Orbit equivalence of global attractors of semilinear parabolic differential equations

Abstract

We consider global attractors $\mathcal {A}_f$ 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, $\mathcal {A}_f$ and $\mathcal {A}_g$, 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.