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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.

MSC:
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
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