Asymptotic symmetry for a class of quasi-linear parabolic problems

TL;DR

This paper investigates the symmetry of solutions to certain quasi-linear elliptic and parabolic problems, showing that solutions tend to radially symmetric forms in spherical domains, with implications for their long-term behavior.

Contribution

It establishes asymptotic symmetry results for a class of quasi-linear problems, linking elliptic and parabolic solution behaviors in symmetric domains.

Findings

Solutions become radially symmetric in spherical domains.

Global solutions tend to stationary radially symmetric solutions.

The study connects elliptic symmetry with parabolic asymptotic behavior.

Abstract

We study the symmetry properties of the weak positive solutions to a class of quasi-linear elliptic problems having a variational structure. On this basis, the asymptotic behaviour of global solutions of the corresponding parabolic equations is also investigated. In particular, if the domain is a ball, the elements of the $\omega$-limit set are nonnegative radially symmetric solutions of the stationary problem.