Compact uniform attractors for dissipative lattice dynamical systems with delays

Caidi Zhao; Shengfan Zhou

doi:10.3934/dcds.2008.21.643

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2008, Volume 21, Issue 2: 643-663. Doi: 10.3934/dcds.2008.21.643

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Compact uniform attractors for dissipative lattice dynamical systems with delays

Caidi Zhao^1, and
Shengfan Zhou^2,

1.
College of Mathematics and Information Science, Wenzhou University, Zhejiang, 325035
2.
Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234

Received: December 31, 2006

Revised: September 30, 2007

Published: April 2008

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Abstract

In this paper, we consider the long time behavior of solutions for dissipative lattice dynamical systems with delays. We first prove a sufficient and necessary condition for the existence of a compact uniform attractor for the family of processes corresponding to the lattice dynamical systems with delays. Then we apply this result to prove the existence of a compact uniform attractor for the process associated to the retarded lattice Zakharov equations. As a consequence, some results for the non-delay lattice dynamical systems are deduced as particular cases.

Keywords:

Lattice dynamical systems; Uniform attractor; Delay systems; Kolmogorov $\varepsilon$-entropy.

Mathematics Subject Classification: 35B40; 37L50; 35Q55.

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