Exponential Attractors for Lattice Dynamical Systems in Weighted Spaces

Abstract

The aim of this paper is to investigate the existence of exponential attractors for lattice reaction-diffusion systems in weighted spaces l 2 σ and for partly dissipative lattice reaction-diffusion systems in weighted spaces l 2 μ × l 2 μ , respectively. In contrast to the previous work by Abdallah in J. Math. Anal. Appl. 339, 217-224 (2008) and Commun. Pure Appl. Anal. 8, 803-818 (2009), we get the existence of exponential attractors for lattice dynamical systems in the weak topology spaces.

Lattice dynamical systems: dissipative mechanism and fractal dimension of global and exponential attractors

“…The above-mentioned results were obtained in the standard spaces of summable sequences, which is the setting chosen in this paper as well. However, lattice systems were also considered in weighted spaces of sequences; see, e.g., [10,21,23,31]. This is a distinct approach, since such phase spaces have different properties from the ones considered here.…”

Pullback attractors of nonautonomous discrete p-Laplacian complex Ginzburg–Landau equations with fast-varying delays

“…For an understanding of the dynamical behavior of dissipative infinite lattice systems, attractors are especially important because they retain most of the dynamical information. The existence of global attractors for lattice systems was initialed by Bates et al [1], followed by extensions in [3,8,13,16,19,24] and the references therein. Of those, the asymptotic behavior of an infinite-dimensional p-Laplacian lattice system was investigated in [8].…”

Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems

“…Note that (19) and (15) imply that d i (t) 0 for i ∈ Z and t ∈ R. We point out that assumption (19) is another advantage of our approach that broadens the class of investigated nonlinear lattice differential equations in comparison with a common simple condition found in the literature of the form (2), (3) (cp. e.g.…”

“…Mostly these systems were studied in the autonomous case, see e.g. [17,19,6] for systems generating semigroups in the weighted spaces of summable double-sided sequences or e.g. [4,25,26,20,28,1,10] in the classical spaces of summable double-sided sequences.…”

“…We further study the asymptotic behavior of the problem (1) under the assumptions formulated in Section 2, i.e. the linear operator A from ( 7) satisfies (10), g is as in ( 14) and f i fulfill ( 15), ( 16), ( 18) and (19). It follows, in particular, that d and g belong to G γ0 , so there exists…”