Do Global Attractors Depend on Boundary Conditions?

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.

The results are based on joint work with Carlos Rocha.

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