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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.
) absorbing set 
,
) 
-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)
the set 
is precompact in 
;
, 
such that
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].
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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
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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
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DOI: https://doi.org/10.1155/2009/916316