Abstract
In this paper, we study global attractors for implicit discretizations of a semilinear parabolic system on the line. It is shown that under usual dissipativity conditions there exists a global (Z u ,Z ρ )-attractor \(A\) in the sense of Babin-Vishik and Mielke-Schneider. Here Z ρ is a weighted Sobolev space of infinite sequences with a weight that decays at infinity, while the space Z u carries a locally uniform norm obtained by taking the supremum over all Z ρ norms of translates. We show that the absorbing set containing \(A\) can be taken uniformly bounded (in the norm of Z u ) for small time and space steps of the discretization. We establish the following upper semicontinuity property of the attractor \(A\) for a scalar equation: if \(A\) N is the global attractor for a discretization of the same parabolic equation on the finite segment [−N,N] with Dirichlet boundary conditions, then the attractors \(A\) N (properly embedded into the space Z u ) tend to \(A\) as N→∞ with respect to the Hausdorff semidistance generated by the norm in Z ρ . We describe a possibility of “embedding” certain invariant sets of some planar dynamical systems into the global attractor \(A\). Finally, we give an example in which the global attractor \(A\) is infinite-dimensional.
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Beyn, WJ., Pilyugin, S.Y. Attractors of Reaction Diffusion Systems on Infinite Lattices. Journal of Dynamics and Differential Equations 15, 485–515 (2003). https://doi.org/10.1023/B:JODY.0000009745.41889.30
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DOI: https://doi.org/10.1023/B:JODY.0000009745.41889.30