Gradient estimates on graphs with the $CD\psi (n,-K)$condition

Abstract

This paper investigates gradient estimates on graphs satisfying the $CD\psi (n,-K)$ condition with positive constants $n,K$, and concave $C^1$ functions $\psi :(0,+\infty )\to \mathbb{R}$. Our study focuses on gradient estimates for positive solutions of the heat equation ${\partial }_{t}u=\Delta u$. Additionally, the estimate is extended to a heat-type equation ${\partial }_{t}u=\Delta u+c{u}^{\sigma }$, where $\sigma$ is a constant and $c$ is a continuous function defined on $[0,+\infty )$. Furthermore, we utilize these estimates to derive heat kernel bounds and Harnack inequalities.