Numerische Approximation und Spektrale Analysis unendlich-dimensionaler Dynamischer Systeme
02-007 Wolf-Jürgen Beyn, Sergei Pilyugin.
Attractors of reaction diffusion systems on infinite
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
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