Global Attractor for Partial Functional Differential Equations with State-Dependent Delay

Abstract

This work aims to investigate the existence of global attractors for a class of partial functional differential equations with state-dependent delay. Using the classic theory about global attractors in infinite dimensional dynamical systems, we obtain some sufficient conditions for guaranteeing the existence of a global attractor.

1. Introduction

Partial functional differential equations with state-dependent delay appear frequently in applications as models of various phenomena, such as biological, chemical, and physical systems, which are characterized by both spatial and temporal variables. For this reason, the study of this kind of equation has received much attention in recent years. For more details, see for instance [1–5] and the references therein. However, it is worth pointing out that all of the papers mentioned above are mainly devoted to the existence of solutions or mild solutions. The literature related to global attractors is limited.

It is known that the global attractor is a very useful tool, which is valid for more general situations than those for stability to study the asymptotical behavior. In the present paper, we are devoted to investigating the existence of a global attractor for a type of partial functional differential equations with state-dependent delay as follows: 𝑢^′(𝑡)=𝐴𝑢(𝑡)+𝐹𝑡,𝑢_{𝜌(𝑡,𝑢𝑡)}𝑢,𝑡≥0,₀=𝜙∈𝐶, (1) where 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)|, and   𝐴 is a linear operator on a Banach space 𝐸 satisfying the following well-known Hille-Yosida condition:(H₁) there exist 𝑀₀≥1 and 𝜔∈ℝ such that (𝜆𝐼−𝐴)^−𝑛≤𝑀₀(𝜆−𝜔)^𝑛for𝜆∈ℛ(𝐴),𝜆>𝜔, (2) where ℛ(𝐴) is the resolvent set of 𝐴. Consider that 𝜌∶[0,+∞)×𝐶→[0,+∞) satisfies the following properties.(H₂) Let ℒ(𝜌)={𝜌(𝑠,𝜙)∶(𝑠,𝜙)∈[0,+∞)×𝐶,𝜌(𝑠,𝜙)≥0}. There exists a continuous and bounded function 𝐽^𝜙∶ℒ(𝜌)→[0,+∞) such that 𝜑_{𝜌(𝑡,𝑥𝑡)}≤𝐽^𝜙(𝑡)𝜙_𝑡,∀𝑡∈ℒ𝜌. (3)

Consider that 𝐹∶[0,+∞)×𝐶→𝐸 satisfies the following properties.(H₃) (i) For every 𝜙∈𝐶, the function 𝑡→𝐹(𝑡,𝜙) is strongly measurable. (ii) For each 𝑡∈[0,+∞],    𝐹(𝑡,⋅)∶𝐶→𝐸 is continuous. (iii) There exist a positive constant 𝑐 and a bounded function 𝑚∶[0,+∞)→[0,+∞) such that  𝐹𝑡,𝜙≤𝑐+𝑚(𝑡)𝜙. (4)

For every 𝑡≥0, the history function 𝑥_𝑡∈𝐶 is defined by 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃),for𝜃∈[−𝑟,0]. (5) In the present paper, we will obtain some sufficient conditions for guaranteeing the existence of a global attractor to (1) with 𝐴 being a Hille-Yosida operator but not necessarily densely defined.

2. Preliminaries

We recall some definitions and results from the integrated semigroup.

Definition 1 (see [6]). Let 𝑇>0. A function 𝑥∶[−𝑟,𝑇]→𝐸 is said to be an integral solution of (1) if (i)∫^𝑡₀𝑥(𝑠)𝑑𝑠∈𝐷(𝐴) for 𝑡≥0,(ii)𝑥(𝑡)=𝜙(0)+𝐴(∫^𝑡₀𝑥(𝑠)𝑑𝑠)+∫^𝑡₀𝐹(𝑠,𝑥_{𝜌(𝑠,𝑥𝑠)})𝑑𝑠, (iii)𝑥₀=𝜙.

Remark 2. Clearly, if 𝑥 is an integral solution of (1), then 𝑥_𝑡(0)=𝑥(𝑡)∈𝐷(𝐴) for 𝑡∈[0,𝑇]. So 𝜙(0)∈𝐷(𝐴), which is a necessary condition for the existence of an integral solution.

Let us introduce the part 𝐴₀ of the operator 𝐴 in 𝐷(𝐴) which is defined by 𝐷𝐴₀=𝑥∈𝐷(𝐴)∶𝐴𝑥∈𝐷(𝐴,𝐴₀)𝑥=𝐴𝑥for𝑥∈𝐷𝐴₀. (6)

Lemma 3 (see [7]). 𝐴₀ generates a strongly continuous semigroup (𝑇₀(𝑡))_𝑡≥0 on 𝐷(𝐴).

Based on the previous abstract results, we give some concrete results for (1); see [6].

Definition 4. Let 𝑇>0. 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 (1) with initial function 𝜙 at 𝑡=0 if 𝑥(𝑡)=𝑇₀(𝑡)𝜙(0)+lim_𝜆→+∞^𝑡₀𝑇₀()()𝑡−𝑠×𝜆𝜆𝐼−𝐴⁻¹𝐹𝑠,𝑥_{𝜌(𝑠,𝑥𝑠)}𝜙(𝑡)𝑑𝑠,0≤𝑡≤𝑇,,−𝑟≤𝑡≤0. (7)

Lemma 5 (see [6]). Under the assumptions (H₁)–(H₃), if 𝜙∈𝐶 with 𝜙(0)∈𝐷(𝐴), then (1) posseses a unique global integral solution 𝑥(⋅,𝜙)∶[−𝑟,+∞)→𝐸 with initial function 𝜙 at 𝑡=0, which can be expressed by (7).

According to Remark 2, denote Σ₀={𝜙∈𝐶∶𝜙(0)∈𝐷(𝐴)}. Then from Lemma 5, for each 𝜙∈Σ₀, we define the following operator on Σ₀ by 𝑈(𝑡)𝜙=𝑥_𝑡⋅,𝜙,𝑡≥0, (8) where 𝑥_𝑡(⋅,𝜙) is a unique global integral solution of (1) in Lemma 5. Clearly, (𝑈(𝑡))_𝑡≥0 is a strongly continuous semigroup on Σ₀.

Definition 6 (see [8]). 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 7 (see [8]). 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 8 (see [9]). 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 attractor 𝒜 in 𝑋.

3. The Global Attractor

In this section, we will obtain the existence of a global attractor to (7) by using Lemma 8. For the convenience of the proof, we give some assumptions and lemmas.(H₄) For 𝐶₀-semigroup 𝑇₀(𝑡),𝑡≥0, there exists positive constant 𝛼 such that 𝑇₀(𝑡)≤𝑒^−𝛼𝑡for𝑡≥0. (9)(H₅)𝑇₀(𝑡) is compact for 𝑡>0.

Lemma 9 (see [10]). If 𝑥(𝑡)≤ℎ(𝑡)+^𝑡_𝑡0𝑘(𝑠)𝑥(𝑠)𝑑𝑠,𝑡∈𝑡₀,𝑇, (10) where all the functions involved are continuous on [𝑡₀,𝑇), 𝑇≤+∞, and 𝑘(𝑡)≥0, then 𝑥(𝑡) satisfies 𝑥(𝑡)≤ℎ(𝑡)+^𝑡_𝑡0ℎ(𝑠)𝑘(𝑠)𝑒^{∫𝑡𝑠𝑘(𝑢)𝑑𝑢}𝑑𝑠,𝑡∈𝑡₀,𝑇. (11)

Lemma 10. Assume that assumptions (H₁)–(H₄) hold. Then, for each 𝜙∈Σ₀, if 𝛼≠𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟, there exists a constant 𝛾>𝛼 such that the integral solution 𝑥(⋅,𝜙) of (1) satisfies the following inequality: 𝑥_𝑡≤𝑐𝑒^𝛾𝑟𝛼−𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟+𝑒^𝛾𝑟𝜙−𝑐𝛼−𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟𝑒^{(𝑚𝑎𝐽𝑎𝑒𝛾𝑟−𝛼)𝑡},𝑡≥0, (12) where 𝑚_𝑎=max_{𝑠∈[0,∞)}𝑚(𝑠) and 𝐽_𝑎=max_{𝑠∈ℒ(𝜌)}𝐽^𝜙(𝑠).

Proof. By (H₃), for each 𝜙∈Σ₀, we have 𝐹𝑡,𝜙≤𝑐+𝑚(𝑡)𝜙. (13)
In the following proof, for simplicity, take 𝑀₀=1,𝑛=1 in (H₁); then (𝜆𝐼−𝐴)⁻¹≤1𝜆−𝜔for𝜆>𝜔. (14) Instead of considering the norm ‖𝑥_𝑡‖ directly, we firstly estimate ‖𝑒^𝛾⋅𝑥_𝑡‖ for some constant 𝛾>𝛼.
Case 1. For 0≤𝑡≤𝑟, by (7), we have sup_{−𝑟≤𝜃≤0}𝑒^𝛾𝜃𝑥_𝑡(𝜃)=maxsup_{−𝑟≤𝜃≤−𝑡}𝑒^𝛾𝜃𝜙(𝑡+𝜃),sup_{−𝑡≤𝜃≤0}𝑒^𝛾𝜃𝑥_𝑡(𝜃𝑒)≤max^−𝛾𝑡𝜙,sup_{−𝑡≤𝜃≤0}𝑒^𝛾𝜃𝑒^{−𝛼(𝑡+𝜃)}𝜙(0)+sup_{−𝑡≤𝜃≤0}𝑒^𝛾𝜃lim_𝜆→+∞0^𝑡+𝜃𝑒^{−𝛼(𝑡+𝜃−𝑠)}𝜆(𝜆𝐼−𝐴)⁻¹×𝑐+𝑚(𝑠)𝐽^𝜙(𝑠)𝑥_𝑠𝑒𝑑𝑠≤max^−𝛾𝑡𝜙,𝑒^−𝛼𝑡𝜙(0)+sup_{−𝑡≤𝜃≤0}𝑐𝑒^{−𝛼(𝑡+𝜃)}𝑒^𝛾𝜃lim_𝜆→+∞0^𝑡+𝜃𝜆𝑒𝜆−𝜔^𝛼𝑠𝑑𝑠+sup_{−𝑡≤𝜃≤0}𝑚_𝑎𝐽_𝑎𝑒^{−𝛼(𝑡+𝜃)}𝑒^𝛾𝜃×lim_𝜆→+∞0^𝑡+𝜃𝜆𝑒𝜆−𝜔^𝛼𝑠𝑥_𝑠𝑑𝑠≤𝑒^−𝛼𝑡𝜙+𝑐𝑒^{−𝛼𝑡𝑡}₀𝑒^𝛼𝑠𝑑𝑠+𝑚_𝑎𝐽_𝑎𝑒^{−𝛼𝑡𝑡}₀𝑒^𝛼𝑠𝑥_𝑠𝑑𝑠=𝑒^−𝛼𝑡𝜙+𝑐𝛼1−𝑒^−𝛼𝑡+𝑚_𝑎𝐽_𝑎𝑒^{−𝛼𝑡𝑡}₀𝑒^𝛼𝑠𝑥_𝑠𝑑𝑠. (15)
Case 2. For 𝑡≥𝑟, we have sup_{−𝑟≤𝜃≤0}𝑒^𝛾𝜃𝑥_𝑡(𝜃)≤sup_{−𝑟≤𝜃≤0}𝑒^𝛾𝜃𝑒^{−𝛼(𝑡+𝜃)}𝜙(0)+sup_{−𝑟≤𝜃≤0}𝑒^𝛾𝜃lim_𝜆→+∞0^𝑡+𝜃𝑒^{−𝛼(𝑡+𝜃−𝑠)}𝜆(𝜆𝐼−𝐴)⁻¹×𝑐+𝑚_𝑎𝐽_𝑎𝑥_𝑠𝑑𝑠≤𝑒^−𝛼𝑡𝜙+𝑐𝛼1−𝑒^−𝛼𝑡+𝑚_𝑎𝐽_𝑎𝑒^{−𝛼𝑡𝑡}₀𝑒^𝛼𝑠𝑥_𝑠𝑑𝑠. (16) Therefore, from (15) and (16), for 𝑡≥0, we get sup_{−𝑟≤𝜃≤0}𝑒^𝛾𝜃𝑥_𝑡(𝜃)≤𝑒^−𝛼𝑡𝜙+𝑐𝛼1−𝑒^−𝛼𝑡+𝑚_𝑎𝐽_𝑎𝑒^{−𝛼𝑡𝑡}₀𝑒^𝛼𝑠𝑥_𝑠𝑑𝑠. (17) On the other hand, we have sup_{−𝑟≤𝜃≤0}𝑒^𝛾𝜃𝑥_𝑡(𝜃)=sup_{−𝑟≤𝜃≤0}𝑒^𝛾𝜃𝑥_𝑡(𝜃)≥sup_{−𝑟≤𝜃≤0}𝑒^−𝛾𝑟𝑥_𝑡(𝜃)=𝑒^−𝛾𝑟𝑥_𝑡, (18) which combines with (17) and yields 𝑒^−𝛾𝑟𝑥_𝑡≤𝑒^−𝛼𝑡𝜙+𝑐𝛼1−𝑒^−𝛼𝑡+𝑚_𝑎𝐽_𝑎𝑒^{−𝛼𝑡𝑡}₀𝑒^𝛼𝑠𝑥_𝑠𝑑𝑠. (19) So we get 𝑒^𝛼𝑡𝑥_𝑡≤𝑒^𝛾𝑟𝜙+𝑐𝛼𝑒^𝛼𝑡−1+𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟𝑡₀𝑒^𝛼𝑠𝑥_𝑠𝑑𝑠. (20) Using Lemma 9, we have 𝑒^𝛼𝑡𝑥_𝑡≤𝑒^𝛾𝑟𝜙+𝑐𝛼𝑒^𝛾𝑟𝑒^𝛼𝑡−1+𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟𝑡₀𝑒^𝛾𝑟𝜙+𝑐𝛼𝑒^𝛾𝑟𝑒^𝛼𝑠−1×𝑒^{∫𝑡𝑠𝑚𝑎𝐽𝑎𝑒𝛾𝑟𝑑𝑢}≤𝑑𝑠𝑐𝑒^𝛾𝑟𝑒^𝛼𝑡𝛼−𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟+𝑒^𝛾𝑟𝜙−𝑐𝛼−𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟𝑒^{𝑚𝑎𝐽𝑎𝑒𝛾𝑟𝑡}. (21) Thus, (12) holds.

Lemma 11. Assume that the conditions of Lemma 10 are satisfied; furthermore, 𝛼>𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟, where 𝛾 is the constant defined by Lemma 10. Then (𝑈(𝑡))_𝑡≥0 is point dissipative.

Proof. From Lemma 10, we find that for each 𝜙∈Σ₀, since 𝛼>𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟, there exits a 𝑡₀∶=𝑡₀(𝜙)>0 such that for 𝑡>𝑡₀, 𝑥_𝑡≤𝑐𝑒^𝛾𝑟𝛼−𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟+1independentof𝜙. (22) Therefore, 𝐵_𝑋00,𝑐𝑒^𝛾𝑟𝛼−𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟+1∩𝑋₀ (23) attracts each point of 𝑋₀, where 𝐵_𝑋0(0,(𝑐𝑒^𝛾𝑟/(𝛼−𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟))+1) denotes the open ball in Σ₀ with center 0 and radius (𝑐𝑒^𝛾𝑟/(𝛼−𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟))+1.

Now, we show the compactness of the operator 𝑈(𝑡).

Lemma 12. Assume that assumptions (H₁)–(H₅) hold. Then, 𝑈(𝑡) is compact for 𝑡>𝑟.

Proof. Let 𝑡>𝑟 and let {𝜙_𝑛} be any bounded sequence of Σ₀. We will use the Ascoli-Arzelà theorem to show that {𝑈(𝑡)𝜙_𝑛∶𝑛∈ℕ} is precompact in Σ₀ by two steps.
Step 1. Show that for any 𝜃∈[−𝑟,0], the set 𝑍(𝜃)=𝑈(𝑡)𝜙_𝑛(𝜃)∶𝑛∈ℕ (24) is precompact. For 𝑡>𝑟 and 𝜃∈[−𝑟,0], by (8), we have 𝑈(𝑡)𝜙_𝑛(𝜃)=𝑇₀(𝑡+𝜃)𝜙_𝑛(0)+lim_𝜆→+∞0^𝑡+𝜃𝑇₀()()𝑡+𝜃−𝑠×𝜆𝜆𝐼−𝐴⁻¹𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠, (25) where 𝑥^𝑛(⋅) is the integer solution of (1) with initial function 𝜙_𝑛. From (H₄) and the boundedness of {𝜙_𝑛}, we know that {𝑇₀(𝑡+𝜃)𝜙_𝑛(0)∶𝑛∈ℕ} are precompact. Now, considering the second term in (25), for sufficiently small 𝜀>0, we have lim_𝜆→+∞0^𝑡+𝜃𝑇₀(𝑡+𝜃−𝑠)𝜆(𝜆𝐼−𝐴)⁻¹𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠=𝑇₀(𝜀)lim_𝜆→+∞0^{𝑡+𝜃−𝜀}𝑇₀(𝑡+𝜃−𝑠−𝜀)𝜆(𝜆𝐼−𝐴)⁻¹×𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠+lim_𝜆→+∞^𝑡+𝜃_{𝑡+𝜃−𝜀}𝑇₀(𝑡+𝜃−𝑠)𝜆(𝜆𝐼−𝐴)⁻¹𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠. (26) Note that from Lemma 10, we have sup_𝑛∈ℕ𝑥^𝑛_𝑠<∞,𝑠∈[0,𝑡]. (27) By (H₂) and (H₃), we get 𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}≤𝑐+𝑚_𝑎𝐽_𝑎𝑥^𝑛_𝑠. (28) Therefore, there exist some constants 𝑀₁,𝑀₂>0 such that lim_𝜆→+∞0^{𝑡+𝜃−𝜀}𝑇₀(𝑡+𝜃−𝑠−𝜀)𝜆(𝜆𝐼−𝐴)⁻¹𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠≤𝑀₁,lim_𝜆→+∞^𝑡+𝜃_{𝑡+𝜃−𝜀}𝑇₀(𝑡+𝜃−𝑠)𝜆(𝜆𝐼−𝐴)⁻¹𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠≤𝑀₂𝜀 (29) which yields 𝑇₀(𝜀)lim_𝜆→+∞0^{𝑡+𝜃−𝜀}𝑇₀()()𝑡+𝜃−𝑠−𝜀×𝜆𝜆𝐼−𝐴⁻¹𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠∶𝑛∈ℕ⊂Γ_𝜀, (30) where Γ_𝜀 is a compact set. Thus, 𝑍(𝜃) is precompact.
Step 2. Show the equicontinuity of {𝑈(𝑡)𝜙_𝑛∶𝑛∈ℕ}. Let −𝑟≤𝜃₁<𝜃₂≤0; we have 𝑈(𝑡)𝜙_𝑛𝜃₂−𝑈(𝑡)𝜙_𝑛𝜃₁=𝑇₀𝑡+𝜃₂−𝑇₀𝑡+𝜃₁𝜙_𝑛(0)+lim_𝜆→+∞^𝑡+𝜃2₀𝑇₀𝑡+𝜃₂−𝑠𝜆(𝜆𝐼−𝐴)⁻¹𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠−lim_𝜆→+∞^𝑡+𝜃1₀𝑇₀𝑡+𝜃₁−𝑠𝜆(𝜆𝐼−𝐴)⁻¹𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠=𝑇₀𝑡+𝜃₁𝑇₀𝜃₂−𝜃₁−𝐼𝜙_𝑛(0)+lim_𝜆→+∞^𝑡+𝜃2_𝑡+𝜃1𝑇₀𝑡+𝜃₂−𝑠𝜆(𝜆𝐼−𝐴)⁻¹𝐹𝑥^𝑛_{𝜌(𝑠,𝑥𝑠)}𝑑𝑠+lim_𝜆→+∞^𝑡+𝜃1₀𝑇₀𝑡+𝜃₂−𝑠−𝑇₀𝑡+𝜃₁()−𝑠×𝜆𝜆𝐼−𝐴⁻¹𝐹𝑥^𝑛_{𝜌(𝑠,𝑥𝑠)}𝑑𝑠, (31) which leads to 𝑈(𝑡)𝜙_𝑛𝜃₂−𝑈(𝑡)𝜙_𝑛𝜃₁≤𝑇₀𝑡+𝜃₁𝑇₀𝜃₂−𝜃₁−𝐼+lim_𝜆→+∞^𝑡+𝜃2_𝑡+𝜃1𝑇₀𝑡+𝜃₂−𝑠𝜆(𝜆𝐼−𝐴)⁻¹×𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠+𝑇₀𝜃₂−𝜃₁−𝐼×lim_𝜆→+∞^𝑡+𝜃1₀𝑇₀𝑡+𝜃₁−𝑠𝜆(𝜆𝐼−𝐴)⁻¹×𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠. (32) Since the mapping 𝑡→𝑇₀(𝑡) is norm continuous for 𝑡>0, for some 𝛿∈(0,𝑡−𝑟), put 𝑇₀𝑡+𝜃₁𝑇₀𝜃₂−𝜃₁−𝐼=𝑇₀𝑡+𝜃₁−𝛿𝑇₀𝜃₂−𝜃₁+𝛿−𝑇₀(𝛿). (33) Then 𝑇₀𝜃₂−𝜃₁+𝛿−𝑇₀(𝛿)⟶0as𝜃₂⟶𝜃₁. (34) Thus 𝑇₀𝑡+𝜃₁𝑇₀𝜃₂−𝜃₁−𝐼⟶0as𝜃₂⟶𝜃₁. (35) By the boundedness of |𝑇₀(𝑡+𝜃₂−𝑠)𝜆(𝜆𝐼−𝐴)⁻¹𝐹(𝑥^𝑛_{𝜌(𝑠,𝑥𝑠)})|, then lim_𝜆→+∞^𝑡+𝜃2_𝑡+𝜃1𝑇₀𝑡+𝜃₂−𝑠𝜆(𝜆𝐼−𝐴)⁻¹×𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠⟶0as𝜃₂⟶𝜃₁. (36) Obviously, lim_𝜆→+∞∫^𝑡+𝜃1₀|𝑇₀(𝑡+𝜃₁−𝑠)𝜆(𝜆𝐼−𝐴)⁻¹𝐹(𝑥^𝑛_{𝜌(𝑠,𝑥𝑠)})|𝑑𝑠 belongs to a compact subset of 𝐸; we have 𝑇₀𝜃₂−𝜃₁−𝐼×lim_𝜆→+∞^𝑡+𝜃1₀𝑇₀𝑡+𝜃₁−𝑠𝜆(𝜆𝐼−𝐴)⁻¹×𝐹𝑥^𝑛_{𝜌𝑠,𝑥𝑠}𝑑𝑠⟶0as𝜃₂⟶𝜃₁. (37) Hence {𝑈(𝑡)𝜙_𝑛∶𝑛∈ℕ} is equicontinuity.

Here, we state our main theorem of this paper, which is an immediate consequence of Lemmas 8, 11, and 12.

Theorem 13. Assume that assumptions (H₁)–(H₅) hold. If 𝛼>𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟, then (1) has a nonempty global attractor 𝒜.

As applications, we give the following example.

Let 𝐸=𝐶([0,𝜋],ℝ) and 𝐶=𝐶([−𝑟,0],𝐸).𝜕𝑢()=𝜕𝜕𝑡𝑡,𝑥²𝜕𝑥²𝑢()()(𝑢(𝑢()()𝑢()(𝜃)𝑡,𝑥−𝜇𝑢𝑡,𝑥+𝑓𝑡−𝑟,𝑥)),0≤𝑥≤𝜋,𝑡≥0,𝑡,0=𝑢𝑡,𝜋=0,𝑡≥0,𝜃,𝑥=𝜙𝑥,0≤𝑥≤𝜋,−𝑟≤𝜃≤0, (38) where 𝜇>0 is a constant, 𝑢(𝑡,⋅)∈𝐸,   and  𝑓∶𝐶→𝐸 satisfies (H₃). Define the operator 𝐴 by 𝐷(𝐴)=𝑞∈𝐶²])(0)(𝜋),𝐴𝑞([0,𝜋,ℝ∶𝑞=𝑞=0=𝑞^′′ (39) and 𝐹∶𝐶→𝐸 with 𝐹𝜑(𝑥)=𝑓𝜑(−𝑟)(𝑥),𝑥∈[0,𝜋]. (40) Then 𝐴 satisfies the Hille-Yosida condition in 𝐸 and (𝜆𝐼−𝐴)⁻¹≤1𝜆for𝜆>0. (41) Moreover, the part 𝐴₀ of 𝐴 in 𝐷(𝐴) is the infinitesimal generator of a strongly continuous semigroup (𝑇₀(𝑡))_𝑡≥0 on 𝐸 such that 𝑇₀(𝑡)≤𝑒^−𝑡for𝑡≥0. (42) According to Theorem 13, if there exist 𝛾>𝜇, 𝑚_𝑎 and 𝐽_𝑎 such that 𝑚_𝑎𝐽_𝑎𝑒^𝛾𝑟<𝜇, then (38) has a global attractor.

Acknowledgments

This paper is supported by the Natural Science Foundation of Jiangsu Education Office (11KJB110002), Postdoctoral Foundation of Jiangsu (1102096C), and Postdoctoral Foundation of China (2012M511296).