Using a difference inequality due to Nakao [14] and suitable estimates, we prove the existence of global at- tractors

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

¨

u_n(t) + (−1)^p∆^p_du_n(t) +αu_n(t) +F(n, u_n(t),∇⁺u_n(t)) +g(n,u˙_n(t)) =f_n, un(0) =u0,n, u˙n(0) =u1,n, (1.1) where n ∈ Z^d 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 ∆_du_n=Pd

i=1(u_n+ei+u_n−ei−2un), where{ei}^d_i=1is the canonical basis ofRⁿ. We assume that the nonlinear termF(n, u_n(t),∇⁺u_n(t)) has the form

F(n, un(t),∇⁺un(t)) =h0(n, un(t))−

d

X

i=1

∂⁻_i hi(∂_i⁺un(t)), (1.2)

2010Mathematics 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

whereh0:Z^d×R→R,hi:R→R,i= 1, . . . , d, are functions satisfying appropriate assumptions stated in Section 2 and∂_i⁺un =un+e_i−un and ∂_i⁻un =un−un−e_i, withi= 1, . . . , d. Moreover,g(n,u˙n(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 casep = 1,h_i ≡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 h_i ≡ 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) whenp= 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

¨

u_n(t) + (−1)^p∆^p_du_n(t) +αu_n(t) +h₀(n, u_n(t)) +g₀( ˙u_n(t)) =f_n. (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 g₀⁰(s)≥c0 >0 for alls∈R. Observe that we do not demand any relation on the parametersαandc0, andg⁰₀(s) need not be bounded above.

The inclusion of the term−Pd

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 choosep= 2,h_i(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 ∈ R^d, u = u(x, t) and ∆_qu = Pd

i=1

∂

∂x_i |_∂x^∂u

i|^q−2_∂x^∂u

i

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

¨

u_n(t) + (−1)^p∆^p_du_n(t) +αu_n(t) +F(u_n(t)) +g(n,u˙_n(t)) =f_n,

un(0) =u0,n, u˙n(0) =u1,n, (1.4) where

F(un(t)) =h⁰(un(t)) X

m∈Z^d

V(n−m)h(um(t)). (1.5) In (1.4) and (1.5), g:Z^d×R→R, h: R→R⁺, and V :Z^d →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 Schr¨odinger and Klein-Gordon equations, see [22, 27].

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 withα= 0 and nonlinear dissipation

¨

un(t) + (−1)^p∆^p₁un(t)−∂⁻₁h(∂₁⁺un(t))−∂₁⁻g(∂⁺₁u˙n(t)) =fn, (1.6) whereh:R→Randg:R→Rsatisfy appropriate conditions stated in Section 4.

Note that when g(s) =µ s, withµ >0, we obtain the strong dampingµ∆₁u˙_n(t).

Whenp= 2,g(s) =µs, andh(s) =α₁s⁵−α₂s³−α₃s, withα_i>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 termsh0(n, s) andg(n, s) more restrictive than those used in this paper. We note that our assumptions do not require any growth condition on the dissipative termg(n, s). In particular, our results apply to cases with nonlinear dissipative terms such asg(n, s) =a_n(s³+s) org(n, s) =b_nsinhs, when the real constantsa_n andb_n are suitably chosen.

This paper is organized as follows. In Section 2, we state the assumptions on the functionsh₀,h_i,i= 1, . . . , d, in (1.2),gin (1.1), andhandV 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 Poincar´e 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`<sup>p</sup> the space of real sequencesu= (un)<sub>n∈Z</sub>d such thatkuk`^p<∞, where

kuk_`p= X

n∈Z^d

|u_n|^p1/p

, if 1≤p <∞, kuk`^∞ = sup

n∈Z^d

|un|, ifp=∞.

Whenp= 2, `<sup>2</sup> is a Hilbert space with the inner product (u, v)<sub>`</sub>2= X

n∈Z^d

unvn, u, v∈`².

In this case, we denote byk · kthe 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:

`<sup>q</sup> ⊂`^p, kuk`<sup>p</sup>≤ kuk`^q, 1≤q≤p≤ ∞.

For the functionsh₀:Z^d×R→R,h_i:R→R, i= 1, . . . , d, andg:Z^d×R→R, we assume that

(A1) For eachs0>0, there exist positive constantsLj=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)−h_i(s₂)| ≤L₃|s1−s₂|,

for all s₁, s₂ in R, |s1| ≤s₀, |s2| ≤s₀, for all n∈Z^d andi = 1, . . . , d. In addition,h₀(n,0) = 0,g(n,0) = 0,h_i(0) = 0, for alln∈Z^dandi= 1, . . . , d.

(A2) There exist sequences of nonnegative real numbers b₁ = (b_1,n) ∈`<sup>1</sup>, b<sub>2</sub> = (b<sub>2,n</sub>)∈`¹, and a positive constantk₁ such that

sh0(n, s) +b1,n≥k1(˜h0(n, s) +b2,n)≥0, ∀s∈Randn∈Z^d, where ˜h0(n, s) =Rs

0 h0(n, σ)dσ.

(A3) There exist positive constantsk0,isuch that

sh_i(s)≥k_0,i˜h_i(s), ∀s∈Randi= 1, . . . , d, where ˜h_i(s) = Z s

0

h_i(σ)dσ≥0.

(A4) There exist constantsk2>0,r≥0 and a positive integern0 satisfying sg(n, s)≥k2|s|^r+2, if|n|0≤n0,

sg(n, s)≥k₂|s|², if|n|₀> n₀,

for alls∈R, where|n|₀= max_1≤i≤d|n_i|, ifn= (n₁, . . . , n_d).

Regarding the LDE (1.1) with the nonlocal term (1.4), we assume thatgsatisfies the same hypotheses above and thath:R→R⁺ andV :Z^d →R⁺ satisfy

(A5) h ∈ C²(R;R⁺), h(0) = h⁰(0) = 0 and there exists a positive constant c₁ such thatsh⁰(s)≥c₁h(s)≥0 for alls∈R.

(A6) V = (V(m))∈`² andV(m) =V(−m) for allm∈Z^d. The following notation will be useful.

∇⁺un = (∂₁⁺un, . . . , ∂⁺_dun), ∇⁻un= (∂₁⁻un, . . . , ∂_d⁻un),

∇⁺u_n· ∇⁺v_n=

d

X

i=1

∂_i⁺u_n∂_i⁺v_n, |∇⁺u_n|²=∇⁺u_n· ∇⁺u_n,

D^pu_n =

(∆^p/2_d un, ifpis even

∇⁺ ∆

p−1 2

d un

, ifpis odd,

(2.1) where ∆⁰_d=I.

Lemma 2.1. If u= (u_n)∈`² thenP

n∈Z^d|D^pu_n|²≤(4d)^pkuk².

Proof. Using the elementary inequality Pd i=1a_i2

≤dPd

i=1a²_i,a_i ∈R, and argu- ing by induction we find that

X

n∈Z^d

|∆^k_dun|²≤(4d)^2k X

n∈Z^d

|un|²,∀k∈N. (2.2) In (2.2) and hereafter N denotes the set of all positive integers. If pis odd then, from (2.1) and (2.2), we have

X

n∈Z^d

|D^pu_n|²= X

n∈Z^d

|∇⁺(∆

p−1 2

d u_n)|²= X

n∈Z^d d

X

i=1

|∂_i⁺∆

p−1 2

d u_n|²

≤4 X

n∈Z^d d

X

i=1

|∆

p−1 2

d un|²= 4d X

n∈Z^d

|∆

p−1 2

d un|²

≤4d(4d)^p−1kuk²= (4d)^pkuk².

Similarly, using (2.1) and (2.2) we can treat the case when p is even.

Lemma 2.2. For any u= (un)andv= (vn)in`² we have (−1)^p X

n∈Z^d

(∆^p_dun)vn = (P

n∈Z^d∆^p/2_d un∆^p/2_d vn, if pis even P

n∈Z^d∇⁺(∆

p−1 2

d un)· ∇⁺(∆

p−1 2

d vn), if pis odd.

Proof. Sinceu= (un) andv= (vn) belong to`², we have X

n∈Z^d

(∆dun)vn=

d

X

i=1

X

n∈Z^d

(∂_i⁺un)vn−

d

X

i=1

X

n∈Z^d

(∂_i⁻un)vn

=

d

X

i=1

X

n∈Z^d

(∂_i⁺u_n)v_n−

d

X

i=1

X

n∈Z^d

(∂_i⁺u_n)v_n+ei

=− X

n∈Z^d d

X

i=1

∂_i⁺un∂⁺_i vn=−X

n∈Z^d

∇⁺un· ∇⁺vn.

This proves Lemma 2.2 ifp= 1. The general case follows by induction onp.

In what follows, given a sequenceu= (un), we will write h0(u) = (h0(n, un)), g(u) = (g(n, un)), Au= ((−1)^p∆^p_dun), B(u) =

−

d

X

i=1

∂_i⁻hi(∂_i⁺un)

. (2.3)

Also, we will use the Hilbert space H = `<sup>2</sup>×`² equipped with the usual inner product and norm,

((u, v),(w, z))H = (u, w)_{`</sub>2+ (v, z)<sub>`}2 and k(u, v)kH= kuk²+kvk^21/2 , for any (u, v) and (w, z) inH.

Lemma 2.3. Under assumption (A1), we have

(i) h0, B, andg are locally Lipschitz continuous maps from `² into itself.

(ii) A:`<sup>2</sup>→`² is a bounded operator and kAuk ≤(4d)^p/2kuk for allu∈`².

Proof. (i) Givenu= (un) in`², using (A1), we have kB(u)k²= X

n∈Z^d

d

X

i=1

∂_i⁻hi(∂_i⁺un)

2

≤d X

n∈Z^d d

X

i=1

∂_i⁻hi(∂_i⁺un)

2

≤d X

n∈Z^d d

X

i=1

|hi(∂_i⁺un)−hi(∂_i⁻un)|²

≤dL₃(2kuk)² X

n∈Z^d d

X

i=1

|∂_i⁺u_n−∂_i⁻u_n|²

≤(4d)²L₃(2kuk)²kuk²<∞.

If u= (un) and v = (vn) belong to `², with kuk ≤ R and kvk ≤ R, then, using (A1) again, we have

kB(u)−B(v)k²≤d X

n∈Z^d d

X

i=1

|∂_i⁻h_i(∂_i⁺u_n)−∂_i⁻h_i(∂⁺_i v_n)|²

= X

n∈Z^d d

X

i=1

|hi(∂_i⁺un)−hi(∂_i⁺vn)−[hi(∂_i⁻un)−hi(∂_i⁻vn)]|²

≤2dL3(2R)² X

n∈Z^d d

X

i=1

|∂_i⁺(un−vn)|²+|∂_i⁻(un−vn)|²

≤(4d)²L3(2R)²ku−vk².

This shows thatB is a locally Lipschitz continuous map from`<sup>2</sup> into itself. Simi- larly, we prove thath<sub>0</sub>, g:`²→`²are 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

¨

u(t) +Au(t) +αu(t) +h0(u(t)) +B(u(t)) +g( ˙u(t)) =f, t >0,

u(0) =u₀, u(0) =˙ u₁, (2.4) where u0 = (u0,n), u1 = (u1,n), f = (fn), u(t) = (un(t)), ˙u(t) = ( ˙un(t)) and

¨

u(t) = (¨un(t)).

Theorem 2.4. Assume that (A1)–(A3) hold and let u0, u1 and f belong to `<sup>2</sup>. Assume also that sg(n, s) ≥ 0 for all n ∈ Z<sup>d</sup> and s ∈ R. Then the initial value problem (2.4) has a unique solution u ∈ C<sup>2</sup>(R<sup>+</sup>;`²). Moreover, for each τ > 0, the map I:H →C([0, τ];H), defined byI(u0, u1)(t) = (u(t),u(t)),˙ 0 ≤t ≤τ, is continuous.

Proof. Introducing the change of variable ˙u= v we can rewrite problem (2.4) in the spaceH as

dw

dt(t) +Bw(t) = 0, t >0, w(0) =w0,

wherew= (u, v), ^dw_dt = ( ˙u,v),˙ w0= (u0, u1), and

B(w) = (−v, Au+αu+h₀(u) +B(u) +g(v)−f). (2.5)

Using Lemma 2.3 we can easily prove that the mapBdefined 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 solutionu∈C²([0, τ_max);`²) such that eitherτ_max=∞ orτ_max<∞and lim_t→τ−

maxk(u(t),u(t))k˙ H =∞.

To extend the solution globally we proceed as follows. Taking the inner product of equation (2.4) with ˙u(t) in`² and using Lemma 2.2 we find

d

dtE(t) =−(g( ˙u(t)),u(t))˙ `² ≤0, ∀0≤t < τmax, (2.6) whereE(t) is the energy associated with the initial value problem (2.4) given by

E(t) = 1

2ku(t)k˙ ²+1

2kD^pu(t)k²+α

2ku(t)k²+ X

n∈Z^d

˜h0(n, un(t))

+ X

n∈Z^d d

X

i=1

˜hi(∂_i⁺un)−(f, u(t))_`2.

(2.7)

In (2.7) and hereafter kD^puk² = P

n∈Z^d|D^pu_n|². Thus, E(t) ≤ E(0), for all 0 ≤t < τmax. Since |(f, u)_`2| ≤ ^α₄kuk²+ _α⁴kfk², using (A2), (A3) and (2.7) we deduce that

k(u(t),u(t))k˙ ²_H ≤α⁻¹₀ E(t),˜ ∀0≤t < τmax, (2.8) whereα₀= min{¹₂,^α₄} and

E(t) =˜ E(t) + 4

αkfk²+kb2k_`1, ∀0≤t < τmax. (2.9) From (2.8) and (2.9) we conclude thatτmax =∞. 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 mapF:`<sup>2</sup>→`² by F(u) =

h⁰(u_n) X

m∈Z^d

V(n−m)h(u_m)

, ∀u= (u_n)∈`<sup>2</sup>. Then, using the above notation, we can write (1.4) in the space`² as

¨

u(t) +Au(t) +αu(t) +F(u(t)) +g( ˙u(t)) =f, t >0,

u(0) =u0, u(0) =˙ u1, (2.10) To see thatFis a locally Lipschitz continuous map from`<sup>2</sup>into iself, letu= (un) and v = (vn) in `² such that kuk ≤ R and kvk ≤ R. Since h ∈ C²(R;R⁺), h(0) = h⁰(0) = 0 by (A5) and V = (V(m)) ∈ `² by (A6) and |un| ≤ R and

|vn| ≤R, for alln∈Z^d, then kF(u)−F(v)k²≤2 X

n∈Z^d

|h⁰(un)−h⁰(vn)|² X

m∈Z^d

V(n−m)|h(um)|² + 2 X

n∈Z^d

|h⁰(v_n)|² X

m∈Z^d

V(n−m)|h(u_m)−h(v_m)|2

≤2M₁²M₂²kVk²kuk²ku−vk²+ 2M₁²M₂²kVk²kvk²ku−vk²

≤4R²M₁²M₂²kVk²ku−vk²,

whereM1= max_|s|≤2R|h⁰(s)|andM2= max_|s|≤2R|h⁰⁰(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∈Z^d,(A5), (A6)and let u0, u1, and f belong to `<sup>2</sup>. Assume also that sg(n, s) ≥0 for all n ∈Z<sup>d</sup> and s∈R. Then the initial value problem (2.10)has a unique solutionu∈C<sup>2</sup>(R<sup>+</sup>;`²).

Moreover, for each τ >0, the mapI:H →C([0, τ];H), defined by I(u0, u1)(t) = (u(t),u(t)),˙ 0≤t≤τ, is continuous.

In this case, as before, we obtain the identity (2.6) with the energy function E(t) =1

2ku(t)k˙ ²+1

2kD^pu(t)k²+α 2ku(t)k² +1

2 X

n∈Z^d

X

m∈Z^d

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 E(t) =˜ E(t) + 4

αkfk², ∀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 onH as follows

S(t)(u₀, u₁) = (u(t),u(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]). Letψ(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 andγ >0. Then

ψ(t)≤

C⁻¹γ(t−1)⁺+ sup

0≤s≤1

ψ(s)−γ−_γ¹

+K^γ+11 , 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 andβ⁺= max{β,0}.

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∈Z^d ands∈R. Let

P(t)²=E(t)−E(t+ 1) and Q(t) = X

n∈Z^d

˙

un(t)g(n,u˙n(t)), ∀t≥0.

According to (2.6),E(t) is a non-increasing function on [0,∞) and P(t)²=

Z 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+¹₄] and t2∈[t+³₄, t+ 1] such that

Z t+¹₄

t

Q(s)ds=1

4Q(t1) and

Z t+1

t+³₄

Q(s)ds= 1

4Q(t2). (3.4) To simplify notation we will write ˆh(u) =h₀(u) +B(u) for allu∈`².

Lemma 3.2. Assume that(A1)–(A4)hold and letu₀,u₁, andf belong to`². Then there exists a positive constant C₀ such that

sup

t≤s≤t+1

E(s)˜ ≤C₀ P(t)^r+24 +P(t)⁴+P(t)²+kfk²+kb1k`¹

,∀t≥0. (3.5) Proof. Taking the inner product of 2.4 withu=u(t) in`², using Lemma 2.2, and integrating the result over [t1, t2] we obtain

Z t₂

t1

kD^pu(s)k²ds+α Z t₂

t1

ku(s)k²ds+ Z t₂

t1

(ˆh(u(s)), u(s))_`2ds

= ( ˙u(t1), u(t1))_{`</sub>2−( ˙u(t2), u(t2))<sub>`}2+ Z t2

t1

ku(s)k˙ ²ds

− Z t2

t₁

(g( ˙u(s)), u(s))_`2ds+ Z t2

t₁

(f, u(s))_`2ds.

(3.6)

Let us estimate the terms in the right hand side of (3.6). We initially write ( ˙u(tj), u(tj))_`2 = X

|n|0≤n0

˙

un(tj)un(tj) + X

|n|0>n0

˙

un(tj)un(tj), j = 1,2, (3.7) withn0 as in (A4). Using (2.8) we have

X

|n|0≤n0

|un(tj)|²1/2

≤α^−1/2₀ sup

t≤s≤t+1

E(s)˜ ^1/2. (3.8) Using H¨older’s inequality, (A4), (3.3), and (3.4) we have

X

|n|0≤n0

|u˙n(tj)|²≤(2n0+ 1)^r+2rd X

|n|0≤n0

|u˙n(tj)|^r+2_r+2²

≤(2n0+ 1)^r+2rd

k₂⁻¹ X

|n|0≤n0

g(n,u˙n(tj)) ˙un(tj)_r+2²

≤(2n₀+ 1)^r+2rd k⁻

2 r+2

2 4^r+22 P(t)^r+24 .

(3.9)

Similarly, we can estimate the second term in (3.7) to find X

|n|0>n₀

|u˙_n(t_j)u_n(t_j)| ≤2α₀^−1/2k₂^−1/2P(t) sup

t≤s≤t+1

E(s)˜ ^1/2. (3.10) From (3.7)-(3.10) we conclude that there is a positive constant C0,1 depending only onn0,α,k2,r, anddsuch that

|(u(t₁),u(t˙ ₁))_{`</sub>2−(u(t<sub>2</sub>),u(t˙ <sub>2</sub>))<sub>`}2| ≤C_0,1

P(t)^r+22 +P(t) sup

t≤s≤t+1

E(s)˜ ^1/2. (3.11) Proceeding as in (3.9), replacingtj bys∈[t1, t2], and noticing that

X

|n|0>n0

|u˙_n(s)|²≤k₂⁻¹Q(s), for anys∈[t₁, t₂] by (A4), we obtain

Z t₂

t₁

ku(s)k˙ ²ds≤(2n0+ 1)^r+2rd k⁻

2 r+2

2

Z t₂

t₁

Q(s)^r+22 ds+k₂⁻¹ Z t₂

t₁

Q(s)ds

≤(2n0+ 1)^r+2rd k⁻

2 r+2

2 P(t)^r+24 +k⁻¹₂ P(t)².

(3.12)

Next, for eacht≥0 fixed, we define the sets I₁(t) =

n∈Z^d; |u˙_n(t)| ≤1 , I₂(t) =Z^d\I1(t).

Note that by (A1),|g(n,u˙_n(s))| ≤L₂(1)|u˙_n(s)|, whenevern∈I₁(s). Then X

n∈I1(s)

un(s)g(n,u˙n(s))≤α

2ku(s)k²+ 1 2α

X

n∈I1(s)

|g(n,u˙n(s))|²

≤α

2ku(s)k²+ 1

2αL₂(1) X

n∈I1(s)

|u˙_n(s)| |g(n,u˙_n(s))|

=α

2ku(s)k²+ 1

2αL2(1) X

n∈I1(s)

˙

un(s)g(n,u˙n(s))

≤α

2ku(s)k²+ 1

2αL₂(1)Q(s).

(3.13)

In addition, using (A4) and (2.8) we have X

n∈I2(s)

un(s)g(n,u˙n(s))≤ X

n∈I2(s)

|un(s)| |u˙n(s)| |g(n,u˙n(s))|

≤ ku(s)k X

n∈I₂(s)

|u˙_n(s)| |g(n,u˙_n(s))|

=ku(s)k X

n∈I2(s)

˙

un(s)g(n,u˙n(s))

≤ ku(s)kQ(s)≤α^−1/2₀ Q(s) sup

t≤s≤t+1

E(s)˜ ^1/2.

(3.14)

It follows from (3.13), (3.14), and (3.3) that Z t₂

t₁

(g( ˙u(s), u(s))`²ds≤ α 2

Z t₂

t₁

ku(s)k²ds+ 1

2αL2(1)P(t)² +α^−1/2₀ P(t)² sup

t≤s≤t+1

E(s)˜ ^1/2.

(3.15)

Finally, in view of (2.8), we easily see that

Z t₂

t₁

(f, u(s))_`2ds

≤α^−1/2₀ kfk sup

t≤s≤t+1

E(s)˜ ^1/2. (3.16) Substituting (3.11), (3.12), (3.15), and (3.16) into (3.6) we obtain the estimate

Z t₂

t1

kD^pu(s)k²ds+α 2

Z t₂

t1

ku(s)k²ds+ Z t₂

t1

J1(s)ds

≤C_0,2h

P(t)^r+22 +P(t)²+P(t) +kfk

× sup

t≤s≤t+1

E(s)˜ ^1/2+P(t)^r+24 +P(t)²i

+kb1k_`1,

(3.17)

whereC0,2 is a positive constant depending only on n0, α, k2, r, d, L2(1), and J₁(t) = X

n∈Z^d

u_n(t)h₀(n, u_n(t)) +b_1,n

+ X

n∈Z^d d

X

i=1

∂⁺_i u_n(t)h_i(∂_i⁺u_n(t))≥0, for allt≥0, because of (A2) and (A3).

On the other hand, using hypotheses (A2) and (A3) we have X

n∈Z^d

˜h₀(n, u_n(t)) + X

n∈Z^d d

X

i=1

h˜_i(∂_i⁺u_n(t))

≤k⁻¹₁ X

n∈Z^d

un(t)h0(n, un(t)) +b1,n

− kb2k`¹

+ X

n∈Z^d d

X

i=1

k_0,i⁻¹

∂⁺_i un(t)hi(∂_i⁺un(t))

≤k0

h X

n∈Z^d

un(t)h0(n, un(t)) +b1,n

+ X

n∈Z^d d

X

i=1

∂_i⁺un(t)hi(∂_i⁺un(t))i

− kb2k_`1,

(3.18)

wherek0= max{k⁻¹₁ , k⁻¹_0,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

Z t₂

t₁

E(s)ds˜

= Z t₂

t₁

E(s)ds+ 4 α

Z t₂

t₁

kfk²ds+ Z t₂

t₁

kb2k_`1ds

≤max{k0,1}hZ t₂ t1

kD^pu(s)k²ds+α 2

Z t₂

t1

ku(s)k²ds+ Z t₂

t1

J1(s)dsi +1

2 Z t2

t1

ku(s)k˙ ²ds+ 4 α

Z t2

t1

kfk²ds

≤C_0,3

P(t)^r+22 +P(t)²+P(t) +kfk sup

t≤s≤t+1

E(s)˜ ^1/2+P(t)^r+24 +P(t)² + 4

αkfk²+ max{k₀,1}kb₁k_`1, (3.19)

for some positive constantC0,3 depending only onn0, α, k0, k2, r, d, andL2(1).

Furthermore, by the Mean Value Theorem for integrals, there existst^∗∈[t1, t2] such that

1 2

E(t˜ ^∗)≤(t2−t1) ˜E(t^∗) = Z t₂

t1

E(s)˜ ds. (3.20)

If ˜t∈[t, t^∗] then using (2.6) and (3.3) we obtain E(˜˜ t) = ˜E(t^∗) +

Z t^∗

˜t

Q(s)ds≤E(t˜ ^∗) +P(t)². Analogously, we obtain the same estimate if ˜t∈[t^∗, t+ 1]. Therefore,

E(s)˜ ≤E(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 writeK₀=kfk²+kb1k`¹ and assume thatK₀>0.

Lemma 3.3. Under the assumptions of Lemma 3.2 there exists ρ0>0 such that B[0;ρ0] ={(w, z)∈H;k(w, z)kH ≤ρ0} is an absorbing set for{S(t)}t≥0 inH. Proof. Let O be any bounded subset of H and let ρ = ρ(O) be a positive con- stant such that k(w, z)kH ≤ ρ,∀(w, z) ∈ O. Assume that (u0, u1) ∈ O. Let us first consider the case r > 0. Since ˜E(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,kfk,kb1k`<sup>1</sup>, andkb2k`¹ such that

P(t)²=E(t)−E(t+ 1) = ˜E(t)−E(t˜ + 1)≤2 ˜E(0)≤µ₀, ∀t≥0. (3.21) It follows from (3.5) and (3.21) that

sup

t≤s≤t+1

E(s)˜ ^1+r2 ≤C1,1[ ˜E(t)−E(t˜ + 1)] + 2C0K0

1+^r₂

, (3.22)

where C_1,1 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) = ˜E(t), γ = ^r₂, C =C1,1, and K =

2C0K0

1+^r₂

we obtain E(t)˜ ≤h

C_1,1⁻¹r

2(t−1)⁺+ sup

0≤s≤1

E(s)˜ −r/2i−2/r

+ 2C0K0. (3.23) From (3.23), using (3.21), we deduce that

E(t)˜ ≤C1(1 +t)^−2/r+ 2C0K0, (3.24) whereC1 is a positive constant depending only onr, C0, andµ0. Combining (3.24) with (2.8) we obtain

kS(t)(u0, u1)k²_H ≤α⁻¹₀ C1(1 +t)^−2r + 2α⁻¹₀ C0K0 ∀t≥0. (3.25) Consequently,kS(t)(u0, u1)kH ≤ρ0, for allt≥τ if we take

ρ₀= 2C0K0

α₀ 1/2

, τ=τ(O) = maxn

0, C1

2C₀K₀ r/2

−1o .

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

E(s)˜ ≤C₀(2 +µ₀)[ ˜E(t)−E(t˜ + 1)] +C₀K₀,∀t≥0,

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

kS(t)(u0, u₁)k²_H≤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) andν = ln

C2+1 C₂

. This implies Lemma 3.3 in the case r= 0 with ρ0= ^2C_α^0K0

0

^1/2

andτ =τ(O) = max

0,_ν¹ln ^µ_2C^0(C2+1)

2C₀K₀ .

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 >0, there existτ()>0 and a positive integerk() such that

X

|n|0≥k()

[( ˙u_n(t))²+ (u_n(t))²]< , for allt≥τ(),

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+e_i+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 toC¹(R⁺;`<sup>2</sup>), and(θn) belong to`². Then (−1)^p X

n∈Z^d

∆^p_dun(t) (θnu˙n(t)) = 1 2

d dt

X

n∈Z^d

θn|D^pun(t)|²+ X

n∈Z^d d

X

i=1

(∂_i⁺θn)z_p,n⁽ⁱ⁾(t), where

X

n∈Z^d d

X

i=1

|z⁽ⁱ⁾_p,n(t)| ≤C(p, d)k(u(t),u(t))k˙ ²_H, ∀t≥0, for some positive constantC(p, d)depending onpandd.

Proof. A proof of this lemma whend= 1 was first presented in [18]. We will argue by induction. Consider first podd. Ifp= 1 then using Lemma 2.2 and (3.27) we have

− X

n∈Z^d

∆dun(θnu˙n) = X

n∈Z^d

∇⁺un· ∇⁺(θnu˙n)

= X

n∈Z^d d

X

i=1

∂_i⁺u_n θ_n∂⁺_i u˙_n+∂_i⁺θ_nu˙_n+ei

= 1 2

d dt

X

n∈Z^d

θn|∇⁺un|²+ X

n∈Z^d d

X

i=1

∂_i⁺θnz_1,n⁽ⁱ⁾,

where z⁽ⁱ⁾_1,n = (∂_i⁺un) ˙un+ei satisfies P

n∈Z^d

Pd

i=1|z_1,n⁽ⁱ⁾| ≤ 2dk(u,u)k˙ ²_H by Lemma 2.1.

Assume now that Lemma 3.4 holds forp= 2k−1 with k ∈N. Using Lemma 2.2 again and (3.30) we have

(−1)^2k+1 X

n∈Z^d

∆^2k+1_d u_n(θ_nu˙_n)

=−X

n∈Z^d

∆^2k_d u_n∆_d(θ_nu˙_n)

=−X

n∈Z^d

∆^2k_d un

(∆dθn) ˙un+θn∆du˙n+∇⁺θn· ∇⁺u˙n+∇⁻θn· ∇⁻u˙n .

(3.31)

By Lemma 2.2 and (3.28) we also see that

− X

n∈Z^d

∆^2k_d un(∆dθn) ˙un

= X

n∈Z^d d

X

i=1

∂_i⁺θn(∆^2k_d un)∂_i⁺u˙n+ X

n∈Z^d d

X

i=1

∂_i⁺θn(∂_i⁺∆^2k_d un) ˙un

+ X

n∈Z^d d

X

i=1

∂⁺_i θ_n(∂_i⁺∆^2k_d u_n)∂_i⁺u˙_n.

(3.32)

In addition,

− X

n∈Z^d

∆^2k_d un∇⁻θn· ∇⁻u˙n

=− X

n∈Z^d d

X

i=1

∆^2k_d u_n+ei∂_i⁺θ_n∂_i⁺u˙_n

=− X

n∈Z^d d

X

i=1

∂_i⁺∆^2k_d un∂_i⁺θn∂⁺_i u˙n− X

n∈Z^d d

X

i=1

∆^2k_d un∂_i⁺θn∂_i⁺u˙n.

(3.33)

Substituting (3.32) and (3.33) into (3.31) results (−1)^2k+1 X

n∈Z^d

∆^2k+1_d un(θnu˙n)

=− X

n∈Z^d

∆^2k_d un(θn∆du˙n)

+ X

n∈Z^d d

X

i=1

∂_i⁺θ_n

(∂_i⁺∆^2k_d u_n) ˙u_n−∆^2k_d u_n(∂_i⁺u˙_n) .

(3.34)