# 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.

## Key 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].

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Source: https://tomesphere.com/paper/1903.05329