Hamilton-Souplet-Zhang's gradient estimates for two types of nonlinear parabolic equations under the Ricci flow

TL;DR

This paper derives Hamilton-Souplet-Zhang type gradient estimates for two classes of nonlinear parabolic equations evolving under the Ricci flow, expanding the understanding of their behavior and properties.

Contribution

The paper introduces new gradient estimates for specific nonlinear parabolic equations under Ricci flow, extending previous results to these particular forms.

Findings

Established gradient bounds for equations with logarithmic and power nonlinearities

Extended Hamilton-Souplet-Zhang estimates to Ricci flow context

Provided a framework for analyzing nonlinear parabolic equations under geometric evolution

Abstract

In this paper, we consider gradient estimates for two type of nonlinear parabolic equations under the Ricci flow: one is the equation $$u_t=\Delta u+au\log u+bu$$ with $a,b$ two real constants, the other is $$u_t=\Delta u+\lambda u^{\alpha}$$ with $\lambda,\alpha$ two real constants. By a suitable scaling for the above two equations, we obtain Hamilton-Souplet-Zhang type gradient estimates.