Pullback attractor for a nonlocal discrete nonlinear Schrödinger equation with delays

Pullback attractor for a nonlocal discrete nonlinear Schrödinger equation with delays

Jardel Morais Pereira

Department of Mathematics, Federal University of Santa Catarina, Florianópolis SC, 88040-900, Brazil Received 15 February 2021, appeared 30 December 2021

Communicated by Christian Pötzsche

Abstract. We consider a nonlocal discrete nonlinear Schrödinger equation with delays.

We prove that the process associated with the non-autonomous model possesses a pull- back attractor. As a consequence of our discussion, the existence of a global attractor for the autonomous model is derived.

Keywords: pullback attractor, discrete nonlinear Schrödinger equation, delay terms, global attractor differential equations, difference equations.

2020 Mathematics Subject Classification: 35Q55, 37L60, 37B55, 37L30.

1 Introduction

Discrete Schrödinger equations are widely used as models in Physics and other branches of science (see, e.g., [3,6,11,12,14,19] and the references therein). These discrete equations belong to a large class of lattice dynamical systems which has been the object of extensive research (see, for example, [4,5,7,9,12,13,19,22] and the references therein). Various properties related to the dynamics of such systems have been studied. Among them, the existence of global attractors is a theme which attracts a great deal of attention. However, most of the contributions in this line of research addressed to discrete Schrödinger models are concerned the discrete nonlinear Schrödinger equation (DNLS). In this paper, our main aim is to prove the existence of a pullback attractor for a nonlocal discrete nonlinear Schrödinger equation when delay terms are considered. The model is written as follows

iu˙_n(t) +

+_∞ m=−

∑

J(n−m)u_m(t) +g_n(t,u_nt) +iγu_n(t) = f_n(t), t>τ,n∈_Z, un(s) =ψn(s−τ), ∀s∈ [τ−h,τ],

(1.1)

where τ,h, and γ are real numbers with h > 0 and γ > 0. In (1.1), u_n(t), f_n(t), and ψ_n are complex functions and unt denotes the translation of un at time t, defined by unt(s) = u_n(t+s),∀s ∈ [−h, 0]. The dispersive coupling parameters J(m) are assumed to be real numbers, symmetric (i.e., J(−m) = J(m), for all positive integerm) and ∑⁺_m^∞=1|J(m)| < +_∞.

BEmail: [email protected]

This includes important special cases asJ(m) =J₀e^−β|m| andJ(m) = J₀|m|^−s, whereJ₀, β, and sare positive real constants suitably chosen [8].

We assume that the nonlinear termg_n(t,u_nt)in (1.1) includes delay terms as follows gn(t,unt) =g0,n(un(t)) +g_1,n(un(t−ρ(t)) +

Z ₀

−hbn(s,un(t+s))ds. (1.2) Appropriate hypotheses on the functions ρ : R → [0,h], g_i,n : C → _C,i = 0, 1, bn : [0,h]×_C→_{C, and} f_n(t)are stated in Section 2.

Specific deterministic cases of equation (1.1) have been used in the study of physical phe- nomena in which long-range dispersive interactions cannot be disregarded (see the physical discussions in [8]). An example is the model proposed in [17] for the description of the non- linear dynamics of the DNA molecule.

A class of discrete Schrödinger equations of great importance is

iu˙_n(t) +_∆^p_du_n(t) +g_n(t,u_nt) +iγu_n(t) = f_n(t), (1.3) where ∆_d^p = _∆d◦ · · · ◦_∆d, p times, and ∆d is the one-dimensional discrete Laplace operator defined by ∆du_n = u_n+1+u_n−1−2u_n. Equation (1.3) can be derived from (1.1) by choosing the coupling parametersJ(m)as

J(m) =

∑

2p j

(−1)^jδ_m,j−p,

where pis any positive integer andδ_m,k is the Kronecker delta.

Many contributions on existence and properties of solutions of the DNLS equation (i.e, (1.3) with p = 1, g_1,n = b_n = 0) and f_n independent of time can be found in the literature (see, e.g., [3,4,11,19] and references therein). For example, the existence and approximation of attractors for the DNLS equation were investigated in [11] while the existence of attractors for the DNLS with retarded terms was studied in [4]. Concerning equation (1.1), in [19], the authors studied the existence of localized solutions for the homogeneous case without delays. Later, also for the autonomous deterministic model, the existence of a global attractor in weighted spaces was established in [20]. For the existence of attractors for some non- autonomous lattice dynamical systems with retarded terms of the type (1.2) and references about related works we refer the reader to the article [2]. Still concerning lattice models with nonlocal terms, we would like to mention the papers [1,10,15,18,21].

In this paper, under suitable conditions on the functions ρ, g_i,n,i = 0, 1, bn, and fn, we prove the existence of a pullback attractor for theprocessassociated with problem (1.1). As a consequence of our discussion, the existence of a global attractor for the autonomous model is derived.

The paper is organized as follows. In Section 2, we prove that the initial value problem (1.1) is globally well posed. In Section 3, we establish the existence of a pullback attractor for the process associated with problem (1.1) using the results in [16]. Finally, in Section 4, we briefly show how the same ideas of the previous sections can be adapted to prove the existence of a global attractor for the autonomous model.

2 Existence of solutions

In this section, we discuss the existence of solutions for the problem (1.1). We denote by `<sup>p</sup> the usual space of complex sequences u= (<sub>un</sub>)<sub>n∈Z such that</sub>kuk<sub>`</sub>^p <_{∞, where}

kuk_`^p =

+_∞ n=−

∑

|un|^p

!¹_p

, if 1≤ p<_∞ andkuk_`∞ =sup

n∈_Z

|un|, if p=_∞.

When p=2,`<sup>2</sup> is a Hilbert space with the inner product given by (u,v)<sub>`</sub>2 =

+_∞ n=−

∑

u_nv_n, u,v∈`², and, in this case, we denote by k · kthe corresponding norm.

For 1 ≤ p < _∞, L^p(−h, 0)denotes the usual Banach space of (class of ) real functions f defined on [−h, 0]such that |f|^p is integrable in sense of Lebesgue and we recall that for the

`^pspaces the following embedding relation holds:

`<sup>q</sup>⊂ `^p, kuk_{`</sub>p ≤ kuk<sub>`}q, 1≤q≤ p≤_∞.

Regarding the functionsg_i,n : C → _C,i= _{0, 1,} b_n : [−h, 0]×_C→ _C , f = (f_n(t))_{n∈Z, and} ρ(t)in (1.1) and (1.2) we assume that

(A1) zg0,n(z)is real for allz ∈_Candn∈_Z.

(A2) There exist a functionκ∈ L²(−h, 0)and functionsb_0,n:C→_Csuch that

|b_n(s,z₁)−b_n(s,z₂)| ≤κ(s)|b_0,n(z₁)−b_0,n(z₂)|,

∀s ∈[−h, 0] and∀z₁,z2∈_{C. We set}κ²₀ :=R0

−h|κ(s)|²ds.

(A3) For every R>0 there exist positive constants L_j(R),j=1, 2, such that

|g_i,n(z₁)−g_i,n(z₂)| ≤L₁(R)|z₁−z₂|, i=0, 1,

|b_0,n(z₁)−b_0,n(z₂)| ≤L₂(R)|z₁−z₂|,

for anyn∈_Zand anyz₁,z2∈_Csuch that|z_j| ≤R, j=1, 2. Moreover,(g0,n(0))_n∈Z∈ `<sup>2</sup>. (A4) There exist sequences of real numbers k<sub>1</sub> = (k1,n)<sub>n∈Z</sub> ∈ `^∞, k2 = (k2,n)_n∈Z ∈ `² and

non-negative real functionsβ_1,n(·)∈ L²(−h, 0)andβ_2,n(·)∈ L¹(−h, 0)such that

|g_1,n(z)| ≤k_1,n|z|+k_2,n and |b_n(s,z)| ≤ β_1,n(s)|z|+β_2,n(s)_, for alln ∈_Z, s∈[−h, 0], and z∈_{C. We set}K₁=kk₁k_`∞,K₂ =kk₂k, and

B₁ =sup

n∈_Z

_{Z 0}

−hβ²_1,n(s)ds 1/2

<_∞, B₂ =

" ₊

∑

_{Z 0}

−hb2,n(s)ds 2#1/2

<_∞.

(A5) f ∈C(_R;`²). (A6) ρ ∈C(_R;[_0,h])_. (A7) Rt

−_∞kf(s)k²ds<_∞, ∀t∈_R.

Example 2.1. Let 0 6= χ = (χ_n)_n∈Z ∈ `^p, for some 1≤ p ≤ _{∞, and} ϕ₁ : R → _R defined by ϕ₁(t) = ^t2

a+bt², whereaandbare positive real constants. Also define the functionsg_1,n:C→_C, b0,n :C→_Candbn :[−h, 0]×_C→_Cby

g_1,n(z) =b_0,n(z) =χ_nϕ₁(|z|)z, b_n(s,z) =χ_nϕ₁(|z|)z1

h(s+h), ∀n ∈_Z, s∈ [−h, 0]and z∈_C.

Then, the hypotheses (A2)–(A4) are satisfied with L₁(R) = L₂(R) =

1 b+√^R

ab

kχk_`p,

κ(s) = ¹

h(s+h)_, k_1,n= ¹

b|χ_n|_, k_2,n=_0, β_1,n(s) = ¹

bh|χ_n|(s+h)_{, and} β_2,n=_0.

Conditions (A1) and (A3) concerningg0,nare satisfied, for example, ifg0,n(z) =χnϕ2(|z|)z, withχ_nas before and any ϕ₂ ∈C¹(_R⁺;R), such that ϕ₂(0) =0.

Now let us write (1.1) as an evolution equation with a retarded term in `². For any u= (un)_n∈Zwe define(Au)_n =_∑⁺_m=−^∞ _∞J(n−m)um, ∀n∈_Z.

Lemma 2.2. A:`<sup>2</sup>→`²is a bounded operator andkAuk ≤4kJk<sub>`</sub>1kuk, ∀u∈`². Proof. See Lemma 2.1 in [20].

We consider the space E_h = C([−h, 0];`<sup>2</sup>) with the usual norm given by kuk<sub>Eh</sub> = max<sub>s∈[−h,0]</sub>ku(s)k and define the map g : R×E<sub>h</sub> → `² by (g(t,v))_n∈Z = g_n(t,v_n), where v(s) = (vn(s))_n∈Z, for anys ∈[−h, 0], and

g_n(t,v_n) =g_0,n(v_n(0)) +g_1,n(v_n(−ρ(t))) +

Z ₀

−hb_n(s,v_n(s))ds.

If we set u_t = (u_nt)_n∈Zfor any t ≥ τ, then we can write the initial value problem (1.1) in

`² as

iu˙(t) +Au(t) +g(t,ut) +iγu(t) = f(t), t >τ,

u(s) =ψ(s−τ), ∀s∈[τ−h,τ], (2.1) whereψ(s) = (ψ_n(s))_n∈Z, for anys∈[−h, 0].

We now define the mapB:R×E_h→ `²by B(t,v) =−i

Av(0) +g(t,v) +iγv(0)− f(t).

Then, problem (2.1) can be rewritten as the following functional equation in`² du

dt +B(t,u_t) =_0, t>τ uτ =ψ.

(2.2)

The following two lemmas are sufficient to ensure the existence of a local solution for (2.1).

Lemma 2.3. Assume that (A2)–(A6) hold. Then the map B is continuous and satisfies the local Lipschitz condition: For any v,w ∈ E_h, with kvk_Eh ≤ R and kwk_Eh ≤ R, there exists a positive constant L= L(R)such that

kB(t,v)− B(t,w)k ≤ Lkv−wk_Eh, ∀t ∈_R.

Proof. Using (A2)–(A6) we see thatB is well defined. Fix(t,v)∈_R×E_hand consider t^m →t inR andv^m →v in E_h. We have that

kB(t^m,v^m)− B(t,v)k ≤ kA(v^m(0)−v(0))k+kg(t^m,v^m)−g(t,v)k

+γkv^m(0)−v(0)k+kf(t^m)− f(t)k. (2.3) Since the sequence(v^m)_m∈N is bounded inE_h, then using the assumptions (A2), (A3), and (A6) we can find a positive constant Ldepending only onkvk_Eh such that

kg(_t^m_,v^m)−g(_t,v)k² ≤4

+_∞ n=−

∑

|g0,n(_v^m_n(₀))−g0,n(_vn(₀))|² +4

+_∞ n=−

∑

|g_1,n(v^m_n(−ρ(t^m)))−g_1,n(v_n(−ρ(t)))|² +4

+_∞ n=−

∑

_{Z 0}

−h

|b_n(s,v^m_n(s))−b_n(s,v_n(s))|ds 2

≤8L²kv^m−vk²_E

h+4L²

+_∞ n=−

∑

Z ₀

−h

|κ(s)| |v^m_n(s)−vn(s)|ds 2

. (2.4)

Using the Cauchy–Schwarz inequality and the fact thatkv^m−vk_Eh < _∞we can estimate the last term in (2.4) as follows

+_∞ n=−

∑

Z ₀

−h

|κ(s)| |v^m_n(s)−vn(s)|ds 2

ds≤κ²₀

+_∞ n=−

∑

Z ₀

−h

|v^m_n(s)−v(s)|²ds

≤κ₀² Z ₀

−h +_∞ n=−

∑

|v^m_n(s)−v(s)|²ds≤κ²₀kv^m−vk²_E

hh.

(2.5)

From (2.3), (2.4), (2.5), (A5), and Lemma 2.2 we deduce the continuity of B. In a similar manner we prove the Lipschitz condition.

Lemma 2.4. Assume that (A2)–(A6) hold. Then the map Bis bounded, i.e., it takes bounded subsets ofR×E_h onto bounded subsets of`².

Proof. LetObe a bounded subset ofR×E_h. Then, there exists a positive constantRsuch that

|t|²+kvk²_E

h ≤ R², ∀(t,v) ∈ O. Using Lemma2.3 we find a positive constant L = L(R)such that

kB(t,v)k ≤ kB(t,v)− B(t, 0)k+kB(t, 0)k

≤ LR+max

|t|≤RkB(t, 0)k<_∞, ∀(t,v)∈ O.

Using Lemmas2.3, 2.4and applying the Theory of Functional Equations to problem (2.2) we deduce the following result of existence of local solution for (2.1).

Theorem 2.5. Assume that (A2)–(A6) hold. Then, for each ψ ∈ E_h, the initial value problem (2.1) has a unique solution u= u(t)defined in[τ−h,T)such that u∈ C([τ−h,T);`<sup>2</sup>)∩C<sup>1</sup>([τ,T);`²). Moreover, if T<_∞thenlim_t→T⁻ku(t)k=_∞.

Next let us show that the local solution obtained in Theorem2.5can be extended globally.

Lemma 2.6. Assume that (A1)–(A6) hold. Then the solution u of (2.1)with u_τ = ψ∈ E_hsatisfies 1

2 d

dtku(t)k²+ ^γ

2ku(t)k²≤ ¹

2γkf(t)k²+ (K₁ku(t−ρ(t))k+K₂)ku(t)k +

"

B₁ Z ₀

−h

ku(t+s)k²ds 1/2

+B2

#

ku(t)k, τ≤t< T.

(2.6)

Proof. Taking the imaginary part of the inner product of equation (2.1) withuin`², we obtain 1

2 d

dtku(t)k²+Im(Au(t),u(t))_{`</sub>2+γku(t)k<sup>2</sup>+Im(g(t,u<sub>t</sub>),u(t))<sub>`}2 =Im(f(t),u(t))_`2, for allτ≤t< T. Since

Im(f(t),u(t))_`2 ≤ ¹

2γkf(t)k²+ ^γ

2ku(t)k², (Au(t)_,u(t))_`2 = J(₀)ku(t)k²+₂

+_∞ m

∑

+_∞ n=−

∑

J(m)_Re(u_n+m(t)u_n(t))_, then, using (A1), we get the inequality

1 2

d

dtku(t)k²+ ^γ

2ku(t)k²≤ ¹

2γkf(t)k²−Im

+_∞ n=−

∑

g_1,n(u_n(t−ρ(t)))u_n

−Im

+_∞ n=−

∑

Z ₀

−hbn(s,un(t+s))ds un, τ≤t <T.

(2.7)

Let us estimate the last two terms in (2.7) using the assumption (A4) and the fact that kutk_Eh < _∞, ∀τ≤t <T. We have that

−Im

+_∞ n=−

∑

g_1,n(un(t−ρ(t)))un ≤

+_∞ n=−

∑

[k_1,n|un(t−ρ(t))|+k2,n]|un|

≤(K₁ku(t−ρ(t))k+K₂)kuk_,

(2.8)

−Im

+_∞ n=−

∑

Z ₀

−hb_n(s,u_n(t+s))ds u_n≤

+_∞ n=−

∑

Z ₀

−h

[β_1,n(s)|u_n(t+s)|+β_2,n(s)]ds|u_n|

≤

" ₊

∑

_{Z 0}

−hβ²_1,n(s)ds Z ₀

−h

|un(t+s)|²ds #1/2

kuk+B₂kuk

≤

"

B_{1 Z 0}

−h

ku(t+s)k²ds 1/2

+B₂

# kuk.

(2.9)

From (2.7)–(2.9) we obtain (2.6).

We now make the following assumptions on the constants B₁, K₁, γ, h, and a suitable positive parameterµ, which will be used in Section 3 to define the universe where the pullback attractor will lie in.

(A8) We assume that there exists a positive real numberµsuch that (i) IfK₁ >0 andB₁ ≥0 then

4B₁²h<e^−µhγ γ

2 −µ

(2.10) and

µ>2K₁e^µh. (2.11)

(ii) IfK₁ =0 andB₁ >0 then

µ< ^γ

2 and µ> ⁴

γB₁²e^2µhh. (2.12)

(iii) IfK₁ =B₁=0 then µ= ^γ₂ andhis arbitrary.

Remark 2.7. Conditions in (A8) will be used in the next theorem to prove an estimate for the solution of (2.1) that allows us to extend it globally and that will be used in the proofs of Lemmas3.1,3.2 and3.3 in Section 3. It is clear from (2.10) that µ< ^γ₂. We also observe that (2.11) holds if and only if 0 < 2K₁ < _he¹, where _he¹ is the maximum value of the real function φ(s) = se^−hs, s≥0. From this we see that 2K₁eh< 1 and µ∈(µ₁,µ₂), where µ_j,j=1, 2, are the two positive solutions of the equation µe^−µh =2K₁.

Theorem 2.8. Assume that (A1)–(A8) hold. Then, the solution u= u(t)of (2.1)with uτ = ψ∈ E_h exists globally. Moreover, for eachτ< T<_{∞, the map}I:E_h →C([τ,T];E_h), defined byI(ψ)(t) = u_t, ∀τ≤t ≤T, is continuous.

Proof. Assume that (A8)(i) holds. Multiplying (2.6) by e^µt and integrating the resulting in- equality over[τ,t]we have, for any positive real constantsεandε⁰,

e^µtku(t)k² ≤e^µτkψk²_E

h+ (µ−γ+ε+ε⁰)

Z _t

τ

e^µsku(s)k²ds+ ¹ γ

Z _t

τ

e^µskf(s)k²ds +

2B²₂ ε

+ ^K

22

ε⁰ e^µt

µ

+2K₁ Z _t

τ

e^µsku_sk²_E

hds + ^2B

12

ε Z _t

τ

Z ₀

−he^µt0ku(t⁰+s)k²ds dt⁰.

(2.13)

Let us estimate the last term in (2.13) using the initial condition in (2.1). We have Z _t

τ

Z ₀

−he^µt0ku(t⁰+s)k²ds dt⁰ =

Z ₀

−h

Z _t

τ

e^−µse^µ(t0+s)ku(t⁰+s)k²dt⁰ds

≤e^µh Z ₀

−h

Z _t

τ−h

e^µσku(σ)k²dσds

=e^µhh _{Z τ}

τ−he^µσku(σ)k²dσ+

Z _t

τ

e^µσku(σ)k²dσ

≤ ^e

µ(τ+h)h µ kψk²_E

h +e^µhh Z _t

τ

e^µσku(σ)k²dσ.

(2.14)

Substituting (2.14) into (2.13) we get e^µtku(t)k²≤ e^µτkψk²_E

h+ µ−γ+ε+ε⁰+^2B

12e^µhh ε

! Z _t

τ

e^µsku(s)k²ds +^2B

2 1e^µhh

µε e^µτkψk²_E

h+

2B²₂ ε + ^K

22

ε⁰ e^µt

µ + ¹

γ Z _t

τ

e^µskf(s)k²ds+2K₁ Z _t

τ

e^µsku_sk²_E

hds.

(2.15)

Using (2.10) we can chooseε= ^γ₂ and ε⁰ = ^γ

2 −µ−^4B

2 1e^µhh

γ (2.16)

in (2.15) to obtain

e^µtku(t)k²≤e^µτ 1+ ^4B

12e^µhh µγ

! kψk²_E

h+

4B²₂ γ + ^K

22

ε⁰ e^µt

µ + ¹

γ Z _t

τ

e^µskf(s)k²ds+2K₁ Z _t

τ

e^µsku_sk²_E

hds.

(2.17)

Since ku(s)k ≤ kψk_Eh, ∀s ∈ [τ−h,τ], then we can replace t in (2.17) by t+_{σ, with} σ∈[−h, 0], to deduce that

e^µtku_tk²_E

h ≤ M(t) +L

Z _t

τ

e^µsku_sk²_E

hds, whereL=2K₁e^µhand

M(t) =e^µ(τ+h) 1+ ^4B

12e^µhh µγ

! kψk²_E

h+

4B₂² γ +^K

22

ε⁰

e^µ(t+h) µ + ^e

µh

γ Z _t

τ

e^µskf(s)k²ds.

The above inequality implies that e^µtku_tk²_E

h ≤e^L(t−τ)M(τ) +e^Lt Z _t

τ

e^−LsM⁰(s)ds. (2.18) Performing the calculations in (2.18) using M(t) above and the fact thatµ > L by (2.11), we find the following estimate for the solution of (2.1)

ku_tk²_E

h ≤c₁kψk²_E

he^(L−µ)te^(µ−L)τ+ ^2µ−L µ−L c₂+ ^e

µh

γ Z _t

−_∞

kf(s)k²ds, (2.19) where

c₁= e^µh 1+ ^4B

12e^µhh µγ

!

and c₂= 4B²₂

γ + ^K

22

ε⁰ e^µh

µ . (2.20)

Now, assume that (A8)(ii) holds. For this case we replace (2.14) by e^µtku(t)k² ≤e^µτkψk²_E

h+ (µ−γ+ε+ε⁰)

Z _t

τ

e^µsku(s)k²ds +^2B

21e^µhh

µε e^µτkψk²_E

h+

2B₂² ε +^K

22

ε⁰ e^µt

µ + ¹

γ Z _t

τ

e^µskf(s)k²ds+^2B

21e^µhh ε

Z _t

τ

e^µskusk²_E

hds.

(2.21)

Since thatµ< ^γ₂, then we can chooseε= ^γ₂ andε⁰ = ^γ₂ −µin (2.21) and proceed as before to obtain

e^µtku(t)k²≤e^µ(τ+h) 1+ ^4B

21e^µhh µγ

! kψk²_E

h +

4B₂² γ +^K

22

ε⁰

e^µ(t+h) µ + ^e

µh

γ Z _t

τ

e^µskf(s)k²ds+L Z _t

τ

e^µskusk²_E

hds,

(2.22)

where L = _γ⁴B²₁e^2µhh. By (2.12) we see that µ > L. Therefore, we can deduce the estimate (2.19) with c₁ andc₂ as in (2.20), with ε⁰ = ^γ₂ −µ. Similarly, we can treat the case (A8)(iii) to obtain the estimate

kutk²_E

h ≤c⁰₁kψk²_E

he^−µte^µτ+2c⁰₂+ ^e

µh

γ Z _t

−_∞kf(s)k²ds, (2.23) where

c⁰₁ =2e^µh and c⁰₂= ^e

µh

µ² B²₂+K²₂

. (2.24)

From (2.19) or (2.23) and Theorem2.5we conclude that the solution of (2.1) exists globally.

Next, let us prove that the map I is continuous. Fix τ < T < _∞,ψ ∈ E_h and consider ψ₁ ∈ E_h such that kψ−ψ₁k_Eh < 1. Let us denote byv = v(t)the solution of (2.1) with initial condition v(_s) = ψ₁(_s−τ)_, ∀s ∈ [τ−h,τ]. Using the estimate (2.19) or (2.23) we can find a positive constantK₀depending onkψk_Eh andTsuch thatkutk_Eh ≤K₀andkvtk_Eh ≤K₀, for all τ≤ t≤T. Then, using the integral representations ofuandv and Lemma2.3it follows that

ku(t)−v(t)k ≤ kψ(0)−ψ₁(0)k+

Z _t

τ

kB(s,us)− B(s,vs)kds

≤ kψ−ψ₁k_Eh +L(K₀)

Z _t

τ

ku_s−v_sk_Ehds.

(2.25)

Replacing t in (2.25) by t+σ, with σ ∈ [−h, 0], taking into account that ku(t+σ)− v(t+σ)k_Eh ≤ kψ−ψ₁k_{Eh if}t+σ≤τ, we obtain

ku_t−v_tk_Eh ≤ kψ−ψ₁k_Eh+L(K₀)

Z _t

τ

ku_s−v_sk_Ehds, ∀τ≤t ≤T.

Then, by Gronwall’s inequality, we conclude that kut−vtk_Eh ≤ e^{L(K0)(T−τ)}kψ−ψ₁k_Eh, which implies the continuity ofI.

3 Existence of a pullback attractor

By Theorem 2.8 we can associate to the initial value problem (2.1) a process{U(t,τ)}_t≥τ of continuous maps U(t,τ) in E_h defined by U(t,τ)ψ = u_t, where τ ≤ t and u = u(t) is the global solution of (2.1). In this section, we establish the existence of a pullback attractor for the process {U(t,τ)}_t≥τ using the results obtained in [16]. We are interested in the existence of a pullback attractor for a family of sets depending on time (see [16, Section 3]). Motivated by the estimate (2.19) we consider the setR_µ of all functionsr:R→(_0,∞)_{such that}

t→−lim_∞e^(µ−L)tr²(t) =_{0. (3.1)}

Let us denote byD_µthe class of all families ˆD={D(t);t ∈_R}of nonempty subsets ofE_hsuch that D(t) ⊂ BE_h[0;rDˆ(t)] := {ψ ∈ E_h;kψk_Eh ≤ rDˆ(t)}, for some radius rDˆ ∈ R_µ. For the case (A8)(iii) we consider in (3.1) L = 0. In what follows, we will assume that (A8)(i) or (A8)(ii) holds. Suitable modifications will be indicated for the case (A8)(iii). We will also consider L as in the proof of Theorem2.8and the constantsc₁,c2, c₁⁰ andc₂⁰ given by (2.20) and (2.24).

Lemma 3.1. Assume that (A1)–(A8) hold. Then, the family Bˆµ of closed balls Bµ(t) =BE_h[0;Rµ(t)], where for each t∈R, the radius Rµ(t)is defined by

R²_µ(t) = ^2µ−L µ−L c₂+^e

µh

γ Z _t

−_∞kf(s)k²ds+1, (3.2) is pullbackD_µ-absorbing for the process{U(t,τ)}_t≥τ.

Proof. Sinceµ> L, then using (A7), we have

t→−lim_∞e^(µ−L)tR²_µ(t) = lim

t→−_∞e^(µ−L)t

2µ−L µ−L c₂+ ^e

µh

γ Z _t

−_∞

kf(s)k²ds+1

=0,

which shows that ˆBµ ∈ D_µ. Now, fixed t ∈ _R and ˆD ∈ D_µ, there exists a τ₀ = τ₀(t, ˆD) ≤ t such that

e^(µ−L)τr²_Dˆ(τ)<_c⁻₁¹_e^(µ−L)t_, for anyτ≤τ₀. Then, for anyψ∈ D(τ), using (2.19) we obtain

kU(t,τ)ψk²_E

h ≤c₁r²_Dˆ(τ)e^(µ−L)τe^(L−µ)t+^2µ−L µ−L c₂+ ^e

µh

γ Z _t

−_∞kf(s)k²ds

≤R²_µ(t).

Therefore,U(t,τ)D(τ)⊂ B_µ(t), for allτ≤ τ₀, which proves that the family ˆB_µ is pullback D_µ-absorbing for the process{U(t,τ)}_t≥τ.

In Lemma3.1, in the case (A8)(iii), we take L = 0 and replace c₂ by c⁰₂ in (3.2). Next, let us prove an estimate for the tails of the solutionsu=u(t)of (2.1) when the initial conditions u_τ = ψbelong to B_µ(τ).

Lemma 3.2. Assume that (A1)–(A8) hold. Let Bˆ_µ be the pullback D_µ-absorbing family defined in Lemma3.1. Then, for anyε>0and any t⁰ < T, there existτ₀=τ₀(ε,t⁰,T, ˆBµ)and a positive integer k=k(ε,T, ˆB_µ), such that

s∈[−maxh,0]

∑

|n|>2k

|un(t+s)|² <ε, ∀τ≤τ₀, t ∈[t⁰,T], for any solution u=u(t)of (2.1)with initial condition uτ ∈ Bµ(τ).

Proof. Assume that (A8)(i) holds. Similarly, we treat the case (A8)(ii). Letu_τ =ψ∈B_µ(τ)and consider the corresponding solution u = u(t) of (2.1) defined in [τ,∞). Let θ ∈ C¹(_R⁺;R) be a function such thatθ ≡ 0 on [0, 1],θ ≡ 1 on[2,∞), 0 ≤ θ ≤ 1, and |θ⁰(t)| ≤ 2,∀t ≥ 0.

Let v = (v_n(t))_n∈Z, where v_n(t) = θ(^|n_k^|)u_n(t), with k > 0 fixed in Z. In order to simplify notation, we will writeθn =θ ^|n_k^|

,kwk_θ =_∑⁺_n=−^∞ _∞θn|wn|²andkutk²_E

h,θ =max_s∈[−h,0]kut(s)k²_θ. Taking the imaginary part of the inner product of equation (2.1) with vin`² we find

1 2

d

dt(u,v)_{`</sub>2 +γ(u,v)<sub>`}2 =Im(f,v)_{`</sub>2 −Im(Au,v)<sub>`}2−Im(g(t,u_t),v)_`2, ∀t≥ τ. (3.3)

Let us estimate the terms on the right-hand side of (3.3). Since ψ ∈ B_µ(τ) then, using (2.19), we see that

ku(t)k ≤r₀, ∀t∈ [τ,T],

withr₀ = (c₁+1)R_µ(T). Moreover, by the definition ofθ, we have that|θ_n+m−θ_n| ≤ ²_kmand

|θn+m−θn| ≤2. Then,

−Im(Au(t),v(t))_`2 =−Im (

J(0)ku(t)k²_θ+

+_∞ n=−

∑

+_∞ m

∑

J(m)(θn+m−θn)un+m(t)un(t) )

≤

+_∞ n=−

∑

+_∞ m

∑

|J(m)| |θn+m−θn| |un+m(t)| |un(t)| ≤ν(T,k,l), whereν(T,k,l) = ²_{k ∑}^l_m=1m|J(m)|+2 ∑⁺_m^∞=l+1|J(m)|r₀², l≥1.

Using the hypotheses (A1) and (A4) and proceeding as in the proof of Lemma 2.6 we obtain the estimate

−_Im(g(t,u_t)_,v(t))_`2 ≤

+_∞ n=−

∑

θ_n|g_1,n(t,u_n(t−ρ(t)))| |u_n(t)|

+

+_∞ n=−

∑

θn

Z ₀

−h

|bn(s,un(t+s))|ds|un(t)|

≤(K₁ku(t−ρ(t))k_θ+K_2,θ)kuk_θ +

"

B₁ Z ₀

−h

ku(t+s)k²_θds 1/2

+B_2,θ

# kuk_θ,

where B_2,θ =_∑⁺_n=−^∞ _∞θ_n R0

−hβ_2,n(s)ds21/2

andK_2,θ = _∑⁺_n=−^∞ _∞θ_nk²_2,n1/2

. In addition, we know that

−Im(f(t),v(t))_`2 ≤ ¹

2γkf(t)k²_θ+ ^γ

2ku(t)k²_θ. Therefore,

d

dtku(t)k²_θ+γku(t)k²_θ ≤ ¹

γkfk²_θ+2(K₁ku(t−ρ(t))k_θ+K_2,θ)ku(t)k_θ +2

"

B₁ Z ₀

−h

ku(t+s)k²_θds 1/2

+B_2,θ

#

ku(t)k_θ+2ν(T,k,l), (3.4)

for all τ≤t≤ T.

Now, we multiply (3.4) bye^µtand use the inequalities 2

"

B₁ Z ₀

−h

ku(t+s)k²_θds 1/2

+B_2,θ

#

kuk_θ ≤ ^4B

21

γ Z ₀

−h

ku(t+s)k²_θds+ ^4B

2,θ2

γ + ^γ 2 kuk²_θ,

2(K₁ku(t−ρ(t))k_θ+K_2,θ)ku(t)k_θ ≤2K₁ku_tk²_E

h,θ+ ^K

22,θ

ε⁰ +ε⁰kuk²_θ,

whereε⁰ >0, to find d

dt e^µtku(t)k²_θ ≤µ− ^γ 2 +ε⁰

e^µtku(t)k²_θ+ ¹

γe^µtkf(t)k²_θ+2K₁e^µtku_tk²_E

h,θ

+ ^4B

2,θ2

γ + ^K

22,θ

ε⁰

!

e^µt+2ν(T,k,l)e^µt +^4B

21e^µt γ

Z ₀

−h

ku(t+s)k²_θds, ∀τ≤ t≤T.

(3.5)

Integrating (3.5) over[τ,t]and using the following estimate analogous to (2.14) Z _t

τ

Z ₀

−he^µt0ku(t+s)k²_θds dt⁰ ≤ ^e

µ(τ+h)h

µ kψk²_Eh+e^µhh Z _t

τ

e^µsku(s)k²_θds, we obtain

e^µtku(t)k²_θ ≤e^µτ 1+^4B

12e^µhh µγ

! kψk²_E

h+ µ−^γ

2 +ε⁰+^4B

21e^µhh γ

! Z _t

τ

e^µsku(s)k²_θds + ^4B

22,θ

γ + ^K

22,θ

ε⁰ +2ν(T,k,l)

!e^µt µ +2K₁

Z _t

τ

e^µsku_sk²_E

h,θds + ¹

γ Z _t

τ

e^µskf(s)k²_θds.

By condition (2.10) we can chooseε⁰ as in (2.16) in the above inequality to obtain e^µtku(t)k²_θ ≤e^µτ 1+^4B

21e^µhh µγ

! kψk²_E

h+ ^4B

22,θ

γ + ^K

2,θ2

ε⁰ +2ν(T,k,l)

!e^µt µ +2K₁

Z _t

τ

e^µsku_sk²_E

h,θds+ ¹ γ

Z _t

τ

e^µskf(s)k²_θds.

(3.6)

Replacing t by t+σ, with σ ∈ [−h, 0] in (3.6) and using the inequality ku(t+σ)k = kψ(t+σ)k ≤ kψk_Eh, valid fort+σ <τ, we deduce that

e^µtku_tk²_E

h,θ ≤ M_θ(t) +L Z _t

τ

e^µsku_sk²_E

h,θds, (3.7)

whereL=2K₁e^µhand

M_θ(t) =e^µ(τ+h) 1+^4B

12e^µhh µγ

! kψk²_E

h+ ^4B

22,θ

γ + ^K

22,θ

ε⁰ +2ν(T,k,l)

!e^µ(t+h) µ +^e

µh

γ Z _t

τ

e^µskf(s)k²_θds.

We know thatµ> L. Then, from (3.7) andψ∈ B_µ(τ), we obtain ku_tk²_E

h,θ ≤c₁R²_µ(τ)e^(L−µ)te^(µ−L)τ+ ^2µ−L

µ−L c_2,θ+²(2µ−L) µ(µ−L)^e

µhν(T,k,l) + ^e

µh

γ Z _t

−_∞kf(s)k²_θds, ∀t ≥τ,

(3.8)