Symmetrization and anti-symmetrization in parabolic equations

TL;DR

This paper investigates symmetry properties of solutions to parabolic equations, providing quantitative estimates of level set shapes and criteria for non-spherical convergence in certain reaction-diffusion models.

Contribution

It introduces new symmetrization techniques and criteria for non-spherical level set behavior in parabolic equations, extending previous results.

Findings

Quantitative estimate of level set sphericity

Criterion for non-spherical level set convergence

Application to Fisher-KPP reaction-diffusion equations

Abstract

We derive some symmetrization and anti-symmetrization properties of parabolic equations. First, we deduce from a result by Jones a quantitative estimate of how far the level sets of solutions are from being spherical. Next, using this property, we derive a criterion providing solutions whose level sets do not converge to spheres for a class of equations including linear equations and Fisher-KPP reaction-diffusion equations.