Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 65, pp. 1–36. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS WITH LOCAL AND NONLOCAL NONLINEARITIES JARDEL MORAIS PEREIRA Abstract. We study the dynamics of some dissipative lattice differential equations with local and nonlocal nonlinearities. Using a difference inequality due to Nakao [14] and suitable estimates, we prove the existence of global at- tractors. In addition, we briefly discuss the dynamics of two periodic lattice differential equations. 1. Introduction Lattice differential equations (LDEs) are used to model various problems that occur in important areas of science and technology, for instance, in biophysics [13], electrical engineering [4], image processing [5], chemical reaction theory [10], etc.. They also naturally arise as spatial discretization of continuous models. The dynamics of nonlinear LDEs is a wide-ranging theme that includes as subjects the existence of solutions, existence and stability of traveling waves, asymptotic behavior, attractors and their properties, etc., see e.g. [1, 2, 3, 8, 20, 25] and the references therein. In particular, the existence of attractors for LDEs is a subject that attracts a great deal of attention. In this article, we study the dynamics of some dissipative LDEs with local and nonlocal nonlinearities in arbitrary spatial dimensions. Our main objective is to prove the existence of global attractors. Firstly, we consider the following class of second order LDEs ün(t) + (−1)p∆p dun(t) + αun(t) + F (n, un(t),∇+un(t)) + g(n, u̇n(t)) = fn, un(0) = u0,n, u̇n(0) = u1,n , (1.1) where n ∈ Zd and t ∈ R+. In (1.1), α is a positive constant, p is any positive integer, ∆p d = ∆d ◦ · · · ◦ ∆d, p times, and ∆d denotes the d-dimensional discrete Laplacian operator defined by ∆dun = ∑d i=1(un+ei +un−ei−2un), where {ei}di=1 is the canonical basis of Rn. We assume that the nonlinear term F (n, un(t),∇+un(t)) has the form F (n, un(t),∇+un(t)) = h0(n, un(t))− d∑ i=1 ∂−i hi(∂ + i un(t)), (1.2) 2010 Mathematics Subject Classification. 37L30, 37L60, 39A12. Key words and phrases. Global attractors; dissipative lattice differential equations; local and nonlocal nonlinearities; periodic lattice differential equations; difference inequality. c©2021. This work is licensed under a CC BY 4.0 license. Submitted June 15, 2020. Published July 19, 2021. 1 2 J. M. PEREIRA EJDE-2021/65 where h0 : Zd×R→ R, hi : R→ R, i = 1, . . . , d, are functions satisfying appropriate assumptions stated in Section 2 and ∂+i un = un+ei − un and ∂−i un = un − un−ei , with i = 1, . . . , d. Moreover, g(n, u̇n(t)) is a nonlinear dissipation. Many papers in the literature deal with the existence of attractors for LDEs. Here, we just mention some works that are related to our model (1.1). The existence of attractor and finite-dimensional approximations for the case p = 1, hi ≡ 0 for i = 1, . . . , d, was studied by Zhou [33]. The existence of a global attractor for the class (1.1) when d = 1 and hi ≡ 0 for i = 1, . . . , d, was first investigated by Oliveira and Pereira [18]. Later, for this last class with a suitable delay term, the existence of a pullback attractor was established by Wang and Bai [29]. Another paper related to (1.1) is [32] in which the authors studied the existence and the upper semi-continuity of attractors for second order lattices with retarded terms in dimension one (d = 1) when p = 1. In addition, the investigation of the existence of attractors for second order LDEs in weighted spaces was considered in [7] by Han and some contributions concerning non-autonomous and stochastic LDEs are provided in [6] and [30], respectively. An important special class of LDEs included in (1.1) is ün(t) + (−1)p∆p dun(t) + αun(t) + h0(n, un(t)) + g0(u̇n(t)) = fn. (1.3) Models of type (1.3) when p = 1 are known as discrete nonlinear Klein-Gordon models and appear in different physical contexts, see e.g. [21, 23]. When p = 2, equation (1.3) can be regarded as discrete versions of beam equations. An example of dissipative term for which our results apply is g0 ∈ C1(R;R), g0(0) = 0, and g′0(s) ≥ c0 > 0 for all s ∈ R. Observe that we do not demand any relation on the parameters α and c0, and g′0(s) need not be bounded above. The inclusion of the term − ∑d i=1 ∂ − i hi(∂ + i un) in (1.2) was motivated by some studies on the dynamics of nonlinear LDEs in periodic spaces [16, 17, 19], and continuous models of beam equations studied in the literature, see e.g. [12, 24, 31]. Note that if we choose p = 2, hi(s) = |s|q−2s, q ≥ 2, i = 1, . . . , d, in (1.1), then we obtain a discrete version of the beam equation utt + ∆2u+ αu+ h0(x, u)−∆qu+ g(x, ut) = f(x), where x ∈ Rd, u = u(x, t) and ∆qu = ∑d i=1 ∂ ∂xi ( | ∂u∂xi | q−2 ∂u ∂xi ) is the usual q- Laplacian operator. Secondly, we study the existence of global attractors for the following class of dissipative LDEs with a nonlocal nonlinearity ün(t) + (−1)p∆p dun(t) + αun(t) + F (un(t)) + g(n, u̇n(t)) = fn, un(0) = u0,n, u̇n(0) = u1,n, (1.4) where F (un(t)) = h′(un(t)) ∑ m∈Zd V (n−m)h(um(t)). (1.5) In (1.4) and (1.5), g : Zd × R → R, h : R → R+, and V : Zd → R+ are functions satisfying suitable conditions stated in Section 2. To the best of our knowledge, the existence of attractor for LDEs of type (1.4) with a nonlocal nonlinearity as in (1.5) has not been considered before. The continuous convolution term corresponding to (1.5) is known as Hartree-type nonlinearity and appears in Schrödinger and Klein-Gordon equations, see [22, 27]. EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 3 In addition to the problems described above we also briefly discuss the existence of global attractors for two periodic LDEs. We first consider a periodic problem for the LDE in (1.1) in arbitrary spatial dimension, then we consider the following one-dimensional periodic LDE with α = 0 and nonlinear dissipation ün(t) + (−1)p∆p 1un(t)− ∂−1 h(∂+1 un(t))− ∂−1 g(∂+1 u̇n(t)) = fn, (1.6) where h : R→ R and g : R→ R satisfy appropriate conditions stated in Section 4. Note that when g(s) = µ s, with µ > 0, we obtain the strong damping µ∆1u̇n(t). When p = 2, g(s) = µs, and h(s) = α1s 5−α2s 3−α3s, with αi > 0, i = 1, 2, 3, model (1.6) can be regarded as a discrete version of the beam equation studied by Racke and Shang in [24]. However, unlike the continuous model treated in [24], here, the global attractor we obtain exists in the whole phase space where the dynamics is considered. Our main purpose in this paper is to prove the existence of global attractors for the LDEs (1.1) and (1.4) by combining the use of a difference inequality due to Nakao [14] with a method to derive “tail estimates of solutions” introduced by Wang in [28]. As far as we know, in the context of the dynamics of discrete models, this approach was only used for LDEs of type (1.1) in dimension one in [18] when hi(s) ≡ 0, i = 1, . . . , d, under conditions on the nonlinear terms h0(n, s) and g(n, s) more restrictive than those used in this paper. We note that our assumptions do not require any growth condition on the dissipative term g(n, s). In particular, our results apply to cases with nonlinear dissipative terms such as g(n, s) = an(s3 + s) or g(n, s) = bn sinh s, when the real constants an and bn are suitably chosen. This paper is organized as follows. In Section 2, we state the assumptions on the functions h0, hi, i = 1, . . . , d, in (1.2), g in (1.1), and h and V in (1.5) that we need to prove the existence of solutions and global attractors. Then, after introducing some notation, we briefly discuss the global well-posedness of problems (1.1) and (1.4). In Section 3, we establish the existence of global attractors for the semi- groups generated by the solutions of (1.1) and (1.4). We first prove the existence of absorbing sets, then we prove the asymptotic compactness of the semigroups. The proofs are based on Nakao’s method [14] and suitable estimates. In Section 4, we show how some arguments used in Section 3 can be adapted to prove the existence of global attractors for the periodic problems described above. In the case of model (1.6), we also used a Poincaré inequality valid for the periodic space where the problem is considered. Our results, in particular, generalize and complement the studies of [16, 17, 18]. Finally, in appendix A, we present examples of functions that satisfy some assumptions used in this paper. 2. Existence of solutions In this section, we briefly discuss the existence of solutions for the initial value problems (1.1) and (1.4). We begin establishing the appropriate assumptions on the functions in (1.1), (1.2), and (1.5) and introducing some notation. We denote by `p the space of real sequences u = (un)n∈Zd such that ‖u‖`p <∞, where ‖u‖`p = ( ∑ n∈Zd |un|p )1/p , if 1 ≤ p <∞, ‖u‖`∞ = sup n∈Zd |un|, if p =∞. 4 J. M. PEREIRA EJDE-2021/65 When p = 2, `2 is a Hilbert space with the inner product (u, v)`2 = ∑ n∈Zd un vn, u, v ∈ `2. In this case, we denote by ‖ · ‖ the corresponding norm. Also, to simplify notation, we denote a sequence (un)n∈Zd by (un). We recall that for the `p spaces the following embedding relation holds: `q ⊂ `p, ‖u‖`p ≤ ‖u‖`q , 1 ≤ q ≤ p ≤ ∞. For the functions h0 : Zd×R→ R, hi : R→ R, i = 1, . . . , d, and g : Zd×R→ R, we assume that (A1) For each s0 > 0, there exist positive constants Lj = Lj(s0), j = 1, 2, 3, such that (i) |h0(n, s1)− h0(n, s2)| ≤ L1|s1 − s2|, (ii) |g(n, s1)− g(n, s2)| ≤ L2|s1 − s2|, (iii) |hi(s1)− hi(s2)| ≤ L3|s1 − s2|, for all s1, s2 in R, |s1| ≤ s0, |s2| ≤ s0, for all n ∈ Zd and i = 1, . . . , d. In addition, h0(n, 0) = 0, g(n, 0) = 0, hi(0) = 0, for all n ∈ Zd and i = 1, . . . , d. (A2) There exist sequences of nonnegative real numbers b1 = (b1,n) ∈ `1, b2 = (b2,n) ∈ `1, and a positive constant k1 such that sh0(n, s) + b1,n ≥ k1(h̃0(n, s) + b2,n) ≥ 0, ∀s ∈ R and n ∈ Zd, where h̃0(n, s) = ∫ s 0 h0(n, σ)dσ. (A3) There exist positive constants k0,i such that shi(s) ≥ k0,ih̃i(s), ∀s ∈ R and i = 1, . . . , d, where h̃i(s) = ∫ s 0 hi(σ)dσ ≥ 0. (A4) There exist constants k2 > 0, r ≥ 0 and a positive integer n0 satisfying sg(n, s) ≥ k2|s|r+2, if |n|0 ≤ n0, sg(n, s) ≥ k2|s|2, if |n|0 > n0, for all s ∈ R, where |n|0 = max1≤i≤d |ni|, if n = (n1, . . . , nd). Regarding the LDE (1.1) with the nonlocal term (1.4), we assume that g satisfies the same hypotheses above and that h : R→ R+ and V : Zd → R+ satisfy (A5) h ∈ C2(R;R+), h(0) = h′(0) = 0 and there exists a positive constant c1 such that sh′(s) ≥ c1h(s) ≥ 0 for all s ∈ R. (A6) V = (V (m)) ∈ `2 and V (m) = V (−m) for all m ∈ Zd. The following notation will be useful. ∇+un = (∂+1 un, . . . , ∂ + d un), ∇−un = (∂−1 un, . . . , ∂ − d un), ∇+un · ∇+vn = d∑ i=1 ∂+i un∂ + i vn, |∇+un|2 = ∇+un · ∇+un, Dpun = { ∆ p/2 d un, if p is even ∇+ ( ∆ p−1 2 d un ) , if p is odd, (2.1) where ∆0 d = I. Lemma 2.1. If u = (un) ∈ `2 then ∑ n∈Zd |Dpun|2 ≤ (4d)p‖u‖2. EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 5 Proof. Using the elementary inequality (∑d i=1 ai )2 ≤ d∑d i=1 a 2 i , ai ∈ R, and argu- ing by induction we find that∑ n∈Zd |∆k dun|2 ≤ (4d)2k ∑ n∈Zd |un|2, ∀k ∈ N. (2.2) In (2.2) and hereafter N denotes the set of all positive integers. If p is odd then, from (2.1) and (2.2), we have∑ n∈Zd |Dpun|2 = ∑ n∈Zd |∇+(∆ p−1 2 d un)|2 = ∑ n∈Zd d∑ i=1 |∂+i ∆ p−1 2 d un|2 ≤ 4 ∑ n∈Zd d∑ i=1 |∆ p−1 2 d un|2 = 4d ∑ n∈Zd |∆ p−1 2 d un|2 ≤ 4d(4d)p−1‖u‖2 = (4d)p‖u‖2. Similarly, using (2.1) and (2.2) we can treat the case when p is even. � Lemma 2.2. For any u = (un) and v = (vn) in `2 we have (−1)p ∑ n∈Zd (∆p dun)vn = {∑ n∈Zd ∆ p/2 d un∆ p/2 d vn, if p is even∑ n∈Zd ∇+(∆ p−1 2 d un) · ∇+(∆ p−1 2 d vn), if p is odd. Proof. Since u = (un) and v = (vn) belong to `2, we have∑ n∈Zd (∆dun)vn = d∑ i=1 ∑ n∈Zd (∂+i un)vn − d∑ i=1 ∑ n∈Zd (∂−i un)vn = d∑ i=1 ∑ n∈Zd (∂+i un)vn − d∑ i=1 ∑ n∈Zd (∂+i un)vn+ei = − ∑ n∈Zd d∑ i=1 ∂+i un∂ + i vn = − ∑ n∈Zd ∇+un · ∇+vn. This proves Lemma 2.2 if p = 1. The general case follows by induction on p. � In what follows, given a sequence u = (un), we will write h0(u) = (h0(n, un)), g(u) = (g(n, un)), Au = ((−1)p∆p dun), B(u) = ( − d∑ i=1 ∂−i hi(∂ + i un) ) . (2.3) Also, we will use the Hilbert space H = `2 × `2 equipped with the usual inner product and norm, ((u, v), (w, z))H = (u,w)`2 + (v, z)`2 and ‖(u, v)‖H = ( ‖u‖2 + ‖v‖2 )1/2 , for any (u, v) and (w, z) in H. Lemma 2.3. Under assumption (A1), we have (i) h0, B, and g are locally Lipschitz continuous maps from `2 into itself. (ii) A : `2 → `2 is a bounded operator and ‖Au‖ ≤ (4d)p/2‖u‖ for all u ∈ `2. 6 J. M. PEREIRA EJDE-2021/65 Proof. (i) Given u = (un) in `2, using (A1), we have ‖B(u)‖2 = ∑ n∈Zd ∣∣∣ d∑ i=1 ∂−i hi(∂ + i un) ∣∣∣2 ≤ d ∑ n∈Zd d∑ i=1 ∣∣∣∂−i hi(∂+i un) ∣∣∣2 ≤ d ∑ n∈Zd d∑ i=1 |hi(∂+i un)− hi(∂−i un)|2 ≤ dL3(2‖u‖)2 ∑ n∈Zd d∑ i=1 ( |∂+i un − ∂ − i un| 2 ) ≤ (4d)2L3(2‖u‖)2‖u‖2 <∞. If u = (un) and v = (vn) belong to `2, with ‖u‖ ≤ R and ‖v‖ ≤ R, then, using (A1) again, we have ‖B(u)−B(v)‖2 ≤ d ∑ n∈Zd d∑ i=1 |∂−i hi(∂ + i un)− ∂−i hi(∂ + i vn)|2 = ∑ n∈Zd d∑ i=1 |hi(∂+i un)− hi(∂+i vn)− [hi(∂ − i un)− hi(∂−i vn)]|2 ≤ 2dL3(2R)2 ∑ n∈Zd d∑ i=1 ( |∂+i (un − vn)|2 + |∂−i (un − vn)|2 ) ≤ (4d)2 L3(2R)2‖u− v‖2. This shows that B is a locally Lipschitz continuous map from `2 into itself. Simi- larly, we prove that h0, g : `2 → `2 are locally Lipschitz continuous maps. (ii) It follows immediately from (2.2). � Using Lemma 2.3 we can write the initial value problem (1.1) in `2 as ü(t) +Au(t) + αu(t) + h0(u(t)) +B(u(t)) + g(u̇(t)) = f, t > 0, u(0) = u0, u̇(0) = u1, (2.4) where u0 = (u0,n), u1 = (u1,n), f = (fn), u(t) = (un(t)), u̇(t) = (u̇n(t)) and ü(t) = (ün(t)). Theorem 2.4. Assume that (A1)–(A3) hold and let u0, u1 and f belong to `2. Assume also that sg(n, s) ≥ 0 for all n ∈ Zd and s ∈ R. Then the initial value problem (2.4) has a unique solution u ∈ C2(R+; `2). Moreover, for each τ > 0, the map I : H → C([0, τ ];H), defined by I(u0, u1)(t) = (u(t), u̇(t)), 0 ≤ t ≤ τ , is continuous. Proof. Introducing the change of variable u̇ = v we can rewrite problem (2.4) in the space H as dw dt (t) + Bw(t) = 0, t > 0, w(0) = w0, where w = (u, v), dw dt = (u̇, v̇), w0 = (u0, u1), and B(w) = (−v,Au+ αu+ h0(u) +B(u) + g(v)− f). (2.5) EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 7 Using Lemma 2.3 we can easily prove that the map B defined by (2.5) is a locally Lipschitz continuous map from H into itself. Then, an application of the Theory of Ordinary Differential Equations in Banach Spaces shows that the initial value problem (2.4) has a unique solution u ∈ C2([0, τmax); `2) such that either τmax =∞ or τmax <∞ and limt→τ−max ‖(u(t), u̇(t))‖H =∞. To extend the solution globally we proceed as follows. Taking the inner product of equation (2.4) with u̇(t) in `2 and using Lemma 2.2 we find d dt E(t) = −(g(u̇(t)), u̇(t))`2 ≤ 0, ∀0 ≤ t < τmax, (2.6) where E(t) is the energy associated with the initial value problem (2.4) given by E(t) = 1 2 ‖u̇(t)‖2 + 1 2 ‖Dpu(t)‖2 + α 2 ‖u(t)‖2 + ∑ n∈Zd h̃0(n, un(t)) + ∑ n∈Zd d∑ i=1 h̃i(∂ + i un)− (f, u(t))`2 . (2.7) In (2.7) and hereafter ‖Dpu‖2 = ∑ n∈Zd |Dpun|2. Thus, E(t) ≤ E(0), for all 0 ≤ t < τmax. Since |(f, u)`2 | ≤ α 4 ‖u‖ 2 + 4 α‖f‖ 2, using (A2), (A3) and (2.7) we deduce that ‖(u(t), u̇(t))‖2H ≤ α−10 Ẽ(t), ∀0 ≤ t < τmax, (2.8) where α0 = min{ 12 , α 4 } and Ẽ(t) = E(t) + 4 α ‖f‖2 + ‖b2‖`1 , ∀0 ≤ t < τmax. (2.9) From (2.8) and (2.9) we conclude that τmax = ∞. Finally, under the assumptions of Theorem 2.4, the continuity of I can be proved using (2.8) and the Gronwall inequality. Since the arguments are well known, we omit the details here. � Now, let us consider the initial value problem (1.4) with the nonlocal term (1.5). By assumptions (A5) and (A6) we can define the map F : `2 → `2 by F (u) = ( h′(un) ∑ m∈Zd V (n−m)h(um) ) , ∀u = (un) ∈ `2. Then, using the above notation, we can write (1.4) in the space `2 as ü(t) +Au(t) + αu(t) + F (u(t)) + g(u̇(t)) = f, t > 0, u(0) = u0, u̇(0) = u1, (2.10) To see that F is a locally Lipschitz continuous map from `2 into iself, let u = (un) and v = (vn) in `2 such that ‖u‖ ≤ R and ‖v‖ ≤ R. Since h ∈ C2(R;R+), h(0) = h′(0) = 0 by (A5) and V = (V (m)) ∈ `2 by (A6) and |un| ≤ R and |vn| ≤ R, for all n ∈ Zd, then ‖F (u)− F (v)‖2 ≤ 2 ∑ n∈Zd |h′(un)− h′(vn)|2 ( ∑ m∈Zd V (n−m)|h(um)| )2 + 2 ∑ n∈Zd |h′(vn)|2 ( ∑ m∈Zd V (n−m)|h(um)− h(vm)| )2 ≤ 2M2 1M 2 2 ‖V ‖2‖u‖2‖u− v‖2 + 2M2 1M 2 2 ‖V ‖2‖v‖2‖u− v‖2 ≤ 4R2M2 1M 2 2 ‖V ‖2‖u− v‖2, 8 J. M. PEREIRA EJDE-2021/65 where M1 = max|s|≤2R |h′(s)| and M2 = max|s|≤2R |h′′(s)|. Proceeding as we did for the initial value problem (1.1), we can prove the fol- lowing results. Theorem 2.5. Assume (A1)-(ii), with g(n, 0) = 0 for all n ∈ Zd, (A5), (A6) and let u0, u1, and f belong to `2. Assume also that sg(n, s) ≥ 0 for all n ∈ Zd and s ∈ R. Then the initial value problem (2.10) has a unique solution u ∈ C2(R+; `2). Moreover, for each τ > 0, the map I : H → C([0, τ ];H), defined by I(u0, u1)(t) = (u(t), u̇(t)), 0 ≤ t ≤ τ , is continuous. In this case, as before, we obtain the identity (2.6) with the energy function E(t) = 1 2 ‖u̇(t)‖2 + 1 2 ‖Dpu(t)‖2 + α 2 ‖u(t)‖2 + 1 2 ∑ n∈Zd ∑ m∈Zd V (n−m)h(um(t))h(un(t))− (f, u(t))`2 . (2.11) The same inequality in (2.8) can be derived with Ẽ(t) = E(t) + 4 α ‖f‖2, ∀0 ≤ t <∞. (2.12) 3. Existence of global attractors Our aim in this section is to prove the existence of global attractors for the semigroups generated by the solutions of the initial value problems (2.4) and (2.10). Let us first consider the initial value problem (2.4). Using Theorem 2.4 we can define a semigroup of continuous operators {S(t)}t≥0 on H as follows S(t)(u0, u1) = (u(t), u̇(t)), ∀(u0, u1) ∈ H. (3.1) To prove the existence of a global attractor for {S(t)}t≥0 in H it is sufficient to prove that {S(t)}t≥0 has an absorbing set in H and that it is asymptotically compact in H, see e.g. [26]. Our proofs are based on a difference inequality by Nakao as stated in Lemma 3.1 below. This difference inequality was introduced in [14] to study the existence of attractors for some nonlinear wave equations with nonlinear dissipation. Some other applications to the study of the dynamics of continuous models can be seen in [9, 11, 15]. In the context of LDEs, it was used in [16, 17, 18]. Lemma 3.1 (Nakao [14]). Let ψ(t) be a nonnegative continuous function on [0, T ), T > 1, possibly T =∞, satisfying sup t≤s≤t+1 ψ(s)1+γ ≤ C[ψ(t)− ψ(t+ 1)] +K, ∀0 ≤ t < T − 1, (3.2) with some C > 0, K > 0 and γ > 0. Then ψ(t) ≤ [ C−1γ(t− 1)+ + ( sup 0≤s≤1 ψ(s) )−γ]− 1 γ +K 1 γ+1 , 0 ≤ t < T. If (3.2) holds with γ = 0, then ψ(t) ≤ sup 0≤s≤1 ψ(s) ( C C + 1 )[t] +K, 0 ≤ t < T, where [t] is the largest integer less than or equal to t and β+ = max{β, 0}. EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 9 Let us now introduce some notation that will be used in this section. We first observe that if (A4) holds then, in particular, we have sg(n, s) ≥ 0,∀n ∈ Zd and s ∈ R. Let P (t)2 = E(t)− E(t+ 1) and Q(t) = ∑ n∈Zd u̇n(t)g(n, u̇n(t)), ∀t ≥ 0. According to (2.6), E(t) is a non-increasing function on [0,∞) and P (t)2 = ∫ t+1 t Q(s) ds, ∀t ≥ 0. (3.3) Also, by the Mean Value Theorem for integrals, there exist real numbers t1 ∈ [t, t+ 1 4 ] and t2 ∈ [t+ 3 4 , t+ 1] such that∫ t+ 1 4 t Q(s) ds = 1 4 Q(t1) and ∫ t+1 t+ 3 4 Q(s) ds = 1 4 Q(t2). (3.4) To simplify notation we will write ĥ(u) = h0(u) +B(u) for all u ∈ `2. Lemma 3.2. Assume that (A1)–(A4) hold and let u0, u1, and f belong to `2. Then there exists a positive constant C0 such that sup t≤s≤t+1 Ẽ(s) ≤ C0 ( P (t) 4 r+2 + P (t)4 + P (t)2 + ‖f‖2 + ‖b1‖`1 ) , ∀t ≥ 0. (3.5) Proof. Taking the inner product of 2.4 with u = u(t) in `2, using Lemma 2.2, and integrating the result over [t1, t2] we obtain∫ t2 t1 ‖Dpu(s)‖2ds+ α ∫ t2 t1 ‖u(s)‖2ds+ ∫ t2 t1 (ĥ(u(s)), u(s))`2ds = (u̇(t1), u(t1))`2 − (u̇(t2), u(t2))`2 + ∫ t2 t1 ‖u̇(s)‖2ds − ∫ t2 t1 (g(u̇(s)), u(s))`2 ds+ ∫ t2 t1 (f, u(s))`2 ds. (3.6) Let us estimate the terms in the right hand side of (3.6). We initially write (u̇(tj), u(tj))`2 = ∑ |n|0≤n0 u̇n(tj)un(tj) + ∑ |n|0>n0 u̇n(tj)un(tj), j = 1, 2, (3.7) with n0 as in (A4). Using (2.8) we have( ∑ |n|0≤n0 |un(tj)|2 )1/2 ≤ α−1/20 sup t≤s≤t+1 Ẽ(s)1/2. (3.8) Using Hölder’s inequality, (A4), (3.3), and (3.4) we have∑ |n|0≤n0 |u̇n(tj)|2 ≤ (2n0 + 1) rd r+2 ( ∑ |n|0≤n0 |u̇n(tj)|r+2 ) 2 r+2 ≤ (2n0 + 1) rd r+2 ( k−12 ∑ |n|0≤n0 g(n, u̇n(tj))u̇n(tj) ) 2 r+2 ≤ (2n0 + 1) rd r+2 k − 2 r+2 2 4 2 r+2P (t) 4 r+2 . (3.9) 10 J. M. PEREIRA EJDE-2021/65 Similarly, we can estimate the second term in (3.7) to find∑ |n|0>n0 |u̇n(tj)un(tj)| ≤ 2α −1/2 0 k −1/2 2 P (t) sup t≤s≤t+1 Ẽ(s)1/2. (3.10) From (3.7)-(3.10) we conclude that there is a positive constant C0,1 depending only on n0, α, k2, r, and d such that | (u(t1), u̇(t1))`2 − (u(t2), u̇(t2))`2 | ≤ C0,1 ( P (t) 2 r+2 +P (t) ) sup t≤s≤t+1 Ẽ(s)1/2. (3.11) Proceeding as in (3.9), replacing tj by s ∈ [t1, t2], and noticing that∑ |n|0>n0 |u̇n(s)|2 ≤ k−12 Q(s), for any s ∈ [t1, t2] by (A4), we obtain∫ t2 t1 ‖u̇(s)‖2ds ≤ (2n0 + 1) rd r+2 k − 2 r+2 2 ∫ t2 t1 Q(s) 2 r+2 ds+ k−12 ∫ t2 t1 Q(s)ds ≤ (2n0 + 1) rd r+2 k − 2 r+2 2 P (t) 4 r+2 + k−12 P (t)2. (3.12) Next, for each t ≥ 0 fixed, we define the sets I1(t) = { n ∈ Zd; |u̇n(t)| ≤ 1 } , I2(t) = Zd\I1(t). Note that by (A1), |g(n, u̇n(s))| ≤ L2(1)|u̇n(s)|, whenever n ∈ I1(s). Then∑ n∈I1(s) un(s)g(n, u̇n(s)) ≤ α 2 ‖u(s)‖2 + 1 2α ∑ n∈I1(s) |g(n, u̇n(s))|2 ≤ α 2 ‖u(s)‖2 + 1 2α L2(1) ∑ n∈I1(s) |u̇n(s)| |g(n, u̇n(s))| = α 2 ‖u(s)‖2 + 1 2α L2(1) ∑ n∈I1(s) u̇n(s)g(n, u̇n(s)) ≤ α 2 ‖u(s)‖2 + 1 2α L2(1)Q(s). (3.13) In addition, using (A4) and (2.8) we have∑ n∈I2(s) un(s)g(n, u̇n(s)) ≤ ∑ n∈I2(s) |un(s)| |u̇n(s)| |g(n, u̇n(s))| ≤ ‖u(s)‖ ∑ n∈I2(s) |u̇n(s)| |g(n, u̇n(s))| = ‖u(s)‖ ∑ n∈I2(s) u̇n(s)g(n, u̇n(s)) ≤ ‖u(s)‖Q(s) ≤ α−1/20 Q(s) sup t≤s≤t+1 Ẽ(s)1/2. (3.14) It follows from (3.13), (3.14), and (3.3) that∫ t2 t1 (g(u̇(s), u(s))`2ds ≤ α 2 ∫ t2 t1 ‖u(s)‖2ds+ 1 2α L2(1)P (t)2 + α −1/2 0 P (t)2 sup t≤s≤t+1 Ẽ(s)1/2. (3.15) EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 11 Finally, in view of (2.8), we easily see that∣∣∣ ∫ t2 t1 (f, u(s))`2 ds ∣∣∣ ≤ α−1/20 ‖f‖ sup t≤s≤t+1 Ẽ(s)1/2. (3.16) Substituting (3.11), (3.12), (3.15), and (3.16) into (3.6) we obtain the estimate∫ t2 t1 ‖Dpu(s)‖2ds+ α 2 ∫ t2 t1 ‖u(s)‖2ds+ ∫ t2 t1 J1(s)ds ≤ C0,2 [( P (t) 2 r+2 + P (t)2 + P (t) + ‖f‖ ) × sup t≤s≤t+1 Ẽ(s)1/2 + P (t) 4 r+2 + P (t)2 ] + ‖b1‖`1 , (3.17) where C0,2 is a positive constant depending only on n0, α, k2, r, d, L2(1), and J1(t) = ∑ n∈Zd ( un(t)h0(n, un(t)) + b1,n ) + ∑ n∈Zd d∑ i=1 ∂+i un(t)hi(∂ + i un(t)) ≥ 0, for all t ≥ 0, because of (A2) and (A3). On the other hand, using hypotheses (A2) and (A3) we have∑ n∈Zd h̃0(n, un(t)) + ∑ n∈Zd d∑ i=1 h̃i(∂ + i un(t)) ≤ k−11 ∑ n∈Zd ( un(t)h0(n, un(t)) + b1,n ) − ‖b2‖`1 + ∑ n∈Zd d∑ i=1 k−10,i ( ∂+i un(t)hi(∂ + i un(t)) ) ≤ k0 [ ∑ n∈Zd ( un(t)h0(n, un(t)) + b1,n ) + ∑ n∈Zd d∑ i=1 ∂+i un(t)hi(∂ + i un(t)) ] − ‖b2‖`1 , (3.18) where k0 = max{k−11 , k−10,i , i = 1, . . . , d}. Then, integrating (2.9) over [t1, t2] and using (2.7), (3.17), (3.12), and (3.18) we deduce that∫ t2 t1 Ẽ(s)ds = ∫ t2 t1 E(s)ds+ 4 α ∫ t2 t1 ‖f‖2ds+ ∫ t2 t1 ‖b2‖`1ds ≤ max{k0, 1} [ ∫ t2 t1 ‖Dpu(s)‖2ds+ α 2 ∫ t2 t1 ‖u(s)‖2ds+ ∫ t2 t1 J1(s)ds ] + 1 2 ∫ t2 t1 ‖u̇(s)‖2ds+ 4 α ∫ t2 t1 ‖f‖2ds ≤ C0,3 [( P (t) 2 r+2 + P (t)2 + P (t) + ‖f‖ ) sup t≤s≤t+1 Ẽ(s)1/2 + P (t) 4 r+2 + P (t)2 ] + 4 α ‖f‖2 + max{k0, 1}‖b1‖`1 , (3.19) 12 J. M. PEREIRA EJDE-2021/65 for some positive constant C0,3 depending only on n0, α, k0, k2, r, d, and L2(1). Furthermore, by the Mean Value Theorem for integrals, there exists t∗ ∈ [t1, t2] such that 1 2 Ẽ(t∗) ≤ (t2 − t1) Ẽ(t∗) = ∫ t2 t1 Ẽ(s) ds. (3.20) If t̃ ∈ [t, t∗] then using (2.6) and (3.3) we obtain Ẽ(t̃) = Ẽ(t∗) + ∫ t∗ t̃ Q(s) ds ≤ Ẽ(t∗) + P (t)2. Analogously, we obtain the same estimate if t̃ ∈ [t∗, t+ 1]. Therefore, Ẽ(s) ≤ Ẽ(t∗) + P (t)2, ∀s ∈ [t, t+ 1]. Using this fact, (3.19) and (3.20), we complete the proof. � In what follows we will write K0 = ‖f‖2 + ‖b1‖`1 and assume that K0 > 0. Lemma 3.3. Under the assumptions of Lemma 3.2 there exists ρ0 > 0 such that B[0; ρ0] = {(w, z) ∈ H; ‖(w, z)‖H ≤ ρ0} is an absorbing set for {S(t)}t≥0 in H. Proof. Let O be any bounded subset of H and let ρ = ρ(O) be a positive con- stant such that ‖(w, z)‖H ≤ ρ,∀(w, z) ∈ O. Assume that (u0, u1) ∈ O. Let us first consider the case r > 0. Since Ẽ(t) is a non-increasing function, then using (A1)-(A3) and Lemma 2.1 we can find a positive constant µ0 depending only on p, d, α, ρ, k1, k0,i, i = 1, . . . , d, ‖f‖, ‖b1‖`1 , and ‖b2‖`1 such that P (t)2 = E(t)− E(t+ 1) = Ẽ(t)− Ẽ(t+ 1) ≤ 2Ẽ(0) ≤ µ0, ∀t ≥ 0. (3.21) It follows from (3.5) and (3.21) that sup t≤s≤t+1 Ẽ(s)1+ r 2 ≤ C1,1[Ẽ(t)− Ẽ(t+ 1)] + ( 2C0K0 )1+ r 2 , (3.22) where C1,1 is a positive constant depending only on r, C0, and µ0. Applying the first part of Lemma 3.1 to (3.22) with ψ(t) = Ẽ(t), γ = r 2 , C = C1,1, and K =( 2C0K0 )1+ r 2 we obtain Ẽ(t) ≤ [ C−11,1 r 2 (t− 1)+ + ( sup 0≤s≤1 Ẽ(s) )−r/2]−2/r + 2C0K0. (3.23) From (3.23), using (3.21), we deduce that Ẽ(t) ≤ C1(1 + t)−2/r + 2C0K0, (3.24) where C1 is a positive constant depending only on r, C0, and µ0. Combining (3.24) with (2.8) we obtain ‖S(t)(u0, u1)‖2H ≤ α−10 C1(1 + t)− 2 r + 2α−10 C0K0 ∀t ≥ 0. (3.25) Consequently, ‖S(t)(u0, u1)‖H ≤ ρ0, for all t ≥ τ if we take ρ0 = 2 (C0K0 α0 )1/2 , τ = τ(O) = max { 0, ( C1 2C0K0 )r/2 − 1 } . This completes the proof of Lemma 3.3 if r > 0. The proof for the case r = 0 is analogous. Indeed, using (3.5) and (3.21) we have sup t≤s≤t+1 Ẽ(s) ≤ C0(2 + µ0)[Ẽ(t)− Ẽ(t+ 1)] + C0K0, ∀t ≥ 0, EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 13 with C0 and µ0 as before. Then, applying the second part of Lemma 3.1 to this inequality and using (2.8) again, we deduce that ‖S(t)(u0, u1)‖2H ≤ α−10 µ0 2 ( C2 C2 + 1 )[t] + α−10 C0K0 ≤ α−10 µ0 2 (C2 + 1 C2 ) e−νt + α−10 C0K0, ∀t ≥ 0, (3.26) where C2 = C0(2 + µ0) and ν = ln ( C2+1 C2 ) . This implies Lemma 3.3 in the case r = 0 with ρ0 = ( 2C0K0 α0 )1/2 and τ = τ(O) = max { 0, 1ν ln (µ0(C2+1) 2C2C0K0 )} . � Next, we prove that the semigroup {S(t)}t≥0 is asymptotically compact in H. Here, the main step consists in using a method introduced by B. Wang in [28] combined with Lemma 3.1 to derive an appropriate estimate for the “tail” of the solution of (2.4). More precisely, we will show that, for all ε > 0, there exist τ(ε) > 0 and a positive integer k(ε) such that∑ |n|0≥k(ε) [(u̇n(t))2 + (un(t))2] < ε, for all t ≥ τ(ε), whenever the initial data (u0, u1) belongs to the absorbing set B[0; ρ0]. To do this, we will need the following auxiliary lemma whose proof relies on the following elementary identities valid for any sequences w = (wn) and z = (zn) and i = 1, . . . , d. ∂+i (wnzn) = (∂+i wn)zn+ei + wn∂ + i zn, (3.27) ∂+i (wnzn) = (∂+i wn)zn + wn∂ + i zn + ∂+i wn∂ + i zn, (3.28) partial−i (wnzn) = (∂−i wn)zn + wn∂ − i zn − ∂ − i wn∂ − i zn, (3.29) ∆d(wnzn) = (∆dwn)zn + wn∆dzn +∇+wn · ∇+zn +∇−wn · ∇−zn. (3.30) Note that (3.30) follows from (3.28) and (3.29). Lemma 3.4. Let u = (un(t)) belong to C1(R+; `2), and (θn) belong to `2. Then (−1)p ∑ n∈Zd ∆p dun(t) (θnu̇n(t)) = 1 2 d dt ∑ n∈Zd θn|Dpun(t)|2 + ∑ n∈Zd d∑ i=1 (∂+i θn)z(i)p,n(t), where ∑ n∈Zd d∑ i=1 |z(i)p,n(t)| ≤ C(p, d)‖(u(t), u̇(t))‖2H , ∀t ≥ 0, for some positive constant C(p, d) depending on p and d. Proof. A proof of this lemma when d = 1 was first presented in [18]. We will argue by induction. Consider first p odd. If p = 1 then using Lemma 2.2 and (3.27) we have − ∑ n∈Zd ∆dun(θnu̇n) = ∑ n∈Zd ∇+un · ∇+(θnu̇n) = ∑ n∈Zd d∑ i=1 ∂+i un ( θn∂ + i u̇n + ∂+i θnu̇n+ei ) 14 J. M. PEREIRA EJDE-2021/65 = 1 2 d dt ∑ n∈Zd θn|∇+un|2 + ∑ n∈Zd d∑ i=1 ∂+i θnz (i) 1,n, where z (i) 1,n = (∂+i un)u̇n+ei satisfies ∑ n∈Zd ∑d i=1 |z (i) 1,n| ≤ 2d‖(u, u̇)‖2H by Lemma 2.1. Assume now that Lemma 3.4 holds for p = 2k − 1 with k ∈ N. Using Lemma 2.2 again and (3.30) we have (−1)2k+1 ∑ n∈Zd ∆2k+1 d un(θnu̇n) = − ∑ n∈Zd ∆2k d un∆d(θnu̇n) = − ∑ n∈Zd ∆2k d un [ (∆dθn)u̇n + θn∆du̇n +∇+θn · ∇+u̇n +∇−θn · ∇−u̇n ] . (3.31) By Lemma 2.2 and (3.28) we also see that − ∑ n∈Zd ∆2k d un(∆dθn)u̇n = ∑ n∈Zd d∑ i=1 ∂+i θn(∆2k d un)∂+i u̇n + ∑ n∈Zd d∑ i=1 ∂+i θn(∂+i ∆2k d un)u̇n + ∑ n∈Zd d∑ i=1 ∂+i θn(∂+i ∆2k d un)∂+i u̇n. (3.32) In addition, − ∑ n∈Zd ∆2k d un∇−θn · ∇−u̇n = − ∑ n∈Zd d∑ i=1 ∆2k d un+ei∂ + i θn∂ + i u̇n = − ∑ n∈Zd d∑ i=1 ∂+i ∆2k d un∂ + i θn ∂ + i u̇n − ∑ n∈Zd d∑ i=1 ∆2k d un∂ + i θn ∂ + i u̇n. (3.33) Substituting (3.32) and (3.33) into (3.31) results (−1)2k+1 ∑ n∈Zd ∆2k+1 d un(θnu̇n) = − ∑ n∈Zd ∆2k d un(θn∆du̇n) + ∑ n∈Zd d∑ i=1 ∂+i θn [ (∂+i ∆2k d un)u̇n −∆2k d un(∂+i u̇n) ] . (3.34) EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 15 Then, using the induction hypothesis with v(t) = (∆dun(t)) ∈ C1(R+; `2), from (3.34), we obtain (−1)2k+1 ∑ n∈Zd ∆2k+1 d un(θnu̇n) = 1 2 d dt ∑ n∈Zd θn|D2k−1vn|2 + ∑ n∈Zd d∑ i=1 (∂+i θn)z (i) 2k−1,n + ∑ n∈Zd d∑ i=1 ∂+i θn [ (∂+i ∆2k d un)u̇n −∆2k d un(∂+i u̇n) ] , (3.35) where, in view of Lemma 2.1, ∑ n∈Zd d∑ i=1 |z(i)2k−1,n| ≤ C(2k − 1, d)‖(v, v̇)‖2H ≤ 16d2C(2k − 1, d)‖(u, u̇)‖2H , ∑ n∈Zd d∑ i=1 |(∂+i ∆2k d un)u̇n −∆2k d un(∂+i u̇n)| ≤ (4d)4k+1‖(u, u̇)‖2H . Therefore, the proof of Lemma 3.4 for p = 2k + 1 can be concluded from (3.35) with z (i) 2k+1,n = z (i) 2k−1,n + (∂+i ∆2k d un)u̇n −∆2k d un(∂+i u̇n). Consider now p even. Since the proof is similar to the one in the previous case, we summarize it as follows. In the case p = 2, using Lemma 2.2 and working with the identity (3.30) we can prove that ∑ n∈Zd ∆2 d(θnu̇n) = 1 2 d dt ∑ n∈Zd θn(∆dun)2 + ∑ n∈Zd d∑ i=1 (∂+i θn)z (i) 2,n, where, by Lemma 2.1, z (i) 2,n = (∆dun)∂+i u̇n − (∂+i ∆dun)u̇n satisfies ∑ n∈Zd d∑ i=1 |z(i)2,n| ≤ (4d)3‖(u, u̇)‖2H . Next, assuming Lemma 3.4 valid for p = 2k, with k ∈ N and proceeding as in (3.33)-(3.35) we can prove that ∑ n∈Zd ∆2k+2 d un(θnu̇n) = 1 2 d dt ∑ n∈Zd θn|D2k+2vn|2 + ∑ n∈Zd d∑ i=1 ∂+i θnz (i) 2k,n + ∑ n∈Zd d∑ i=1 ∂+i θn [ ∆2k+1 d un(∂+i u̇n)− (∂+i ∆2k+1 d un)u̇n ] , with z (i) 2k+2,n = z (i) 2k,n + ∆2k+1 d un(∂+i u̇n)− (∂+i ∆2k+1 d un)u̇n satisfying ∑ n∈Zd d∑ i=1 |z(i)2k+2,n| ≤ [16d2C(2k, d) + (4d)4k+3]‖(u, u̇)‖2H , by Lemma 2.1. This completes the proof. � 16 J. M. PEREIRA EJDE-2021/65 Remark 3.5. Along the proof of Lemma 3.4 if we replace θnu̇n(t) by θnun(t) then we can easily check that (−1)p ∑ n∈Zd ∆p dun(t) (θnun(t)) = ∑ n∈Zd θn|Dpun(t)|2 + ∑ n∈Zd d∑ i=1 (∂+i θn)z̃(i)p,n(t), (3.36) with z̃ (i) p,n(t) satisfying the same estimate in Lemma 3.4. Let us now introduce some notation analogous to those used in the proof of Lemma 3.2. Let u = (un(t)) be the global solution of (2.4) obtained by Theorem 2.4. We consider a function θ ∈ C1(R+;R) satisfying θ ≡ 0 in [0, 1], θ ≡ 1 in [2,∞), 0 ≤ θ(t) ≤ 1 and |θ′(t)| ≤ 2 for all t ≥ 0. Let w = (wn(t)), where wn(t) = θnun(t), with θn = θ ( |n|0 k ) and k fixed in N. We define the modified energy Eθ(t) = 1 2 ∑ n∈Zd θn(u̇n(t))2 + 1 2 ∑ n∈Zd θn|Dpun(t)|2 + α 2 ∑ n∈Zd θn(un(t))2 + ∑ n∈Zd θnh̃0(n, un(t)) + ∑ n∈Zd d∑ i=1 θnh̃i(∂ + i un(t))− ∑ n∈Zd θnfnun(t), (3.37) for t ≥ 0. Differentiating Eθ(t) with respect to t, using (2.4) and Lemma 3.4, we have d dt Eθ(t) = − ∑ n∈Zd θng(n, u̇n(t))u̇n(t)− ∑ n∈Zd d∑ i=1 (∂+i θn)z(i)p,n(t), ∀t ≥ 0, (3.38) with z (i) p,n(t) as in Lemma 3.4. From (3.38), for all t ≥ 0, we can write Eθ(t)− Eθ(t+ 1) = ∫ t+1 t Qθ(s)ds+ ∫ t+1 t Rθ(s)ds, where Qθ(t) = ∑ n∈Zd θng(n, u̇n(t))u̇n(t), Rθ(t) = ∑ n∈Zd d∑ i=1 (∂+i θn)z(i)p,n(t). Also, observe that from (3.37) and the hypotheses (A2) and (A3) we have Eθ(t) ≥ α0 ∑ n∈Zd θn[u̇n(t)2 + un(t)2]− 4 α ∑ n∈Zd θnf 2 n − ∑ n∈Zd θnb2,n, ∀t ≥ 0, (3.39) where, as before, α0 = min{ 12 , α 4 }. The above considerations motivate us to define on [0,∞) the following nonneg- ative functions Pθ(t) 2 = Eθ(t)− Eθ(t+ 1)− ∫ t+1 t Rθ(s)ds = ∫ t+1 t Qθ(s)ds, (3.40) Ẽθ(t) = Eθ(t) + 4 α ∑ n∈Zd θnf 2 n + ∑ n∈Zd θnb2,n. (3.41) Finally, to simplify notation, for any sequences w = (wn) and z = (zn), we will write (w, z)θ = ∑ n∈Zd θnwnzn, ‖w‖2θ = ∑ n∈Zd θn|wn|2, EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 17 ‖w‖`1θ = ∑ n∈Zd θn|wn|, ‖Dpw‖2θ = ∑ n∈Zd θn|Dpwn|2, (ĥ(w), w)θ = ∑ n∈Zd θnwnh0(n,wn)− ∑ n∈Zd d∑ i=1 θnwn∂ − i hi(∂ + i wn). Lemma 3.6. Under the assumptions of Lemma 3.2 and (u0, u1) ∈ B[0; ρ0], there exist positive constants C2,i, i = 1, 2, 3, with C2,1, C2,2 depending on ρ0 and C2,3 depending only on k1, k0,i, i = 1, . . . , d, and α, such that sup t≤s≤t+1 Ẽθ(s) ≤ C2,1 ( Pθ(t) 4 r+2 + Pθ(t) 2 ) + 1 k C2,2 + C2,3(‖f‖2θ + ‖b1‖`1θ ), ∀t ≥ 0. Proof. We first observe that (u0, u1) ∈ B[0; ρ0] implies the existence of a positive constant r0 depending on ρ0 and some parameters of the problem (see (3.24) and (3.26)) such that ‖(u(t), u̇(t))‖ ≤ r0, ∀t ≥ 0. (3.42) By the Mean Value Theorem for integrals there exist t1 ∈ [t, t + 1 4 ] and t2 ∈ [t+ 3 4 , t+ 1] such that∫ t+ 1 4 t Qθ(s) ds = 1 4 Qθ(t1) and ∫ t+1 t+ 3 4 Qθ(s) ds = 1 4 Qθ(t2). (3.43) Taking the inner product of equation (2.4) with w(t) in `2, using (3.36) and inte- grating the result over [t1, t2] we find∫ t2 t1 ‖Dpu(s)‖2θds+ α ∫ t2 t1 ‖u(s)‖2θds+ ∫ t2 t1 (ĥ(u(s)), u(s))θds = (u(t1), u̇(t1))θ − (u(t2), u̇(t2))θ + ∫ t2 t1 ‖u̇(s)‖2θds − ∫ t2 t1 (g(u̇(s)), u(s))θds+ ∫ t2 t1 (f, u(s))θds− ∫ t2 t1 R̃θ(s)ds, (3.44) where R̃θ(t) = ∑ n∈Zd d∑ i=1 ∂+i θnz̃ (i) p,n(t), (3.45) with z̃ (i) p,n(t) satisfying ∑ n∈Zd d∑ i=1 |z̃(i)p,n(t)| ≤ C(p, d)r20, ∀t ≥ 0. (3.46) Observe that by the definition of θn we have that |∂+i θn| ≤ 2 k , i = 1, . . . , d. Since (u0, u1) ∈ B[0; ρ0], from (3.46) we see that∣∣∣ ∫ t2 t1 R̃θ(s)ds ∣∣∣ ≤ ∑ n∈Zd d∑ i=1 ∫ t2 t1 |∂+i θn| |z̃ (i) p,n(s)|ds ≤ 2 k C(p, d)r20. (3.47) Next, we follow the same steps of the proof of Lemma 3.2. We first write (u(tj), u̇(tj))θ = ∑ |n|0≤n0 θnun(tj)u̇n(tj) + ∑ |n|0>n0 θnun(tj)u̇n(tj), j = 1, 2, (3.48) 18 J. M. PEREIRA EJDE-2021/65 with n0 as in (A4). Using (3.39) and (3.41) we have ( ∑ |n|0≤n0 θn|un(tj)|2 )1/2 ≤ α−1/20 sup t≤s≤t+1 Ẽθ(s) 1/2. (3.49) Using Hölder’s inequality, (A4), (3.43), and (3.40), we also have ∑ |n|0≤n0 θn|u̇n(tj)|2 ≤ (2n0 + 1) rd r+2 ( ∑ |n|0≤n0 θ r+2 2 n |u̇n(tj)|r+2 ) 2 r+2 ≤ (2n0 + 1) rd r+2 ( k−12 ∑ |n|0≤n0 θng(n, u̇n(tj))u̇n(tj) ) 2 r+2 ≤ (2n0 + 1) rd r+2 k − 2 r+2 2 4 2 r+2Pθ(t) 4 r+2 . (3.50) Similarly, we can estimate the second term in (3.48) to find that∑ |n|0>n0 |u̇n(tj)un(tj)| ≤ 2α −1/2 0 k −1/2 2 Pθ(t) sup t≤s≤t+1 Ẽθ(s) 1/2. (3.51) Thus, from (3.48)-(3.51) we conclude that there is a positive constant C1,1 depend- ing only on n0, α, k2, r, and d such that |(u(t1), u̇(t1))θ− (u(t2), u̇(t2))θ| ≤ C1,1 ( Pθ(t) 2 r+2 +Pθ(t) ) sup t≤s≤t+1 Ẽθ(s) 1/2. (3.52) In the same way we obtain the estimate∫ t2 t1 ‖u̇(s)‖2θds ≤ (2n0 + 1) rd r+2 k − 2 r+2 2 Pθ(t) 4 r+2 + 4k−12 Pθ(t) 2. (3.53) Now, to estimate the third term in right hand side of (3.44), we define the sets I1(t) = { n ∈ Zd; |u̇n(t)| ≤ 1 } and I2(t) = Zd\I1(t), with t ≥ 0 fixed. Using the assumptions (A1) and (A4 ) we see that∑ n∈I1(s) θnun(s)g(n, u̇n(s)) ≤ α 2 ∑ n∈I1(s) θn|un(s)|2 + 1 2α ∑ n∈I1(s) θn|g(n, u̇n(s))|2 ≤ α 2 ‖u(s)‖2θ + L2(1) 2α ∑ n∈I1(s) θn|u̇n(s)| |g(n, u̇n(s))| = α 2 ‖u(s)‖2θ + L2(1) 2α ∑ n∈I1(s) θnu̇n(s)g(n, u̇n(s)) ≤ α 2 ‖u(s)‖2θ + L2(1) 2α Qθ(s). (3.54) EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 19 Moreover, using (A4) and (3.42) we have∑ n∈I2(s) θnun(s)g(n, u̇n(s)) ≤ ∑ n∈I2(s) θn|un(s)| |u̇n(s)| |g(n, u̇n(s))| ≤ ‖u(s)‖ ∑ n∈I2(s) θn|u̇n(s)| |g(n, u̇n(s))| = ‖u(s)‖ ∑ n∈I2(s) θnu̇n(s)g(n, u̇n(s)) ≤ r0Qθ(s). (3.55) From (3.54), (3.55), and (3.40) it follows that∫ t2 t1 (g(u̇(s), u(s))θds ≤ α 2 ∫ t2 t1 ‖u(s)‖2θds+ ( 1 2α L2(1) + r20 ) Pθ(t) 2. (3.56) Finally, using (3.39) and (3.41) we also have∣∣∣ ∫ t2 t1 (f, u(s))θds ∣∣∣ ≤ ∫ t2 t1 ‖f‖θ ‖u(s)‖θds ≤ α−1/20 ‖f‖θ sup t≤s≤t+1 Ẽθ(s) 1/2. (3.57) Therefore, using (3.47), (3.52), (3.53), (3.56), and (3.57) in (3.44) we conclude that there exists a positive constant C1,2, which depends only on n0, α, k2, r, d, L2(1), and r0, such that∫ t2 t1 ‖Dpu(s)‖2θds+ α 2 ∫ t2 t1 ‖u(s)‖2θds+ ∫ t2 t1 (ĥ(u(s)), u(s))θds ≤ C1,2 [( Pθ(t) 2 r+2 + Pθ(t) ) sup t≤s≤t+1 Ẽθ(s) 1/2 + Pθ(t) 4 r+2 + Pθ(t) 2 ] + α −1/2 0 ‖f‖θ sup t≤s≤t+1 Ẽθ(s) 1/2 + 2 k C(p, d)r20. (3.58) On the other hand, integrating (3.37) over [t1, t2] we obtain∫ t2 t1 Eθ(s)ds = 1 2 ∫ t2 t1 ‖Dpu(s)‖2θds+ α 2 ∫ t2 t1 ‖u(s)‖2θds+ ∫ t2 t1 J1,θ(s)ds + 1 2 ∫ t2 t1 ‖u̇(s)‖2θds− ∫ t2 t1 (f, u(s))θds, (3.59) where J1,θ(s) = ∑ n∈Zd θnh̃0(n, un(s)) + ∑ n∈Zd d∑ i=1 θnh̃i(∂ + i un(s)). Let us compare (3.58) and (3.59). Using the identity (3.27) we can write (ĥ(u(t)), u(t))θ = ∑ n∈Zd θnun(t)h0(n, un(t)) + ∑ n∈Zd d∑ i=1 θn∂ + i un(s)hi(∂ + i un(t)) +R1,θ(t) = J2,θ(t) +R1,θ(t)− ‖b1‖`1θ , 20 J. M. PEREIRA EJDE-2021/65 where R1,θ(t) = ∑ n∈Zd d∑ i=1 (∂+i θn)un+ei(t)hi(∂ + i un(t)), J2,θ(t) = ∑ n∈Zd θn[un(t)h0(n, un(t)) + b1,n] + ∑ n∈Zd d∑ i=1 θn∂ + i un(t)hi(∂ + i un(t)) ≥ 0, for all t ≥ 0 because of (A2) and (A3). Substituting the above expression into (3.58) yields∫ t2 t1 ‖Dpu(s)‖2θds+ α 2 ∫ t2 t1 ‖u(s)‖2θds+ ∫ t2 t1 J2,θ(s)ds ≤ C1,2 [( Pθ(t) 2 r+2 + Pθ(t) ) sup t≤s≤t+1 Ẽθ(s) 1/2 + Pθ(t) 4 r+2 + Pθ(t) 2 ] + α −1/2 0 ‖f‖θ sup t≤s≤t+1 Ẽθ(s) 1/2 + 2 k C(p, d)r20 + ∫ t2 t1 ‖b1‖`1θds − ∫ t2 t1 R1,θ(s)ds. (3.60) Since (u0, u1) ∈ B[0; ρ0] and |∂+i θn| ≤ 2 k , i = 1, . . . , d, then using (3.42) and (A1) we have ∣∣∣ ∫ t2 t1 R1,θ(s)ds ∣∣∣ ≤ 4d k L3(2r0)r20. (3.61) In addition, by (A2) and (A3) we see that J1,θ(s) ≤ k−11 ∑ n∈Zd d∑ i=1 θn[un(s)h0(n, un(s)) + b1,n]− ‖b2‖`1θ + ∑ n∈Zd d∑ i=1 k−10,i θn∂ + i un(s)hi(∂ + i un(s)) ≤ k0J2,θ(s)− ‖b2‖`1θ , (3.62) where k0 = max{k−11 , k−10,i , i = 1, . . . , d}. Therefore, from (3.41), (3.53), (3.57), (3.59), (3.60), (3.61), and (3.62) we deduce that∫ t2 t1 Ẽθ(s)ds ≤ C1,3 [( Pθ(t) 2 r+2 + Pθ(t) ) sup t≤s≤t+1 Ẽθ(s) 1/2 + Pθ(t) 4 r+2 + Pθ(t) 2 ] + 2α −1/2 0 ‖f‖θ sup t≤s≤t+1 Ẽθ(s) 1/2 + 1 k C1,4 + max{k0, 1}‖b1‖`1θ + 4 α ‖f‖2θ, (3.63) where C1,3 and C1,4 are positive constants depending on r0. Finally, as in the end of the proof of Lemma 3.2, we can find t∗ ∈ [t1, t2] so that 1 2 Ẽθ(t ∗) ≤ (t2 − t1)Ẽθ(t ∗) = ∫ t2 t1 Ẽθ(s)ds. (3.64) If t̃ ∈ [t, t∗], then, using (3.41), (3.38), and (3.40), we have Ẽθ(t̃)− Ẽθ(t∗) = Eθ(t̃)− Eθ(t∗) EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 21 = ∫ t̃ t∗ Qθ(s)ds+ ∫ t∗ t̃ Rθ(s)ds ≤ Pθ(t)2 + ∫ t+1 t |Rθ(s)|ds. Similarly, if t̃ ∈ [t∗, t+ 1] then Ẽθ(t̃)− Ẽθ(t∗) = − ∫ t̃ t∗ Qθ(s)ds− ∫ t∗ t̃ Rθ(s)ds ≤ ∫ t+1 t |Rθ(s)|ds. Therefore, sup t≤s≤t+1 Ẽθ(s) ≤ Ẽθ(t∗) + Pθ(t) 2 + ∫ t+1 t |Rθ(s)|ds, (3.65) where, as in (3.47), we have∫ t+1 t |Rθ(s)|ds ≤ 2 k C(p, d)r20. (3.66) The conclusion of the proof of Lemma 3.6 now follows from (3.63)-(3.66). � Lemma 3.7. Under the assumptions of Lemma 3.6, for each ε > 0, there exist τ(ε) > 0 and a positive integer k(ε) such that∑ |n|0≥k(ε) [(u̇n(t))2 + (un(t))2] < ε, ∀t ≥ τ(ε). Proof. We will denote by C3,i, i = 1, . . . , 6, the positive constants depending on ρ0 that appear along the proof. Let us first assume that r > 0. Since (3.42) holds, using (A1), we have Pθ(t) 2 = ∫ t+1 t Qθ(s)ds ≤ ∫ t+1 t ∑ n∈Zd θn|g(n, u̇n(s))||u̇n(s)|ds ≤ L2(r0) ∫ t+1 t ‖u̇(s)‖2ds ≤ L2(r0)r20. (3.67) By (3.66) we also have Pθ(t) 2 = Ẽθ(t)− Ẽθ(t+ 1)− ∫ t+1 t Rθ(s)ds ≤ Ẽθ(t)− Ẽθ(t+ 1) + 2 k C(p, d)r20. (3.68) Using Lemma 3.6, (3.67), and (3.68) we obtain sup t≤s≤t+1 Ẽθ(s) 1+ r 2 ≤ 21+ r 2C 1+ r 2 2,1 Pθ(t) 2 ( 1 + Pθ(t) 2r r+2 )1+ r 2 + 21+ r 2 [1 k C2,2 + C2,3(‖f‖2θ + ‖b1‖`1θ ) ]1+ r 2 ≤ C3,1[Ẽθ(t)− Ẽθ(t+ 1)] + 21+ r 2 1 k C3,2 + [2 k C2,2 + 2C2,3(‖f‖2θ + ‖b1‖`1θ ) ]1+ r 2 . (3.69) 22 J. M. PEREIRA EJDE-2021/65 Applying the first part of Lemma 3.1 to inequality (3.69) we deduce that Ẽθ(t) ≤ [ C−13,1 r 2 (t− 1)+ + ( sup 0≤s≤1 Ẽθ(s) )− r2 ]− 2 r + C3,3k − 2 k+2 + 2C2,3(‖f‖2θ + ‖b1‖`1θ ), ∀t ≥ 0. (3.70) Since (u0, u1) ∈ B[0; ρ0] then proceeding as in (3.21) we can bound the term sup0≤s≤1 Ẽθ(s) in (3.70) by a positive constant that depends on ρ0. Thus, from (3.70), (3.39), and (3.41) we obtain∑ n∈Zd θn[u̇n(t)2 + un(t)2] ≤ α−10 C3,4(1 + t)− 2 r + α−10 C3,3k − 2 r+2 + 2α−10 C2,3(‖f‖2θ + ‖b1‖`1θ ), ∀t ≥ 0. (3.71) Since f ∈ `2 and b1 ∈ `1 then, given ε > 0, there exists a positive integer k(ε) such that α−10 C3,3k − 2 r+2 + 2α−10 C2,3 ( ∑ |n|0>k |fn|2 + ∑ |n|0>k b1,n ) < ε 2 , ∀k ≥ k(ε) 2 . Then, from (3.71) it follows that∑ n∈Zd θn[u̇n(t)2 + un(t)2] ≤ α−10 C3,4(1 + t)− 2 r + ε 2 , for all t ≥ 0 and k ≥ k(ε) 2 . Choosing τ(ε) = max{0, τ1(ε)}, with τ1(ε) = ( 2C3,4 α0ε ) r 2−1 we obtain ∑ |n|0≥2k [u̇n(t)2 + un(t)2] < ε, ∀t ≥ τ(ε) and ∀k ≤ k(ε) 2 . This estimate implies Lemma 3.6 if r > 0. Suppose now that r = 0. By Lemma 3.6 and (3.68) we have sup t≤s≤t+1 Ẽθ(s) ≤ 2C2,1Pθ(t) 2 + 1 k C2,2 + C2,3(‖f‖2θ + ‖b1‖`1θ ) ≤ 2C2,1[Ẽθ(t)− Ẽθ(t+ 1)] + C3,5 1 k + C2,3(‖f‖2θ + ‖b1‖`1θ ). (3.72) Applying the second part of Lemma 3.1 to inequality (3.72) yields Ẽθ(t) ≤ sup 0≤s≤1 Ẽθ(s) ( 2C2,1 2C2,1 + 1 )[t] +C3,5 1 k +C2,3(‖f‖2θ +‖b1‖`1θ ), ∀t ≥ 0. (3.73) Then, estimating the term sup0≤s≤1 Ẽθ(s) in (3.73) by a positive constant depend- ing on ρ0 and using (3.39) and (3.41) again we obtain∑ n∈Zd θn[u̇n(t)2 + un(t)2] ≤ α−10 C3,6e −νt + α−10 C3,5 1 k + α−10 C2,3(‖f‖2θ + ‖b1‖`1θ ), ∀t ≥ 0, with ν = ln ( 2C2,1+1 2C2,1 ) , which implies Lemma 3.7 if r = 0. � Lemma 3.8. Under the assumptions of Lemma 3.6, the semigroup {S(t)}t≥0 is asymptotically compact in H. EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 23 In view of Lemmas 3.3 and 3.7, the proof of the above lemma is similar to that of Lemma 3.9 in [18], so we omit its proof here. Using the notation of Lemma 3.3 and denoting by dist(A,B) = sup a∈A inf b∈B ‖a− b‖H the Hausdorff semi-distance between two subsets A and B of H we can state our first main result in this section. Theorem 3.9. Assume that (A1)–(A4) hold and let u0, u1, and f belong to `2. Then the semigroup {S(t)}t≥0 associated with the initial value problem (2.4) pos- sesses a unique global attractor A in H. Moreover, A ⊂ B[0; ρ0] and for any bounded set B0 in H there exist positive constants C(B0) and ν depending on B0 such that dist ( S(t)B0, B[0; ρ0] ) ≤ C(B0) { (1 + t)− 1 r , if r > 0 e−νt, if r = 0 . (3.74) Proof. In view of Lemmas 3.3, 3.7 and [26, Theorem 1.1], the ω-limit set A = ω(B[0; ρ0]) is the unique global attractor for {S(t)}t≥0 in H. Clearly, A ⊂ B[0; ρ0] because A is a bounded set of H and invariant under {S(t)}t≥0. Let us prove the absorbing rate in (3.74) when r > 0. By (3.25) for any (u0, u1) ∈ B0 we can find a positive constant C(B0) depending on B0 such that ‖S(t)(u0, u1)‖H ≤ C(B0)(1 + t)− 1 r + 2−1/2ρ0, ∀t ≥ 0. (3.75) Now, for any (u0, u1) ∈ B0 and t ≥ 0 fixed, from (3.75) we easily see that inf (w0,w1)∈B[0;ρ0] ‖S(t)(u0, u1)− (w0, w1)‖H ≤ C(B0)(1 + t)−1/r. Therefore, dist ( S(t)B0, B[0; ρ0] ) ≤ C(B0)(1 + t)−1/r. The exponential absorbing rate in (3.74), i.e., when r = 0, is proved in the same manner using (3.26). � Next, we will prove the existence of a global attractor for the semigroup {S(t)}t≥0 associated with the initial value problem (2.10). Thus, in the rest of this section, u = u(t) denotes the global solution of (2.10) obtained by Theorem 2.5 and the semigroup {S(t)}t≥0 is defined as in (3.1). We also consider E(t) and Ẽ(t) de- fined by (2.11) and (2.12), respectively, and the functions P (t)2 and Q(t) as before satisfying (3.3) and (3.4). Lemma 3.10. Assume that (A1), (A4)–(A6) hold and let u0, u1, and f belong to `2. Then there exists ρ1 > 0 such that B[0; ρ1] = {(w, z) ∈ H; ‖(w, z)‖H ≤ ρ1} is an absorbing set for {S(t)}t≥0 in H. 24 J. M. PEREIRA EJDE-2021/65 Proof. The proof of this lemma is similar to the proofs of Lemmas 3.2 and 3.3. We first proceed as in the proof of Lemma 3.2 to obtain∫ t2 t1 ‖Dpu(s)‖2ds+ α 2 ∫ t2 t1 ‖u(s)‖2ds+ ∫ t2 t1 J̃2(s)ds ≤ C̃0,2 [( P (t) 2 r+2 + P (t)2 + P (t) + ‖f‖ ) sup t≤s≤t+1 Ẽ(s)1/2 + P (t) 4 r+2 + P (t)2 ] , (3.76) where C̃0,2 is a positive constant that depends only on n0, α, k2, r, d, L2(1) and J̃2(t) = ∑ n∈Zd ∑ m∈Zd V (n−m)h(um(t))h′(un(t))un(t), ∀t ≥ 0. Using hypotheses (A5) and (A6) we have that∑ n∈Zd ∑ m∈Zd V (n−m)h(um(t))h(un(t)) ≤ c−11 J̃2(t), ∀t ≥ 0. (3.77) Integrating (2.12) over [t1, t2] and using (2.11), (3.76), and (3.77) we obtain∫ t2 t1 Ẽ(s)ds ≤ max{c−11 , 1}C̃0,2 [( P (t) 2 r+2 + P (t)2 + P (t) + ‖f‖ ) sup t≤s≤t+1 Ẽ(s)1/2 + P (t) 4 r+2 + P (t)2 ] + 1 2 ∫ t2 t1 ‖u̇(s)‖2ds+ ∣∣∣ ∫ t2 t1 (f, u(s))`2ds ∣∣∣+ 4 α ‖f‖2. Then, still arguing as in the proof of Lemma 3.2, we prove the following inequality analogous to (3.5), sup t≤s≤t+1 Ẽ(s) ≤ C̃0 [ P (t) 4 r+2 + P (t)4 + P (t)2 + ‖f‖2 ] , ∀t ≥ 0, (3.78) for some positive constant C̃0. Now, let O be a bounded subset of H and choose ρ = ρ(O) > 0 so that ‖(w, z)‖H ≤ ρ, for all (w, z) ∈ O. If (u0, u1) ∈ O, then, as in (3.21), now us- ing Lemma 2.1, (A5), and (A6), we can bound P (t)2 by a positive constant µ̃0 depending on p, d, α, ρ, ‖V ‖, and ‖f‖. Therefore, the proof of Lemma 3.10 can be concluded as in the proof o Lemma 3.3, using (3.78) and Lemma 3.1. � Next, let us modify the proofs of Lemmas 3.6 and 3.7. We will keep the same notation used in Lemma 3.6. Under the assumptions of Lemma 3.10, we define Eθ(t) = 1 2 ∑ n∈Zd θn(u̇n(t))2 + 1 2 ∑ n∈Zd θn|Dpun(t)|2 + α 2 ∑ n∈Zd θn(un(t))2 + J̃1,θ(t)− ∑ n∈Zd θnfnun(t), ∀t ≥ 0, (3.79) where J̃1,θ(t) = 1 2 ∑ n∈Zd ∑ m∈Zd θnV (n−m)h(um(t))h(un(t)). (3.80) EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 25 Differentiating Eθ(t) with respect to t, using (2.10), (A5), (A6) and Lemma 3.4, we find that d dt Eθ(t) = − ∑ n∈Zd θng(n, u̇n(t))u̇n(t)−Rθ(t), ∀t ≥ 0, (3.81) with Rθ(t) = R1,θ(t) +R2,θ(t), where R1,θ(t) = ∑ n∈Zd d∑ i=1 (∂+i θn)z(i)p,n(t), (3.82) R2,θ(t) = −1 2 ∑ n∈Zd ∑ m∈Zd (θn − θm)V (n−m)h(un(t))h′(um(t))u̇m(t), (3.83) and the functions z (i) p,n(t), i = 1, . . . , d are as in Lemma 3.4. Thus, defining on [0,∞) the functions Pθ(t) 2 = Eθ(t)− Eθ(t+ 1)− ∫ t+1 t Rθ(s)ds, (3.84) Ẽθ(t) = Eθ(t) + 4 α ∑ n∈Zd θnf 2 n, (3.85) from (3.79) and (3.81) we obtain Pθ(t) 2 = ∫ t+1 t Qθ(s)ds ≥ 0, ∀t ≥ 0, (3.86) Ẽθ(t) ≥ α0 ∑ n∈Zd θn[u̇n(t)2 + un(t)2], ∀t ≥ 0, (3.87) where, as before, in (3.86), Qθ(t) = ∑ n∈Zd θng(n, u̇n(t)u̇n(t)) for all t ≥ 0. Lemma 3.11. Under the assumptions of Lemma 3.10 and also (u0, u1) ∈ B[0; ρ1], there exist positive constants C̃2,i, i = 1, 2, 3, with C̃2,1, C̃2,2 depending on ρ1 and C̃2,3 depending only on α, such that sup t≤s≤t+1 Ẽθ(s) ≤ C̃2,1 ( Pθ(t) 4 r+2 + Pθ(t) 2 ) + C̃2,2 [ 1 k2 ( 1 + ∑ |j|0≤l |j|20|V (j)|2 ) + ∑ |j|0>l |V (j)|2 ]1/2 + C̃2,3‖f‖2θ, for any positive integer l and any t ≥ 0. Proof. Since (u0, u1) ∈ B[0; ρ1], as in Lemma 3.6, there exists a positive constant r1 depending on ρ1 such that ‖(u(t), u̇(t))‖H ≤ r1, ∀t ≥ 0. (3.88) 26 J. M. PEREIRA EJDE-2021/65 Let t1, t2 satisfy (3.43) with Qθ(t) as above. Taking the inner product of equation (2.10) with w(t) in `2 and integrating the result over [t1, t2] we find that ∫ t2 t1 ‖Dpu(s)‖2θds+ α ∫ t2 t1 ‖u(s)‖2θds+ ∫ t2 t1 J̃2,θ(s)ds = (u̇(t1), u(t1))θ − (u̇(t2), u(t2))θ + ∫ t2 t1 ‖u̇(s)‖2θds − ∫ t2 t1 (g(u̇(s)), u(s))θds+ ∫ t2 t1 (f, u(s))θds− ∫ t2 t1 R̃θ(s)ds, (3.89) where J̃2,θ(t) = ∑ n∈Zd ∑ m∈Zd θnV (n−m)h(um(t))h′(un(t))un(t), (3.90) R̃θ(t) = ∑ n∈Zd d∑ i=1 (∂+i θn)z̃(i)p,n(t), and the functions z̃ (i) p,n(t), i = 1, . . . , d satisfy the estimate (3.46) with r0 = r1. Substituting the estimates obtained in the proof of Lemma 3.6 into (3.89) yields ∫ t2 t1 ‖Dpu(s)‖2θds+ α 2 ∫ t2 t1 ‖u(s)‖2θds+ ∫ t2 t1 J̃2,θ(s)ds ≤ C̃1,2 [( Pθ(t) 2 r+2 + Pθ(t) ) sup t≤s≤t+1 Ẽθ(s) 1/2 + Pθ(t) 4 r+2 + Pθ(t) 2 ] + α −1/2 0 ‖f‖θ sup t≤s≤t+1 Ẽθ(s) 1/2 + 2 k C(p, d)r21, (3.91) where C̃1,2 is a positive constant that depends on n0, α, k2, r, d, L2(1), and r1. By (A5) and (A6) we know that J̃2,θ(t) ≥ 2c1J̃1,θ(t), ∀t ≥ 0. Then, integrating (3.79) over [t1, t2], using (3.53), (3.57), and (3.91), we obtain ∫ t2 t1 Ẽθ(s)ds ≤ C̃1,3 [( Pθ(t) 2 r+2 + Pθ(t) ) sup t≤s≤t+1 Ẽθ(s) 1/2 + Pθ(t) 4 r+2 + Pθ(t) 2 ] + α −1/2 0 ‖f‖θ sup t≤s≤t+1 Ẽθ(s) 1/2 + 1 k C̃1,4 + 4 α ‖f‖2θ, (3.92) with some positive constants C̃1,3 and C̃1,4 depending on r1. To complete the proof we need to estimate the term Rθ(t). Observe that for any m, j ∈ Zd, by the definition of θn, we know that |θm+j − θm| ≤ 2 k |j|0 and |θm+j − θm| ≤ 2. EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 27 Since (u0, u1) ∈ B[0; ρ1], for any integer l > 1, using (3.88), (A5), and (A6) we can estimate R2,θ(t) as follows |R2,θ(t)| ≤ 1 2 ∑ m∈Zd ( |h′(um(t))| |u̇m(t)| ∑ n∈Zd |θn − θm|V (n−m)|h(un(t))| ) ≤ 1 2 β1 ( ∑ m∈Zd |h′(um(t))|2 |u̇m(t)|2 )1/2( ∑ j∈Zd |θm+j − θm|2|V (j)|2 )1/2 ≤ β2 1r1 ( 1 k2 ∑ |j|0≤l |j|20|V (j)|2 + ∑ |j|0>l |V (j)|2 )1/2 , (3.93) where β1 = max|s|≤r1{|h(s)|, |h′(s)|}. Estimating the term (3.82) as in (3.47) we obtain∫ t+1 t |R1,θ(s)|ds ≤ 2 k C(p, d)r21. (3.94) Now, as in the proof of Lemma 3.6, by the Mean-Value Theorem for integrals there exists t∗ ∈ [t1, t2] such that 1 2 Ẽθ(t ∗) ≤ (t2 − t1)Ẽθ(t ∗) = ∫ t2 t1 Ẽθ(s)ds (3.95) and, as in (3.65), we have sup t≤s≤t+1 Ẽθ(s) ≤ Ẽθ(t∗) + Pθ(t) 2 + ∫ t+1 t |Rθ(s)|ds. (3.96) Finally, from (3.92)-(3.96) we conclude the proof. � Lemma 3.12. Under the assumptions of Lemma 3.10, for each ε > 0, there exist τ(ε) > 0 and a positive integer k(ε) such that∑ |n|0≥k(ε) [(u̇n(t))2 + (un(t))2] < ε, ∀t ≥ τ(ε). Proof. Let us consider the case r > 0. We will denote by C̃3,i, i = 1, . . . , 5, the positive constants depending on ρ1 which will appear along the proof. By Lemma 3.11 we have sup t≤s≤t+1 Ẽθ(s) 1+ r 2 ≤ 21+ r 2Pθ(t) 2 [ C̃2,1 ( 1 + Pθ(t) 2r r+2 )]1+ r 2 + 21+ r 2 [ C̃2,2 ϑ(k, l) + C̃2,3‖f‖2θ ]1+ r 2 , ∀t ≥ 0, (3.97) where ϑ(k, l) = 1 k2 ( 1 + ∑ |j|0≤l |j|20|V (j)|2 ) + ∑ |j|0>l |V (j)|2. In (3.97) we can bound the term Pθ(t) 2r r+2 by a positive constant depending on r1 using (3.86), (A1) and (3.88) (see (3.67)). Then, using (3.84) and the estimates (3.93) and (3.94) again, from (3.97) it follows that sup t≤s≤t+1 Ẽθ(s) 1+ r 2 ≤ C̃3,1[Ẽθ(t)− Ẽθ(t+ 1)] + 21+ r 2 C̃3,2ϑ(k, l) + 21+ r 2 [ C̃2,2ϑ(k, l) + C̃2,3‖f‖2θ ]1+ r 2 . 28 J. M. PEREIRA EJDE-2021/65 Applying the first part of Lemma 3.1 to this inequality and using the elementary inequality (a+ b)s ≤ as + bs, valid for all real numbers a, b > 0 and 0 < s ≤ 1, we deduce that Ẽθ(t) ≤ [ C̃−13,1 r 2 (t− 1)+ + ( sup 0≤s≤1 Ẽθ(s) )− r2 ]− 2 r + C̃3,3 [ ϑ1(k, l) + ϑ2(l) ] + 2C̃2,3‖f‖2θ, ∀t ≥ 0, (3.98) where ϑ1(k, l) = 1 k2 ( 1 + ∑ |j|0≤l |j|20|V (j)|2 ) + [ 1 k2 ( 1 + ∑ |j|0≤l |j|20|V (j)|2 )] 2 r+2 , ϑ2(l) = ∑ |j|0>l |V (j)|2 + ( ∑ |j|0>l |V (j)|2 ) 2 r+2 . Since (u0, u1) ∈ B[0; ρ1], using (3.79), (3.85), Lemma 2.1, and (A5) we can bound the term sup0≤s≤1 Ẽθ(s) in (3.98) by a positive constant depending on ρ1, p, d, α, ‖V ‖, and ‖f‖. Therefore, from (3.98) and (3.87), we obtain the inequality∑ n∈Zd θn[u̇n(t)2 + un(t)2] ≤ α−10 C̃3,4(1 + t)− 2 r + α−10 ˜C3,3[ϑ1(k, l) + ϑ2(l)] + 2α−10 C̃2,3‖f‖2θ, ∀t ≥ 0, Now, given ε > 0, using (A6) we choose l = l(ε) > 1 such that α−10 C̃3,3ϑ2(l) < ε/4. Thus,∑ n∈Zd θn[u̇n(t)2 + un(t)2] ≤ α−10 C̃3,4(1 + t)− 2 r + α−10 C̃3,3ϑ1(k, l(ε)) + 2α−10 C̃2,3‖f‖2θ + ε 4 , ∀t ≥ 0. Since f ∈ `2, then there exists a positive integer k(ε) such that α−10 C̃3,3ϑ1(k, l(ε)) + 2α−10 C̃2,3‖f‖2θ < ε 4 , for all k ≥ k(ε) 2 . Consequently, the above inequality reduces to∑ n∈Zd θ ( |n|0 k )[ u̇n(t)2 + un(t)2 ] ≤ α−10 C̃3,4(1 + t)− 2 r + ε 2 , (3.99) for all t ≥ 0 and k ≥ k(ε) 2 . Now, choosing τ(ε) = max { 0, ( 2C̃3,4 α0ε )r/2 − 1 } , from (3.99) we obtain∑ |n|≥2k [u̇n(t)2 + un(t)2] ≤ ε, ∀t ≥ τ(ε), ∀k ≥ k(ε) 2 , which implies Lemma 3.12 if r > 0. If r = 0 then by Lemma 3.11, (3.93) and (3.94) we have sup t≤s≤t+1 Ẽθ(s) ≤ 2C̃2,1[Ẽθ(t)− Ẽθ(t+ 1)] + C̃3,5ϑ(k, l) + C̃2,3‖f‖2θ, with ϑ(k, l) as before. Applying the second part of Lemma 3.1 to this inequality we can conclude the proof as we did in the case r > 0. � EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 29 Lemma 3.13. Under the same assumptions of Lemma 3.10 the semigroup {S(t)}t≥0 associated with the initial value problem (2.10) is asymptotically com- pact in H. The above lemma is a consequence of Lemma 3.12 and well known arguments (see [18, Lemma 3.9]). As a direct consequence of Lemmas 3.10, 3.13 and [26, Theorem 1.1 ] we obtain our second main result in this section. Theorem 3.14. Assume that (A1), (A4)–(A6) hold and let u0, u1, and f belong to `2. Then the semigroup {S(t)}t≥0 associated with the initial value problem (2.10) possesses a unique global attractor A em H. Moreover, A ⊂ B[0; ρ1] and the absorbing rates (3.74) hold with ρ0 replaced by ρ1. 4. Periodic lattice differential equations Let N be a fixed positive integer (the period). We denote by `2per the Hilbert space of all real sequences u = (un)n∈Zd such that un+Nei = un, i = 1, . . . , d, n ∈ Zd, (4.1) equipped with the inner product and norm given by (u, v)`2per = N∑ n=1 unvn, ‖u‖ = ( N∑ n=1 |un|2 )1/2 , for all u = (un)n∈Zd and v = (vn)n∈Zd . Throughout this section we donote by∑N n=1 un the sum ∑N n1=1 · · · ∑N nd=1 u(n1,...,nd), whenever u = (un)n∈Zd satisfies (4.1). We also consider the Hilbert space ˙̀2 per of all sequences u = (un)n∈Zd in `2per such that ∑N n=1 un = 0, equipped with the inner product and norm above. As in the previous sections we will denote a sequence (un)n∈Zd by (un). Also, we will refer to a sequence u = (un)n∈Zd satisfying (4.1) as a periodic sequence. Using (4.1) it is easy to check that for any u = (un) and v = (vn) in `2per, Lemma 2.1, inequality (2.2), and Lemma 2.2 are still valid with ∑ n∈Zd replaced by ∑N n=1 and |Dpun|2 defined by (2.1). We assume that the functions h0 : Zd × R → R, hi : R → R, i = 1, . . . , d, and g : Zd × R → R satisfy (A1)–(A3) stated in Section 2, where in (A2), b1 = (b1,n), and b2 = (b2,n) are nonnegative periodic sequences. In addition, we assume that (A4’) For each s ∈ R, (h0(n, s)), (g(n, s)) are periodic sequences and there exist real constants k2 > 0, r ≥ 0 such that sg(n, s) ≥ k2 |s|r+2, for all n ∈ Zd and s ∈ R. Let us first consider the periodic problem ün(t) + (−1)p∆p dun(t) + αun(t) + F (n, un(t),∇+un(t)) + g(n, u̇n(t)) = fn, un(0) = u0,n, u̇n(0) = u1,n, un+Nei(t) = un(t), (4.2) where n ∈ Zd, i = 1, . . . , d, t ≥ 0, (u0,n), (u1,n), (fn) belong to `2per, and the term F (n, un(t),∇+un(t)) is as in (1.2). We introduce the Hilbert space H = `2per × `2per equipped with the natural inner product given by ( (u, v), (w, z) ) H = (u,w)`2per + (v, z)`2per , (4.3) 30 J. M. PEREIRA EJDE-2021/65 for all (u, v) and (w, z) in H. Proceeding as in Sections 2 and 3 we can study the dynamics of the problem (4.2) in the space H and extend the results of Theorems (2.4) and (3.9) to this case. Since the proofs are entirely analogous, here, we will just indicate what should be done. • We first observe that, under the above assumptions, we can prove that the maps h0, B, g defined in (2.3) are locally Lipschitz continuous from `2per into itself and that the linear operator A, also defined in (2.3), is bounded in `2per. Then we can write the problem (4.2) in `2per as the initial value problem (2.4). As a result, we can use the Theory Ordinary Differential Equations in Banach Spaces and the energy equation associated with the problem (4.2), which is given by (2.7), with ∑ n∈Zd replaced by ∑N n=1 and `2 by `2per, to obtain the same results of Theorem 2.4 for problem (4.2) in the space H above. • The second step consists in defining the semigroup {S(t)}t≥0 of continuous operator associate with problem (4.2) as in (3.1). Since the dimension of H = `2per × `2per is finite, in order to prove the existence of a global attractor in H we only need to prove the existence of an absorbing set for the semigroup {S(t)}t≥0 in H. This can be done as in Lemmas 3.2 and 3.3 using Lemma 3.1. Next, let us discuss the existence of a global attractor for the one-dimensional periodic problem ün(t) + (−1)p∆p 1un(t)− ∂−1 h(∂+1 un(t))− ∂−1 g(∂+1 u̇n(t)) = fn, un(0) = u0,n, u̇n(0) = u1,n, un+N (t) = un(t), (4.4) where n ∈ Z and t ≥ 0. In (4.4) (u0,n), (u1,n), (fn) belong to ˙̀2 per, and we assume that the functions h : R→ R and g : R→ R satisfy the assumptions (A7) For each s0 > 0, there exist positive constants Lj = Lj(s0), j = 1, 2, such that (i) |h(s1)− h(s2)| ≤ L1|s1 − s2|, (ii) |g(s1)− g(s2)| ≤ L2|s1 − s2|, for all s1, s2 in R, |s1| ≤ s0, |s2| ≤ s0 and h(0) = 0, g(0) = 0. (A8) There exist constants k0 > 0, k1 ≥ 0 and k2 ≥ 0 such that sh(s) + k1 ≥ k0(h̃(s) + k2) ≥ 0, ∀s ∈ R, where h̃(s) = ∫ s 0 h(σ)dσ. (A9) There exist constants k3 > 0, r ≥ 0 such that sg(s) ≥ k3|s|r+2 for all s ∈ R. Examples of functions satisfying the above assumptions are given in Appendix A. We want to study the dynamics of problem (4.4) in the Hilbert space Ḣ = ˙̀2 per × ˙̀2 per, equipped with the inner product (·, ·)Ḣ defined as in (4.3), with H replaced by Ḣ. We denote by ‖ · ‖Ḣ the corresponding norm. Under assumptions (A7)–(A9), the well-posedness of problem (4.4) can be proved following the same steps of Section 2. Indeed, introducing the notation Au = ((−1)p∆p 1un), B(u) = ( − ∂−1 h(∂+1 un) ) , G(u) = ( − ∂−1 g(∂+1 un) ) , EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 31 for any u = (un) in ˙̀2 per, we can write (4.4) in ˙̀2 per as ü(t) +Au(t) +B(u(t)) +G(u̇(t)) = f, t > 0, u(0) = u0, u̇(0) = u1, (4.5) where u0 = (u0,n), u1 = (u1,n), f = (fn), u(t) = (un(t)), u̇(t) = (u̇n(t)), and ü(t) = (ün(t)). Therefore, we can prove the existence of a unique solution u = u(t) of (4.5) belonging to C2([0, τmax); ˙̀ per) with the property that either τmax =∞ or τmax <∞ and lim t→τ−max ‖(u(t), u̇(t))‖Ḣ =∞. To extend the solution u = u(t) globally we will need the following consequence of Poincaré inequality valid in ˙̀2 per. Lemma 4.1. Let u = (un) belong to ˙̀2 per. Then N∑ n=1 u2n ≤ C p 0 N∑ n=1 |Dpun|2, for all p ∈ N, where C0 = 4N2. Proof. By Poincaré inequality (see [17]), we know that N∑ n=1 u2n ≤ C0 N∑ n=1 |∂+1 un|2, (4.6) where C0 = 4N2. Since ∂+1 un ∈ ˙̀2 per, using (4.6) and observing that N∑ n=1 |∂+1 vn|2 = N∑ n=1 |∂−1 vn|2 for any v = (vn) ∈ ˙̀2 per because of the periodicity, we obtain N∑ n=1 u2n ≤ C2 0 N∑ n=1 |∆1un|2. Then by induction we can conclude the proof. � Theorem 4.2. Assume that (A7)–(A9) hold and let u0, u1, and f belong to ˙̀2 per. Then the initial value problem (4.5) has a unique solution u ∈ C2(R+; ˙̀2 per). More- over, for each τ > 0, the map I : Ḣ → C([0, τ ]; Ḣ), defined by I(u0, u1)(t) = (u(t), u̇(t)), 0 ≤ t ≤ τ , is continuous. Proof. Taking the inner product of equation 4.5 with u̇ = u̇(t) in ˙̀2 per we find d dt E(t) = − N∑ n=1 ∂+1 u̇n(t)g(∂+1 u̇n(t)) ≤ 0, ∀0 ≤ t < τmax, (4.7) because sg(s) ≥ 0 for all s ∈ R by (A9), where E(t) = 1 2 ‖u̇(t)‖2 + 1 2 ‖Dpu(t)‖2 + N∑ n=1 h̃(∂+1 un(t))− (f, u(t))`2per . (4.8) 32 J. M. PEREIRA EJDE-2021/65 Recall that ‖Dpu(t)‖2 = ∑N n=1 |Dpun(t)|2. From (4.7) and (4.8), using Lemma 4.1 and (A8) we obtain ‖(u(t), u̇(t))‖2 Ḣ ≤ α−10 Ẽ(t) ≤ α−10 Ẽ(0) <∞, ∀0 ≤ t < τmax, (4.9) where Ẽ(t) = E(t) + k2N + 4CP0 ‖f‖2, ∀0 ≤ t < τmax, (4.10) and α0 = min{ 12 , 1 Cp0 }. Hence τmax =∞. Finally, the proof of the continuity of I is standard using Gronwall inequality and so we omit it here. � Let us denote by {S(t)}t≥0 the semigroup of continuous operator in Ḣ associated with problem (4.5), which is defined in the same manner as in (3.1). Lemma 4.3. Under the assumptions of Theorem 4.2 there exists ρ0 > 0 such that B[0; ρ0] = { (w, z) ∈ Ḣ : ‖(w, z)‖Ḣ ≤ ρ0 } is an absorbing set for {S(t)}t≥0 in Ḣ. Proof. The proof is similar to that of Lemma 3.2. Let us indicate the appropriate modifications. Define P (t)2 = E(t)− E(t+ 1) and Q(t) = N∑ n=1 ∂+1 u̇n(t)g(∂+1 u̇n(t)), for all t ≥ 0 and let t1, t2 be chosen as in (3.4). Also note that by (4.7) P (t)2 = ∫ t+1 t Q(s)ds ≥ 0, ∀t ≥ 0. Using (4.6), Hölder’s inequality, (4.9), and (A9) we have |(u̇(tj), u(tj))| ≤ α−1/20 C 1/2 0 sup t≤s≤t+1 Ẽ(s)1/2 ( N∑ n=1 |∂+1 u̇n(tj)|2 )1/2 ≤ N r 2(r+2)α −1/2 0 C 1/2 0 sup t≤s≤t+1 Ẽ(s)1/2 ( N∑ n=1 |∂+1 u̇n(tj)|r+2 ) 1 r+2 ≤ C0,1P (t) 2 r+2 sup t≤s≤t+1 Ẽ(s)1/2, j = 1, 2, for some positive constant C0,1. By Lemma 4.1, N∑ n=1 |∂+1 un(t)|2 ≤ 4 N∑ n=1 |un(t)|2 ≤ 4Cp0‖Dpu(t)‖2, ∀t ≥ 0. (4.11) For each t fixed, define the sets I1(t) = {n ∈ {1, . . . , N}; |∂+1 u̇n(t)| ≤ 1} and I2(t) = {1, . . . , N}\I1(t). Using Lemma 4.1, (4.9), (4.11), and (A8) we see that, for any t ≤ s ≤ t+ 1,∑ n∈I1(s) ∂+1 un(s)g(∂+1 u̇n(s)) ≤ 1 2 ‖Dpu(s)‖2 + 2Cp0L2(1)Q(s) and ∑ n∈I2(s) ∂+1 un(s)g(∂+1 u̇n(s)) ≤ ∑ n∈I2(s) |∂+1 un(s)| |∂+1 u̇n(s)| |g(∂+1 u̇n(s))| ≤ 2‖u(s)‖ ∑ n∈I2(s) |∂+1 u̇n(s)| |g(∂+1 u̇n(s))| EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 33 ≤ 2α −1/2 0 Q(s) sup t≤s≤t+1 Ẽ(s)1/2. Therefore,∫ t2 t1 N∑ n=1 ∂+1 un(s)g(∂+1 u̇n(s))ds ≤ 1 2 ∫ t2 t1 ‖Dpu(s)‖2ds+ [ 2Cp0L2(1) + 2α −1/2 0 sup t≤s≤t+1 Ẽ(s)1/2 ] P (t)2. The other estimates are analogous to those of Lemma 3.2 and we can prove the inequality sup t≤s≤t+1 Ẽ(s) ≤ C0 ( P (t) 4 r+2 + P (t)4 + P (t)2 + ‖f‖2 + k1 ) , ∀t ≥ 0, for some positive constant C0, depending on N, p, r, L2(1), k0, and k3. Then, ar- guing as in the proof of Lemma 3.3 and using Lemma 3.1 we can conclude the proof. � Since Ḣ has finite dimension, it follows from Lemma 4.3 that the semigroup {S(t)}t≥0 is asymptotically compact in Ḣ. Thus, we obtain the following result. Theorem 4.4. Assume that (A7)–(A9) hold and let u0, u1, and f belong to ˙̀2 per. Then the semigroup {S(t)}t≥0 associated with problem (4.5) possesses a unique global attractor A in Ḣ. Moreover, A ⊂ B[0; ρ0] and absorbing rates analogous to those in (3.74) are valid. Remark 4.5. Theorem 4.4 remains valid if we replace the dissipative term −∂−1 g(∂+1 u̇n(t)) in (4.4) by +µu̇n(t), with µ > 0. Remark 4.6. The restriction on the dimension in (4.4) is due to the fact that we used Lemma 4.1 which we do not know if it is valid when the operator ∆p 1 is replaced by ∆p d, with p > 1. The case with ∆d when d is arbitrary was considered in [17]. 5. Appendix A In this appendix we give examples of functions satisfying the assumptions (A1)– (A9). Example 5.1. We start with functions satisfying (A1)–(A4). The following func- tions h0 : Zd × R→ R satisfy (A1) and (A2). (a) h0(n, s) = an|s|γs−λn|s|σs, with γ > σ ≥ 0, λ = (λn) ∈ `1, a = (an) ∈ `∞, and an ≥ α > 0 for all n ∈ Zd. (b) h0(n, s) = an sinh s, with a = (an) ∈ `∞ and an ≥ 0,∀n ∈ Zd. (c) h0(n, s) = { an|s|s, |s| ≤ 1 an s |s| , |s| > 1 , where a = (an) is as in b). Note that in (a) assumption (A2) is satisfied with k1 = γ + 2, b1,n = (γ + 2)b2,n, b2,n = µλq+1 n , if λn ≥ 0 k1 = σ + 2, b1,n = 0, b2,n = 0 if λn ≤ 0, 34 J. M. PEREIRA EJDE-2021/65 where µ = α σ+2 σ−γ ( 1 σ + 2 − 1 γ + 2 ) , q = σ + 2 γ − σ . Also (b) and (c) fulfill (A2) with k1 = 1, b1,n = 0, and b2,n = 0. The following functions hi : R→ R, i = 1, . . . , d, satisfy (A1) and (A3). (d) hi(s) = µi|s|q−2s, µi > 0, and q ≥ 2. (e) hi(s) = µis e λis 2 , µi, and λi > 0. (f) hi(s) = ∑mi j=0 ai,mi−js 2mi+1−2j , wheremi is a positive integer and ai,mi−j > 0, i = 1, . . . , d, j = 0, . . . ,mi. In addition, we observe that assumptions (A1) and (A3) are fulfilled if the functions hi : R → R, i = 1, . . . , d, are non-decreasing and locally Lipschitz continuous and satisfy shi(s) ≥ 0 for all s ∈ R. Some examples of functions g(n, s) satisfying (A1) and (A4) are as follows. (g) g(n, s) = g0(s), with g0 ∈ C1(R,R), g0(0) = 0 and g′0(s) ≥ c0 > 0. (h) g(n, s) = { an|s|rs+ bng1(s), |n|0 ≤ n0, s ∈ R cng0(s), |n|0 > n0, s ∈ R , where n0 is a positive integer, g0, g1 are locally Lipschitz functions satisfying g0(0) = g1(0) = 0, sg0(s) ≥ c0s 2, sg1(s) ≥ 0, with c0 > 0, for all s ∈ R, (an), (bn), and (cn) belong to `∞, an > 0, bn ≥ 0, cn > 0, for all n ∈ Zd and cn ≥ c∗ > 0 if |n|0 > n0. Example 5.2. We now give examples of functions h : R→ R+ and V : Zd → R+ that satisfy (A5) and (A6). (a) h(s) = µ|s|γ , µ > 0, γ ≥ 2. (b) h(s) = µ(cosh s− 1), µ > 0. (c) V (n) = e−λ|n| |n|α , if n 6= 0 and V (0) = 0, where λ > 0, α ≥ 0 and |n| =(∑d i=1 |ni|2 )1/2 . Example 5.3. Finally, we present functions h : R → R and g : R → R satisfying (A7)–(A9). Other examples can be built using some functions from the preceding examples. (a) h(s) = µ arctan s, µ > 0. (b) h(s) = se−s 2 . (c) h(s) = ∑m j=0 am−js 2m+1−2j , where m ≥ 1 is a positive integer, am > 0 and am−j < 0, j = 1, . . . ,m. Concerning example (c), clearly k2 = −mins∈R h̃(s) ≥ 0. Note that sh(s) = h̃(s) + h1(s), where h1(s) = ( 1− 1 2m+ 2 ) am−1s 2m+2 + ( 1− 1 2m ) am−1s 2m + · · ·+ 1 2 a0s 2. Let s0 > 0 be such that h1(s) ≥ k2 for |s| > s0 and set k1 = k2 − ( 1− 1 2m ) am−1s 2m 0 − · · · − 1 2 a0s 2 0. Then it is easy to check that assumption (A8) is satisfied with k0 = 1. EJDE-2021/65 ATTRACTORS FOR DISSIPATIVE LATTICE DIFFERENTIAL EQUATIONS 35 (d) g(s) = g0(s)|s|rs+ g1(s), where r ≥ 0, g0, g1 are locally Lipschitz functions satisfying g0(s) ≥ c0 > 0, g1(0) = 0 and sg1(s) ≥ 0, for all s ∈ R. A function of this type is g(s) = s3 + s. (e) g(s) = sinh s. Acknowledgments. The content of this paper, except Section 4, is part of a the- sis submitted by the author to the Center of Physical and Mathematical Sciences of Federal University of Santa Catarina as part of the requirements to get the ti- tle of Full Professor. 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Periodic lattice differential equations 5. Appendix A Acknowledgments References