Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 12, pp. 1–19.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.12

GLOBAL ATTRACTOR AND `p SOLUTIONS TO INITIAL

VALUE PROBLEMS OF DISCRETE NONLINEAR

SCHRÖDINGER EQUATIONS COMPLEX POTENTIAL

GUOPING ZHANG, GHDER ABURAMYAH

Abstract. In this article, we investigate the global well-posedness of initial

value problems of the time-dependent discrete nonlinear Schrödinger equation
with a complex potential and sufficiently general nonlinearity on a multidimen-

sional lattice in weighted `p spaces for 1 < p < ∞. Thanks to our improved
estimates we are able to prove the existence of global attractor for `p solutions

to the initial value problem.

1. Introduction

The discrete nonlinear Schrödinger equation (DNLS) describes the evolution of
complex-valued wave amplitudes at discrete sites, capturing the interplay between
discreteness and nonlinearity. This discreteness can arise in systems where wave
propagation is confined to a lattice or network, as observed in optical waveguide
arrays and certain crystalline structures.

The presence of a complex potential in the DNLS equation introduces additional
complexities and nonlinearity, leading to rich dynamics and novel phenomena. The
complex potential can arise from various sources, such as an external field or a
spatially varying refractive index in optics. It can significantly influence the propa-
gation characteristics of waves, including wave localization, soliton formation, and
wave scattering.

Understanding the properties and dynamics of the DNLS equation is crucial for
gaining insights into the behavior of discrete wave systems and exploring nonlinear
effects in different physical systems. For instance, we mention nonlinear wave trans-
mission in discrete media, propagation of localized pulses in coupled waveguides
and optical fibers, and modeling Bose-Einstein condensates (see, e.g., [8, 11, 12]
and references therein).

Research activity in this area mainly focuses on the so-called “breathers” which
are standing waves. The profile function of such a wave solves an appropriate
stationary DNLS equation. Most works in this area deal with (discrete) translation-
invariant DNLS on a one-dimensional lattice and employ perturbation techniques,

2020 Mathematics Subject Classification. 37L60, 35B41, 35Q55.
Key words and phrases. Discrete nonlinear Schrödinger equation; semigroup; `p solution;

global attractor; Lipschitz continuous; initial value problem; complex potential.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted November 1, 2023. Published January 31, 2024.

1



2 G. ZHANG, G. ABURAMYAH EJDE-2024/12

two-dimensional discrete-time dynamical systems, and numerical simulation (see,
e.g., [6, 7, 8] and references therein).

On the other hand, the series of papers [3, 15, 16, 17, 18, 19, 25, 26, 29, 30, 31]
applies the theory of critical points of smooth functionals to the study of breathers
for DNLS with various types of nontrivial potentials. In this context, we also
mention the remarkable paper [24].

The initial value problem (IVP) associated with the DNLS equation with a
complex potential deals with determining the evolution of the wave function over
time when its initial configuration is known. In other words, given the initial
values of the wave function and its derivative at a specific time, the IVP seeks to
find a solution that satisfies the DNLS equation with the given complex potential.
The DNLS equation with a complex potential on a one-dimensional lattice can be
written as

i(dψn/dt) +Anψn +Bn|ψn|2ψn + Cnψn+1 +Dnψn−1 = 0,

where ψn is the complex-valued wave function at the discrete lattice site n, and
An, Bn, Cn, and Dn represent the coefficients associated with the linear and non-
linear interactions between adjacent lattice sites.

In [20] we investigated the weighted `2 solution of the following initial value prob-
lem for the time-dependent d-dimensional discrete nonlinear Schrödinger equation

iu̇ = −∆u+Wu− f(n, u) + b(t, n), (1.1)

u(0, n) = u0(n) , (1.2)

where the potential W = V + iδ is a complex function of

n = (n1, n2, . . . , nd) ∈ Zd ,

where u̇ stands for the time derivative and −∆ is the d-dimensional discrete Lapla-
cian defined by

∆u(n) =

d∑
j=1

[
u(n+ ej) + u(n− ej)

]
− 2du(n), n ∈ Zd, (1.3)

where ej ∈ Zd has 1 at the j-th component and 0 elsewhere.
Note that if δ(n) is negative for all n ∈ Zd, the part δ of the potential represents

dissipation effects. Additionally, our Assumption (iii) below allows the nonlinearity
to contain a dissipative term. This DNLS (1.1) is the space discretization of the
nonlinear Schrödinger equation in continuous media.

Only a few papers [9, 10, 13, 14] are devoted to equations of the form (1.1).
The paper [14] focuses on the initial value problem for the DNLS with a zero
potential and power nonlinearity on a one-dimensional lattice with weighted `2

initial value. The main result provides global well-posedness in weighted `2 spaces
with power weights. In [9] and [10], the authors consider the DNLS with V = 0 and
δ = const. The main results are global well-posedness in the conservative (δ = 0)
and dissipative (δ < 0) cases, as well as the existence of attractors in weighted `2

spaces in the conservative case, on one-dimensional and multidimensional lattices,
respectively. In the paper [13], the well-posedness in weighted spaces is studied for
the DNLS on a one-dimensional lattice in the case when W = V is a general real
potential and b = 0.



EJDE-2024/12 GLOBAL ATTRACTOR AND `p SOLUTIONS TO DNLS 3

In [20], we extended those results to the multidimensional case, allowing a suffi-
ciently general, not necessarily bounded potential W with weighted `2 initial value.
In [27], we used the integral equation defining the mild solution of the DNLS in [20]
to prove the existence of global solution for the DNLS with a weighted `p initial
value when 1 ≤ p < 2 by leveraging the existing `2 global solutions obtained in
[20]. In [28], we investigated the global solutions in the weighted `p space when
2 < p <∞.

Note that the study of attractors for spatially discrete systems goes back to the
paper [1] and has been continued in [32, 33, 34, 35, 20].

We observe that, in contrast to [20, Theorem 3.1], the proof of [27, Theorem 4.1]
necessitates the inclusion of following additional assumptions

δ(n) ≤ 0, for all n ∈ Zd and b ∈ L1([0,∞), `2(Zd)). (1.4)

In this paper, we aim to enhance the existing results regarding the global solu-
tion for DNLS with a weighted `p initial value, where 1 < p < ∞. Our approach
involves optimal estimates and minimal assumptions, contributing to a substan-
tial improvement in results. Specifically, we obtain the global weighted `p solution
without relying on the aforementioned assumption (1.4). Additionally, we rigor-
ously establish the existence of a global attractor for solutions in the weighted `p

space to the initial value problem.
The organization of this paper is as follows: For readers’ convenience, we provide

a reminder of some preliminaries on the semigroup theory of abstract differential
equations in Section 2. The previous results in [27] and [28] will be reviewed in
Section 3. Section 4 is devoted to the existence of weighted `p global solutions for
1 < p <∞. We prove the existence of the global attractor in the last section 5.

2. Semigroup theory and abstract initial value problem

The content of this section can be found in [20]. For reader’s convenience, we
include it here. We treat (1.1) as an abstract differential equation of the form

u̇ = Au+N(t, u) (2.1)

in a complex Banach space. We always assume that A is a closed operator in a
Banach space E with the domain D(A), and N : [0,∞) × E → E is continuous.
Let us provide a reminder of some elementary facts related to such equations.

A family U(t), t ∈ [0,∞), of bounded linear operators in E is a strongly contin-
uous semigroup of operators if

(1) U(t)v is a continuous function on [0,∞) with values in E for every v ∈ E;
(2) U(0) = I is the identity operator in E;
(3) U(t+ s) = U(t)U(s) for all t, s ∈ [0,∞).

If the family U(t) is defined for all t ∈ R and satisfies (1)–(3) above on the whole
real line, we say that U(t) is a strongly continuous group of operators.

If U(t) is a strongly continuous semigroup of operators, then its generator A is
defined by

Av = lim
t→0+

t−1(U(t)− I)v, (2.2)

where the domain D(A) consists of those v ∈ E for which the limit in (2.2) exists.
The following result is well known (see [5, 21]).



4 G. ZHANG, G. ABURAMYAH EJDE-2024/12

Proposition 2.1. If A is a generator of a strongly continuous semigroup in a
Banach space E and B is a bounded linear operator in E, then A+B is a generator
of a strongly continuous semigroup.

If A is a bounded linear operator, then it generates a one-parameter group etA.
In general, if A is a generator of a strongly continuous semigroup, we still use the
same exponential notation etA for the semigroup generated by A.

Now we discuss the abstract initial value problem for equation (2.1), with initial
data

u(0) = u0 ∈ E. (2.3)

If A is a bounded operator, then it is sufficient to consider classical solutions, i.e.
continuously differentiable functions with values in E that satisfy (2.1) and (2.3).
In general, when the operator A is unbounded, we consider mild solutions to (2.1)
and (2.3).

A continuous function u on [0, T ] with values in E is a mild solution of the initial
value problem (2.1) and (2.3) if it satisfies the integral equation

u(t) = etAu0 +

∫ t

0

e(t−s)AN(s, u(s))ds. (2.4)

In the case when the operator A is bounded, these are classical solutions. We need
the following well-known result (see [2, 21]).

Proposition 2.2. Let A be a generator of a strongly continuous semigroup in a
Banach space E, and N(t, u) : [0,∞) × E → E be continuous in t and locally
Lipschitz continuous in u with the Lipschitz constant being bounded on bounded
intervals of t. That is, for any T > 0 and R > 0, there exists C = C(T,R) > 0
such that

max
0≤t≤T

‖N(t, w)−N(t, w′)‖ ≤ C‖w − w′‖ (2.5)

whenever ‖w‖ ≤ R and ‖w′‖ ≤ R.

(a) For every u0 ∈ E, there exists a unique local mild solution of the initial
value problem (2.1) and (2.3) defined on the maximal interval [0, τmax).

(b) If τmax <∞, then limt↗τmax
‖u(t)‖ =∞.

(c) The solution u(t) depends continuously on u0 in the topology of uniform
convergence on bounded closed subintervals of [0, τmax).

(d) Assume, in addition, that the map N : [0,∞)× E → E is locally Lipschitz
continuous, i.e., for any T > 0 and R > 0, there exists C = C(T,R) > 0
such that

‖N(t, w)−N(t′, w′)‖ ≤ C(|t− t′|+ ‖w − w′‖) (2.6)

whenever t ∈ [0, T ], t′ ∈ [0, T ], ‖w‖ ≤ R and ‖w′‖ ≤ R. If u0 ∈ D(A), then
the mild solution of the initial value problem (2.1) and (2.3) is a classical
solution.

Note that (2.5) implies automatically that N is bounded on bounded sets.

Remark 2.3. If N(t, u) is globally Lipschitz continuous in u, i.e. there exists a
constant C = C(T ) > 0 such that

max
0≤t≤T

‖N(t, w)−N(t, w′)‖ ≤ C‖w − w′‖, ∀w,w′ ∈ E, (2.7)



EJDE-2024/12 GLOBAL ATTRACTOR AND `p SOLUTIONS TO DNLS 5

then the initial value problem (2.1) and (2.3) possesses a unique global mild solution
defined on [0,∞). Moreover, the solution u(t) depends continuously on u0 in the
topology of uniform convergence on bounded closed subintervals of [0,∞).

Remark 2.4. Let

N(t, u) = M(u) + f(t) .

Then assumption (2.6) holds if and only if M and f are locally Lipschitz continuous
on E and [0,∞), respectively.

3. Assumptions and review

We review local solution to the equation (1.1) under the following assumptions:

(i) The complex potential W = V +iδ is such that both V and δ are real-valued
functions on Zd, and

δ = sup{δ(n)|n ∈ Zd} <∞ .

(ii) The nonlinearity f : Zd × C→ C satisfies the following conditions:
(1) f(n, 0) = 0,
(2) f(n, z) = o(z) as z → 0 uniformly with respect to n ∈ Zd,
(3) f is uniformly locally Lipschitz continuous, that is, for every R > 0,

there exists a constant C = C(R) independent of n ∈ Zd such that

|f(n, z)− f(n, z′)| ≤ C|z − z′|

for all n ∈ Zd whenever |z| ≤ R and |z′| ≤ R.
(iii) The nonlinearity f(n, z) is of the form f(n, z) = g(n, |z|)z, where g(n, r) is

a function and its imaginary part is nonnegative.

Let Θ = (θn)n∈Zd be a sequence of positive numbers (weights). The space `pΘ(Zd)
consists of all two-sided sequences of complex numbers such that the norm

‖u‖`pΘ =
( ∑
n∈Zd

|u(n)θn|p
)1/p

is finite, where 1 ≤ p < ∞. We notice that u ∈ `pΘ(Zd) if and only if uΘ ∈ `p(Zd)
and

‖u‖`pΘ = ‖uΘ‖`p .

Therefore for 1 ≤ p < q ≤ ∞ we have

‖u‖`qΘ ≤ ‖u‖`pΘ and `pΘ(Zd) ⊂ `qΘ(Zd).

We always assume that the weight Θ is regular in the sense that

(iv) The sequence Θ is bounded below by a positive constant, and there exists
a constant c0 ≥ 1 such that

c−1
0 ≤ θn+ei

θn
≤ c0

for all n ∈ Zd and i = 1, . . . , d, where ei ∈ Zd has 1 at the i-th component
and 0 elsewhere.



6 G. ZHANG, G. ABURAMYAH EJDE-2024/12

From Assumption (iv), we obtain

‖u‖lp(Zd) ≤ C0‖u‖lpΘ , (3.1)

which implies that `pΘ(Zd) is densely and continuously embedded into `p(Zd). Set-
ting Θ0 as the constant weight with unit components, we have that

`pΘ0
(Zd) = `p(Zd)

From the perspective of functional analysis, Assumption (iv) means that the space
`pΘ(Zd) is translation invariant. More precisely, let Si and Ti be the operators
defined by

(Siw)(n) = w(n− ei), (Tiw)(n) = w(n+ ei), i = 1, . . . , d.

To understand the equation (1.1) in the framework of evolution equations, we
interpret it as an evolution equation of the form (2.1), where A = −iH and H is
the Schrödinger operator defined as

H = −∆ +W (3.2)

and the operator N is defined as

N(t, u)(n) = if(n, u(t, n))− ib(t, n). (3.3)

To establish a precise interpretation, we need to analyze certain properties of the
Schrödinger operator H in the space `pΘ(Zd). First, we observe that the operator
(of multiplication by) −iW = −iV + δ is a diagonal operator. Since V is real
and δ(n) ≤ δ for all n ∈ Zd, the operator −iW generates a strongly continuous
semigroup in `pΘ(Zd) given by

(e−itWu)(n) = e−iV (n)teδ(n)tu(n), n ∈ Zd .
The domain of this operator in `pΘ(Zd) is defined as

DΘ = {u ∈ `pΘ(Zd) : Wu ∈ `pΘ(Zd)}. (3.4)

where we use the notation D to represent the domain of the operator W in `p(Zd).
It is clear that DΘ ⊂ D.

Based on Proposition 2.1, we derived the following lemma in [27].

Lemma 3.1. The operator A = −iH is a generator of strongly continuous group
etA in the space `pΘ(Zd), where 1 ≤ p < ∞. Moreover, there exist two constants
M ≥ 1 and ω such that for all t ≥ 0

‖etA‖ ≤Meωt. (3.5)

We define the operator

N(t, u)(n) = if(n, u(n))− ib(t, n).

Then equation (1.1) can be expressed in the form of equation (2.1). The following
local well-posedness result is proved in [27].

Theorem 3.2. (1) Under Assumptions (i), (ii), (iv), if b ∈ C([0,∞), `pΘ(Zd)),
where 1 ≤ p <∞, then for every u0 ∈ `pΘ(Zd), problem (1.1) and (1.2) has
a unique local mild solution u ∈ C([0, τmax), `pΘ(Zd)) which is defined on the
maximal interval [0, τmax).

(2) If τmax < ∞, then limt↗τmax ‖u(t)‖ = ∞. The solution u(t) depends con-
tinuously on u0 in the topology of uniform convergence on bounded closed
subintervals of [0, τmax).



EJDE-2024/12 GLOBAL ATTRACTOR AND `p SOLUTIONS TO DNLS 7

(3) The mild solution u(t) ∈ C([0, τmax), `pΘ(Zd)) of problem (1.1) and (1.2)
obtained in part (1) is a classical solution if one of the following conditions
holds
(a) u0 ∈ `pΘ(Zd) and W is bounded;
(b) u0 ∈ D(A) = DΘ and b : [0,∞)→ `pΘ(Zd) is locally Lipschitz continu-

ous.

4. Global solutions for 1 < p <∞

In [27] we utilized the fact `pΘ(Zd) ⊂ `2Θ(Zd) for the case 1 ≤ p < 2, and we relied
on the established global existence of `2 solutions in [20] to prove the following
theorem.

Theorem 4.1. (1) Let assumptions (i)–(iv) be satisfied, and 1 ≤ p ≤ 2, if
δ ≤ 0 and b ∈ C([0,∞), `pΘ(Zd)) ∩ L1([0,∞), `2(Zd)), then for every u0 ∈
`pΘ(Zd), problem (1.1) and (1.2) has a unique global mild solution u ∈
C([0,∞), `pΘ(Zd)) which continuously depends on u0 in the topology of uni-
form convergence on bounded closed subintervals of [0,∞). Moreover, for
any t ≥ 0

‖u(t)‖`pΘ ≤ (‖u0‖`pΘ +B(ω + CM, t))e(ω+CM)t , (4.1)

where

B(ω + CM, t) =

∫ t

0

e−(ω+CM)s‖b(s)‖`pΘ ds,

C is the Lipschitz constant independent of t, ω and M are the constants in
Lemma 3.1.

(2) The global mild solution u(t) ∈ C([0,∞), `pΘ(Zd)) of problem (1.1) and (1.2)
obtained in (1) is a classical solution if one of the following conditions holds
(a) u0 ∈ `pΘ(Zd) and W is bounded;
(b) u0 ∈ D(A) = DΘ and b : [0,∞)→ `pΘ(Zd) is locally Lipschitz continu-

ous.

We observed that, when comparing with Theorem 3.1 in [20], the proof of The-
orem 4.1 required the following additional assumption:

δ ≤ 0, and b ∈ L1([0,∞), `2(Zd)).
However, using a different method, we proved the following theorem in the case
2 ≤ p <∞ without the aforementioned additional assumption (refer to [28]).

Theorem 4.2. (1) Assume that assumptions (i)–(iv) are satisfied and 2 ≤ p <∞,
if b ∈ C([0,∞), `pΘ(Zd)), then for every u0 ∈ `pΘ(Zd), problem (1.1) and (1.2) has
a unique global mild solution u ∈ C([0,∞), `pΘ(Zd)) which continuously depends on
u0 in the topology of uniform convergence on bounded closed subintervals of [0,∞).
Moreover, for any t ≥ 0,

‖u(t)‖`pΘ ≤ (‖u0‖`pΘ +B(δ̄ + 2dc0, t))e
(δ̄+2dc0)t , (4.2)

where

B(δ̄ + 2dc0, t) =

∫ t

0

e−(δ̄+2dc0)s‖b(s)‖`pΘds,

δ̄ and c0 are constants in assumption (i) and (iv) respectively.
(2) The global mild solution u(t) ∈ C([0,∞), `pΘ(Zd)) of problem (1.1) and (1.2)

obtained in (1) is a classical solution if one of the following conditions holds



8 G. ZHANG, G. ABURAMYAH EJDE-2024/12

(a) u0 ∈ `pΘ(Zd) and W is bounded;
(b) u0 ∈ D(A) = DΘ and b : [0,∞)→ `pΘ(Zd) is locally Lipschitz continuous.

In this section, we reconsider about the global solution in the case of 1 < p < 2
without those additional assumptions. To do so, we need some preparations.

Lemma 4.3. If u ∈ C1([0, T ], lpΘ(Zd)), 1 < p < ∞, then for each n ∈ Zd and
t ∈ [0, T ],

d

dt
|u(t, n)|p = Re Big(p|u(t, n)|p−2ū(t, n)

d

dt
u(t, n)

)
(4.3)

Proof. Fix t0 ∈ (0, T ). If u(t0, n) 6= 0, then there exists δ > 0 such that

|u(t, n)| > |u(t0, n)|
2

> 0, whenever |t− t0| < δ,

by the chain rule, we obtain

d

dt
|u(t, n)|p

∣∣
t=t0

=
d

dt
[u(t, n)ū(t, n)]p/2

∣∣
t=t0

= p|u(t0, n)|p−2 Re
(
ū(t0, n)

d

dt
u(t0, n)

)
.

If u(t0, n) = 0, by definition

d

dt
|u(t, n)|p

∣∣
t=t0

= lim
t→t0

|u(t, n)|p − |u(t0, n)|p

t− t0
= lim
t→t0

|u(t, n)|p

t− t0

=
∣∣ lim
t→t0

u(t, n)− u(t0, n)

t− t0
∣∣p lim
t→t0

(t− t0)p−1

= | d
dt
u(t, n)

∣∣
t=t0
|p lim
t→t0

(t− t0)p−1 = 0.

Notice that

|p|u(t0, n)|p−2 Re
(
ū(t0, n)

d

dt
u(t0, n)

)
| ≤ p|u(t0, n)|p−1| d

dt
u(t0, n)|,

we obtain (4.3) for 0 < t < T . Similarly we can prove (4.3) for t = 0 or T if we
replace two-sided limit by one-sided limit. �

For each positive integer k we define

χk(n) =

{
1, when |n| ≤ k
0, otherwise.

Lemma 4.4. If u ∈ C1([0, T ], lpΘ(Zd)), 1 < p <∞, then

w(t) ≡ ‖u(t)‖p
lpΘ

=
∑
n∈Zd

|u(t, n)θn|p

is differentiable and

dw

dt
=

d

dt

∑
n∈Zd

|u(t, n)θn|p =
∑
n∈Zd

d

dt
|u(t, n)|pθpn

= Re
( ∑
n∈Zd

pθpn|u(t, n)|p−2ū(t, n)
d

dt
u(t, n)

)
.



EJDE-2024/12 GLOBAL ATTRACTOR AND `p SOLUTIONS TO DNLS 9

Proof. For each k ≥ 1 we define the partial sum sequence

wk(t) =
∑
|n|≤k

|u(t, n)θn|p =
∑
n∈Zd

|u(t, n)χk(n)θn|p.

By Lemma 4.3 we obtain

d

dt
wk(t) = Re

( ∑
|n|≤k

pθpn|u(t, n)|p−2ū(t, n)
d

dt
u(t, n)

)
= Re

( ∑
n∈Zd

pθpn|u(t, n)χk(n)|p−2ū(t, n)χk(n)
d

dt
u(t, n)

)
.

u ∈ C1([0, T ], lpΘ(Zd)) implies wk(t) converges uniformly to w(t) on [0, T ], and
sup0≤t≤T ‖u′(t)‖lpΘ <∞. We denote

s(t) = Re
( ∑
n∈Zd

pθpn|u(t, n)|p−2ū(t, n)
d

dt
u(t, n)

)
sk(t) = Re

( ∑
n∈Zd

pθpn|u(t, n)χk(n)|p−2ū(t, n)χk(n)
d

dt
u(t, n)

)
.

By Hölder’s inequality,

sup
0≤t≤T

|sk(t)− s(t)| ≤ sup
0≤t≤T

∑
n∈Zd

pθpn|u(t, n)(1− χk(n))|p−1| d
dt
u(t, n)|

≤ p sup
0≤t≤T

‖u(t)(1− χk)‖p−1
lpΘ

sup
0≤t≤T

‖u′(t)‖lpΘ ,

which implies that sk(t) = d
dtwk(t) converges uniformly to s(t) on [0, T ] and s(t) ∈

C([0, T ],R). By [22, Theorem 316] we obtain

s(t) = lim
k→∞

d

dt
wk(t) =

d

dt
lim
k→∞

wk(t) =
d

dt
w(t). �

Lemma 4.5. Let 1 < p <∞ and u ∈ `pΘ(Zd)). If Θ satisfies the assumption (iv),
then ∑

n∈Zd

Im[−∆u(n)u(n)]|u(n)|p−2θpn ≤ 2dc0‖u‖p`pΘ .

Proof. Since for all n ∈ Zd we have

−∆u(n)u(n) = −
d∑
j=1

[
u(t, n+ ej) + u(t, n− ej)

]
u(t, n) + 2d|u(n)|2 (4.4)

which implies∑
n∈Zd

Im[−∆u(n)u(n)]|u(n)|p−2θpn

= −
d∑
j=1

∑
n∈Zd

Im
[
[u(t, n+ ej) + u(t, n− ej)]u(n)

]
|u(n)|p−2θpn

≤
d∑
j=1

∑
n∈Zd

|u(n)θn|p−1|u(t, n+ ej)θn+ej |
θn

θn+ej



10 G. ZHANG, G. ABURAMYAH EJDE-2024/12

+

d∑
j=1

∑
n∈Zd

|u(n)θn|p−1|u(t, n− ej)θn−ej |
θn

θn−ej
.

Since Θ is a regular weight by assumption (iv), then by Hölder’s inequality we
obtain ∑

n∈Zd

Im[−∆u(n)u(n)]|u(n)|p−2θpn ≤ 2dc0‖u‖p`pΘ . �

For 1 < p <∞ and Θ satisfying assumption (iv), we define

ρΘ(p) ≡ sup
‖u‖`p

Θ
=1

∑
n∈Zd

Im[−∆u(n)u(n)]|u(n)|p−2θpn. (4.5)

By Lemma 4.5 we know that ρΘ(p) ≤ 2dc0.
Let

N(t, u)(n) = if(n, u(t, n))− ib(t, n), A = −iH = −i(−∆ +W ).

For u0 ∈ `pΘ, 1 < p < ∞, by Theorem 3.2, there is a unique mild solution u(t)
defined on the maximal interval [0, τmax) and if u0 ∈ D(A) and b(t) is locally
Lipschitz in [0, τmax), then u ∈ C1([0, τmax), `pΘ) is a classical solution satisfying

d

dt
u(t, n) = Au(t, n) +N(t, u)(n), ∀n ∈ Zd. (4.6)

Lemma 4.6. Let assumptions (i)–(iv) be satisfied and b ∈ C([0,∞), `pΘ(Zd)) with
1 < p < ∞. If u0 ∈ `pΘ(Zd), then there exists a constant C̄ = δ̄ + ρΘ(p), such that
the mild solution u ∈ C([0, τmax), `pΘ(Zd)) satisfies

‖u(t)‖`pΘ ≤ e
C̄t
(
‖u0‖`pΘ +B(C̄, t)

)
(4.7)

for all 0 ≤ t < τmax.

Proof. First of all we assume that u0 ∈ D(A) and b(t) is locally Lipschitz in
[0, τmax), then u ∈ C1([0, τmax), lpΘ) is a classical solution by Theorem 3.2. By
Lemma 4.3 we obtain

d

dt
|u(t, n)|p = Re

[
p
∣∣u(t, n)

∣∣p−2
u(t, n)

d

dt
u(t, n).

]
Multiplying (4.6) by p

∣∣u(t, n)
∣∣p−2

u(t, n), and taking the real part we obtain

d

dt
|u(t, n)|p

= Re
[
p|u(t, n)|p−2u(t, n)[(−iH)u(t, n)] + p|u(t, n)|p−2u(t, n)N(t, u)(n)

]
= Re

[
p|u(t, n)|p−2u(t, n)[(i∆u(t, n)− iWu(t, n)]

+ p|u(t, n)|p−2u(t, n)
(
if(n, u(t, n)

)
− p|u(t, n)|p−2u(t, n)ib(t, n).

]
From assumption (i) we have −iW = −iV +δ and from assumption (iii) we have

f(n, u) = g(n, |u|)u(t, n) = (Re g + i Im g)u(t, n),

d

dt
|u(t, n)|p = Re

[
p|u(t, n)|p−2u(t, n)[(i∆u(t, n)− iV (n)u(t, n) + δ(n)u(t, n)]

+ p|u(t, n)|p−2|u(t, n)|2
(
iRe g − Im g)

)
− p|u(t, n)|p−2u(t, n) ib(t, n).

]



EJDE-2024/12 GLOBAL ATTRACTOR AND `p SOLUTIONS TO DNLS 11

∆u(t, n) =

d∑
j=1

[
u(t, n+ ej) + u(t, n− ej)

]
− 2du(t, n),

i∆u(t, n)u(t, n) = i

d∑
j=1

[
u(t, n+ ej) + u(t, n− ej)

]
u(t, n)− i2d|u(t, n)|2, (4.8)

δ(n)u(t, n)u(t, n) = δ(n)|u(t, n)|2. (4.9)

Using (4.4) and (4.9), we have

d

dt
|u(t, n)|p = pδ(n)|u(t, n)|p − p|u(t, n)|p−2 Img

d∑
j=1

[
u(t, n+ ej)u(t, n)

+ u(t, n− ej)u(t, n)
]
− p|u(t, n)|p

(
Img

)
+ p|u(t, n)|p−2 Im(u(t, n)b).

]
Multiplying both sides by θpn, then by assumption (iii), Im g ≥ 0 implies that
−p|u(t, n)|p(Im g) ≤ 0. Thus by assumption (i) we have

d

dt
|u(t, n)|pθpn ≤ pδ̄ |u(t, n)|pθpn − p|u(t, n)|p−2 Im

d∑
j=1

[
u(t, n+ ej)θ

p
nu(t, n)

+ u(t, n− ej)θpnu(t, n)
]

+ p|u(t, n)|p−2 Im(u(t, n)b)θpn.

Taking summation over n ∈ Zd, by Lemma 4.4 we obtain

d

dt

( ∑
n∈Zd

|u(t, n)θn|p
)

≤ pδ̄
∑
n∈Zd

|u(t, n)θn|p − p Im

d∑
j=1

∑
n∈Zd

|u(t, n)|p−2u(t, n)
[
u(t, n+ ej)θ

p
n

+ u(t, n− ej)θpn
]

+ p|u(t, n)|p−2 Im(u(t, n)b(t, n)θpn).

Since

Im(u(t, n)b(t, n)) ≤ |u(t, n)b(t, n)| = |u(t, n)||b(t, n)|,
by Lemma 4.5 we have

d

dt
‖u‖p

`pΘ
≤ pδ‖u(t)‖p

`pΘ
+ pρΘ(p)‖u(t)‖p

`pΘ
+ p

∑
n∈Zd

|u(t, n)θn|p−1|b(t, n)||θn|.

Then by Hölder’s inequality we obtain

d

dt
‖u(t)‖p

`pΘ
≤ p(δ + ρΘ(p))‖u(t)‖p

`pΘ
+ p‖u(t)‖p−1

`pΘ
‖b(t)‖`pΘ .

Let C = δ + ρΘ(p) and α = p−1
p ∈ (0, 1), and by the definition of w(t) = ‖u(t)‖p

`pΘ
we obtain

d

dt
w(t) ≤ pCw(t) + p‖b(t)‖`pΘw(t)α. (4.10)

Since the inequality (4.7) holds if w(t) = 0, we only need to consider the case
w(t) > 0. Then Z(t) = w(t)1−α is differentiable and by the chain rule we obtain

d

dt
Z(t) = (1− α)w(t)−α

d

dt
w(t) ≤ (1− α)pCZ(t) + (1− α)p‖b(t)‖`pΘ .



12 G. ZHANG, G. ABURAMYAH EJDE-2024/12

Notice that (1 − α)p = 1 and Z(t) = ‖u(t)‖`pΘ and by using Grönwall’s inequality
we obtain

‖u(t)‖`pΘ ≤ e
Ct
(
‖u0‖`pΘ +B(C, t)

)
.

Now we assume that u0 ∈ `pΘ(Zd) and b is a continuous function with values in
`pΘ(Zd). Let u be a solution of problem (1.1)-(1.2) with maximal interval of existence
[0, τmax). Choose u0,k ∈ D(A) and bk ∈ C1([0, τmax), `pΘ(Zd)) such that u0,k → u0

in `pΘ(Zd) and bk → b uniformly on compact intervals. Let uk be a solution of
problem (1.1)- (1.2). By Theorem 3.2, for any T ∈ (0, τmax) the solution uk is
defined on [0, T ] for all k large enough and uk → u uniformly on [0, T ]. Applying
inequality (4.7) to uk and passing to the limit, we extend these inequalities to the
general case. �

Now, we are ready to prove the following theorem regarding the existence of
global weighted `p solution to the initial value problem(1.1) and (1.2) for 1 < p <∞.

Theorem 4.7. (1) Assume that assumptions (i)–(iv) are satisfied and 1 < p <∞,
if b ∈ C([0,∞), `pΘ(Zd)), then for every u0 ∈ `pΘ(Zd), problem (1.1) and (1.2) has
a unique global mild solution u ∈ C([0,∞), `pΘ(Zd)) which continuously depends on
u0 in the topology of uniform convergence on bounded closed subintervals of [0,∞).
Moreover, for any t ≥ 0

‖u(t)‖`pΘ ≤ (‖u0‖`pΘ +B(δ + ρΘ(p), t))e(δ+ρΘ(p))t , (4.11)

where ρΘ(p) ≤ 2dc0 defined in (4.5) and

B(δ + ρΘ(p), t) =

∫ t

0

e−(δ+ρΘ(p))s‖b(s)‖`pΘds,

δ and c0 are constants in assumption (i) and (iv) respectively.
(2) The global mild solution u(t) ∈ C([0,∞), `pΘ(Zd)) of problem (1.1) and (1.2)

obtained in (1) is a classical solution if one of the following conditions holds

(a) u0 ∈ `pΘ(Zd) and W is bounded;
(b) u0 ∈ D(A) = DΘ and b : [0,∞)→ `pΘ(Zd) is locally Lipschitz continuous.

Proof of Theorem 4.7. (1) From Theorem 3.2 we know u is the unique mild solution
to initial value problem(1.1) and (1.2) on [0, τmax). By Lemma 4.6, for all 0 ≤ t <
τmax we have

‖u(t)‖`pΘ ≤ (‖u0‖`pΘ +B(δ + ρΘ(p), t))e(δ+ρΘ(p))t.

If τmax <∞, the limit of the right-hand side of the inequality is finite as t→ τmax

since all functions of t involved are continuous on [0,∞) which implies

lim
t→τmax

‖u(t)‖`pΘ ≤ (‖u0‖`pΘ +B(δ + ρΘ(p), τmax))e(δ+ρΘ(p))τmax <∞,

which contradicts with Theorem 3.2 (2). Therefore we must have τmax = ∞ and
the mild solution u(t) is global.

(2) The proof follows from a similar reasoning in [20]. �

Remark 4.8. (1) Theorem 4.7 holds for all 1 < p < ∞, which represents an
improvement over Theorem 4.1 and 4.2. However, its proof is not valid for the case
p = 1, which was covered in Theorem 4.1.

(2) There are no results concerning global solutions in `∞Θ .



EJDE-2024/12 GLOBAL ATTRACTOR AND `p SOLUTIONS TO DNLS 13

(3) When p = 2, the constant ρΘ(p) is similar to ρΘ defined in [20]. We can
easily prove that ρΘ0(p) = 0 if and only if p = 2. Therefore, with a slight different
definition of the weighted lp space, our results generalize the theorems in [20].

If we set f = b = 0 in equation (1.1), we obtain the following corollary, which is
an enhanced version of Lemma 3.1 for the case 1 < p <∞.

Corollary 4.9. Let W satisfy assumption (i) and Θ satisfy assumption (iv) and
H = −∆ + W , then the operator −iH generates a strongly continuous semigroup
e−itH in the space `pΘ(Zd) for 1 ≤ p < ∞. Moreover if 1 < p < ∞, then for each
u0 ∈ `pΘ(Zd) we have the following estimate for all t ≥ 0

‖e−itHu0‖`pΘ ≤ e
(δ+ρΘ(p))t‖u0‖`pΘ , (4.12)

where ρΘ(p) ≤ 2dc0 defined in (4.5) and δ and c0 are constants in assumption (i)
and (iv) respectively.

5. Global attractor for 1 < p <∞

Now, we investigate the long-term behavior of global solutions to the initial value
problem for the autonomous DNLS equation. In equation (1.1), we introduce the
additional assumption that

(v) The forcing term b(t, n) ≡ b(n) is independent of t.

Let us denote by S(t) = SΘ(t) the solution operator to

`pΘ(Zd) S(t)→ `pΘ(Zd)
u0 7→ S(t)u0 = u(t)

, (5.1)

where t → u(t) represents the solution to the initial value problem (1.1), (1.2). It
is important to note that under assumptions (i)–(v), Theorem 4.2 is applicable,
and u(t) is a globally defined classical solution. In this scenario, the path t→ S(t)
constitutes a nonlinear semigroup of operators as described in [21], operating within
the space `pΘ(Zd).

Theorem 5.1. Assume that assumptions (i)–(v) hold, and that 1 < p < ∞. If
δ + ρΘ(p) < 0 and b ∈ `pΘ(Zd), then the semigroup of operators S(t) in (5.1)
has a compact, and connected, global attractor set AΘ that is maximal among all
functional invariant sets in `pΘ(Zd).

For the proof of the existence of this global attractor in `pΘ(Zd) we follow the
strategy outlined in [20], which is in turn quite similar to that in [23, Theorem
1.1.1] (see also [4, Chapter 2, Theorem 3.1]). We proceed in steps, starting with
the following Lemma on the existence of an absorbing set for S(t).

Lemma 5.2. Assume that assumptions (i)–(v) hold, and that 1 < p < ∞. If
δ + ρΘ(p) < 0 and b ∈ lpΘ, then the solution operator S(t) in (5.1) possesses a
bounded absorbing set B0 in `pΘ(Zd), that is to say, for every bounded set B ⊂
`pΘ(Zd) there exists t0(B) such that S(t)B ⊂ B0 for all t ≥ t0(B).

Proof. If u0 ∈ `pΘ, then u(t) ∈ `pΘ satisfies (4.2). For convenience, we set κ =

−(δ + ρΘ(p)), which is positive by hypothesis. Since (v) holds, we obtain that

‖u(t)‖`pΘ ≤ ‖u
0‖`pΘe

−κt + (1− e−κt)
‖b‖`pΘ
κ
→
‖b‖`pΘ
k

, (5.2)



14 G. ZHANG, G. ABURAMYAH EJDE-2024/12

and so

lim
t→∞

sup ‖u(t)‖`pΘ ≤
‖b‖`pΘ
κ

:= r ,

where we use the expression on the right side to define r.
Let B be any bounded subset in `pΘ(Zd). Then, B is contained within a ball with

its center at the origin of radius 2r+ l. To demonstrate its absorption in finite time,
we can assume, without loss of generality, that B is the ball of radius R ≥ 2r + 1
centered at the origin.

We choose R0 in the interval (r, 2r + 1) and let B0 be the ball of radius R0

centered at (0, 0). Then, for all u0 ∈ B, the solution u(t) = S(t)u0 satisfies

‖u(t)‖`pΘ ≤ ‖u
0‖`pΘe

−κt +
1− e−κt

κ
‖b‖`pΘ ≤ (‖u0‖`pΘ − r)e

−κt + r ≤ (R− r)e−κt + r .

It follows that if

t0 = t0(B) =
1

κ
ln

R− r
R0 − r

,

for each t ≥ t0 we have that ‖u(t)‖`pΘ ≤ R0 and so u(t) ⊂ B0. Thus, B0 is an
absorbing set. �

As in [20], we introduce the following piecewise line Lipschitz continuous function

ϕ(r) =


0 if 0 ≤ r ≤ 1,

r − 1 if 1 ≤ r < 2,

1 if r ≥ 2,

with Lipschitz constant 1. When R > 0, we use the notation φ(n) = ϕ(|n|/R) for
all n ∈ Zd. We suppress the explicit dependency of φ on R in the notation for
convenience. It is worth noting that for any n ∈ Zd, we have

0 ≤ φ(n) ≤ 1, |φ(n± ej)− φ(n)| ≤ 1

R
, 1 ≤ j ≤ d .

We continue our study of the semigroup S(t) by establishing the following lemma,
which provides a tail estimate.

Lemma 5.3. Assume that assumptions (i)–(v) hold, and that 1 < p < ∞. If
δ + ρΘ(p) < 0, b ∈ `pΘ, and B is a bounded subset of `pΘ(Zd), then for any ε > 0,
there exist positive constants R0 = R0(ε), and T = T (ε) such that for all u0 ∈ B
the solution u(t) = S(t)u0 of (1.1)-(1.2) is such that∑

|n|>2R

|u(t, n)|pθpn ≤
( ε

|δ + ρΘ(p)|

)p
, (5.3)

for all t ≥ T and R ≥ R0.

Proof. By Theorem 4.2, if u0 ∈ u0 ∈ DΘ∩B, then u(t) = S(t)u0 is a global classical
solution of the initial value problem (1.1)-(1.2). With the help of Lemma 4.4, we
can then obtain

d

dt
‖φu(t)‖p

`pΘ
=

d

dt

∑
n∈Zd

|u(t, n)φ(n)θn|p

=
∑
n∈Zd

d

dt
|u(t, n)φ(n)|pθpn



EJDE-2024/12 GLOBAL ATTRACTOR AND `p SOLUTIONS TO DNLS 15

= Re
( ∑
n∈Zd

pθpn|u(t, n)φ(n)|p−2φ(n)2u(t, n)
d

dt
u(t, n)

)
,

and if we multiply both sides of (1.1) by pθpnφ(n)p|u(t, n)|p−2u(t, n), and then use
assumptions (i) and (iii) to find the real parts of both sides both sides of the
resulting expression, we obtain that

d

dt
‖φu(t)‖p

`pΘ

≤ pδ
∑
n∈Zd

|(φu)(n)|pθpn + p
∑
n∈Zd

Im[−∆u(n)u(n)]|u(n)|p−2θpnφ(n)p

+ p
∑
n∈Zd

|(φu)(n)|p−2 Im(φu(n)b(n)φ(n))θpn

≤ pδ‖φu‖p
`pΘ

+ p‖φu‖p−1
`pΘ
‖φb‖`pΘ + p

∑
n∈Zd

Im[−∆u(n)u(n)]|u(n)|p−2θpnφ(n)p.

By the Lipschitz property of φ, condition (iv), Lemma 4.5, and Hölder’s inequality,
we obtain that

p
∑
n∈Zd

Im[−∆u(n)u(n)]|u(n)|p−2θpnφ(n)p

= −p
d∑
i=1

∑
n∈Zd

|(θφu)(n)|p−2 Im
[
(u(n+ ej)φ(n)θn + u(n− ej)φ(n)θn)θφu(n)

]
= −p

d∑
i=1

∑
n∈Zd

|(θφu)(n)|p−2 Im
[
u(n+ ej)φ(n+ ej)θnθφu(n)

]
− p

d∑
i=1

∑
n∈Zd

|(θφu)(n)|p−2 Im
[
u(n− ej)φ(n− ej)θnθφu(n)

]
+ p

d∑
i=1

∑
n∈Zd

|(θφu)(n)|p−2 Im
[
u(n+ ej)(φ(n+ ej)− φ(n))θnθφu(n)

]
+ p

d∑
i=1

∑
n∈Zd

|(θφu)(n)|p−2 Im
[
u(n− ej)(φ(n− ej)− φ(n))θnθφu(n)

]
= p

∑
n∈Zd

Im[−∆(φu)(n)φu(n)]|(φu)(n)|p−2θpn

+ p

d∑
i=1

∑
n∈Zd

|(θφu)(n)|p−2 Im
[
u(n+ ej)(φ(n+ ej)− φ(n))θnθφu(n)

]
+ p

d∑
i=1

∑
n∈Zd

|(θφu)(n)|p−2 Im
[
u(n− ej)(φ(n− ej)− φ(n))θnθφu(n)

]
≤ pρΘ(p)‖φu(t)‖p

`pΘ
+ p(2d/R)c0‖u(t)‖`pΘ‖φu(t)‖p−1

`pΘ
,

which implies that

d

dt
‖φu(t)‖p

`pΘ
≤ p(δ + ρΘ(p))‖φu‖p

`pΘ
+ p(2d/R)c0‖u(t)‖`pΘ‖φu(t)‖p−1

`pΘ
.



16 G. ZHANG, G. ABURAMYAH EJDE-2024/12

Since κ = δ + ρΘ(p) < 0, by (5.2) we obtain

sup
t≥0
‖u(t)‖`pΘ ≤ max{‖u0‖`pΘ ,

‖b‖`pΘ
|δ + ρΘ(p)|

} := M . (5.4)

Therefore,

d

dt
‖φu(t)‖p

`pΘ
≤ p(δ + ρΘ(p))‖φu‖p

`pΘ
+

2pdc0M

R
‖φu‖p−1

`pΘ
+ p‖φu‖p−1

`pΘ
‖φb‖`pΘ

= p(δ + ρΘ(p))‖φu(t)‖p
`pΘ

+ p
(2dc0M

R
+ ‖φb‖`pΘ

)
‖φu‖p−1

`pΘ
.

By Grönwall’s inequality, we conclude that

‖φu(t)‖`pΘ ≤ ‖φu
0‖`pΘe

(δ+ρΘ(p))t +
‖φb‖`pΘ + 2dc0M

R

|δ + ρΘ(p)|
(1− e(δ+ρΘ(p))t) . (5.5)

If we now consider an initial condition such that u0 ∈ B, we take any sequence
u0
k ∈ DΘ∩B such that u0

k → u0 in `pΘ(Zd). Then, as functions with values in `pΘ(Zd),
the solutions uk(t) = S(t)u0

k converge to u(t) = S(t)u0 uniformly on bounded
intervals, and applying inequality (5.5) to uk(t) and passing to the limit, we can
see (5.5) holds also if u(t) is the solution corresponding to the initial condition u0

also.
Let us set

r = sup {‖u0‖`pΘ : u0 ∈ B} .
Given any ε > 0, we define

T (ε) = max
{

0,
ln 2r|δ+ρΘ(p)|

ε

|δ + ρΘ(p)|
}
.

Then for all t > T (ε), we readily obtain that

‖φu0‖`pΘe
(δ+ρΘ(p))t <

ε

2|δ + ρΘ(p)|
.

On the other hand, since φ(n) is supported on n > R, we see easily that

lim
R→∞

(
‖φb‖`pΘ +

2dc0M

R

)
= lim
R→∞

‖φb‖`pΘ = 0 ,

and so each ε > 0, there exists R0 > 0 such that, for any R > R0 and t ≥ 0, we
have (

‖φb‖`pΘ +
2dc0M

R

) (1− e(δ+ρΘ(p))t)

|δ + ρΘ(p)|
≤
‖φb‖`pΘ + 2dc0M

R

|δ + ρΘ(p)|
<

ε

2|δ + ρΘ(p)|
,

and the estimate (5.3) follows. �

Lemma 5.4. Let Θ = {θn} be a sequence of positive terms, and p and q conjugate
numbers such that 1 ≤ p <∞, 1 < q ≤ ∞. Then

(`pΘ(Zd))∗ = `qΘ(Zd) ,

and if 1 < p <∞ then `pΘ(Zd) is a reflexive Banach space, that is to say,

(`pΘ(Zd))∗∗ = `pΘ(Zd) .



EJDE-2024/12 GLOBAL ATTRACTOR AND `p SOLUTIONS TO DNLS 17

Proof. The identification (`p(Zd))∗ = `q(Zd) is well known. Let v ∈ `qΘ(Zd). By the
definition of `qΘ(Zd), the multiplier operator Θ : `qΘ(Zd)→ `q(Zd) is an isomorphism.
Therefore, Θv ∈ `q(Zd), and this element induces the linear functional

`pΘ(Zd) 3 u→ (u, Θv) =
∑
n∈Zd

un(θnvn) ,

associated with the quadratic form

`pΘ(Zd)× `qΘ(Zd) 3 u, v → (u, v)Θ = (Θu, Θv) =
∑
n∈Zd

(θnun)(θnvn) .

This proves that the dual of `pΘ(Zd) is `qΘ(Zd), which addresses the first part of the
statement. The second part, regarding the double dual, follows by applying the
first part to the dual of `qΘ(Zd). �

Since bounded sets in a reflexive Banach space are precompact in the weak*
topology, we now obtain the following.

Lemma 5.5. Assume that assumptions (i)–(v) hold, and that 1 < p < ∞. If
δ + ρΘ(p) < 0 and b ∈ `pΘ(Zd), then the semigroup S(t) is asymptotically compact
in `pΘ(Zd), that is, if the sequence ψk is bounded in `pΘ(Zd), and tk →∞ as k →∞,
then S(tk)ψk is precompact in `pΘ(Zd).

Proof. Let ψk ∈ `pΘ(Zd) be a bounded sequence. According to Lemma 5.2, there
exist an absorbing set B0 and a T > 0 such that S(t)ψk ⊂ B0 for all t ≥ T .
Therefore, if tk is a sequence converging to infinity, the sequence wk = S(tk)ψk has
only finitely many terms outside of B0 and is bounded in the reflexive Banach space
`pΘ(Zd) (as shown in Lemma 5.4). Consequently, we can pass to a subsequence, if
necessary, and assume that for some w0, wk → w0 in the weak* topology of `pΘ(Zd).

For any R > 0, we consider the characteristic function of the R ball in Zd
centered at the origin,

χR(n) =

{
1 if n ∈ Zd, |n| ≤ R,
0 otherwise.

According to Lemma 5.3, for any ε > 0 there exist constants R = R(ε) and N =
N(ε) such that

‖(1− χR)wk‖`pΘ ≤ ε,
for all k ≥ N , and enlarging R if necessary, we also have that

‖(1− χR)w0‖`pΘ ≤ ε .

Therefore,

‖wk − w0‖`pΘ ≤ ‖χR(wk − w0)‖`pΘ + ‖(1− χR)(wk − w0)‖`pΘ
≤ ‖χR(wk − w0)‖`pΘ + ‖(1− χR)wk‖`pΘ + ‖(1− χR)w0‖`pΘ
≤ ‖χR(wk − w0)‖`pΘ + 2ε,

for k ≥ N .
Since all functions χR(wk−w0) belong to a finite-dimensional subspace of `pΘ(Zd),

and on a finite-dimensional subspace of a reflexive Banach space, the weak* con-
vergence implies the strong convergence. Therefore, we can conclude that χR(wk−



18 G. ZHANG, G. ABURAMYAH EJDE-2024/12

w0)→ 0 strongly in `pΘ(Zd). Consequently, we can find N such that

‖χR(wk − w0)‖`pΘ < ε ,

which, when used in the previous estimate, yields:

‖wk − w0‖`pΘ ≤ 3ε

for any k ≥ max{N,N}. Since ε is an arbitrary positive number, this proves that
wk → w0 strongly in `pΘ(Zd). �

Proof of Theorem 5.1. Under the stated hypothesis, the initial value problem has
a global classical solution (1.1),(1.2) in `pΘ(Zd). We let B0 = B0,Θ be the absorbing
set produced by Lemma 5.2, and set AΘ to be its ω-limit,

AΘ := ω(B0,Θ) = ∩s≥0∪t≥sS(t)B0,Θ,

where S(t) is the semigroup of operators in `pΘ(Zd) in (5.1), and the closure is
taken with respect to the `pΘ-norm. According to [23, Theorem 1.1.1] (see also [4,
Chapter 2, Theorem 3.1]), this set is compact and connected, and by definition, it
is maximal among all invariant sets in `pΘ(Zd). �

When Θ = Θ0, we have `pΘ0
(Zd) = `p(Zd). Theorem 5.1 then demonstrates the

existence of a global attractor in `p(Zd) space.

Acknowledgments. The authors want to thank the reviewers and editors for their
valuable comments.

References

[1] P. W. Bates, K. Liu, B. Wang; Attractors for lattice dynamical systems, J. Bifur. Chaos

Appl. Sci. Eng., 11 (2001), 143–153.
[2] T. Cazenave, A. Haraux; An Introduction to Semilinear Evolution Equations, Translation

Oxford University Press, 1998

[3] M. Cheng, A. Pankov; Gap solitons in periodic nonlinear Schrödinger equations with nonlinear
hopping, Electr. J. Differential Equat., 2016 (2016), no. 287, 1–14.

[4] V. V. Chepyzhov, M. I. Vishik; Attractors for Equations of Mathematical Physics, Collo-

quium Publication, vol. 49, American Math Soc., Providence, RI, 2002.
[5] K.-J. Engel, R. Nagel; A Short Course on Operator Semigroups, Springer, New York, 2006.

[6] S. Flach, A. V. Gorbach; Discrete breathers–advances in theory and applications, Phys. Repts,

467 (2008), 1–116.
[7] S. Flach, C. R. Willis; Discrete breathers, Phys. Repts, 295 (1998), 181–264.

[8] D. Hennig, G. P. Tsironis; Wave transmission in nonlinear lattices, Physics Repts, 309 (1999),
333–432.

[9] N. I. Karachalios, A. N. Yannacopoulos; Global existence and compact attractors for the
discrete nonlinear Schrödinger equations, J. Differential Equat., 217 (2005), 88–123.

[10] N. I. Karachalios, A. N. Yannacopoulos; The existence of global attractor for the discrete
nonlinear Schrödinger equation II. Compactness without tail estimates in ZN , N ≥ 1, lattices,

Proc. Roy. Soc. Edinburgh, 137A (2007), 63–76.
[11] P. G. Kevrekidis (ed.); The Discrete Nonlinear Schrödinger Equation: Mathematical Analy-

sis, Numerical Computations and Physical Perspectives, Springer, Berlin, 2009.
[12] P. G. Kevrekidis, K. O. Rasmussen, A. R. Bishop; The discrete nonlinear Schrödinger equa-

tion: a survey of recent results, Intern. J. Modern. Phys. B, 15 (2001), 2833–2900.
[13] G. N’Guérékata, A. Pankov; Global well-posedness for discrete nonlinear Schrödinger equa-

tion, Applicable Anal., 89 (2010), 1513–1521.
[14] P. Pacciani, V. V. Konotop, G. Perla Menzala; On localized solutions of discrete nonlinear

Schrödinger equation: an exact result, Physica D, 204 (2005), 122–133.
[15] A. Pankov; Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity,

19 (2006), 27–40.



EJDE-2024/12 GLOBAL ATTRACTOR AND `p SOLUTIONS TO DNLS 19

[16] A. Pankov; Gap solitons in periodic discrete nonlinear Schrödinger equations, II: a generalized

Nehari manifold approach, Discr. Cont. Dyn. Syst. A, 19 (2007), 419–430.

[17] A. Pankov; Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable
nonlinearities, J. Math. Anal. Appl., 371 (2010), 254–265.

[18] A. Pankov, V. Rothos; Periodic and decaying solutions in discrete nonliinear Schrödinger

equations with saturable nonlinearity, Proc. Roy. Soc. A, 464 (2008), 3219–3236.
[19] A. Pankov, G. Zhang; Standing wave solutions for discrete nonlinear Schrödinger equations

with unbounded potentials and saturable nonlinearities, J. Math. Sci., 177 (2011), 71–82.

[20] A. Pankov, G. Zhang; Initial value problem of the discrete nonlinear Schrödinger equation
with complex potential, Applicable Analysis, Volume 101, Issue 16 (2022), pp. 5760-5774.

[21] A. Pazy; Semigroups of Linear Operators and Applications, Springer, New York, 1983.

[22] H. Sedaghat; Real Analysis and Infinity, Oxford University Press, 2022.
[23] R. Temam; Infinite-Dimensional Dynamical Systems in Mathematics and Physics, Springer,

New York, 1997.
[24] M. I. Weinstein; Excitation threshold for nonlinear localized modes on lattices, Nonlinearity,

19 (1999), 673–691.

[25] G. Zhang; Breather solutions of the discrete nonlinear Schrödinger equation with unbounded
potential, J. Math.Phys., 50 (2009), 013505.

[26] G. Zhang; Breather solutions of the discrete nonlinear Schrödinger equation with sign chang-

ing nonlinearity, J. Math.Phys., 52 (2011), 043516.
[27] G. Zhang, G. Aburamyah; lp Solution to the initial value problem of the discrete nonlinear

Schrödinger equation with complex potential, Nonlinear and Modern Mathematical Physics:

NMMP-2022, Springer Proceedings in Mathematics and Statistics, in press 2023.
[28] G. Zhang, G. Aburamyah; `p solution to the initial value problem of the discrete nonlinear

Schrödinger equation with complex potential, II, Journal of Nonlinear Evolution Equations

and Applications (JNEEA), in press 2024
[29] G. Zhang, F. Liu; Existence of breather solutions of the DNLS equation with unbounded

potential, Nonlin. Anal., 71 (2009), e786–e792.
[30] G. Zhang, A. Pankov; Standing waves of the discrete nonlinear Schrödinger equations with

growing potentials, Commun. Math. Analysis, 5(2)(2008), 38–49.

[31] G. Zhang, A. Pankov; Standing wave solutions for the discrete nonlinear Schrödinger equa-
tions with unbounded potentials, II, Applicable Anal., 89 (2011), 1541–1557.

[32] S. Zhou; Attractors for second order lattice dynamical systems, J. Differential Equat., 179

(2002), 605–624.
[33] S. Zhou; Attractors for first order dissipative lattices, Physica D, 178 (2003), 51–61.

[34] S. Zhou; Attractors and approximations for lattice dynamical systems, J. Differential Equat.,

200 (2004), 342–368.
[35] S. Zhou, W. Shi; Attractors and dimension for dissipative lattice systems, J. Differential

Equat., 224 (2006), 172–204.

Guoping Zhang
Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA

Email address: guouping.zhang@morgan.edu

Ghder Aburamyah
Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA

Email address: ghabu2@morgan.edu


	1. Introduction
	2. Semigroup theory and abstract initial value problem
	3. Assumptions and review
	4. Global solutions for 1<p<
	5. Global attractor for 1<p<
	Acknowledgments

	References