Sturm attractors for quasilinear parabolic equations

TL;DR

This paper explicitly constructs the global attractors for quasilinear parabolic equations, extending previous semilinear results, and provides conditions for heteroclinic connections, illustrated by the Chafee-Infante attractor example.

Contribution

It generalizes the construction of global attractors and heteroclinic connections from semilinear to quasilinear parabolic equations, with explicit conditions based on permutations.

Findings

Explicit construction of global attractors for quasilinear equations

Necessary and sufficient conditions for heteroclinic connections

Application to the Chafee-Infante attractor example

Abstract

The goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations, as it was done for the semilinear case by Brunovsk\'y and Fiedler (1986), and generalized by Fiedler and Rocha (1996). In particular, we construct heteroclinic connections between hyperbolic equilibria, stating necessary and sufficient conditions for heteroclinics to occur. Such conditions can be computed through a permutation of the equilibria. Lastly, an example is computed yielding the well known Chafee-Infante attractor.