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.
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
The first result of this paper is stated in the following, which will be proved in Section 4.
Theorem 2 A.
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.
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.
4. Uniform Attractor
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 .
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 .
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 .
In the following, we give some properties of locally asymptotic smallness function.
The proof is completed.
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.
Proof of Theorem B.
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
- 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.2004.02.038View ArticleGoogle Scholar
- 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)00301-8View ArticleGoogle Scholar
- 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.MathSciNetMATHGoogle Scholar
- Chua LO, Yang L: Cellular neural networks: theory. IEEE Transactions on Circuits and Systems 1988,35(10):1257-1272. 10.1109/31.7600MathSciNetView ArticleMATHGoogle Scholar
- 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/S0308210500031498MathSciNetView ArticleMATHGoogle Scholar
- 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.1011MathSciNetView ArticleMATHGoogle Scholar
- 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.10068MathSciNetView ArticleMATHGoogle Scholar
- 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.2000.7122MathSciNetView ArticleMATHGoogle Scholar
- Temam R: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences. Volume 68. Springer, New York, NY, USA; 1988:xvi+500.View ArticleGoogle Scholar
- Wang B: Attractors for reaction-diffusion equations in unbounded domains. Physica D 1999,128(1):41-52. 10.1016/S0167-2789(98)00304-2MathSciNetView ArticleMATHGoogle Scholar
- Carvalho AN, Dlotko T: Partially dissipative systems in locally uniform space. Cadernos De Matematica 2001, 02: 291-307.Google Scholar
- 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.Google Scholar
- Chepyzhov VV, Vishik MI: Attractors of non-autonomous dynamical systems and their dimension. Journal de Mathématiques Pures et Appliquées 1994, 73: 279-333.MathSciNetMATHGoogle Scholar
- 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.MathSciNetView ArticleMATHGoogle Scholar
- 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/S0218127401002031MathSciNetView ArticleMATHGoogle Scholar
- 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.2004.09.014MathSciNetView ArticleMATHGoogle Scholar
- Zhou S: Attractors for second order lattice dynamical systems. Journal of Differential Equations 2002,179(2):605-624. 10.1006/jdeq.2001.4032MathSciNetView ArticleMATHGoogle Scholar
- Wang B: Dynamics of systems on infinite lattices. Journal of Differential Equations 2006,221(1):224-245. 10.1016/j.jde.2005.01.003MathSciNetView ArticleMATHGoogle Scholar
- Zhou S, Shi W: Attractors and dimension of dissipative lattice systems. Journal of Differential Equations 2006,224(1):172-204. 10.1016/j.jde.2005.06.024MathSciNetView ArticleMATHGoogle Scholar
- Wang B: Asymptotic behavior of non-autonomous lattice systems. Journal of Mathematical Analysis and Applications 2007,331(1):121-136. 10.1016/j.jmaa.2006.08.070MathSciNetView ArticleMATHGoogle Scholar
- 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.2006.11.027MathSciNetView ArticleMATHGoogle Scholar
- 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.MathSciNetView ArticleMATHGoogle Scholar
- 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/S0036141000374002MathSciNetView ArticleMATHGoogle Scholar
- 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.1998.3478MathSciNetView ArticleMATHGoogle Scholar
- 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-AMathSciNetView ArticleMATHGoogle Scholar
- 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.MathSciNetView ArticleMATHGoogle Scholar
- 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.2006.01.054MathSciNetView ArticleMATHGoogle Scholar
- 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.2002.51.2255MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.