Uniform Attractor for the Partly Dissipative Nonautonomous Lattice Systems
© X. Li and H. Lv. 2009
Received: 25 March 2009
Accepted: 17 June 2009
Published: 20 July 2009
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.
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  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  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 , authors in  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 , 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
Denote by the space endow with the local weak convergence topology.
where , .
In the following, we give some assumption on nonlinear function , and :
The first result of this paper is stated in the following, which will be proved in Section 4.
Theorem 2 A.
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.
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.
where , .
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.
Lemma 3.1 also shows that the family of processes possesses a uniform absorbing set in .
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.
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 .
has a bounded uniformly (w.r.t. ) absorbing set ,
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 .
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 .
for any the set is precompact in ;
- (ii)there exists a function , such that(42)
Now, one introduces a class of function.
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 .
For every , ,
Here, denote the positive integer set.
For every positive integer , , and for ,
From Proposition 4.4, is not translation compact in .
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.
is a closed subspace of .
Therefore, . This completes the proof.
Every translation compact function in is locally asymptotic smallness.
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 .
The proof is completed.
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
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.
In particular, is independent of and .
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.
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
Next, we show that is the solution of (2.8)–(2.10). It follows from (5.10) that
The proof is complete.
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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