Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains

TL;DR

This paper proves that solutions to certain nonlinear parabolic equations in bounded radial domains become asymptotically axially symmetric over time, extending symmetry results to equilibria and periodic solutions.

Contribution

It establishes the asymptotic axial symmetry of solutions in bounded radial domains, including equilibria and periodic solutions, under general conditions.

Findings

Solutions become foliated Schwarz symmetric asymptotically

Symmetry holds for equilibria and periodic solutions

Results apply to a broad class of nonlinear parabolic problems

Abstract

We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfy a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega limit set are axially symmetric with respect to a common axis passing through the origin and nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e. solutions of the corresponding elliptic problem) and time periodic solutions.