∈ (0,t−r) Then

kT0(θ2− θ1+ δ) −T0(δ)kL→0 asθ2→ θ1.

The third term approaches to 0 as θ → θ due to the fact that T(t)x is continuous in t for each x ∈ E and

limλ→+∞R0t+θ1|T0(t+ θ1− τ)λ(λI−A)− 1F(x nτ)|dτbelongs to a compact subset of E Regarding the second term, thanks

to the boundedness of|T0(t + θ2− τ)λ(λI−A)− 1f(x n

τ)|, it also approaches to 0 ifθ2 → θ1 Thus we have proved the equicontinuity on the right A similar argument deduces the equicontinuity on the left, which yields that{U(t)φn:n∈N}

is equicontinuous in[−r,0] 

Up until now, we can state our main theorem of this paper, which is an immediate consequence ofLemmas 2.3,3.2and

3.3 Therefore, we omit its proof

Theorem 3.1 Assume that(H1)–(H4)hold true If α > Leγr , whereγ is the constant given in Proposition 3.1 , then Eq.(1.1)

has a nonempty global attractorA.

Remark 3.2 The conclusion ofTheorem 3.1holds under small delays, because of the relationsγ > α >Leγr

CombineRemark 3.1andTheorem 3.1, we obtain the following result

Corollary 3.1 Under the assumptions of Theorem 3.1 , in addition, F(0) =0, thenA= {0}is the unique global attractor of Eq.

(1.1).

Remark 3.3. Theorem 3.1is also valid for r=0 Indeed, let us consider the following Cauchy problem



x0(t) =Ax(t) +F(x(t)), t ≥0,

where A : D(A) ⊂ E → E satisfies (H1), F : E → E satisfies (H2) Then Eq.(3.9)has a unique global integral solution

x(·) : [0, +∞) →E with initial value x0, which is given by

x(t) =T0(t)x0+ lim

λ→+∞

Z t

0

T0(t− τ)λ(λI−A)− 1F(x(τ))dτ, t≥0.

Define U(t) :E→E as following

Then U(t), t ≥0, is a strongly continuous semigroup on E.

Furthermore, assume that (H3) and (H4) are also fulfilled A similar argument as that inProposition 3.1can give an

estimate of x(·)as following

|x(t)| ≤ c1



|x0| − c1



which implies the point dissipativeness of U(t), where c1= |F(0)|,α 6=L.

On the other hand, we can also obtain with a similar proof as that inLemma 3.3that

U(t), which is defined in(3.10), is compact for t >0. (3.12) Therefore, as a result ofLemma 2.3,(3.11)and(3.12), we give the following theorem to end this section

Theorem 3.2 Assume that(H1)–(H4)hold true Ifα >L, then Eq.(3.9)has a nonempty global attractorA.

4 An example

Let E=C([0, π],R), the Banach space of continuous functions on[0, π]with the supreme norm, and C=C([−r,0] ,E) Consider the following reaction–diffusion equation

( wt(x,t) = wxx(x,t) − µw(x,t) +f(w(x,t−r)), 0≤x≤ π,t ≥0,

whereµ >0 is a constant,w(·,t) ∈E, f :C →E is globally Lipschitz continuous with Lipschitzian constant L For more

information about global attractors of reaction-diffusion equations, we refer to [27] and its bibliographies

Define



Ay=y00−dy, y∈D(A),

D(A) = {y∈C2([0, π] :R) :y(0) =y(π) =0} ,

and F:C→E with

F(ψ)(x) =f(ψ(−r)(x)), ψ ∈C, x∈ [0, π].

We can see that

D(A) = {y∈E:y(0) =y(π) =0} 6=E.

Setting u(t) = w(·,t) ∈ E and u t(θ) =u(t+ θ), u t ∈C , then Eq.(4.1)can be rewritten as the following abstract Cauchy problem



u0(t) =Au(t) +F(u t), t ≥0,

In [19], the authors proved that, for the operator By=y00with D(B) =D(A),

(0, +∞) ⊂ ρ(B) and k (λI−B)− 1kL≤ 1

λ forλ >0.

Therefore,

which implies that A is a Hille–Yosida operator.

Furthermore, denote B0as the part of B on D(B), i.e.,

B0(y) =B(y), for y∈D(B0) = {y∈D(B) :y00(0) =y00(π) =0}

Then B0 is a densely defined Hille–Yosida operator and generates a compact C0-semigroup T B0(t), t ≥ 0, on D(B)with

kT B0(t)kL≤1; see [28] Consequently, A0, the part of A on D(A), generates a compact C0-semigroup T0(t), t≥0, such that

According toTheorem 3.1, if there exists a constantγ > µsuch that Leγr < µ, then Eq.(4.1)has a global attractor

Acknowledgements

The authors would like to thank the referees for the careful reading of the manuscript, whose useful suggestions are highly appreciated

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