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Dynamics of systems on infinite lattices. (English) Zbl 1085.37056
Summary: The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction-diffusion equation has a global attractor in a weighted \(l^{2}\) space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction-diffusion equation is approached by finite-dimensional systems.

MSC:
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
35B41 Attractors
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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