Gradient estimates for the weighted porous medium equation on graphs

TL;DR

This paper develops gradient estimates for positive solutions of a nonlinear weighted porous medium equation on graphs, leading to Harnack inequalities and kernel estimates, extending previous heat equation results.

Contribution

It introduces gradient estimates for the weighted porous medium equation on graphs, a nonlinear extension of prior heat equation analyses.

Findings

Established gradient estimates for solutions on graphs

Derived Harnack inequalities for the equation

Provided estimates for the porous medium kernel on graphs

Abstract

In this paper, we study the gradient estimates for the positive solutions of the weighted porous medium equation $$\Delta u^{m}=\delta(x)u_{t}+\psi u^{m}$$ on graphs for $m>1$, which is a nonlinear version of the heat equation. Moreover, as applications, we derive a Harnack inequality and the estimates of the porous medium kernel on graphs. The obtained results extend the results of Y. Lin, S. Liu and Y. Yang for the heat equation [8, 9].