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Orbit equivalence of global attractors of semilinear parabolic differential equations

Authors:	Bernold Fiedler Carlos Rocha

Institution:	Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2-6, D-14195 Berlin, Germany Carlos Rocha ; Departamento de Matemática, Instituto Superior Técnico, Avenida Rovisco Pais, 1096 Lisboa Codex, Portugal

Abstract:	We consider global attractors ${\cal A}_f$ of dissipative parabolic equations $\begin{equation}u_t=u_{xx}+f(x,u,u_x) \end{equation}$ on the unit interval $0\leq x\leq 1$ with Neumann boundary conditions. A permutation $\pi _f$ is defined by the two orderings of the set of (hyperbolic) equilibrium solutions $u_t\equiv 0$ according to their respective values at the two boundary points and We prove that two global attractors, ${\cal A}_f$ and ${\cal A}_g$ , are globally orbit equivalent, if their equilibrium permutations $\pi _f$ and $\pi _g$ coincide. In other words, some discrete information on the ordinary differential equation boundary value problem $u_t\equiv 0$ characterizes the attractor of the above partial differential equation, globally, up to orbit preserving homeomorphisms.

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