Preprint des Projektes: DFG Forschergruppe Spektrale Analysis, asymptotische Verteilungen und stochastische Prozesse

Numerische Approximation und Spektrale Analysis unendlich-dimensionaler Dynamischer Systeme

02-007 Wolf-Jürgen Beyn, Sergei Pilyugin.
Attractors of reaction diffusion systems on infinite lattices

In this paper, we study global attractors for implicit discretizations of a semilinear parabolic equation on the line.
It is shown that under usual dissipativity conditions there exists a global (Zu,Zrho)-attractor A, where Zrho is a Sobolev space of infinite sequences with a norm which decays at infinity, while the space Zu carries a locally uniform norm obtained by taking the supremum over all Zrho norms of translates. We show that the size of an absorbing set containing A can be taken uniformly bounded (in the norm of Zu) for small time and space steps of the discretization.
We establish the following upper semicontinuity property of the attractor A: if A is the global attractor for a discretization of the same parabolic equation on the finite segment [-N,N] with the Dirichlet boundary conditions, then the attractors AN (properly embedded into the space Zu) tend to A as N tends to infinity with respect to the Hausdorff semidistance generated by the norm in Zrho.
We describe a possibility of "embedding" ofinvariant 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.