Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Orbit equivalence of global attractors
of semilinear parabolic differential equations

Authors: Bernold Fiedler and Carlos Rocha
Journal: Trans. Amer. Math. Soc. 352 (2000), 257-284
MSC (1991): Primary 58F39, 35K55, 58F12
Published electronically: September 21, 1999
MathSciNet review: 1475682
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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 $x=0$ and $x=1.$ We prove that two global attractors, ${\cal A}_f$ and ${\cal A}_g$, are globally $C^0$ 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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Additional Information

Bernold Fiedler
Affiliation: Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2-6, D-14195 Berlin, Germany

Carlos Rocha
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Avenida Rovisco Pais, 1096 Lisboa Codex, Portugal

Received by editor(s): September 17, 1996
Received by editor(s) in revised form: June 12, 1997
Published electronically: September 21, 1999
Article copyright: © Copyright 1999 American Mathematical Society