Exponential Attractor for a First-Order Dissipative Lattice Dynamical System

Abstract

We construct an exponential attractor for a first-order dissipative lattice dynamical system arising from spatial discretization of reaction-diffusion equations in ℝ^𝑘. And we obtain fractal dimension of the exponential attractor.

1. Introduction

Lattice systems arise in many applications, for example, in chemical reaction theory, image processing, pattern recognition, material science, biology, electrical engineering, laser systems, and so forth. A lattice dynamical system (LDS) is an infinite system of ordinary differential equations (lattice ODEs) or of difference equations. In some cases, they arise from spatial discretizations of partial differential equations (PDEs), but they possess their own form.

Let 𝑘∈ℕ be a fixed positive integer. Denoteℓ²={𝑢∣𝑢=(𝑢_𝑖)_𝑖∈ℤ^𝑘,𝑖=(𝑖₁,𝑖₂,…,𝑖_𝑘)∈ℤ^𝑘,𝑢_𝑖∈ℝ,_{𝑖∈ℤ𝑘}𝑢²_𝑖<+∞}, (1.1)where ℤ is the set of integers. Define a linear operator 𝐴 acting on ℓ² in the following way: for any 𝑢=(𝑢_𝑖)_𝑖∈ℤ^𝑘∈ℓ², 𝑖=(𝑖₁,𝑖₂,…,𝑖_𝑘)∈ℤ^𝑘,(𝐴𝑢_𝑖)_𝑖∈ℤ^𝑘=2𝑘𝑢_{(𝑖1,𝑖2,…,𝑖𝑘)}−𝑢_{(𝑖1−1,𝑖2,…,𝑖𝑘)}−𝑢_{(𝑖1,𝑖2−1,…,𝑖𝑘)}−…−𝑢_{(𝑖1,𝑖2,…,𝑖𝑘−1)}−𝑢_{(𝑖1+1,𝑖2,…,𝑖𝑘)}−𝑢_{(𝑖1,𝑖2+1,…,𝑖𝑘)}−…−𝑢_{(𝑖1,𝑖2,…,𝑖𝑘+1)}. (1.2)

In this paper, we will consider the following first-order lattice dynamical system:𝑢̇+𝐴𝑢+𝜆𝑢+𝑔(𝑢)=𝑞,𝑡>0,𝑢(0)=(𝑢_0𝑖)_𝑖∈ℤ^𝑘=𝑢₀, (1.3)where 𝜆>0, 𝑢=(𝑢_𝑖)_𝑖∈ℤ^𝑘, 𝐴𝑢=(𝐴𝑢_𝑖)_𝑖∈ℤ^𝑘, 𝑢̇=(𝑢̇_𝑖)_𝑖∈ℤ^𝑘 denote the first-order derivative, and 𝑔(𝑢)=(𝑔(𝑢_𝑖))_𝑖∈ℤ^𝑘, 𝑞=(𝑞_𝑖)_𝑖∈ℤ^𝑘ℓ². Then, problem (1.3) can be regarded as a discrete analogue of the following reaction diffusion equation in ℝ^𝑘:𝜕_𝑡𝑢−Δ𝑢+𝜆𝑢+𝑔(𝑢)=𝑞(𝑥),𝑡≥0. (1.4)One example is the Chafee-Infante equation.

Bates [1] and his collaborators made some results on a global attractor for lattice dynamical system (LDS). Zhou [2] applied them to a first-order dissipative lattice dynamical systems analogue to problem (1.3), proved the existence of the global attractor for the LDS, and considered the finite-dimensional approximation of the attractor. Wang [3] and Zhao and Zhou [4] studied asymptotic behavior of nonautonomous lattice systems. In standard definition of exponential attractor, a compact and positively invariant 𝑀 set is needed for the semigroup 𝑆(𝑡), and the system (𝑆(𝑡),𝑀) possesses a global attractor 𝒜. More specifically, the semigroup 𝑆(𝑡) is not compact for all positive 𝑡. So, it is difficult to find a compact and positively invariant 𝑀 which is not the attractor 𝒜. The first-order dissipative lattice dynamical systems analogue to problem (1.3) is such an example. Babin and Nicolaenko [5] consider reaction-diffusion systems in unbounded domains, prove the existence of exponential attractors for such systems, and estimate their fractal dimension. In [5], the compactness assumption plays a relatively minor role in the whole construction. In [6], Eden et al. provide constructions of exponential attractor for a Lipschitz 𝛼-contraction 𝑆 on a closed bounded 𝐵 that satisfies the discrete squeezing property, where 𝐵 is not assumed to be compact.

The main novelty of this work is that we make an improvement in the constructions of exponential attractors is indicated in [6] that if a map 𝑆 is asymptotically compact on a closed bounded 𝐵 that satisfies the discrete squeezing property, then 𝑆 possesses an exponential attractor. 𝑆 is not assumed to be 𝛼-contraction in the result. We apply the result to study an exponential attractor for a first-order dissipative lattice dynamical system. We not only construct an exponential attractor for the lattice dynamical system and consider its finite-dimensional approximation, but also obtain an upper bound of its fractal dimension.

2. A Key Theorem

Let 𝐸 be a separable Hilbert space with the norm ‖⋅‖, 𝐵⊆𝐸 be nonempty closed bounded set, and 𝑆∶𝐵→𝐵 be a Lipschitz continuous map with Lipschitz constant 𝐿. In this paper, we will always denote dist the Hausdorff semi-distance of sets as follows:dist(𝐵,𝐶)=sup_𝑥∈𝐵inf_𝑦∈𝐶‖𝑥−𝑦‖,forany𝐵,𝐶⊂𝐸. (2.1)

Definition 2.1. 𝑆 is asymptotically compact on 𝐵 if for any {𝑥_𝑛}_𝑛≥1⊆𝐵, there is a convergent subsequence of {𝑆^𝑛𝑥_𝑛} in 𝐸.

Remark 2.2. If 𝑆 is an 𝛼-contraction on 𝐵, then 𝑆 is asymptotically compact on 𝐵.

Definition 2.3. 𝑆 is said to satisfy the discrete squeezing property on 𝐵 if there exists an orthogonal projection 𝑃_𝑁 of rank 𝑁 such that for every 𝑢 and 𝑣 in 𝐵,‖𝑃_𝑁(𝑆𝑢−𝑆𝑣)‖≤‖(𝐼−𝑃_𝑁1)(𝑆𝑢−𝑆𝑣)‖⇒‖𝑆𝑢−𝑆𝑣‖≤8‖𝑢−𝑣‖. (2.2)

Definition 2.4. A compact set 𝑀 is called as an exponential attractor for (𝑆,𝐵) if (i)𝐴⊆𝑀⊆𝐵, where 𝐴 is the global attractor;(ii)𝑆𝑀⊆𝑀, that 𝑀 is positively invariant under 𝑆;(iii)𝑀 has finite fractal dimension; and(iv) there exist universal constants 𝑐₁, 𝑐₂ such that for every 𝑢∈𝐵, for every natural number 𝑛, dist(𝑆^𝑛𝑢,𝑀)≤𝑐₁𝑒^−𝑐2𝑛.

Let 𝑃=𝑃_𝑁 be the orthogonal projection chosen as in the definition of the squeezing property. Denoteℱ=max{𝐹∣all𝑢,𝑣∈𝐹satisfying‖𝑢−𝑣‖≤√2‖𝑃𝑢−𝑃𝑣‖} (2.3)for the inclusion relation. From the definition of ℱ, we know 𝑃|_ℱ is one-to-one on ℱ. Clearly, 𝑃ℱ is a bounded closed set of a finite dimensional vector space, and therefore, it is compact. So, ℱ as the preimage under the continuous map 𝑃|_ℱ must also be compact.

Let 𝐸^(𝑘) be a subset of the set 𝑆^𝑘+1𝐵, which is formed by a finite union of exceptional sets of the form ℱ, which is described above, hence all 𝐸^(𝑘) are compact.

Lemma 2.5. If 𝑆 is asymptotically compact on 𝐵, then 𝑀=^+∞_𝑗,𝑘=1𝑆^𝑗(𝐸^(𝑘)) (2.4)is relatively compact.

Proof. Let {𝑦_𝑛}_𝑛≥1 be a sequence in 𝑀. Then, two cases will appear as follows. Case 1. There exists a natural number 𝑁₀ such that all 𝑦_𝑛 are in ⋃^𝑁0_𝑗,𝑘=1𝑆^𝑗(𝐸^(𝑘));Case 2. There exists a subsequence (still denoted by {𝑦_𝑛}_𝑛≥1) satisfying for every 𝑦_𝑛, there exists 𝑥_𝑛∈𝐵 such that 𝑦_𝑛=𝑆^𝑛𝑥_𝑛. In Case 1, since 𝐸^(𝑘) is compact and 𝑆 is continuous, there exists a convergent subsequence of {𝑦_𝑛}_𝑛≥1 that converges in 𝑀. In Case 2, since 𝑆 is asymptotically compact on 𝐵, it is immediate that we can extract from {𝑦_𝑛}_𝑛≥1 a subsequence that converges in 𝐻. So, 𝑀 is relatively compact.

Theorem 2.6. Let 𝐻 be a separable Hilbert space and let 𝐵 be a nonempty closed bounded subset of 𝐸. Assume that (i)𝑆 is a Lipschitz continuous map with Lipschitz constant 𝐿 on 𝐵;(ii)𝑆 is asymptotically compact on 𝐵;(iii)𝑆 satisfies the discrete squeezing property on 𝐵 (with rank 𝑁₀), then 𝑆 has an exponential attractor on 𝐵:ℳ=𝒜∪^+∞_𝑗,𝑘=1𝑆^𝑗(𝐸^(𝑘)), (2.5)where 𝒜 is a global attractor for 𝑆 on 𝐵, 𝐸^(𝑘) is as the above-mentioned. Moreover, the fractal dimension of ℳ satisfies𝑑_𝐹(ℳ)≤𝑁₀max{1,𝐿𝑜𝑔(16𝐿+1)2𝐿𝑜𝑔2}. (2.6)

Proof. Note that all the limits point of ⋃^+∞_𝑗,𝑘=1𝑆^𝑗(𝐸^(𝑘)) belong to 𝒜. Together with Lemma 2.5, the proof follows exactly in the same way as the proof of Theorem 2.1 in [7].

Remark 2.7. In Theorem 2.6, there are two advantages than all the previous results on the existence of exponential attractor for 𝑆:(i)𝐵 is not assumed to be compact;(ii) if 𝑆 possesses a global attractor, then 𝑆 is at least asymptotically compact. So, we only check that if 𝑆 satisfies the Lipschitz property and the discrete squeezing property to obtain the existence of an exponential attractor for 𝑆.

3. Exponential Attractor

For 𝑖=(𝑖₁,𝑖₂,…,𝑖_𝑘)∈ℤ^𝑘, we will always denote ‖𝑖‖=max_{1≤𝑗≤𝑘}|𝑖_𝑗| in the following discussion. For any 𝑢=(𝑢_𝑖)_𝑖∈ℤ^𝑘∈ℓ², 𝑖=(𝑖₁,𝑖₂,…,𝑖_𝑘)∈ℤ^𝑘, define the operators 𝐵_𝑗, 𝐵_𝑗, and 𝐴_𝑗, 𝑗∈{1,2,…,𝑘} from ℓ² to itself as follows: 𝑢_𝑖=𝑢_{(𝑖1,𝑖2,…,𝑖𝑘)}∈ℓ², 𝑗=1,2,…,𝑘,(𝐵_𝑗𝑢)_𝑖=𝑢_{(𝑖1,…,𝑖𝑗+1,…,𝑖𝑘)}−𝑢_{(𝑖1,…,𝑖𝑗,…,𝑖𝑘)},(𝐵_𝑗𝑢)_𝑖=𝑢_{(𝑖1,…,𝑖𝑗,…,𝑖𝑘)}−𝑢_{(𝑖1,…,𝑖𝑗−1,…,𝑖𝑘)},(𝐴_𝑗𝑢)_𝑖=2𝑢_{(𝑖1,…,𝑖𝑗,…,𝑖𝑘)}−𝑢_{(𝑖1,…,𝑖𝑗+1,…,𝑖𝑘)}−𝑢_{(𝑖1,…,𝑖𝑗−1,…,𝑖𝑘)}. (3.1)Then, we have𝐴=𝐴₁+𝐴₂+…+𝐴_𝑘,𝐴_𝑗=𝐵_𝑗𝐵_𝑗=𝐵_𝑗𝐵_𝑗,𝑗=1,2,…,𝑘. (3.2)

For any 𝑢=(𝑢_𝑖)_𝑖∈ℤ^𝑘, 𝑣=(𝑣_𝑖)_𝑖∈ℤ^𝑘∈ℓ², we define inner product and norm of ℓ² as follows:(𝑢,𝑣)=∑_{𝑖∈ℤ𝑘}𝑢_𝑖𝑣_𝑖,‖𝑢‖=(∑_{𝑖∈ℤ𝑘}|𝑢_𝑖|²)^1/2, (3.3)then ℓ²=(ℓ²,(⋅,⋅),‖⋅‖) is a Hilbert space. It is obvious that any 𝑢=(𝑢_𝑖)_𝑖∈ℤ^𝑘, 𝑣=(𝑣_𝑖)_𝑖∈ℤ^𝑘∈ℓ²,(𝐴𝑢,𝑣)=∑^𝑘_𝑗=1(𝐵_𝑗𝑢,𝐵_𝑗𝑣)=∑^𝑘_𝑗=1(𝐵_𝑗𝑢,𝐵_𝑗𝑣),∑^𝑘_𝑗=1‖𝐵_𝑗𝑢‖²≤4𝑘‖𝑢‖². (3.4)

We always make the following assumptions on 𝑔(𝑠)∈𝐶(ℝ):(H1)𝑔(0)≡0 and 𝑔(𝑠)𝑠≥0.(H2)There exists an increasing continuous function 𝐾(𝑟)∶ℝ⁺→ℝ⁺ with 𝐾(0)=0 such thatsup_{|𝑠|≤𝑟}|𝑔^′(𝑠)|≤𝐾(𝑟²), (3.5)where ℝ⁺=[0,+∞).

Similar to [2, Theorem 1], we have.

Theorem 3.1. For any initial data 𝑢₀∈ℓ², there exists a unique local solution 𝑢(𝑡) of problem (1.3) with 𝑢(0)=𝑢₀ such that 𝑢(𝑡)∈𝐶¹([0,𝑇],ℓ²) for any 𝑇>0.

In fact, it will be showed in Lemma 3.2 below that the local solution 𝑢(𝑡) of problem (1.3) exists globally, that is, 𝑢(𝑡)∈𝐶¹([0,+∞),ℓ²). It implies that the map𝑆(𝑡)∶𝑢(0)=𝑢₀↦𝑢(𝑡),ℓ²↦ℓ², (3.6)generates a continuous semigroup from ℓ² to itself.

Lemma 3.2. Let 𝐵₀=𝐵(0,𝑟₀) be a closed bounded ball of ℓ², centered at 0 with radius 𝑟₀ where 𝑟²₀=1𝜆²‖𝑞‖². (3.7)For any bounded set 𝐵 of ℓ², there exists 𝑇(𝐵)≥0 such that𝑆(𝑡)𝐵⊆𝐵₀,∀𝑡≥𝑇(𝐵). (3.8)

Proof. The proof is easily obtained.

Corollary 3.3. For any 𝑡≥0, 𝑆(𝑡)𝐵₀⊆𝐵₀.

We obtain the following lemma after some simple computation.

Lemma 3.4. Let 𝑢(𝑡)=(𝑢_𝑖)_𝑖∈ℤ^𝑘∈ℓ² be a solution of problem (1.3) with initial data 𝑢₀=(𝑢_0𝑖)_𝑖∈ℤ^𝑘∈𝐵₀. Then, for any 𝑡≥0, _{‖𝑖‖>𝑁}|𝑢_𝑖(𝑡)|²≤𝑟²₀𝑒^−𝜆𝑡+8𝐶₀𝑘𝑟²₀+1𝜆𝑁𝜆²_{‖𝑖‖≥𝑁/2}|𝑞_𝑖|². (3.9)

From Lemmas 3.2 and 3.4, we have the following.

Theorem 3.5. The semigroup {𝑆(𝑡)}_𝑡≥0 is asymptotically compact in ℓ² and possesses a nonempty compact global attractor 𝒜. Furthermore, 𝒜⊆𝐵₀.
Let 𝑆(𝑡)𝑢₀=𝑈(𝑡) and 𝑆(𝑡)𝑤₀=𝑊(𝑡). Since 𝑢₀, 𝑤₀∈𝐵₀, by Corollary 3.3, 𝑈(𝑡), 𝑊(𝑡)∈𝐵₀, for 𝑡≥0. Let 𝑍(𝑡)=𝑆(𝑡)𝑢₀−𝑆(𝑡)𝑤₀=𝑈(𝑡)−𝑊(𝑡). Then, 𝑍(𝑡) satisfies𝑍̇+𝐴𝑍+𝜆𝑍+𝑔(𝑈)−𝑔(𝑊)=0,𝑍(0)=𝑢₀−𝑤₀. (3.10)

After some simple computation, we obtain the following.

Lemma 3.6 (Lipschitz Property). For any 𝑢₀, 𝑤₀∈𝐵₀ and any 𝑇>0, ‖𝑆(𝑇)𝑢₀−𝑆(𝑇)𝑤₀‖≤𝑒^{(𝐾(𝑟20)−𝜆)𝑇}‖𝑢₀−𝑤₀‖. (3.11)

Let 𝑛∈ℕ be a positive integer. Set𝜔𝜔=^{(−𝑛,−𝑛,…,−𝑛,−𝑛)}𝜔_{(−𝑛,−𝑛,…,−𝑛,−𝑛+1)}…𝜔_{(−𝑛,−𝑛,…,−𝑛,𝑛)}𝜔_{(−𝑛,−𝑛,…,−𝑛+1,−𝑛)}𝜔_{(−𝑛,−𝑛,…,−𝑛+1,−𝑛+1)}…𝜔_{(−𝑛,−𝑛,…,−𝑛+1,𝑛)}𝜔…………_{(𝑛,𝑛,…,𝑛,−𝑛)}𝜔_{(𝑛,𝑛,…,𝑛,−𝑛+1)}…𝜔_{(𝑛,𝑛,…,𝑛,𝑛)}. (3.12)For convenience, we always denote𝐸_𝑛={𝜔=(𝜔_𝑖)_𝑖∈ℤ^𝑘∈ℓ²∣𝜔_𝑖withsubscriptsofcomponentsof𝜔whichareorderedasin(3.12)and𝜔_𝑖=0,‖𝑖‖>𝑛}, (3.13)with the same inner product and norm as those of ℓ².

Let 𝒦(𝑥) be the inverse function of 𝐾(𝑥) in (H2). Set𝑇₀4=max{𝜆1𝐿𝑜𝑔2,𝜆(𝐿𝑜𝑔2+2𝐿𝑜𝑔‖𝜆𝑞‖−2𝐿𝑜𝑔𝜆−𝐿𝑜𝑔𝒦(2𝑁))},₀=min{𝑁∈ℕ∶8𝐶₀𝑘‖𝑞‖²+1𝜆𝑁𝜆²_{‖𝑖‖≥𝑁/2}|𝑞_𝑖|²≤12𝜆𝒦(2)}. (3.14)(3.15)

Suppose 𝑃_𝑁 is an orthogonal projection of rank (2𝑁+1)^𝑘 on ℓ² such that 𝑃_𝑁ℓ²=𝐸_𝑁.

Lemma 3.7 (Discrete Squeezing Property). For any 𝑢₀, 𝑤₀∈𝐵₀, if ‖𝑃_𝑁0(𝑆(𝑇₀)𝑢₀−𝑆(𝑇₀)𝑤₀)‖≤‖(𝐼−𝑃_𝑁0)(𝑆(𝑇₀)𝑢₀−𝑆(𝑇₀)𝑤₀)‖, (3.16)then‖𝑆(𝑇₀)𝑢₀−𝑆(𝑇₀)𝑤₀1‖≤8‖𝑢₀−𝑤₀‖. (3.17)

Proof. Denote 𝑈_𝑁0(𝑡)=(𝐼−𝑃_𝑁0)𝑢(𝑡), 𝑊_𝑁0(𝑡)=(𝐼−𝑃_𝑁0)𝑤(𝑡) and 𝑍_𝑁0(𝑡)=(𝐼−𝑃_𝑁0)(𝑢(𝑡)−𝑤(𝑡))=(𝐼−𝑃_𝑁0)𝑍(𝑡). Taking the inner product (⋅,⋅) in (3.10) with 𝑍_𝑁0, we have𝑑𝑑𝑡‖𝑍_𝑁0‖²+2𝜆‖𝑍_𝑁0‖²+2(𝑔(𝑈)−𝑔(𝑊),𝑍_𝑁0)≤0, (3.18)where ‖𝑍_𝑁0‖²=∑_{‖𝑖‖>𝑁0}|𝑍_𝑖|². By the mean value theorem,|(𝑔(𝑈)−𝑔(𝑊),𝑍_𝑁0)|≤_{‖𝑖‖>𝑁0}|𝑔^′(𝑈_𝑖+𝜃_𝑖(𝑊_𝑖−𝑈_𝑖))|𝑍_𝑖|², (3.19)where 𝜃_𝑖∈(0,1), ‖𝑖‖>𝑁₀. By (H2) and Lemma 3.4, for 𝑡≥𝑇₀,|𝑔^′(𝑈_𝑖+𝜃_𝑖(𝑊_𝑖−𝑈_𝑖𝜆))|≤2, (3.20)which implies|(𝑔(𝑈)−𝑔(𝑊),𝑍_𝑁0𝜆)|≤2_{‖𝑖‖>𝑁0}|𝑍_𝑖|². (3.21)By (3.18), (3.21), and the Gronwall inequality, we have‖𝑍_𝑁0(𝑡)‖²≤𝑒^{−𝜆(𝑡−𝑇0)}‖𝑢₀−𝑤₀‖², (3.22)for all 𝑡≥𝑇₀. So, for any 𝑢₀, 𝑤₀∈𝐵₀, if‖𝑃_𝑁0(𝑆(𝑇₀)𝑢₀−𝑆(𝑇₀)𝑤₀)‖≤‖(𝐼−𝑃_𝑁0)(𝑆(𝑇₀)𝑢₀−𝑆(𝑇₀)𝑤₀)‖, (3.23)then‖𝑆(𝑇₀)𝑢₀−𝑆(𝑇₀)𝑤₀‖≤2‖(𝐼−𝑃_𝑁0)(𝑆(𝑇₀)𝑢₀−𝑆(𝑇₀)𝑤₀1)‖≤8‖𝑢₀−𝑤₀‖. (3.24)

From Theorems 2.6 and 3.5, Lemmas 3.6 and 3.7 in this article, and [7, Theorem 3.1], we obtain.

Theorem 3.8. The semigroup 𝑆(𝑡) determined by problem (1.3) with (H1)-(H2) possesses anexponential attractor on 𝐵₀: 𝔐=_{0≤𝑡≤𝑇0}𝑆(𝑡)(𝒜∪^+∞_𝑗,𝑘=1𝑆^𝑗₀(𝐸^(𝑘))), (3.25)whose fractal dimension satisfies𝑑_𝐹(𝔐)≤𝑐₀(2𝑁₀+1)^𝑘+1, (3.26)where 𝑇₀ is as (3.14), 𝑆₀=𝑆(𝑇₀), 𝐸^(𝑘) is defined as in Section 2 and 𝑁₀ is as (3.15), 𝑐₀=max{1,𝐿𝑜𝑔(16𝑒^{(𝐾(𝑟0)−𝜆)𝑇0}+1)/2𝐿𝑜𝑔2}.

Remark 3.9. Indeed, when 𝐾(𝑟²₀)<𝜆, by Lemma 3.6, we easily know that 𝑆(𝑡) has an exponential attractor of dimension zero on 𝐵₀, which is an equilibrium point of problem (1.3) (the global attractor for 𝑆(𝑡)).