Gradient estimates of Hamilton - Souplet - Zhang type for a general heat equation on Riemannian manifolds

TL;DR

This paper derives gradient estimates of Hamilton-Souplet-Zhang type for a general heat equation on noncompact Riemannian manifolds, leading to Harnack inequalities and Liouville theorems, extending previous results in the field.

Contribution

It extends and improves existing gradient estimate results for a broad class of heat equations on Riemannian manifolds, including new applications like Harnack inequalities and Liouville theorems.

Findings

Established gradient estimates for the general heat equation.

Proved a Harnack inequality for heat solutions.

Derived a Liouville type theorem for nonlinear elliptic equations.

Abstract

The purpose of this paper is to study gradient estimate of Hamilton - Souplet - Zhang type for the general heat equation $$ u_t=\Delta_V u + au\log u+bu $$ on noncompact Riemannian manifolds. As its application, we show a Harnak inequality for the heat solution and a Liouville type theorem for a nonlinear elliptic equation. Our results are an extention and improvement of the work of Souplet - Zhang (\cite{SZ}), Ruan (\cite{Ruan}), Yi Li (\cite{Yili}) and Huang-Ma (\cite{HM}).