Some gradient estimates for a diffusion equation on Riemannian manifolds

TL;DR

This paper derives gradient estimates for a diffusion equation on Riemannian manifolds, extending previous results for the heat equation to more general diffusion processes involving a potential function.

Contribution

It generalizes existing gradient estimates for the heat equation to a broader class of diffusion equations with a potential term on Riemannian manifolds.

Findings

Established new gradient bounds for the diffusion equation with potential.

Extended Hamilton's and Zhang's estimates to more general diffusion equations.

Provided tools for analyzing diffusion processes on curved spaces.

Abstract

In this note we present some gradient estimates for the diffusion equation $\partial_t u=\Delta u-\nabla \phi \cdot \nabla u $ on Riemannian manifolds, where $\phi $ is a C^2 function, which generalize estimates of R. Hamilton's and Qi S. Zhang's on the heat equation.