Aronson-B\'enilan estimates for the porous medium equation under the Ricci flow

TL;DR

This paper derives Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for the porous medium equation coupled with Ricci flow on complete manifolds with nonnegative curvature, advancing understanding of nonlinear PDEs in geometric contexts.

Contribution

It introduces new differential Harnack estimates for the porous medium equation under Ricci flow, extending classical results to a geometric setting with nonnegative curvature.

Findings

Established Aronson-Bénilan estimates under Ricci flow.

Derived Li-Yau-Hamilton type inequalities for PME solutions.

Applied results to manifolds with bounded nonnegative curvature.

Abstract

In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete manifolds with bounded nonnegative curvature operator. In particular, we derive Aronson-B\'enilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions to the PME, with a linear forcing term, under the Ricci flow.