Do global attractors depend on boundary conditions?

Abstract

We consider global attractors of infinite dimensional dynamical systems given by dissipative partial differential equations

on the unit interval $0<x<1$ under separated, linear, dissipative boundary conditions. Global attractors are called orbit equivalent, if there exists a homeomorphism between them which maps orbits to orbits. The global attractor class is the set of all equivalence classes of global attractors arising for dissipative nonlinearities $f$. We show that the global attractor class does not depend on the choice of boundary conditions. In particular, Dirichlet and Neumann boundary conditions yield the same global attractor class.