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Attractors and dimension of dissipative lattice systems. (English) Zbl 1091.37023
Summary: By using the argument in [Q. Ma, S. Wang and C. Zhong, Indiana Univ. Math. J. 51(6), 1541–1559 (2002; Zbl 1028.37047)], we prove that the condition given in [S. Zhou, J. Differ. Equations 200, 342–368 (2004; Zbl 1173.37331)] for the existence of a global attractor for the semigroup associated with general lattice systems on a discrete Hilbert space is a sufficient and necessary condition. As an application, we consider the existence of a global attractor for a second-order lattice system in a discrete weighted space containing all bounded sequences. Finally, we show that the global attractor for first-order and partly dissipative lattice systems corresponding to (partly dissipative) reaction-diffusion equations and second-order dissipative lattice systems corresponding to the strongly damped wave equations have finite fractal dimension if the derivative of the nonlinear term is small at the origin.

37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35K57 Reaction-diffusion equations
28A80 Fractals
Full Text: DOI
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