Sturm attractors for fully nonlinear parabolic equations

TL;DR

This paper explicitly constructs global attractors for fully nonlinear parabolic equations, decomposing them into equilibria and heteroclinic orbits, and provides conditions for heteroclinic connections.

Contribution

It introduces a method to compute necessary and sufficient conditions for heteroclinic orbits between hyperbolic equilibria in nonlinear parabolic equations.

Findings

Explicit construction of global attractors

Decomposition into equilibria and heteroclinic orbits

Conditions for heteroclinic connections

Abstract

We explicitly construct global attractors of fully nonlinear parabolic equations. The attractors are decomposed as equilibria (time independent solutions) and heteroclinic orbits (solutions that converge to distinct equilibria backwards and forwards in time). In particular, we state necessary and sufficient conditions for the occurrence of heteroclinics between hyperbolic equilibria, which is accompanied by a method that compute such conditions.