Schwarz symmetrizations in parabolic equations on complete manifolds

TL;DR

This paper extends classical symmetrization and comparison results for elliptic equations to parabolic equations on complete Riemannian manifolds, providing sharp estimates using isoperimetric inequalities.

Contribution

It generalizes Bandle's and Talenti's comparison results from Euclidean spaces to the setting of complete Riemannian manifolds with various Ricci curvature conditions.

Findings

Established sharp estimates for parabolic solutions on manifolds.

Extended Bandle's comparison to noncompact manifolds with nonnegative Ricci curvature.

Generalized Talenti's comparison to parabolic equations on manifolds.

Abstract

In this article, we prove a sharp estimate for the solutions to parabolic equations on manifolds. Precisely, using symmetrization techniques and isoperimetric inequalities on Riemannian manifold, we obtain a Bandle's comparison on complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature respectively. Our results generalize Bandle's result [6] to Riemannian setting, and Talenti's comparison for elliptic equation on manifolds by Colladay-Langford-McDonald [12] and Chen-Li [9] to parabolic equations.