Fiedler, Bernold

Do global attractors depend on boundary conditions?

Doc. Math., J. DMV 1, 215-228 (1996)


Summary: We consider global attractors of infinite dimensional dynamical systems given by dissipative partial differential equations $$u_t=u_{xx}+f(x,u,u_x) $$ 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.