Advances in Difference Equations
Volume 2009 (2009), Article ID 916316, 21 pages
doi:10.1155/2009/916316
Research Article

Uniform Attractor for the Partly Dissipative Nonautonomous Lattice Systems

Xiaojun Li and Haishen Lv

Department of Applied Mathematics, Hohai University, Nanjing, Jiangsu 210098, China

Received 25 March 2009; Accepted 17 June 2009

Academic Editor: Toka Diagana

Copyright © 2009 Xiaojun Li and Haishen Lv. 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.

Abstract

The existence of uniform attractor in 𝑙 2 × 𝑙 2 is proved for the partly dissipative nonautonomous lattice systems with a new class of external terms belonging to 𝐿 2 l o c ( 𝑅 , 𝑙 2 ) , which are locally asymptotic smallness and translation bounded but not translation compact in 𝐿 2 l o c ( 𝑅 , 𝑙 2 ) . 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: ̇ 𝑢 𝑖 + 𝜈 𝑖 ( 𝐴 𝑢 ) 𝑖 + 𝜆 𝑖 𝑢 𝑖 + 𝑓 𝑖 𝑢 𝑖 , ( 𝐵 𝑢 ) 𝑖 + 𝛼 𝑖 𝑣 𝑖 = 𝑘 𝑖 ̇ 𝑣 ( 𝑡 ) , 𝑖 ∈ ℤ , 𝑡 > 𝜏 , ( 1 . 1 ) 𝑖 + 𝛿 𝑖 𝑣 𝑖 − 𝛽 𝑖 𝑢 𝑖 = 𝑔 𝑖 ( 𝑡 ) , 𝑖 ∈ ℤ , 𝑡 > 𝜏 , ( 1 . 2 ) with initial conditions 𝑢 𝑖 ( 𝜏 ) = 𝑢 𝑖 , 𝜏 , 𝑣 𝑖 ( 𝜏 ) = 𝑣 𝑖 , 𝜏 , 𝑖 ∈ ℤ , 𝜏 ∈ ℝ , ( 1 . 3 ) where ℤ is the integer lattice; 𝜈 𝑖 , 𝜆 𝑖 , 𝛿 𝑖 > 0 , 𝛼 𝑖 𝛽 𝑖 > 0 , 𝑓 𝑖 is a nonlinear function satisfying 𝑓 𝑖 ∈ 𝐶 1 ( 𝑅 × 𝑅 , 𝑅 ) , 𝑖 ∈ ℤ ; 𝐴 is a positive self-adjoint linear operator; 𝑘 ( 𝑡 ) = ( 𝑘 𝑖 ( 𝑡 ) ) 𝑖 ∈ ℤ , 𝑔 ( 𝑡 ) = ( 𝑔 𝑖 ( 𝑡 ) ) 𝑖 ∈ ℤ 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 𝐶 𝑏 ( 𝑅 , 𝑙 2 ) and to be almost periodic function. By Bochner-Amerio criterion, the set of this external term's translation is precompact in 𝐶 𝑏 ( 𝑅 , 𝑙 2 ) . Based on ideas of [28], authors in [14] introduce uniformly 𝜔 -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 𝜔 -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 𝑙 2 × 𝑙 2 , 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 𝐿 2 l o c ( 𝑅 , 𝑙 2 ) . We also show that locally asymptotic functions are translation bounded in 𝐿 2 l o c ( 𝑅 , 𝑙 2 ) , but not translation compact (tr.c.) in 𝐿 2 l o c ( 𝑅 , 𝑙 2 ) . Since the locally asymptotic smallness functions are not necessary to be translation compact in 𝐶 𝑏 ( 𝑅 , 𝑙 2 ) , 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 ℋ 𝑤 ( 𝑘 0 ) × ℋ 𝑤 ( 𝑔 0 ) ) 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 𝑙 2 the Hilbert space defined by 𝑙 2 = 𝑢 𝑢 = 𝑖 𝑖 ∈ ℤ ∣ 𝑢 𝑖 ∈ ℝ , 𝑖 ∈ ℤ 𝑢 2 𝑖 < + ∞ , ( 2 . 1 ) with the inner product ⟨ ⋅ , ⋅ ⟩ and norm ‖ ⋅ ‖ given by ⟨ 𝑢 , 𝑣 ⟩ = 𝑖 ∈ ℤ 𝑢 𝑖 𝑣 𝑖 , ‖ 𝑢 ‖ 2 = ⟨ 𝑢 , 𝑢 ⟩ = 𝑖 ∈ ℤ 𝑢 2 𝑖 . ( 2 . 2 ) For 𝑙 2 × 𝑙 2 , we endow with the inner and norm as. For 𝜓 𝑗 = ( 𝑢 ( 𝑗 ) , 𝑣 ( 𝑗 ) ) = ( 𝑢 𝑖 ( 𝑗 ) , 𝑣 𝑖 ( 𝑗 ) ) 𝑖 ∈ ℤ ∈ 𝑙 2 × 𝑙 2 , 𝑗 = 1 , 2 , ⟨ 𝜓 1 , 𝜓 2 ⟩ 𝑙 2 × 𝑙 2 = 𝑢 ( 1 ) , 𝑢 ( 2 ) 𝑙 2 + 𝑣 ( 1 ) , 𝑣 ( 2 ) 𝑙 2 = 𝑖 ∈ ℤ 𝑢 𝑖 ( 1 ) 𝑢 𝑖 ( 2 ) + 𝑣 𝑖 ( 1 ) 𝑣 𝑖 ( 2 ) , ‖ 𝜓 ‖ 2 𝑙 2 × 𝑙 2 = ⟨ 𝜓 , 𝜓 ⟩ 𝑙 2 × 𝑙 2 , ∀ 𝜓 ∈ 𝑙 2 × 𝑙 2 . ( 2 . 3 ) Denote by 𝐿 2 l o c ( 𝑅 , 𝑙 2 ) the space of function 𝜙 ( 𝑠 ) , 𝑠 ∈ ℝ with values in 𝑙 2 that locally 2-power integrable in the Bochner sense, that is, 𝑡 2 𝑡 1 ‖ 𝜙 ( 𝑠 ) ‖ 2 𝑙 2 𝑡 𝑑 𝑠 < + ∞ , ∀ 1 , 𝑡 2 ⊂ ℝ . ( 2 . 4 ) It is equipped with the local 2-power mean convergence topology. Then, 𝐿 2 l o c ( 𝑅 , 𝑙 2 ) is a metrizable space. Let 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) be a space of functions 𝜙 ( 𝑡 ) from 𝐿 2 l o c ( 𝑅 , 𝑙 2 ) such that ‖ 𝜙 ( 𝑡 ) ‖ 2 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) = s u p 𝑡 ∈ ℝ 𝑡 𝑡 + 1 ‖ 𝜙 ( 𝑠 ) ‖ 2 𝑙 2 𝑑 𝑠 < ∞ . ( 2 . 5 ) Denote by 𝐿 2 , 𝑤 l o c ( 𝑅 , 𝑙 2 ) the space 𝐿 2 l o c ( 𝑅 , 𝑙 2 ) endow with the local weak convergence topology.

For each sequence 𝑢 = ( 𝑢 𝑖 ) 𝑖 ∈ ℤ , define linear operators on 𝑙 2 by ( 𝐵 𝑢 ) 𝑖 = 𝑢 𝑖 + 1 − 𝑢 𝑖 , 𝐵 ∗ 𝑢 𝑖 = 𝑢 𝑖 − 1 − 𝑢 𝑖 ( , 𝑖 ∈ ℤ , 𝐴 𝑢 ) 𝑖 = − 𝑢 𝑖 + 1 + 2 𝑢 𝑖 − 𝑢 𝑖 − 1 , 𝑖 ∈ ℤ . ( 2 . 6 ) Then 𝐴 = 𝐵 𝐵 ∗ = 𝐵 ∗ 𝐵 𝐵 , ∗ 𝑢 , 𝑣 = ( 𝑢 , 𝐵 𝑣 ) , ∀ 𝑢 , 𝑣 ∈ 𝑙 2 . ( 2 . 7 )

For convenience, initial value problem (1.1)–(1.3) can be written as ̇ ̇ 𝑢 + 𝜈 ( 𝐴 𝑢 ) + 𝜆 𝑢 + 𝑓 ( 𝑢 , 𝐵 𝑢 ) + 𝛼 𝑣 = 𝑘 ( 𝑡 ) , 𝑡 > 𝜏 , ( 2 . 8 ) 𝑣 + 𝛿 𝑣 − 𝛽 𝑢 = 𝑔 ( 𝑡 ) , 𝑡 > 𝜏 , ( 2 . 9 ) with initial conditions 𝑢 ( 𝜏 ) = 𝑢 𝜏 = 𝑢 𝑖 , 𝜏 𝑖 ∈ ℤ , 𝑣 ( 𝜏 ) = 𝑣 𝜏 = 𝑣 𝑖 , 𝜏 𝑖 ∈ ℤ , 𝜏 ∈ ℝ , ( 2 . 1 0 ) where 𝑢 = ( 𝑢 𝑖 ) 𝑖 ∈ ℤ , 𝑣 = ( 𝑣 𝑖 ) 𝑖 ∈ ℤ , 𝜈 ( 𝐴 𝑢 ) = ( 𝜈 𝑖 ( 𝐴 𝑢 𝑖 ) ) 𝑖 ∈ ℤ , 𝑓 ( 𝑢 , 𝐵 𝑢 ) = ( 𝑓 ( 𝑢 𝑖 , ( 𝐵 𝑢 ) 𝑖 ) ) 𝑖 ∈ ℤ , 𝑘 ( 𝑡 ) = ( 𝑘 𝑖 ( 𝑡 ) ) 𝑖 ∈ ℤ , 𝑔 ( 𝑡 ) = ( 𝑔 𝑖 ( 𝑡 ) ) 𝑖 ∈ ℤ .

In the following, we give some assumption on nonlinear function 𝑓 𝑖 ∈ 𝐶 1 ( 𝑅 × 𝑅 , 𝑅 ) , and 𝜈 𝑖 , 𝜆 𝑖 , 𝛼 𝑖 , 𝛽 𝑖 , 𝛿 𝑖 ∈ ℝ :

( 𝐻 1 ) 𝑓 𝑖 𝑢 𝑖 = 0 , ( 𝐵 𝑢 ) 𝑖 = 0 = 0 , 𝑓 𝑖 𝑢 𝑖 , ( 𝐵 𝑢 ) 𝑖 𝑢 𝑖 ≥ 0 . ( 2 . 1 1 ) ( 𝐻 2 ) There exists a positive-value continuous function 𝑄 ∶ 𝑅 + ↦ 𝑅 + such that s u p 𝑖 ∈ ℤ m a x 𝑢 𝑖 , ( 𝐵 𝑢 ) 𝑖 ∈ [ − 𝑟 , 𝑟 ] | | 𝑓 𝑖 , 𝑢 𝑖 𝑢 𝑖 , ( 𝐵 𝑢 ) 𝑖 | | + s u p 𝑖 ∈ ℤ m a x 𝑢 𝑖 , ( 𝐵 𝑢 ) 𝑖 ∈ [ − 𝑟 , 𝑟 ] | | | 𝑓 𝑖 , ( 𝐵 𝑢 ) 𝑖 𝑢 𝑖 , ( 𝐵 𝑢 ) 𝑖 | | | ≤ 𝑄 ( 𝑟 ) . ( 2 . 1 2 ) ( 𝐻 3 ) There exist positive constants 𝜈 0 , 𝜈 0 , 𝜆 0 , 𝜆 0 , 𝛼 0 , 𝛼 0 , 𝛽 0 , 𝛽 0 , 𝜎 0 , 𝜎 0 such that 0 < 𝜈 0 𝜈 = m i n 𝑖 , ∶ 𝑖 ∈ ℤ , 𝜈 0 𝜈 = m a x 𝑖 , ∶ 𝑖 ∈ ℤ < + ∞ , 0 < 𝜆 0 𝜆 = m i n 𝑖 , ∶ 𝑖 ∈ ℤ , 𝜆 0 𝜆 = m a x 𝑖 , ∶ 𝑖 ∈ ℤ < + ∞ , 0 < 𝛼 0 𝛼 = m i n 𝑖 , ∶ 𝑖 ∈ ℤ , 𝛼 0 𝛼 = m a x 𝑖 , ∶ 𝑖 ∈ ℤ < + ∞ , 0 < 𝛽 0 𝛽 = m i n 𝑖 , ∶ 𝑖 ∈ ℤ , 𝛽 0 𝛽 = m a x 𝑖 , ∶ 𝑖 ∈ ℤ < + ∞ , 0 < 𝛿 0 𝛿 = m i n 𝑖 , ∶ 𝑖 ∈ ℤ , 𝛿 0 𝛿 = m a x 𝑖 , ∶ 𝑖 ∈ ℤ < + ∞ . ( 2 . 1 3 )

Let the external term ℎ ( 𝑡 ) , 𝑔 ( 𝑡 ) belong to 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) , it follows from the standard theory of ordinary differential equations that there exists a unique local solution ( 𝑢 , 𝑣 ) ∈ 𝐶 ( [ 𝜏 , 𝑡 0 ) , 𝑙 2 × 𝑙 2 ) for problem (2.8)–(2.10) if (H1)–(H3) hold. For a fixed external term ( 𝑘 0 ( 𝑡 ) , 𝑔 0 ( 𝑡 ) ) ∈ 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) × 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) , take the symbol space 𝚺 = { 𝑘 0 ( 𝑠 + ℎ ) ∣ ℎ ∈ ℝ } × { 𝑔 0 ( 𝑠 + ℎ ) ∣ ℎ ∈ ℝ } = ℋ ( 𝑘 0 ) × ℋ ( 𝑔 0 ) , the set contains all translations of ( 𝑘 0 ( 𝑠 ) , 𝑔 0 ( 𝑠 ) ) in 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) × 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) . Take the [ 𝚺 ] 𝑤 = ℋ 𝑤 ( 𝑘 0 ) × ℋ 𝑤 ( 𝑔 0 ) the closure of 𝚺 in 𝐿 2 , 𝑤 l o c ( 𝑅 , 𝑙 2 ) × 𝐿 2 , 𝑤 l o c ( 𝑅 , 𝑙 2 ) . Denote by 𝑇 ( ℎ ) the translation semigroup, 𝑇 ( ℎ ) ( 𝑘 ( 𝑠 ) , 𝑔 ( 𝑠 ) ) = ( 𝑘 ( 𝑠 + ℎ ) , 𝑔 ( 𝑠 + ℎ ) ) for all ( 𝑘 , 𝑔 ) ∈ 𝚺 or [ 𝚺 ] 𝑤 , 𝑠 ∈ ℝ , ℎ ≥ 0 . It is evident that { 𝑇 ( ℎ ) } ℎ ≥ 0 is continuous on 𝚺 in the topology of 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) and on [ 𝚺 ] 𝑤 in the topology of 𝐿 2 , 𝑤 l o c ( 𝑅 , 𝑙 2 ) , respectively, 𝑇 𝑘 ( ℎ ) 𝚺 = 𝚺 = ℋ 0 𝑔 × ℋ 0 [ 𝚺 ] , 𝑇 ( ℎ ) 𝑤 = [ 𝚺 ] 𝑤 = ℋ 𝑤 𝑘 0 × ℋ 𝑤 𝑔 0 , ∀ ℎ > 0 . ( 2 . 1 4 )

In Section 3, we will show that for every ( 𝑘 ( 𝑡 ) , 𝑔 ( 𝑡 ) ) ∈ ℋ 𝑤 ( 𝑘 0 ) × ℋ 𝑤 ( 𝑔 0 ) , and ( 𝑢 𝜏 , 𝑣 𝜏 ) = ( 𝑢 𝑖 , 𝜏 , 𝑣 𝑖 , 𝜏 ) 𝑖 ∈ ℤ ∈ 𝑙 2 × 𝑙 2 , 𝜏 ∈ ℝ , problem (2.8)–(2.10) has a unique global solution ( 𝑢 , 𝑣 ) ( 𝑡 ) = ( 𝑢 𝑖 , 𝑣 𝑖 ) 𝑖 ∈ ℤ ( 𝑡 ) ∈ 𝐶 ( [ 𝜏 , ∞ ) , 𝑙 2 × 𝑙 2 ) . Thus, there exists a family of processes { 𝑈 ( 𝑘 , 𝑔 ) ( 𝑡 , 𝜏 ) } from 𝑙 2 × 𝑙 2 to 𝑙 2 × 𝑙 2 . 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 𝐸 be a Banach space which the processes acting in, for a given symbol space Ξ , the uniform (w.r.t. 𝜎 ∈ Ξ ) 𝜔 -limit set 𝜔 𝜏 , Ξ ( 𝐵 ) of 𝐵 ⊂ 𝐸 is defined by 𝜔 𝜏 , Ξ ( 𝐵 ) = 𝑡 ≥ 𝜏 𝜎 ∈ Ξ 𝑠 ≥ 𝑡 𝑈 𝜎 ( 𝑠 , 𝜏 ) 𝐵 𝐸 . ( 2 . 1 5 ) The first result of this paper is stated in the following, which will be proved in Section 4.

Theorem 2 A. Assume that ( k 0 ( s ) , g 0 ( s ) ) ∈ L 2 l o c ( R , l 2 ) × L 2 l o c ( R , l 2 ) be locally asymptotic smallness and ( H 1 ) - ( H 3 ) hold. Then the process { U ( k 0 , g 0 ) } corresponding to problems (2.8)–(2.10) with external term ( k 0 ( s ) , g 0 ( s ) ) possesses compact uniform ( w . r . t . 𝜏 ∈ ℝ ) attractor 𝒜 0 in l 2 × l 2 which coincides with uniform (w.r.t. ( k ( s ) , g ( s ) ) ∈ ℋ w ( k 0 ) × ℋ w ( g 0 ) ) attractor 𝒜 ℋ w ( k 0 ) × ℋ w ( g 0 ) for the family of processes { U ( k , g ) ( t , 𝜏 ) } , ( k , g ) ∈ ℋ w ( k 0 ) × ℋ w ( g 0 ) , that is, 𝒜 0 = 𝒜 ℋ 𝑤 ( 𝑘 0 ) × ℋ 𝑤 ( 𝑔 0 ) = 𝜔 0 , 𝒜 ℋ 𝑤 0 𝑤 0 ) ( 𝑘 ) × ℋ ( 𝑔 𝐵 0 = ( 𝑘 , 𝑔 ) ∈ ℋ 𝑤 ( 𝑘 0 ) × ℋ 𝑤 ( 𝑔 0 ) 𝒦 ( 𝑘 , 𝑔 ) ( 0 ) , ( 2 . 1 6 ) where 𝐵 0 is the uniform ( w . r . t . ( 𝑘 , 𝑔 ) ∈ ℋ 𝑤 ( 𝑘 0 ) × ℋ 𝑤 ( 𝑔 0 ) ) absorbing set in 𝑙 2 × 𝑙 2 , and 𝒦 ( 𝑘 , 𝑔 ) is kernel of the process { 𝑈 ( 𝑘 , 𝑔 ) ( 𝑡 , 𝜏 ) } . The uniform attractor uniformly ( w . r . t . ( 𝑘 , 𝑔 ) ∈ ℋ 𝑤 ( 𝑘 0 ) × ℋ 𝑤 ( 𝑔 0 ) ) attracts the bounded set in 𝑙 2 × 𝑙 2 .

We also consider finite-dimensional approximation to the infinite-dimensional systems (1.2)-(1.3) on finite lattices. For every positive integer 𝑛 > 0 , let 𝑍 𝑛 = 𝑍 ∩ { − 𝑛 ≤ 𝑖 ≤ 𝑛 } , consider the following ordinary equations with initial data in 𝑅 2 𝑛 + 1 × 𝑅 2 𝑛 + 1 :

̇ 𝑢 𝑖 + 𝜈 𝑖 ( 𝐴 𝑢 ) 𝑖 + 𝜆 𝑖 𝑢 𝑖 + 𝑓 𝑖 𝑢 𝑖 , ( 𝐵 𝑢 ) 𝑖 + 𝛼 𝑖 𝑣 𝑖 = 𝑘 𝑖 ( 𝑡 ) , 𝑖 ∈ 𝑍 𝑛 ̇ 𝑣 , 𝑡 > 𝜏 , 𝑖 + 𝛿 𝑖 𝑣 𝑖 − 𝛽 𝑖 𝑢 𝑖 = 𝑔 𝑖 ( 𝑡 ) , 𝑖 ∈ 𝑍 𝑛 𝑢 , 𝑡 > 𝜏 , 𝑢 ( 𝜏 ) = 𝑖 ( 𝜏 ) | 𝑖 | ≤ 𝑛 = 𝑢 𝑖 , 𝜏 | 𝑖 | ≤ 𝑛 𝑣 , 𝑣 ( 𝜏 ) = 𝑖 ( 𝜏 ) | 𝑖 | ≤ 𝑛 = 𝑣 𝑖 , 𝜏 | 𝑖 | ≤ 𝑛 , 𝜏 ∈ ℝ . ( 2 . 1 7 ) In Section 5, we will show that the finite-dimensional approximation systems possess a uniform attractor 𝒜 𝑛 0 in 2 𝑛 + 1 × 𝑅 2 𝑛 + 1 , and these uniform attractors are upper semicontinuous when 𝑛 → ∞ . More precisely, we have the following theorem.

Theorem 2 B. Assume that ( k 0 ( s ) , g 0 ( s ) ) ∈ L 2 b ( R , l 2 ) × L 2 b ( R , l 2 ) and ( H 1 ) - ( H 3 ) hold. Then for every positive integer n , systems (2.17) possess compact uniform attractor 𝒜 n 0 . Further, 𝒜 n 0 is upper semicontinuous to 𝒜 0 as n → ∞ , that is, l i m 𝑛 → ∞ 𝑑 𝑙 2 × 𝑙 2 𝒜 𝑛 0 , 𝒜 0 = 0 , ( 2 . 1 8 ) where 𝑑 𝑙 2 × 𝑙 2 𝒜 𝑛 0 , 𝒜 0 = s u p 𝑎 ∈ 𝒜 𝑛 0 i n f 𝑏 ∈ 𝒜 0 ‖ 𝑎 − 𝑏 ‖ 𝑙 2 × 𝑙 2 . ( 2 . 1 9 )

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 k 0 , g 0 ∈ L 2 b ( R , l 2 ) and ( H 1 ) - ( H 3 ) hold. Let ( k ( s ) , g ( s ) ) ∈ ℋ w ( k 0 ) × ℋ w ( g 0 ) , and ( u 𝜏 , v 𝜏 ) ∈ l 2 × l 2 , 𝜏 ∈ ℝ . Then the solution of (2.8)–(2.10) satisfies ‖ ( 𝑢 , 𝑣 ) ( 𝑡 ) ‖ 2 𝑙 2 × 𝑙 2 ≤ ‖ ( 𝑢 , 𝑣 ) ( 𝜏 ) ‖ 2 𝑙 2 × 𝑙 2 𝑒 − 𝛾 0 / 𝜂 0 ( 𝑡 − 𝜏 ) + 1 𝜂 0 𝛽 0 𝜆 0 ‖ ‖ 𝑘 0 ‖ ‖ ( 𝑠 ) 2 𝐿 2 𝑏 𝑅 , 𝑙 2 + 𝛼 0 𝛿 0 ‖ ‖ 𝑔 0 ‖ ‖ ( 𝑠 ) 2 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) 𝜂 1 + 0 𝛾 0 , ( 3 . 1 ) where 𝜂 0 = m i n { 𝛼 0 , 𝛽 0 } , 𝛾 0 = m i n { 𝜆 0 𝛽 0 , 𝛼 0 𝛿 0 } .

Proof. Taking the inner product of (2.8) with 𝛽 𝑢 in 𝑙 2 , by ( 𝐻 1 ) , we get 1 2 𝑑 𝑑 𝑡 𝑖 ∈ ℤ 𝛽 𝑖 𝑢 2 𝑖 + 𝑖 ∈ ℤ 𝛽 𝑖 𝜈 𝑖 | | ( 𝐵 𝑢 ) 𝑖 | | 2 + 𝑖 ∈ ℤ 𝜆 𝑖 𝛽 𝑖 𝑢 2 𝑖 + 𝑖 ∈ ℤ 𝛽 𝑖 𝛼 𝑖 𝑢 𝑖 𝑣 𝑖 ≤ 𝑖 ∈ ℤ 𝛽 𝑖 𝑢 𝑖 𝑘 𝑖 ( 𝑡 ) . ( 3 . 2 ) Similarly, taking the inner product of (2.9) with 𝛼 𝑣 in 𝑙 2 , we get 1 2 𝑑 𝑑 𝑡 𝑖 ∈ ℤ 𝛼 𝑖 𝑣 2 𝑖 + 𝑖 ∈ ℤ 𝛼 𝑖 𝛿 𝑖 𝑢 2 𝑖 − 𝑖 ∈ ℤ 𝛽 𝑖 𝛼 𝑖 𝑢 𝑖 𝑣 𝑖 = 𝑖 ∈ ℤ 𝛼 𝑖 𝑣 𝑖 𝑔 𝑖 ( 𝑡 ) . ( 3 . 3 ) Note that 𝑖 ∈ ℤ 𝛽 𝑖 𝑢 𝑖 𝑘 𝑖 1 ( 𝑡 ) ≤ 2 𝑖 ∈ ℤ 𝜆 𝑖 𝛽 𝑖 𝑢 2 𝑖 + 1 2 𝑖 ∈ ℤ 𝛽 𝑖 𝜆 𝑖 𝑘 2 𝑖 ( 𝑡 ) , 𝑖 ∈ ℤ 𝛼 𝑖 𝑣 𝑖 𝑔 𝑖 1 ( 𝑡 ) ≤ 2 𝑖 ∈ ℤ 𝛼 𝑖 𝛿 𝑖 𝑣 2 𝑖 + 1 2 𝑖 ∈ ℤ 𝛼 𝑖 𝛿 𝑖 𝑔 2 𝑖 ( 𝑡 ) . ( 3 . 4 ) Summing up (3.2) and (3.3), from (3.4), we get 𝑑 𝑑 𝑡 𝑖 ∈ ℤ 𝛽 𝑖 𝑢 2 𝑖 + 𝛼 𝑖 𝑣 2 𝑖 + 𝑖 ∈ ℤ 𝜆 𝑖 𝛽 𝑖 𝑢 2 𝑖 + 𝛼 𝑖 𝛿 𝑖 𝑣 2 𝑖 ≤ 𝑖 ∈ ℤ 𝛽 𝑖 𝜆 𝑖 𝑘 2 𝑖 𝛼 ( 𝑡 ) + 𝑖 𝛿 𝑖 𝑔 2 𝑖 ( 𝑡 ) . ( 3 . 5 ) Thus, by ( 𝐻 3 ) , 𝜂 0 𝑑 𝑑 𝑡 ‖ ( 𝑢 , 𝑣 ) ( 𝑡 ) ‖ 2 𝑙 2 × 𝑙 2 + 𝛾 0 ‖ ( 𝑢 , 𝑣 ) ( 𝑡 ) ‖ 2 𝑙 2 × 𝑙 2 ≤ 𝛽 0 𝜆 0 ‖ 𝑘 ( 𝑡 ) ‖ 2 𝑙 2 + 𝛼 0 𝛿 0 ‖ 𝑔 ( 𝑡 ) ‖ 2 𝑙 2 . ( 3 . 6 ) Since ( 𝑘 ( 𝑡 ) , 𝑔 ( 𝑡 ) ) ∈ ℋ 𝑤 ( 𝑘 0 ) × ℋ 𝑤 ( 𝑔 0 ) , from [12, Proposition V.4.2.], we have ‖ 𝑘 ( 𝑡 ) ‖ 2 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) ≤ ‖ ‖ 𝑘 0 ‖ ‖ ( 𝑡 ) 2 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) , ‖ 𝑔 ( 𝑡 ) ‖ 2 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) ≤ ‖ ‖ 𝑔 0 ‖ ‖ ( 𝑡 ) 2 𝐿 2 𝑏 ( 𝑅 , 𝑙 2 ) . ( 3 . 7 ) 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 ( 𝑢 , 𝑣 ) of problem (2.8)–(2.10) is defined for all 𝑡 ≥ 𝜏 . Therefore, there exists a family processes acting in the space 𝑙 2 × 𝑙 2 ∶ { 𝑈 ( 𝑘 , 𝑔 ) } ∶ 𝑈 ( 𝑘 , 𝑔 ) ( 𝑡 , 𝜏 ) ( 𝑢 𝜏 , 𝑣 𝜏 ) = ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) , 𝑈 ( 𝑘 , 𝑔 ) ( 𝑡 , 𝜏 ) ∶ 𝑙 2 × 𝑙 2 → 𝑙 2 × 𝑙 2 , 𝑡 ≥ 𝜏 , 𝜏 ∈ ℝ , where ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) is the solution of (2.8)–(2.10), and the time symbol ( 𝑘 ( 𝑠 ) , 𝑔 ( 𝑠 ) ) belongs to ℋ ( 𝑘 0 ) × ℋ ( 𝑔 0 ) and ℋ 𝑤 ( 𝑘 0 ) × ℋ 𝑤 ( 𝑔 0 ) , respectively. The family of processes { 𝑈 ( 𝑘 , 𝑔 ) } satisfies multiplicative properties: 𝑈 ( 𝑘 , 𝑔 ) ( 𝑡 , 𝑠 ) ∘ 𝑈 ( 𝑘 , 𝑔 ) ( 𝑠 , 𝜏 ) = 𝑈 ( 𝑘 , 𝑔 ) 𝑈 ( 𝑡 , 𝜏 ) , ∀ 𝑡 ≥ 𝑠 ≥ 𝜏 , 𝜏 ∈ ℝ , ( 𝑘 , 𝑔 ) ( 𝜏 , 𝜏 ) = I d i s t h e i d e n t i t y o p e r a t o r , 𝜏 ∈ ℝ . ( 3 . 8 ) Furthermore, the following translation identity holds: 𝑈 ( 𝑘 , 𝑔 ) ( 𝑡 + ℎ , 𝜏 + ℎ ) = 𝑈 𝑇 ( ℎ ) ( 𝑘 , 𝑔 ) ( 𝑡 , 𝜏 ) , ∀ 𝑡 ≥ 𝜏 , 𝜏 ∈ ℝ , ℎ ≥ 0 . ( 3 . 9 ) The kernel 𝒦 of the processes 𝑈 ( 𝑘 , 𝑔 ) ( 𝑡 , 𝜏 ) consists of all bounded complete trajectories of the process 𝑈 ( 𝑘 , 𝑔 ) ( 𝑡 , 𝜏 ) , that is, 𝒦 ( 𝑘 , 𝑔 ) = ( 𝑢 ( ⋅ ) , 𝑣 ( ⋅ ) ) ∣ ‖ ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) ‖ 𝑙 2 × 𝑙 2 ≤ 𝐶 ( 𝑢 , 𝑣 ) , 𝑈 ( 𝑘 , 𝑔 ) . ( 𝑡 , 𝜏 ) ( 𝑢 ( 𝜏 ) , 𝑣 ( 𝜏 ) ) = ( 𝑢 ( 𝑡 ) , 𝑣 ( 𝑡 ) ) , ∀ 𝑡 ≥ 𝜏 , 𝜏 ∈ ℝ ( 3 . 1 0 ) 𝒦 ( 𝑠 ) denotes the kernel section at a times moment 𝑠 ∈ ℝ : 𝒦 ( 𝑘 , 𝑔 ) ( 𝑠 ) = ( 𝑢 ( 𝑠 ) , 𝑣 ( 𝑠 ) ) ∣ ( 𝑢 ( ⋅ ) , 𝑣 ( ⋅ ) ) ∈ 𝒦 ( 𝑘 , 𝑔 ) . ( 3 . 1 1 )

Lemma 3.1 also shows that the family of processes possesses a uniform absorbing set in 𝑙 2 × 𝑙 2 .

Lemma 3.2. Assume that k 0 , g 0 ∈ L 2 b ( R , l 2 ) and ( H 1 ) - ( H 3 ) hold. Let ( k ( s ) , g ( s ) ) ∈ ℋ w ( k 0 ) × ℋ w ( g 0 ) . Then, there exists a bounded uniform absorbing set B 0 in l 2 × l 2 for the family of processes { U ( k , g ) } ℋ w ( k 0 ) × ℋ w ( g 0 ) , that is, for any bounded set B ⊂ l 2 × l 2 , there exists t 0 = t 0 ( 𝜏 , B ) ≥ 𝜏 , ( 𝑘 , 𝑔 ) ∈ ℋ 𝑤 ( 𝑘 0 ) × ℋ 𝑤 ( 𝑔 0 ) 𝑈 ( 𝑘 , 𝑔 ) ( 𝑡 , 𝜏 ) 𝐵 ⊂ 𝐵 0 , ∀ 𝑡 ≥ 𝑡 0 . ( 3 . 1 2 )

Proof. Let ‖ ( 𝑢 𝜏 , 𝑣 𝜏 ) ‖ 𝑙 2 × 𝑙 2 ≤ 𝑅 , from (3.1) we have ‖ ( 𝑢 , 𝑣 ) ( 𝑡 ) ‖ 2 𝑙 2 × 𝑙 2 ≤ 𝑅 2 𝑒 − 𝛾 0 / 𝜂 0 ( 𝑡 − 𝜏 ) + 1 𝜂 0 𝛽 0 𝜆 0 ‖ ‖ 𝑘 0 ‖ ‖ ( 𝑠 ) 2 𝐿 2 𝑏 𝑅 , 𝑙 2 + 𝛼 0 𝛿 0 ‖ ‖ 𝑔 0 ‖ ‖ ( 𝑠 ) 2 𝐿 2 𝑏 𝑅 , 𝑙 2 𝜂 1 + 0 𝛾 0 ≤ 2 𝜂 0 𝛽 0 𝜆 0 ‖ 𝑘 0 ( 𝑠 ) ‖ 2 𝐿 2 𝑏 𝑅 , 𝑙 2 + 𝛼 0 𝛿 0 ‖ 𝑔 0 ( 𝑠 ) ‖ 2 𝐿 2 𝑏 𝑅 , 𝑙 2 𝜂 1 + 0 𝛾 0 , ∀ 𝑡 ≥ 𝑡 0 , ( 3 . 1 3 ) where 𝑡 0 = 𝜂 0 𝛾 0 𝑅 l n 2 𝑋 1 + 𝜏 , 𝑋 = 𝜂 0 𝛽 0 𝜆 0 ‖ ‖ 𝑘 0 ‖ ‖ ( 𝑠 ) 2 𝐿 2 𝑏 𝑅 , 𝑙 2 + 𝛼 0 𝛿 0 ‖ ‖ 𝑔 0 ‖ ‖ ( 𝑠 ) 2 𝐿 2 𝑏 𝑅 , 𝑙 2 𝜂 1 + 0 𝛾 0 . ( 3 . 1 4 ) Let 𝐵 0 = { ( 𝑢 , 𝑣 ) ( 𝑡 ) ∈ 𝑙 2 × 𝑙 2 ∣ ‖ ( 𝑢 , 𝑣 ) ( 𝑡 ) ‖ 2 𝑙 2 × 𝑙 2 ≤ 2 𝑋 2 } . 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 𝐸 be a Banach space, and let Ξ be a subset of some Banach space.

Definition 4.1. { 𝑈 𝜎 ( 𝑡 , 𝜏 ) } , 𝜎 ∈ Ξ is said to be   ( 𝐸 × Ξ , 𝐸 ) weakly continuous, if for any 𝑡 ≥ 𝜏 , 𝜏 ∈ ℝ , the mapping ( 𝑢 , 𝜎 ) → { 𝑈 𝜎 ( 𝑡 , 𝜏 ) 𝑢   is weakly continuous from 𝐸 × Ξ to 𝐸 .

A family of processes 𝑈 𝜎 ( 𝑡 , 𝜏 ) , 𝜎 ∈ Ξ is said to be uniformly ( w . r . t . 𝜎 ∈ Ξ ) 𝜔 -limit compact if for any 𝜏 ∈ ℝ and bounded set 𝐵 ⊂ 𝐸 , the set ⋃ 𝜎 ∈ Ξ ⋃ 𝑠 ≥ 𝑡 𝑈 𝜎 ( 𝑠 , 𝜏 ) 𝐵 is bounded for every 𝑡 and ⋃ 𝜎 ∈ Ξ ⋃ 𝑠 ≥ 𝑡 𝑈 𝜎 ( 𝑠 , 𝑡 ) 𝐵 is precompact set as 𝑡 → + ∞ . We need the following result in [14].

Theorem 4.2. Let Ξ be the weak closure of Ξ 0 . Assume that { U 𝜎 ( t , 𝜏 ) } , 𝜎 ∈ Ξ is ( E × Ξ , E ) weakly continuous, and (i)has a bounded uniformly (w.r.t. 𝜎 ∈ Ξ ) absorbing set 𝐵 0 ,(ii)is uniformly (w.r.t. 𝜎 ∈ Ξ ) 𝜔 -limit compact.
Then the families of processes { 𝑈 𝜎 ( 𝑡 , 𝜏 ) } , 𝜎 ∈ Ξ 0 , 𝜎 ∈ Ξ possess, respectively, compact uniform (w.r.t. 𝜎 ∈ Ξ 0 , 𝜎 ∈ Ξ , resp.) attractors 𝒜 Ξ 0 and 𝒜 Ξ satisfying
𝒜 Ξ 0 = 𝒜 Ξ = 𝜔 0 , Ξ 𝐵 0 = 𝜎 ∈ Ξ 𝒦 𝜎 ( 0 ) . ( 4 . 1 ) Furthermore,