Nonlinear Analysis 72 (2010) 3566–3574 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Global attractor for some partial functional differential equations with finite delay✩ Honglian You ∗ , Rong Yuan School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China article abstract info Article history: Received 27 June 2009 Accepted 10 December 2009 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 condition 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 MSC: 35B41 34K30 35B40 Keywords: Global attractor Hille–Yosida Finite delay Point dissipative Introduction The purpose of this paper is to investigate the existence of a global attractor for the following partial differential equations with finite delay x (t ) = Ax(t ) + F (xt ), x0 = φ ∈ C t ≥ 0, (1.1) 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 norm φ = sup−r ≤θ ≤0 |φ(θ )|; the linear operator A : D(A) ⊂ E → E satisfies the following condition (H1) there exist two constants M ≥ and ω ∈ R such that (ω, +∞) ⊂ ρ(A) and (λI − A)−n L ≤ M (λ − ω)n , where ρ(A) is the resolvent set of A, λ > ω, · L denotes the operator norm; F : C → E is globally Lipschitz continuous, i.e., there exists a constant L > such that (H2) |F (φ1 ) − F (φ2 )| ≤ L φ1 − φ2 for any φ1 , φ2 ∈ C ✩ 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 doi:10.1016/j.na.2009.12.027 H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 3567 For every t ≥ 0, the history function xt ∈ C is defined by xt (θ ) = x(t + θ ), for θ ∈ [−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 = in Eq (1.1), i.e., x (t ) = f (xt ), (1.2) 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., x (t ) = f (t , xt ), (1.3) 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 A = 0, i.e., x (t ) = Ax + f (t , xt ) (1.4) 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 C0 contraction 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 A = 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 U (t )φ ≤ Q ( φ )e−β t + C∗ , t ≥ 0, (1.5) where the monotonic function Q and the positive constants β and C∗ are independent of φ ∈ C ; see [20] To get this estimate as (1.5) for U (t ), it is sufficient to give some reasonable assumptions on A; see (H3) and (H4) below The content of this paper is organized as follows In Section 2, we present some basic definitions and results about dissipative dynamics and integrated semigroup theory Section is 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 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 > A continuous function x : [−r , T ] → E is called an integral solution of Eq (1.1) if 3568 (i) H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 t x(s)ds ∈ D(A) for t ≥ 0; (ii) x(t ) = φ(0) + A( (iii) x0 = φ t t x(s)ds) + F (xs )ds; Remark 2.1 From (i) we know that if x is an integral solution of (1.1), then xt (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) t → S (t ) is strongly continuous; s (iii) S (s)S (t ) = (S (t + r ) − S (r )) dr, for all t , s ≥ Definition 2.3 ([24]) An integrated semigroup S (t ), t ≥ 0, is called locally Lipschitz continuous, if for all τ > there exists a constant l(τ ) > such that S (t ) − S (s) L ≤ 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 exists ω ∈ R such that (ω, +∞) ⊂ ρ(A) and there exists a strongly continuous exponentially bounded family S (t ), t ≥ 0, of bounded operators such that S (0) = and +∞ R(λ, A) := (λI − A)−1 = λ e−λt S (t )dt exists for all λ > 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 A0 of A in D(A): A0 = A on D(A0 ) = {x ∈ D(A); Ax ∈ D(A)} Proposition 2.1 ([11]) The part A0 of 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 ≥ Moreover, the author in [11] gives the relationship between S (t ) and T0 (t ): t S (t )x = lim λ→+∞ T0 (s)λ(λI − A)−1 xds, for x ∈ E , t ≥ (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 > For any given φ ∈ C with φ(0) ∈ D(A), the function x(·) := x(·, φ) : [−r , T ] → E is said to be an integral solution of Eq (1.1) with initial function φ at time t = 0, if x(t ) = T0 (t )φ(0) + φ(t ), t d dt S (t − τ )λ(λI − A)−1 F (xτ )dτ , 0 ≤ t ≤ T, (2.2) −r ≤ t ≤ Remark 2.3 From (2.1), we can rewrite (2.2) as following x(t ) = t T0 (t )φ(0) + lim φ(t ), λ→+∞ T0 (t − τ )λ(λI − A)−1 F (xτ )dτ , ≤ t ≤ T, (2.3) −r ≤ t ≤ Lemma 2.2 ([22, Proposition 2.3]) Under the assumption (H1) and (H2), if φ ∈ C with φ(0) ∈ D(A), then Eq (1.1) possesses a unique global integral solution x(·, φ) : [−r , +∞) → E with initial function φ at time t = 0, which can be expressed by (2.3) H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 3569 According to Remark 2.1, let us denote C0 = {φ ∈ C : φ(0) ∈ D(A)} Then, from Lemma 2.2, for any φ ∈ C0 , we define the following operator U (t ) on C0 by U (t )φ = xt (·, φ), t ≥ 0, (2.4) where x(·, φ) is the unique global integral solution of Eq (1.1) in Lemma 2.2 Moreover, U (t ), t ≥ 0, is a strongly continuous semigroup on C0 ; 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 set A is said to be a global attractor if A is 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 ≥ such that U (t ) is compact for t > t0 , (ii) U (t ) is point dissipative in X , then there exists a nonempty global attractor A in X The global attractor In this section, we apply Lemma 2.3 to 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 ≥ (H3) T0 ( t ) L ≤ e−αt for some constant α > 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 t x(t ) ≤ h(t ) + k(s)x(s)ds, t ∈ [t0 , T ), t0 where all the functions involved are continuous on [t0 , T ), T ≤ +∞, and k(t ) ≥ 0, then x(t ) satisfies t x(t ) ≤ h(t ) + h(s)k(s)e t s k(u)du ds, t ∈ [t0 , T ) t0 Proposition 3.1 Suppose that (H1)–(H3) hold Then, for any φ ∈ C0 , there exists a constant γ > α such that the integral solution x(·, φ) of Eq (1.1) satisfies the following inequality xt ≤ c1 eγ r α− Leγ r + eγ r φ − c1 α − Leγ r γr e(Le −α)t , t ≥ 0, (3.1) where c1 = |F (0)|, α = Leγ r Proof By (H2), for any φ ∈ C0 , we have |F (φ)| = |F (φ) − F (0) + F (0)| ≤ |F (0)| + |F (φ) − F (0)| ≤ c1 + L φ Instead of considering the norm xt directly, let us firstly estimate eγ · xt for some constant γ > α Case For ≤ t ≤ r, from the expression of the integral solution in (2.3), we have sup |eγ θ xt (θ )| = max |eγ θ φ(t + θ )|, sup |eγ θ xt (θ )| sup −r ≤θ≤−t −r ≤θ≤0 −t ≤θ≤0 ≤ max e−γ t φ , sup eγ θ e−α(t +θ ) |φ(0)| −t ≤θ≤0 + sup eγ θ lim −t ≤θ≤0 λ→+∞ t +θ e−α(t +θ−τ ) λ(λI − A)−1 L (c1 + L xτ )dτ 3570 H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 In the following, for simplicity, we take M = in (H1), i.e., (λI − A)−1 ≤ for some λ > ω λ−ω In fact, this can be done if we employ the renorming lemma in [26, Page 17] to introduce an equivalent norm in E Therefore, t +θ sup |eγ θ xt (θ )| ≤ max e−γ t φ , e−α t |φ(0)| + sup c1 e−α(t +θ ) eγ θ lim −r ≤θ≤0 λ→+∞ −t ≤θ≤0 t +θ sup Le−α(t +θ) eγ θ lim + λ→+∞ −t ≤θ≤0 λ eατ xτ dτ λ−ω t +θ = max e−γ t φ , e−αt |φ(0)| + sup c1 e−α(t +θ ) eγ θ −t ≤θ≤0 t +θ sup Le−α(t +θ) eγ θ + −t ≤θ≤0 λ eατ dτ λ−ω eατ dτ eατ xτ dτ t ≤ e−αt φ + c1 e−αt t eατ dτ + Le−α t = e−αt φ + c1 α eατ xτ dτ t (1 − e−αt ) + Le−αt eατ xτ dτ Case For t ≥ r, the integral solution does not include the initial part Thus, the estimate of eγ · xt is as following sup |eγ θ xt (θ )| = −r ≤θ≤0 |eγ θ x(t + θ )| sup 0≤t +θ≤t ≤ eγ θ e−α(t +θ) |φ(0)| sup 0≤t +θ≤t + sup t +θ eγ θ lim λ→+∞ 0≤t +θ≤t e−α(t +θ−τ ) λ(λI − A)−1 t +θ ≤ e−αt |φ(0)| + sup c1 e−α(t +θ ) eγ θ 0≤t +θ≤t t t eατ dτ + Le−α t c1 α + L xτ )dτ sup 0≤t +θ≤t ≤ e−αt φ + c1 e−αt = e−αt φ + eατ dτ + L (c1 t +θ Le−α(t +θ ) eγ θ eατ xτ dτ eατ xτ dτ t (1 − e−αt ) + Le−αt eατ xτ dτ Therefore, for t ≥ 0, we obtain t c1 sup |eγ θ xt (θ )| ≤ e−α t φ + (1 − e−α t ) + Le−α t α −r ≤θ≤0 eατ xτ dτ On the other hand, we have sup |eγ θ xt (θ )| = −r ≤θ≤0 sup eγ θ |xt (θ )| −r ≤θ≤0 −γ r sup |xt (θ )| ≥e −r ≤θ≤0 = e−γ r xt , which combines with (3.2) yields that e−γ r xt ≤ e−αt φ + c1 α t (1 − e−αt ) + Le−αt eατ xτ dτ So we get eα t x t ≤ eγ r φ + c1 α t (eαt − 1) + Leγ r eατ xτ dτ By the generalized Gronwall inequality in Lemma 3.1, we obtain that (3.2) H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 eα t x t ≤ eγ r φ + = eγ r φ + + = = c1 γ r α t e (e − 1) + Leγ r α α− Leγ r t γr s Le dτ ds ds t t γr e(α−Le )s ds − γrs e−Le ds c1 γ r α t γr e (e − 1) + eγ r φ (eLe t − 1) α α(α − Leγ r ) c1 eγ r γrs 3571 c1 Leγ r eγ r α e−Le α + α t c1 γ r α t γr e (e − 1) + Leγ r eγ r φ eLe t α c1 eγ r c1 γ r α s e (e − 1) e eγ r φ + c1 L γ r γ r Leγ r t e e e = eγ r φ + + t eα t − eLe c1 Leγ r eγ r α(α − Leγ r ) eα t + eγ r γrt − eα t + φ − c1 Leγ r eγ r eγ r φ − c1 α− γrt eLe α Leγ r Leγ r eLe −1 c1 Leγ r eγ r α(α − γrt Leγ r ) − c1 eγ r α eLe γrt Consequently, we get xt ≤ c1 eγ r α − Leγ r + eγ r φ − c1 γr e(Le −α)t , α − Leγ r t ≥ Remark 3.1 If F (0) = 0, then is an equilibrium of Eq (1.1) In this case, from 3.1 we know that, for any φ ∈ C0 , the integral solution x(·, φ) satisfies xt ≤ eγ r φ e(Le γ r −α)t , t ≥ Furthermore, if we require that α > Leγ r , then is globally exponentially asymptotically stable Lemma 3.2 Under the hypotheses in Proposition 3.1, further more, assume α > Leγ r , where γ is the constant given in Proposition 3.1 Then U (t ), t ≥ 0, is point dissipative Proof From Proposition 3.1, we know that, for any φ ∈ C0 , since α > Leγ r > 0, there exists a t0 = t0 (φ) > such that for t ≥ t0 , xt ≤ c1 eγ r α − Leγ r + (independent of φ) c eγ r c eγ r Therefore, BX0 0, α−1 Leγ r + ∩ X0 attracts each point of X0 , where BX0 0, α−1 Leγ r + denotes the open ball in C0 with c eγ r center and radius α−1 Leγ r + 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 ≥ (H4) T0 (t ) is compact for t > 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 of C0 In the following we use Ascoli–Arzelà theorem to show that {U (t )φn : n ∈ N} is pre-compact in C0 We achieve this by two steps: first, for any fixed θ ∈ [−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 and θ ∈ [−r , 0], by Lemma 2.2 and (2.3), we have t +θ (U (t )φn ) (θ ) = T0 (t + θ )φn (0) + lim λ→+∞ T0 (t + θ − τ )λ(λI − A)−1 F (xnτ )dτ , (3.3) where xn (·) is the integral solution of Eq (1.1) with initial function φn By (H4) and the boundedness of {φn }, we know that {T0 (t + θ )φn (0) : n ∈ N} (3.4) 3572 H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 > 0, we have is pre-compact With regard to the second term in (3.3), for sufficiently small t +θ T0 (t + θ − τ )λ(λI − A)−1 F (xnτ )dτ lim λ→+∞ t +θ− = T0 ( ) lim T0 (t + θ − τ − )λ(λI − A)−1 F (xnτ )dτ λ→+∞ t +θ + lim λ→+∞ t +θ− T0 (t + θ − τ )λ(λI − A)−1 F (xnτ )dτ Note that as {φn } is bounded in C0 , by Proposition 3.1, we have sup xnτ < ∞, n∈N τ ∈ [0, t ] (3.5) By (H2), we get |F (xnτ )| ≤ |F (0)| + L xnτ (3.6) Therefore, combining (3.5) and (3.6), (H3) and (H4), there exist some constants M1 , M2 > 0, such that t +θ lim λ→+∞ t +θ− T0 (t + θ − τ )λ(λI − A)−1 F (xnτ )dτ ≤ M1 , (3.7) and t +θ − lim λ→+∞ T0 (t + θ − τ − )λ(λI − A)−1 F (xnτ )dτ ≤ M2 , which implies that t +θ− T0 ( ) lim λ→+∞ T0 (t + θ − τ − )λ(λI − A)−1 F (xnτ )dτ : n ∈ N ⊂K , (3.8) where K is 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) t +θ2 + lim λ→+∞ t +θ1 − lim λ→+∞ T0 (t + θ2 − τ )λ(λI − A)−1 F (xnτ )dτ T0 (t + θ1 − τ )λ(λI − A)−1 F (xnτ )dτ = T0 (t + θ1 ) (T0 (θ2 − θ1 ) − I ) φn (0) t +θ2 + lim T0 (t + θ2 − τ )λ(λI − A)−1 F (xnτ )dτ + lim (T0 (t + θ2 − τ ) − T0 (t + θ1 − τ )) λ(λI − A)−1 F (xnτ )dτ , λ→+∞ t +θ t +θ1 λ→+∞ which leads to |(U (t )φn )(θ2 ) − (U (t )φn )(θ1 )| ≤ T0 (t + θ1 ) (T0 (θ2 − θ1 ) − I ) t +θ2 + lim λ→+∞ t +θ L |φn (0)| T0 (t + θ2 − τ )λ(λI − A)−1 F (xnτ ) dτ + (T0 (θ2 − θ1 ) − I ) lim λ→+∞ t +θ1 T0 (t + θ1 − τ )λ(λI − A)−1 F (xnτ ) dτ Since T0 (t ) is compact for t > 0, we know that the mapping t → T0 (t ) is norm-continuous for t > Putting T0 (t + θ1 ) (T0 (θ2 − θ1 ) − I ) = T0 (t + θ1 − δ)(T0 (θ2 − θ1 + δ) − T0 (δ)) for some δ ∈ (0, t − r ) Then T0 (θ2 − θ1 + δ) − T0 (δ) L → as θ2 → θ1 The third term approaches to as θ2 → θ1 due to the fact that T (t )x is continuous in t for each x ∈ E and H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 3573 t +θ limλ→+∞ |T0 (t + θ1 − τ )λ(λI − A)−1 F (xnτ )|dτ belongs to a compact subset of E Regarding the second term, thanks to the boundedness of |T0 (t + θ2 − τ )λ(λI − A)−1 f (xnτ )|, it also approaches to if θ2 → 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 of Lemmas 2.3, 3.2 and 3.3 Therefore, we omit its proof Theorem 3.1 Assume that (H1)–(H4) hold true If α > Leγ r , where γ is the constant given in Proposition 3.1, then Eq (1.1) has a nonempty global attractor A Remark 3.2 The conclusion of Theorem 3.1 holds under small delays, because of the relations γ > α > Leγ r Combine Remark 3.1 and Theorem 3.1, we obtain the following result Corollary 3.1 Under the assumptions of Theorem 3.1, in addition, F (0) = 0, then A = {0} is the unique global attractor of Eq (1.1) Remark 3.3 Theorem 3.1 is also valid for r = Indeed, let us consider the following Cauchy problem x (t ) = Ax(t ) + F (x(t )), x(0) = x0 ∈ E , t ≥ 0, (3.9) 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 t x(t ) = T0 (t )x0 + lim T0 (t − τ )λ(λI − A)−1 F (x(τ ))dτ , λ→+∞ t ≥ Define U (t ) : E → E as following U (t )x0 = x(t ), t ≥ (3.10) 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 in Proposition 3.1 can give an estimate of x(·) as following |x(t )| ≤ c1 α−L + |x0 | − c1 α−L e(L−α)t , t ≥ 0, (3.11) which implies the point dissipativeness of U (t ), where c1 = |F (0)|, α = L On the other hand, we can also obtain with a similar proof as that in Lemma 3.3 that U (t ), which is defined in (3.10), is compact for t > (3.12) Therefore, as a result of Lemma 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 If α > L, then Eq (3.9) has a nonempty global attractor A 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 )), w(0, t ) = w(π , t ) = 0, w(x, θ ) = φ(θ )(x), ≤ x ≤ π , t ≥ 0, t ≥ 0, ≤ x ≤ π , −r ≤ θ ≤ 0, (4.1) where µ > 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 = y − dy, y ∈ D(A), D(A) = {y ∈ C ([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] 3574 H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 We can see that D(A) = {y ∈ E : y(0) = y(π ) = 0} = E Setting u(t ) = w(·, t ) ∈ E and ut (θ ) = u(t + θ ), ut ∈ C , then Eq (4.1) can be rewritten as the following abstract Cauchy problem u (t ) = Au(t ) + F (ut ), u0 = ϕ ∈ C t ≥ 0, (4.2) In [19], the authors proved that, for the operator By = y with D(B) = D(A), (0, +∞) ⊂ ρ(B) and (λI − B)−1 L ≤ for λ > λ Therefore, (−µ, +∞) ⊂ ρ(A) and (λI − A)−1 L ≤ λ+µ for λ > −µ, (4.3) which implies that A is a Hille–Yosida operator Furthermore, denote B0 as the part of B on D(B), i.e., B0 (y) = B(y), for y ∈ D(B0 ) = {y ∈ D(B) : y (0) = y (π ) = 0} Then B0 is a densely defined Hille–Yosida operator and generates a compact C0 -semigroup TB0 (t ), t ≥ 0, on D(B) with TB0 (t ) L ≤ 1; see [28] Consequently, A0 , the part of A on D(A), generates a compact C0 -semigroup T0 (t ), t ≥ 0, such that T0 (t ) L ≤ e−µt (4.4) According to Theorem 3.1, if there exists a constant γ > µ such that Leγ r < 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 References [1] M Adimy, H Bouzahirb, K Ezzinbi, Existence for a class of partial functional differential equations with infinite delay, Nonlinear Anal 46 (2001) 91–112 [2] M Adimy, H Bouzahirb, K Ezzinbi, Local existence and stability for some partial functional differential equations with infinite delay, Nonlinear Anal 48 (2002) 323–348 [3] M Adimy, K Ezzinbi, Local existence and linearized stability for partial functional-differential equations, Dynam Systems Appl (3) (1998) 389–403 [4] M Adimy, M Laklach, K Ezzinbi, Non-linear semigroup of a class of abstract semilinear functional differential equations with a non-dense domain, Acta Math Sinica, Engl Ser Mar (Engl Ser.) 20 (5) (2004) 933–942 [5] M Alia, K Ezzinbi, Strong solutions for some nonlinear partial functional differential equations with infinite delay, Electron J Differential Equations (91) (2008) 19 [6] H Bouzahir, Semigroup approach to semilinear partial functional differential equations with infinite delay, J Inequal Appl (2007) 13 Art ID 49125 [7] K Ezzinbi, S Lalaoui Rhali, Positivity and stability for some partial functional differential equations, NoDEA Nonlinear Differential Equations Appl 10 (2003) 15–32 [8] L Hatvani, On the asymptotic stability for functional differential equations by Lyapunov functionals, Nonlinear Anal 40 (2000) 251–263 [9] L Hatvani, On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals, Trans Amer Math Soc 354 (9) (2002) 3555–3571 [10] J.M Tchuenche, M.A Liadi, Asymptotic behavior of an abstract delay-differential equation, Res J Appl Sci (1–4) (2006) 11–15 [11] H.R Thieme, Semiflows generated by Lipschtz perturbations of non-densely defined operators, Differential Integral Equations (6) (1990) 1035–1066 [12] C.C Travis, G.F Webb, Existence and stability for partial functional differential equations, Trans Amer Math Soc 200 (1974) 395–418 [13] T Caraballo, P.E Kloeden, Non-autonomous attractor for integro-differential evolution equations, Discrete Contin Dyn Syst Ser S (1) (2009) 17–36 [14] T Caraballo, P Marín-Rubio, J Valero, Autonomous and non-autonomous attractors for differential equations with delays, J Differential Equations 208 (1) (2005) 9–41 [15] T Caraballo, P Marín-Rubio, J Valero, Attractors for differential equations with unbounded delays, J Differential Equations 239 (2007) 311–342 [16] J.K Hale, Asymptotic Behavior of Dissipative Systems, in: Mathematical Surveys and Monographs, vol 25, American Mathematical Society, Providence, RI, 1988 [17] T Caraballo, M.J Garrido-Atienza, B Schmalfuss, J Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin Dyn Syst 21 (2) (2008) 415–443 [18] H Bouzahir, K Ezzinbi, Global attractor for a class of partial functional differential equations with infinite delay, in: T Faria, P Freitas, Cisbon (Eds.), Topics in Functional Difference Equations, in: Fields Institute Communications, vol 29, American Mathematical Society, Procidence, RI, 1999, pp 63–71 March, 2001 [19] G.D Prato, E Sinestrari, Differential operators with non dense domain, Ann Sc Norm Super Pisa Cl Sci 14 (2) (1987) 285–344 [20] A Miranville, S Zelik, Attractors for dissipative partial functional differential equations in bounded and unbounded domains, in: Handbook of Differential Equations: Evolutionary Equations, in: Handb Differ Equ., vol IV, Elsevier/North–Holland, Amsterdam, 2008, pp 103–200 [21] J Billotti, J.P LaSalle, Dissipative periodic processes, Bull Amer Math Soc 77 (1971) 1082–1088 [22] M Adimy, K Ezzinbi, J Wu, Center manifold and stability in critical cases for some partial functional differential equations, Int J Evol Equ (1) (2006) 69–95 [23] W Arendt, Resolvent positive operators and integrated semigroup, Proc London Math Soc 54 (3) (1987) 321–349 [24] H Kellermann, M Hieber, Integrated semigroup, J Funct Anal 15 (1989) 160–180 [25] C Corduneanu, Principles of Differential and Integral Equations, Allyn and Bacon, Boston, 1971 [26] A Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, in: Applied Mathematical Sciences, vol 44, Springer-Verlag, New York, 1983 [27] L Wang, D Xu, Asymptotic behavior of a class of reactionCdiffusion equations with delays, J Math Anal Appl 281 (2003) 439–453 [28] K.-J Engle, R Nagel, One Parameter Semigroups for Linear Evolution Equations, in: GTM, vol 194, Springer-Verlag, 2000 ...H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 3567 For every t ≥ 0, the history function xt... continuous function x : [−r , T ] → E is called an integral solution of Eq (1.1) if 3568 (i) H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 t x(s)ds ∈ D(A) for t ≥ 0; (ii) x(t ) = φ(0) +... φ) : [−r , +∞) → E with initial function φ at time t = 0, which can be expressed by (2.3) H You, R Yuan / Nonlinear Analysis 72 (2010) 3566–3574 3569 According to Remark 2.1, let us denote C0