Aronson-B\'enilan estimates for the fast diffusion equation under the Ricci flow

TL;DR

This paper establishes Aronson-Bénilan and Li-Yau-Hamilton type estimates for the fast diffusion equation under Ricci flow on complete manifolds, extending classical results to noncompact settings with bounded curvature.

Contribution

It introduces new differential Harnack estimates for the FDE under Ricci flow and extends Li-Yau-Hamilton estimates to noncompact manifolds for heat equations.

Findings

Proved Aronson-Bénilan estimates for FDE under Ricci flow.

Extended Li-Yau-Hamilton estimates to noncompact manifolds.

Derived differential inequalities for heat and conjugate heat equations.

Abstract

We study the fast diffusion equation (FDE) with a linear forcing term under the Ricci flow on complete manifolds with bounded curvature and nonnegative curvature operator. We prove Aronson-B\'enilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of the FDE. In addition, we use similar method to prove certain Li-Yau-Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained by X. Cao and R. Hamilton, X. Cao, and S. Kuang and Q. Zhang to noncompact setting.