Sturm attractors for quasilinear parabolic equations with singular coefficients

TL;DR

This paper explicitly constructs global attractors for quasilinear parabolic equations with singular boundary coefficients, extending previous work to include singular diffusion and analyzing heteroclinic connections.

Contribution

It introduces a method to construct global attractors for singular quasilinear parabolic equations and characterizes heteroclinic connections via permutation conditions.

Findings

Constructed explicit global attractors with singular coefficients.

Derived necessary and sufficient conditions for heteroclinic connections.

Applied the method to recover the Chafee-Infante attractor.

Abstract

The goal of this paper is to construct explicitly the global attractors of parabolic equations with singular diffusion coefficients on the boundary, as it was done without the singular term for the semilinear case by Brunovsk'y and Fiedler (1986), generalized by Fiedler and Rocha (1996) and later for quasilinear equa- tions by the author (2017). 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.