- Research Article
- Open access
- Published:
Uniform Attractor for the Partly Dissipative Nonautonomous Lattice Systems
Advances in Difference Equations volume 2009, Article number: 916316 (2009)
Abstract
The existence of uniform attractor in is proved for the partly dissipative nonautonomous lattice systems with a new class of external terms belonging to
, which are locally asymptotic smallness and translation bounded but not translation compact in
. It is also showed that the family of processes corresponding to nonautonomous lattice systems with external terms belonging to weak topological space possesses uniform attractor, which is identified with the original one. The upper semicontinuity of uniform attractor is also studied.
1. Introduction
This paper is concerned with the long-time behavior of the following non-autonomous lattice systems:


with initial conditions

where is the integer lattice;
is a nonlinear function satisfying
;
is a positive self-adjoint linear operator;
belong to certain metric space, which will be given in the following.
Lattice dynamical systems occur in a wide variety of applications, where the spatial structure has a discrete character, for example, chemical reaction theory, electrical engineering, material science, laser, cellular neural networks with applications to image processing and pattern recognition; see [1–4]. Thus, a great interest in the study of infinite lattice systems has been raising. Lattice differential equations can be considered as a spatial or temporal discrete analogue of corresponding partial differential equations on unbounded domains. It is well known that the long-time behavior of solutions of partial differential equations on unbounded domains raises some difficulty, such as well-posedness and lack of compactness of Sobolev embeddings for obtaining existence of global attractors. Authors in [5–7] consider the autonomous partial equations on unbounded domain in weighted spaces, using the decaying of weights at infinity to get the compactness of solution semigroup. In [8–10], asymptotic compactness of the solutions is used to obtain existence of global compact attractors for autonomous system on unbounded domain. Authors in [11] consider them in locally uniform space. For non-autonomous partial differential equations on bounded domain, many studies on the existence of uniform attractor have been done, for example [12–14].
For lattice dynamical systems, standard theory of ordinary differential equations can be applied to get the well-posedness of it. "Tail ends" estimate method is usually used to get asymptotic compactness of autonomous infinite-dimensional lattice, and by this the existence of global compact attractor is obtained; see [15–17]. Authors in [18, 19] also prove that the uniform smallness of solutions of autonomous infinite lattice systems for large space and time variables is sufficient and necessary conditions for asymptotic compactness of it. Recently, "tail ends" method is extended to non-autonomous infinite lattice systems; see [20–22]. The traveling wave solutions of lattice differential equations are studied in [23–25]. In [18, 26, 27], the existence of global attractors of autonomous infinite lattice systems is obtained in weighted spaces, which do not exclude traveling wave.
In this paper, we investigate the existence of uniform attractor for non-autonomous lattice systems (1.1)–(1.3). The external term in [20] is supposed to belong to and to be almost periodic function. By Bochner-Amerio criterion, the set of this external term's translation is precompact in
. Based on ideas of [28], authors in [14] introduce uniformly
-limit compactness, and prove that the family of weakly continuous processes with respect to (w.r.t.) certain symbol space possesses compact uniform attractors if the process has a bounded uniform absorbing set and is uniformly
-limit compact. Motivated by this, we will prove that the process corresponding to problem (1.1)–(1.3) with external terms being locally asymptotic smallness (see Definition 4.5) possesses a compact uniform attractor in
, which coincides with uniform attractor of the family of processes with external terms belonging to weak closure of translation set of locally asymptotic smallness function in
. We also show that locally asymptotic functions are translation bounded in
, but not translation compact (tr.c.) in
. Since the locally asymptotic smallness functions are not necessary to be translation compact in
, compared with [20], the conditions on external terms of (1.1)–(1.3) can be relaxed in this paper.
This paper is organized as follows. In Section 2, we give some preliminaries and present our main result. In Section 3, the existence of a family of processes for (1.1)–(1.3) is obtained. We also show that the family of processes possesses a uniformly (w.r.t ) absorbing set. In Section 4, we prove the existence of uniform attractor. In Section 5, the upper semicontinuity of uniform attractor will be studied.
2. Main Result
In this section, we describe our main result. Denote by the Hilbert space defined by

with the inner product and norm
given by

For , we endow with the inner and norm as. For

Denote by the space of function
with values in
that locally 2-power integrable in the Bochner sense, that is,

It is equipped with the local 2-power mean convergence topology. Then, is a metrizable space. Let
be a space of functions
from
such that

Denote by the space
endow with the local weak convergence topology.
For each sequence define linear operators on
by

Then

For convenience, initial value problem (1.1)–(1.3) can be written as


with initial conditions

where ,
.
In the following, we give some assumption on nonlinear function , and
:

There exists a positive-value continuous function such that

There exist positive constants such that

Let the external term belong to
, it follows from the standard theory of ordinary differential equations that there exists a unique local solution
for problem (2.8)–(2.10) if (H
1
)–(H
3
) hold. For a fixed external term
, take the symbol space
, the set contains all translations of
in
. Take the
the closure of
in
. Denote by
the translation semigroup,
for all
or
,
,
. It is evident that
is continuous on
in the topology of
and on
in the topology of
, respectively,

In Section 3, we will show that for every and
,
, problem (2.8)–(2.10) has a unique global solution
Thus, there exists a family of processes
from
to
. In order to obtain the uniform attractor of the family of processes, we suppose the external term is locally asymptotic smallness (see Definition 4.5). Let
be a Banach space which the processes acting in, for a given symbol space
, the uniform (w.r.t.
)
-limit set
of
is defined by

The first result of this paper is stated in the following, which will be proved in Section 4.
Theorem 2 A.
Assume that be locally asymptotic smallness and
hold. Then the process
corresponding to problems (2.8)–(2.10) with external term
possesses compact uniform
attractor
in
which coincides with uniform (w.r.t.
attractor
for the family of processes
,
, that is,

where is the uniform
absorbing set in
, and
is kernel of the process
. The uniform attractor uniformly
attracts the bounded set in
.
We also consider finite-dimensional approximation to the infinite-dimensional systems (1.2)-(1.3) on finite lattices. For every positive integer , let
, consider the following ordinary equations with initial data in
:

In Section 5, we will show that the finite-dimensional approximation systems possess a uniform attractor in
, and these uniform attractors are upper semicontinuous when
. More precisely, we have the following theorem.
Theorem 2 B.
Assume that and
hold. Then for every positive integer
, systems (2.17) possess compact uniform attractor
. Further,
is upper semicontinuous to
as
, that is,

where

3. Processes and Uniform Absorbing Set
In this section, we show that the process can be defined and there exists a bounded uniform absorbing set for the family of processes.
Lemma 3.1.
Assume that and
hold. Let
, and
. Then the solution of (2.8)–(2.10) satisfies

where ,
.
Proof.
Taking the inner product of (2.8) with in
, by
we get

Similarly, taking the inner product of (2.9) with in
, we get

Note that

Summing up (3.2) and (3.3), from (3.4), we get

Thus, by ,

Since , from [12, Proposition V.4.2.], we have

From (3.6)-(3.7), applying Gronwall's inequality of generalization (see [12, Lemma II.1.3]), we get (3.1). The proof is completed.
It follows from Lemma 3.1 that the solution of problem (2.8)–(2.10) is defined for all
. Therefore, there exists a family processes acting in the space
,
,
,
, where
is the solution of (2.8)–(2.10), and the time symbol
belongs to
and
, respectively. The family of processes
satisfies multiplicative properties:

Furthermore, the following translation identity holds:

The kernel of the processes
consists of all bounded complete trajectories of the process
, that is,

denotes the kernel section at a times moment
:

Lemma 3.1 also shows that the family of processes possesses a uniform absorbing set in .
Lemma 3.2.
Assume that and
hold. Let
. Then, there exists a bounded uniform absorbing set
in
for the family of processes
, that is, for any bounded set
, there exists
,

Proof.
Let , from (3.1) we have

where

Let . The proof is completed.
4. Uniform Attractor
In this section, we establish the existence of uniform attractor for the non-autonomous lattice systems (2.8)–(2.10). Let be a Banach space, and let
be a subset of some Banach space.
Definition 4.1.
is said to be  
weakly continuous, if for any
, the mapping
  is weakly continuous from
to
.
A family of processes ,
is said to be uniformly
-limit compact if for any
and bounded set
, the set
is bounded for every
and
is precompact set as
. We need the following result in [14].
Theorem 4.2.
Let be the weak closure of
. Assume that
,
is
weakly continuous, and
-
(i)
has a bounded uniformly (w.r.t.
) absorbing set
,
-
(ii)
is uniformly (w.r.t.
)
-limit compact.
Then the families of processes ,
,
possess, respectively, compact uniform (w.r.t.
,
, resp.) attractors
and
satisfying

Furthermore, is nonempty for all
.
Let be a Banach space and
, denote the space
of functions
,
with values in
that are locally
-power integral in the Bochner sense, it is equipped with the local
-power mean convergence topology. Recall the Propositions in [12].
Proposition 4.3.
A set is precompact in
if and only if the set
is precompact in
for every segment
. Here,
denotes the restriction of the set
to the segment
.
Proposition 4.4.
A function is tr.c. in
if and only if
-
(i)
for any
the set
is precompact in
;
-
(ii)
there exists a function
,
such that
(42)
Now, one introduces a class of function.
Definition 4.5.
A function is said to be locally asymptotic smallness if for any
, there exists positive integer
such that

Denote by the set of all locally asymptotic smallness functions in
. It is easy to see that
. The next examples show that there exist functions in
but not in
, and a function belongs to
is not necessary a tr.c. function in
.
Example 4.6.
Let ,

For every ,
,

Thus,

and . However, for every positive integer
, and for any positive
,

Therefore, .
Example 4.7.
,

for ,

Here, denote the positive integer set.
For every positive integer ,
, and for
,

which implies that

Therefore, . Note that for any
,

From Proposition 4.4, is not translation compact in
.
Remark 4.8.
Example 4.7 shows that a locally asymptotic function is not necessary translation compact in .
In the following, we give some properties of locally asymptotic smallness function.
Lemma 4.9.
is a closed subspace of
.
Proof.
Let such that

Then, for any , there exists positive integer
such that for every
,

Since , there exist
such that for all
,

Let , we get that

Therefore, . This completes the proof.
Lemma 4.10.
Every translation compact function in
is locally asymptotic smallness.
Proof.
Since is tr.c. in
, we get that
is precompact in
. By Proposition 4.3, we get that
is precompact in
. Thus, for any
, there exists finite number
such that for every
, there exist some
,
, such that

For the given above,
implies that there exists positive integer
such that

Therefore,

which implies is locally asymptotic smallness. This completes the proof.
We now establish the uniform estimates on the tails of solutions of (2.8)–(2.10) as .
Lemma 4.11.
Assume that hold and
is locally asymptotic smallness. Then for any
, there exist positive integer
and
such that if
,
,
satisfies

Proof.
Choose a smooth function such that
for
, and

and there exists a constant such that
for
. Let
be a suitable large positive integer,
. Taking the inner product of (2.8) with
and (2.9) with
in
, we have

From , we have

where is same as in Lemma 3.1

as in (3.10)

Summing up (4.22), from (4.23)–(4.25) we get

Thus,

We now estimate the integral term on the right-hand side of (4.27).

Similarly,

Since is locally asymptotic smallness, from (4.27)–(4.29) we get that for any
, if
, there exist
and sufficient large positive integer
such that

The proof is completed.
Lemma 4.12.
Assume that hold, let
,
. If
in
and
weakly in
, then for any
,

Proof.
Let ,
. Since
is bounded in
, by Lemma 3.2, we get that

Therefore, for all ,
,

Note that is the solution of (2.8) and (2.9) with time symbol
, it follow from (4.32) that

In the following, we show that . By the fact that
is the solution of (2.8) and (2.9), for any
, we get that

Note that weakly in
. Let
in (4.35), by (4.34) we get that
is the solution of (2.8) and (2.9) with the initial data
. By the unique solvability of problem (2.8)–(2.10), we get that
. This completes the proof.
Proof of Theorem A.
From Lemmas 3.2, 4.11 and 4.12, and Theorem 4.2, we get the results.
5. Upper Semicontinuity of Attractors
In this section, we present the approximation to the uniform attractor obtained in Theory A by the uniform attractor of following finite-dimensional lattice systems in
:

with the initial data

and the periodic boundary conditions

Similar to systems (2.8)–(2.10), under the assumption , the approximation systems (5.1)–(5.2) with
possess a unique solution
, which continuously depends on initial data. Therefore, we can associate a family of processes
which satisfy similar properties (3.8)–(3.9). Similar to Lemma 3.2, we have the following result.
Lemma 5.1.
Assume that , and
hold. Let
. Then, there exists a bounded uniform absorbing set
for the family of processes
, that is, for any bounded set
, there exists
, for
,

In particular, is independent of
and
.
Since (5.1) is finite-dimensional systems, it is easy to know that under the assumption of Lemma 5.1, the family of processes ,
is uniformly (w.r.t.
)
-limit compact. Similar to Lemma 4.12, if
in
,
weakly in
, then for any
,
,

Lemma 5.2.
Assume that and
hold. Then the process
corresponding to problems (5.1)-(5.2) with external term
possesses compact uniform (w.r.t.
) attractor
in
which coincides with uniform (w.r.t.
attractor
for the family of processes
,
, that is,

where is the uniform
absorbing set in
, and
is kernel of the process
. The uniform attractor uniformly
attracts the bounded set in
.
Proof of Theorem B.
If , it follows from Lemma 5.2 that there exist
and a bounded complete solution
such that

Since , there exist
and a subsequence of
, which is still denote by
, such that

From Lemma 5.1, we get that

which imply that

Thus,

Let be a sequence of compact intervals of
such that
and
. From (5.9) and (5.11), using Ascoli's theorem, we get that for each
, there exists a subsequence of
(still denoted by
) and
such that

Proceeding as in the proof of Lemma 4.11, we get that the weak convergence is actually strong convergence, and therefore is precompact in
for each
. Then we infer that there exists a subsequence
of
and
such that
converges to
. Using Ascoli's theorem again, we get, by induction, that there is a subsequence
of
such that
converges to
in
, where
is an extension of
to
. Finally, taking a diagonal subsequence in the usual way, we find that there exist a subsequence
of
and
such that for any compact interval

From (5.9) we get that

Next, we show that is the solution of (2.8)–(2.10). It follows from (5.10) that

For fixed , let
. Since
is the solution of (5.1)–(5.2) with
, we have

Thus, for each , we have

Letting , by (5.8), (5.13), (5.15) and (5.17) we find that
satisfies

Since is arbitrary, we note that (5.18) are valid for all
. From (5.14) we find that
is a bounded complete solution of (2.8)–(2.10). Therefore,
. By (5.13) we get that

The proof is complete.
Remark 5.3.
All the result of this paper is valid for the systems in [20, 21].
References
Arima T, Fukuyo K, Idemitsu K, Inagaki Y: Molecular dynamic simulations of yttri-astabilized zirconia between 300 and 200 K. Journal of Molecular Liquids 2004, 113: 67-73. 10.1016/j.molliq.2004.02.038
Callan JP, Kim AM-T, Huang L, Mazur E: Ultrafast electron and lattice dynamics in semiconductors at high excited carrier densities. Chemical Physics 2000, 251: 167-179. 10.1016/S0301-0104(99)00301-8
Chow S-N, Mallet-Paret J, Van Vleck ES: Pattern formation and spatial chaos in spatially discrete evolution equations. Random & Computational Dynamics 1996,4(2-3):109-178.
Chua LO, Yang L: Cellular neural networks: theory. IEEE Transactions on Circuits and Systems 1988,35(10):1257-1272. 10.1109/31.7600
Babin AV, Vishik MI: Attractors of partial differential evolution equations in an unbounded domain. Proceedings of the Royal Society of Edinburgh. Section A 1990,116(3-4):221-243. 10.1017/S0308210500031498
Efendiev MA, Zelik SV: The attractor for a nonlinear reaction-diffusion system in an unbounded domain. Communications on Pure and Applied Mathematics 2001,54(6):625-688. 10.1002/cpa.1011
Zelik SV: Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity. Communications on Pure and Applied Mathematics 2003,56(5):584-637. 10.1002/cpa.10068
Rodriguez-Bernal A, Wang B:Attractors for partly dissipative reaction diffusion systems in
. Journal of Mathematical Analysis and Applications 2000,252(2):790-803. 10.1006/jmaa.2000.7122
Temam R: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences. Volume 68. Springer, New York, NY, USA; 1988:xvi+500.
Wang B: Attractors for reaction-diffusion equations in unbounded domains. Physica D 1999,128(1):41-52. 10.1016/S0167-2789(98)00304-2
Carvalho AN, Dlotko T: Partially dissipative systems in locally uniform space. Cadernos De Matematica 2001, 02: 291-307.
Chepyzhov VV, Vishik MI: Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications. Volume 49. American Mathematical Society, Providence, RI, USA; 2002:xii+363.
Chepyzhov VV, Vishik MI: Attractors of non-autonomous dynamical systems and their dimension. Journal de Mathématiques Pures et Appliquées 1994, 73: 279-333.
Lu S, Wu H, Zhong C: Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces. Discrete and Continuous Dynamical Systems. Series A 2005,13(3):701-719.
Bates PW, Lu K, Wang B: Attractors for lattice dynamical systems. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 2001,11(1):143-153. 10.1142/S0218127401002031
Li X-J, Zhong C:Attractors for partly dissipative lattice dynamic systems in
. Journal of Computational and Applied Mathematics 2005,177(1):159-174. 10.1016/j.cam.2004.09.014
Zhou S: Attractors for second order lattice dynamical systems. Journal of Differential Equations 2002,179(2):605-624. 10.1006/jdeq.2001.4032
Wang B: Dynamics of systems on infinite lattices. Journal of Differential Equations 2006,221(1):224-245. 10.1016/j.jde.2005.01.003
Zhou S, Shi W: Attractors and dimension of dissipative lattice systems. Journal of Differential Equations 2006,224(1):172-204. 10.1016/j.jde.2005.06.024
Wang B: Asymptotic behavior of non-autonomous lattice systems. Journal of Mathematical Analysis and Applications 2007,331(1):121-136. 10.1016/j.jmaa.2006.08.070
Zhao C, Zhou S: Compact kernel sections of long-wave–short-wave resonance equations on infinite lattices. Nonlinear Analysis: Theory, Methods & Applications 2008,68(3):652-670. 10.1016/j.na.2006.11.027
Zhou S, Zhao C, Liao X: Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure and Applied Analysis 2007,6(4):1087-1111.
Bates PW, Chen X, Chmaj AJJ: Traveling waves of bistable dynamics on a lattice. SIAM Journal on Mathematical Analysis 2003,35(2):520-546. 10.1137/S0036141000374002
Chow S-N, Mallet-Paret J, Shen W: Traveling waves in lattice dynamical systems. Journal of Differential Equations 1998,149(2):248-291. 10.1006/jdeq.1998.3478
Zinner B: Existence of traveling wavefront solutions for the discrete Nagumo equation. Journal of Differential Equations 1992,96(1):1-27. 10.1016/0022-0396(92)90142-A
Beyn W-J, Pilyugin SY: Attractors of reaction diffusion systems on infinite lattices. Journal of Dynamics and Differential Equations 2003,15(2-3):485-515.
Li X-J, Wang D: Attractors for partly dissipative lattice dynamic systems in weighted spaces. Journal of Mathematical Analysis and Applications 2007,325(1):141-156. 10.1016/j.jmaa.2006.01.054
Ma Q, Wang S, Zhong C: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana University Mathematics Journal 2002,51(6):1541-1559. 10.1512/iumj.2002.51.2255
Acknowledgments
The authors are extremely grateful to the anonymous reviewers for their suggestion, and with their help, the version has been improved. This research was supported by the NNSF of China Grant no. 10871059
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, X., Lv, H. Uniform Attractor for the Partly Dissipative Nonautonomous Lattice Systems. Adv Differ Equ 2009, 916316 (2009). https://doi.org/10.1155/2009/916316
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/916316