Lattice dynamical systems: dissipative mechanism and fractal dimension of global and exponential attractors

In this work, we examine first-order lattice dynamical systems, which are discretized versions of reaction–diffusion equations on the real line. We prove the existence of a global attractor in ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^2$$\end{document}, and using the method by Chueshov and Lasiecka (Dynamics of quasi-stable dissipative systems, Springer, Berlin, 2015; Memoirs of the American Mathematical Society, vol 195(912), AMS, 2008), we estimate its fractal dimension. We also show that the global attractor is contained in a finite-dimensional exponential attractor. The approach relies on the interplay between the discretized diffusion and reaction, which has not been exploited as yet for the lattice systems. Of separate interest is a characterization of positive definiteness of the discretized Schrödinger operator, which refers to the well-known Arendt and Batty’s result (Differ Int Equ 6:1009–1024, 1993).


Introduction
The asymptotic behavior of nonlinear reaction-diffusion equations in unbounded domains has been investigated by many authors; see, e.g., [5][6][7]16,17,27,30] and references therein. A simple illustrative example is a scalar reaction-diffusion equation on the real line x ∈ R, t > 0, which can be viewed as an archetype model for the first-order lattice systems. Taking an equi-spaced discretization of (1.1) in the space variable with a step h > 0 and letting ν := μ h 2 > 0 lead to the first-order system of the form which will be of our interest in this paper. Such systems, involving infinite countable number of ordinary differential equations, were broadly studied under various assumptions on the nonlinear term. On the one hand, for a suitable cubic polynomial nonlinearity, traveling wave solutions were established in [36] (see also [12]). On the other hand, for (1.2) with the right-hand side of the form where λ > 0 and F(s)s ≥ 0 for all s ∈ R, (1.4) the strong stability properties of the system in 2 were established in [8], which involved in particular the concept of a global attractor developed in [7,20,22,29]. A similar strong stability result was proved in [33] for a generalized lattice system and in [24] for a partly dissipative system with λ = λ i and F = F i in (1.3) satisfying a variant of (1.4) (cf. [33, p. 53] and [24, p. 162]; also [34, p. 366]). Specifically, finite fractal dimensionality of the global attractor was shown in [35] assuming, in particular, that 0 < λ 0 ≤ λ i ≤ λ 0 and F i (s)s ≥ 0 for all s ∈ R. (1.5) A nonautonomous case was considered separately in [32] with g i in (1.3) depending on time and λ, F as in (1.4), in which case a uniform attractor in 2 was constructed for the associated family of processes. Invariant regions were additionally studied in [28].
In the further development of the theory, sufficient conditions for the existence of an exponential attractor were obtained. This was originated in [1], where such sufficient conditions were exhibited for the right-hand side (1.3) considered with λ = λ i and F = F i satisfying (1.5), and F i being an odd-degree polynomial with nonnegative coefficients (see [1, p. 219]). In [19] an estimate of fractal dimension of exponential attractor for the lattice system was given, which in the similar vein involved the righthand side as in (1.3) and (1.4) above.
The above-mentioned results were obtained in the standard spaces of summable sequences, which is the setting chosen in this paper as well. However, lattice systems were also considered in weighted spaces of sequences; see, e.g., [10,21,23,31]. This is a distinct approach, since such phase spaces have different properties from the ones considered here. Specifically exponential attractors were studied in [21] and the righthand side (1.3) was considered in such setting with F = F i satisfying F i (s)s ≥ −b 2 i and (g i ) ∈ 2 ρ , (b i ) ∈ 2 ρ for some suitable weight ρ. We remark that therein λ = λ i preserved the estimate as in (1.5) (cf. [21, p. 451]).
In this paper, we will consider attracting properties of the lattice system in the strong topology of phase spaces of summable sequences assuming dissipativity conditions under which it does not fall in general into the class of problems covered by the previous results (see Example 5.4 and Remark 5.5). Namely, our approach will be based on a cooperation between the discretized diffusion and the discretized reaction.
Hence we assume for some (m i ) ∈ ∞ , (d i ) ∈ 2 the structure condition requiring simultaneously that the solutions of the associated linear problem are uniformly exponentially decaying as t → ∞. Note that this is a counterpart of a dissipative mechanism already described in a large capacity for the space continuous models (see [5,6,11]). On the other hand, observe via examples below that this essentially generalizes dissipativity assumptions used in the presented literature for proving strong stability properties of (1.2) and in particular for the existence of global and exponential attractor in 2 and estimating its fractal dimension. Our main goal will be to achieve these results based on the interplay between discretized diffusion and reaction.
A brief description of the contents of the paper is as follows. In Sect. 2 we include various technicalities concerning the discretized diffusion operator and review the existence of local solutions. In Sect. 3 we construct the semigroup of global solutions following a suitable structure condition on the nonlinear term mentioned above in (1.6). In Sect. 4 the structure condition combined with the decaying properties of (1.7) described in terms of positive definiteness of the discretized Schrödinger operator will constitute a general framework for the existence of a global attractor. A discussion concerning broader applicability of such approach is given in Sect. 5. In Sect. 6, we will then estimate from above fractal dimension of the attractor using the method developed by Chueshov and Lasiecka [13,14]. Furthermore, in Sect. 7, we prove the existence of a finite dimensional exponential attractor, which contains the global attractor.
Some crucial tools used in the main body of the paper have been included in "Appendix". Of separate interest therein is a characterization of positive definiteness of the discretized Schrödinger operator, which refers to the well-known Arendt and Batty's result [4]. This allows us, in particular, to exhibit variety of sequences (m i ) which can enter conditions (1.6) and (1.7) and thus be admissible in our approach. Namely, a nonnegative (−m i ) ∈ ∞ moves to the right spectrum of the operator given by the linear main part in (1.2) if and only if for some or, equivalently, − i∈G m i = ∞ for each set G ⊂ Z possessing the property that, given any n ∈ N, {i ∈ Z : |i − j n | < n} ⊂ G for some j n ∈ Z (see Theorem A.3 of "Appendix"). We emphasize that for the right-hand side as in (1.3) satisfying (1.4) or (1.5) we can simply take m i = −λ < 0 or m i = −λ i ≤ −λ 0 < 0 for which these criteria apply immediately. An example of an admissible sequence (m i ) such that m i is nonzero only on some rarely distributed in Z subset of indices i is constructed in Remark 5.2. Moreover, some sequences with infinitely many positive terms are admissible (for details, see again Remark 5.2). Our results are, nonetheless, much more involved. Specifically Sect. 5.2 exhibits several nonlinearities for which proposed dissipative mechanism applies, which in turn generalizes previous considerations concerning global and exponential attractors for lattice dynamical systems in spaces of summable sequence. For example, following Corollary 5.8, our approach applies to if there is a decomposition (1.8) and and coefficients a(ρ j ) i , i ∈ Z, with 1 ≤ j < κ can have unspecified sign provided We finally remark that the main results rely on the estimates of solutions in 2 and therefore the existence of attractor is established in 2 . Nevertheless, due to embedding properties of the scale (see (6.20)), the attractor is compact in p for any p ∈ [2, ∞]. Also the semigroup on the attractor is Lipschitz with values in p , p ∈ [2, ∞]. Hence fractal dimension of the attractor is finite in p and the estimate from above established in the Hilbert setting extends respectively (see Corollaries 6.7 and 6.10).

Notation and local existence result
We will use the Banach spaces with the norms u p = i∈Z |u i | p 1 p and u ∞ = sup i∈Z |u i |, respectively. In case p = 2 the space 2 is a Hilbert space with the inner product and the norm We also remark that u ∞ ≤ u p for u ∈ p and that, given v ∈ ∞ and u ∈ p , we write vu ∈ p to denote the sequence In the p spaces we will use isometries τ, τ * , to define a linear operator (2.1)
In what follows, we will view (1.2) in the abstract form Observe that if then the map F takes p into p and is Lipschitz continuous on bounded sets. Indeed, given a ball B = {u ∈ p : u p ≤ R} and using the mean value theorem with Due to the properties of A and F, the general theory of abstract differential equations leads to the existence result as stated below.
Remark 2.3. Besides the p setting as above, in the following sections, we will rely on the estimates of the solution in 2 . Hence we observe that A : 2 → 2 is a self-adjoint nonnegative definite operator with spectrum in [0, ∞). Indeed, since using this and (2.2) we get and we also obtain in a similar manner that (Au, v) = (u, Av), u, v ∈ 2 .

Global existence
We now restrict our considerations to the Hilbert case p = 2. Therefore, we assume that the nonlinearity satisfies (2.7) and (3.1) The aim of the section is to show that the solutions in 2 exist globally in time, i.e., t max (u 0 ) = ∞ for any u 0 ∈ 2 . To this end, we introduce the structure condition for the nonlinearity The global existence of solutions of (2.5) is a consequence of the following lemma.
Proof. We take the inner product of the first equation in (2.5) with u in 2 and get 1 2 we obtain by the nonnegativity of A By the Gronwall inequality, we get (3.3).
Therefore we define on 2 the semigroup {S(t) : t ≥ 0} of global solutions of (2.5) by This is a C 0 semigroup in the sense that then (3.2) holds with d i = 0 for i ∈ Z, but the condition (3.2) is much more flexible than (3.6) (see Remarks 5.6, 5.7).

Dissipativity and global attractor
In this section for the semigroup of global solutions in 2 , we prove the existence of a bounded absorbing set and establish asymptotic compactness. As mentioned in the introduction our approach will be based on a cooperation between the discretized diffusion and the discretized reaction terms. To this end, simultaneously with (3.2), we require for the linear problem that the linear semigroup e (−A+m)t on 2 generated by −A + m is uniformly exponentially stable ([18, Definition I. 3.11]). This is equivalent to the requirement that the spectrum of the self-adjoint operator A − m in 2 is positive This also means that the operator A − m is positive definite, i.e., there exists a positive constant α 0 > 0 such that To find suitable estimates of solutions observe that with the above-mentioned assumptions, the nonlinear term after taking a product with u cannot be dropped, which would be the case for the right-hand side as in (1.3) and (1.4). Differently than in Sect. 3 the linear main part after taking a product with u cannot be dropped either. Actually, both the discretized diffusion and discretized reaction must intervene to produce desired dissipativity and asymptotic compactness effects.
Proof. We return to (3.4) and get 1 2 By assumption (4.2) and the Cauchy inequality, we get and by the Gronwall inequality we get the result.  [24]) to estimate the tail ends of solutions. Since our assumptions are different from the ones used in the mentioned references we proceed with the direct proofs of the two lemmas below. Then standard argument will ensure the asymptotic compactness used in Theorem 4.4.
Let θ : [0, ∞) → [0, 1] be a C 1 function 1 such that and θ (s) ≤ 2 for s ≥ 0. We fix k ∈ N and consider a sequence η k ∈ ∞ given by Proof. Using the formula we get We have by the Cauchy-Schwarz inequality which ends the proof.

Lemma 4.3.
Suppose that the assumptions of Theorem 4.1 hold. Given ε > 0 there exist k ε ∈ N and T ε > 0 such that for any t ≥ T ε and any u 0 in a bounded positively invariant absorbing set B 0 from Theorem 4.1, we have Proof. Taking the inner product of the first equation in (2.5) with η 2 k u in 2 , we get 1 2 Note that by (2.9) and Lemma 4.2 whereas by the structure condition (3.2) and the fact that η 2 k ≤ η k , we get Since (4.2) holds with α 0 > 0, using in the last term the Cauchy inequality with α 0 , we obtain Combining (4.5), (4.6) and the assumption (4.2) with (4.4), we obtain for k ∈ N d dt Fix ε > 0 and let I ε ∈ N be such that where M > 0 is as in Theorem 4.1. For any t > 0 and u 0 ∈ B 0 , we have u(t; u 0 ) ∈ B 0 and d dt Applying the Gronwall inequality, we get the claim with k ε = 2I ε .
Proof. Since by Theorem 4.1 the semigroup is bounded dissipative, it suffices to show that it is asymptotically compact (see [22]). With Theorem 4.1 and Lemma 4.3, the proof of asymptotic compactness is standard, and hence, it will be omitted (for details see, e.g., [8, pp. 147-148] or [24, proof of Lemma 3.3]).

Remarks concerning dissipative mechanism
In this section, we make a few remarks on the assumptions (3.2) and (4.2) which are at the core of the dissipative mechanism.

Bottom spectrum of discretized Schrödinger operator
Recall that the spectrum of the operator 3) and observe that if, for example, the sequence m is a negative constant as in [8,35], that is,

then (4.2) is satisfied and Theorem 4.4 applies.
To show a broader applicability of Theorem 4.4, it is crucial to give a characterization of the sequences which move the spectrum of the operator A to the right. The result below is a consequence of Theorem A.3 proved in "Appendix" by modifying Arendt and Batty's [4, Theorem 1.2]. Proposition 5.1. A necessary and sufficient condition for 0 ≤ −m ∈ ∞ to satisfy (4.1) is that − i∈G m i = ∞ for each set G ⊂ Z possessing the property that, given any n ∈ N, {i ∈ Z : |i − j n | < n} ⊂ G for some j n ∈ Z.
Hence observe that in (3.2) and (4.1), we can actually have sequences m ∈ ∞ with infinitely many positive terms (see Remark 5.2). Consequently, our argument for the existence of the global attractor for (1.2) essentially generalizes [8,35]   Denote by m = m r,k,n ∈ ∞ the periodic extension of m r,k,n to Z. Then, given G ∈ G as in Definition A.1 and an integer p ≥ 1, the interval ( j pn − pn, j pn + pn) contains 2 p − 1 intervals of length n having the form [ln, ln + n) for some integer l and, since Observe that, due to Proposition 5.1, m = m r,k,n satisfies (4.1) and thus also (4.2) with some α 0 > 0. Furthermore, number of zeros between nonzero components of m = m r,k,n equals n − k − 1, and therefore, it can be as large as we intend. The property that − i∈G m i = ∞ for any G ∈ G, and hence (4.1), will also hold for m = χ N m r,k,n , where and N is an arbitrarily large positive integer.
(ii) Given m ∈ ∞ and α 0 > 0 as in (4.2) observe further that ifm ∈ ∞ is any sequence with m ∞ < α 0 , then (Au, In particular, if zero components of the sequence m from (i), which satisfies (4.1), are replaced by a positive number less than α 0 , then the resulting sequence will still maintain (4.1).

Sample nonlinearities
Let us now make a few remarks concerning nonlinearities for which a general mechanism described in the previous section applies. To proceed observe that C 1 maps can be written as for some λ i ∈ R, i ∈ Z, and functions h i : R → R of the class C 1 satisfying h i (0) = 0. A simple application of Theorem 4.4 is given in the corollary below in a situation when (λ i ) moves spectrum of A to the right. Example 5.4. We present an illustrative example of a lattice dynamical system, which is a special case of problems considered in Corollary 5.3.

Corollary 5.3. If
For arbitrarily chosen r > 0, n ∈ N and N ∈ N let m = χ N m r,0,n ∈ ∞ be the sequence defined in Remark 5.2 (i). Therefore, letting Z nN = {i ∈ Z : |i| ≤ N or |i| > N and i = jn for all j ∈ Z}, we see that and by Proposition 5.1 there exists α > 0 such that and set m = m + m; hence m = (m i ) ∈ ∞ is given by and we have (4.2) for α 0 = α 2 > 0, that is, We emphasize that the sequence m = (m i ) has positive values except for the indices i being integer multiples of n, which have modulus larger than N . Now we fix arbitrary sequences a = (a i ) ∈ 6 and g = (g i ) ∈ 2 and define to consider the corresponding nonlinear system because using Young's inequality, we get Thus (5.5) generates a semigroup {S(t) : t ≥ 0} on 2 , which has a global attractor A in 2 .
For i ∈ Z nN this would yield by (5.3), (5.4) that which is absurd for large |s|.
We now remark that our approach also applies to nonlinearities (5.2) with sequences (λ i ), which may not shift the spectrum of the operator A to the right.

Remark 5.6. (i) If h i from (5.2) satisfies
and if there is a decomposition with m as in (4.1), then (3.2) will be satisfied with and coefficients a(ρ j ) i , i ∈ Z, with 1 ≤ j < κ can have unspecified sign provided that The following corollary, which builds upon Remarks 5.6 and 5.7, shows a broader applicability of Theorem 4.4.

Estimate of fractal dimension of attractor
Assuming additionally to (2.7) and (3.2) that we will now prove that the attractor in Theorem 4.4 has finite fractal dimension. The proof of Theorem 6.2 will follow in a sequence of lemmas. This includes, in particular, the estimate of dim f (A) in 2 given in (6.4) and (6.19) (or (6.10)).
, (6.4) where m B η is the maximal number of z j in B η := {z ∈ Z : z Z ≤ 4D 1 2 κ 1−η } possessing the property that Proof. Note that by the Arzela-Ascoli theorem, the seminorm . Thus we apply Proposition B.1 specifying therein In particular, if α 0 is as in (4.2) then for some integer k 0 ≥ 0. (6.6)

Proof. Define maps
Observe that h i are all of the class C 1 , satisfy h i (0) = 0 for i ∈ Z and due to the mean value theorem, (3.2) and (6.7), we have for each Writing now (6.8) with s = d i + 1 |i| , i = 0, and using (6.1), (6.9) we obtain that R i ( d i + 1 |i| ) → 0 as |i| → ∞, which leads to (6.5).  We remark that such interval exists because the space 2 is embedded in ∞ .
If h i is as in (6.7), then by (6.1), there exists δ ∈ (0, M) such that we have from (1.2) that for each t > 0 We get by (2.7) and nonnegativity of A that with someθ i ∈ [0, 1] which by the Gronwall inequality implies We also obtain from (1.2) that for each t > 0 Then, using that Concerning J 1 we replace f i (0) by f i (0) − m i + m i , and with the aid of (6.6), we estimate as follows To estimate J 2 we consider δ as in (6.11) and apply Lemma 4.3 with ε = δ 2 to obtain numbers k δ 2 ∈ N and T = T δ 2 > 0 such that for k ≥ max{k δ 2 , k 0 } (6.14) we have Using (6.15) we infer that |u i | ≤ δ and |v i | ≤ δ for every |i| > k and t ≥ T.
Considerations in the proof of Lemma 6.5 ensure that assumption (6.1) is a bit too strong and the result can be formulated as follows. Corollary 6.9. Besides the assumptions of Theorem 4.4, that is (2.7), (3.1), (3.2) and (4.2), assume also (6.6) and (6.11). Under these assumptions, fractal dimension of the global attractor in Theorem 4.4 is finite in p for any p ∈ [2, ∞] and its estimate from above is given by (6.4) and (6.19).
Sometimes an ∞ sequence m = (m i ), which moves the spectrum of A to the right so that (4.2) holds, can be present in the equation. Then, instead of (6.7), a slightly different decomposition of nonlinearities is useful: with h i satisfying (6.11). In this case we obtain the following result. Proof. Following the proof of Lemma 6.5 note that in this case the left-hand side of (6.13) now is (mU, U ) i . Estimating then the quantity as in the proof of Lemma 6.5 we will have the first line in (6.16) bounded from above by sup |i|≤k sup |s|≤M |h i (s)| |i|≤k U 2 i , with k = k δ 2 from Lemma 4.3, which implies (6.17) and (6.18) with constant D equal now 2 sup |i|≤k sup |s|≤M |h i (s)|. Remark 6.11. Observe that (6.11) is straightforward for the map h defined in (5.4). Applying Corollary 6.10, we thus obtain an estimate of fractal dimension of the attractor from Example 5.4 in p for any p ∈ [2, ∞]. Remark 6.12. If besides (2.7) and (3.1) one assumes that (3.6) holds together with (4.2), then the semigroup {S(t) : t ≥ 0} is a contraction for positive times, that is, In such case, following [13, Corollary 3.1.18, Exercise 3.1.19], the attractor consists of a single point and thus has zero fractal dimension.

Exponential attractor
We use below the Hausdorff semidistance d(·, ·) as in Theorem 4.4. Recalling that conditions (6.2) and (6.3) were verified in Lemma 6.5 for u 0 , v 0 in the closed positively invariant set B 0 from Theorem 4.1, we now use [13,Theorem. 3.2.1] to conclude the following result.
for some r > 0, and also such that its fractal dimension in 2 is finite and estimated by Following [13,Theorem 3.4.7], we now obtain the existence of an exponential attractor. Proof. Recall from Theorem 4.1 and Lemma 6.5 that the C 0 semigroup {S(t) : t ≥ 0} on 2 (see (3.5)) is both dissipative with positively invariant absorbing set B 0 , and quasi-stable on B 0 at time t * = N T in the sense of [13,Definition 3.4.1]. Observe here that (6.2) and (6.3) hold for u 0 , v 0 ∈ B 0 with T 1 = T and T 2 = N T as in Lemma 6.5. Given M as in Theorem 4.1 and denoting u = S(·)u 0 observe also that for any u 0 ∈ B 0 we get from (2.4), (2.8) which ensures that , n t ∈ N, and using that (see (6.12)), we get where ξ = − ln 1+η 2 > 0 and C B = re L M t * +ξ + ξ T B Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Discretized Schrödinger operator
In this section, we obtain a discretized version of the well-known Arendt and Batty's result reported in [4,Theorem 1.2]. To this purpose, we define a suitable family G of subsets of Z being a counterpart of a family of sets in [4] which contain arbitrarily large balls.
Definition A.1. We say that G ⊂ Z belongs to G if and only if for any n ∈ N there exists j n ∈ Z such that {i ∈ Z : |i − j n | < n} ⊂ G. Proof. We have I n (0) = 0, n = 0, I 0 (0) = 1, and

Consider the bounded linear operator
We denote the right-hand side of (A.1) by u i (t). By evaluating it at t = 0 and computing its derivative we obtain which yields the claim. Now we are ready to give equivalent conditions for the discrete Schrödinger operator −A − c with 0 ≤ c ∈ ∞ to generate a uniformly exponentially stable semigroup.
Since for the operator A considered on ∞ we have A1 = 0, it follows that e −At 1 = 1 and  (i)⇔(ii). If (ii) holds, the above considerations imply that (iii) holds for p = 2, which in turn yields (iv) and (i). The converse implication is straightforward.
Corollary B.2. Let X be a Banach space and let M = ∅ be a bounded and closed subset of X and let a map V : M → X be such that M ⊂ V M. Assume that Y , Z are normed spaces, P : Z → Y is a linear compact operator and there exists a map K : M → Z satisfying for some η ∈ [0, 1) and μ, κ > 0 Then M is a compact set in X and for any δ ∈ (0, 1 − η) where m Z 2μκδ −1 is the maximal number of points z j in the ball {z ∈ Z : z Z ≤ 2μκδ −1 } possessing the property that P(z j − z l ) Y > 1 whenever j = l. , (7.4) where N Y θ μκ (B 1 ) denotes the minimal number of closed balls in Y of radius θ μκ necessary to cover the closed unit ball B 1 in Z . In this case, one can apply Corollary B.2 with 2η in the role of η and δ = 2θ making the denominators in (7.3) and (7.4) equal ln 1 2(η+θ) . Observe that the quantity m Z 2μκδ −1 appearing in (7.3) can be viewed as the maximal number of points z j in the closed unit ball in Z having the property that z j − z l Y > θ μκ for j = l. Then, given such points, there exists a cover of the closed unit ball B 1 in Z by m Z μκθ −1 closed balls in Y of radius θ μκ . Concerning estimate of fractal dimension involving discrete exponential attractor, we finally recall the following result by Chueshov [13, Theorem 3.2.1].

Proposition B.4. Let X be a Banach space and M be a bounded closed set in X .
Assume that there is a map V : M → M and there also exists a map K from M into an auxiliary normed space Z such that (B.1) and (B.2) hold with some constants η ∈ [0, 1), κ > 0, and with n Z being a compact seminorm on Z . Then, given δ ∈ (0, 1−η), there exists a compact set A δ ⊂ M satisfying V A δ ⊂ A δ , d(V n M, A δ ) ≤ r (η + δ) n , n ∈ N,