Aronson-B\'{e}nilan and Harnack estimates for the discrete porous medium equation

TL;DR

This paper extends Aronson-Bénilan and Harnack estimates to the discrete porous medium equation on graphs by establishing curvature-dimension conditions, enabling analysis similar to continuous cases.

Contribution

It introduces new curvature-dimension conditions for graphs that ensure Aronson-Bénilan and Harnack estimates for the discrete porous medium equation.

Findings

Discrete Aronson-Bénilan estimates established under new CD conditions.

Harnack inequalities derived for positive solutions on graphs.

Applicable to various graph structures like complete and chain graphs.

Abstract

We consider the porous medium equation (PME) on a locally finite graph and identify suitable curvature-dimension (CD) conditions under which a discrete version of the fundamental Aronson-B\'enilan estimate holds true for positive solutions of the PME. We also show that these estimates allow to prove Harnack inequalities which are structurally similar to the continuous case. The new CD conditions are illustrated with several concrete examples, e.g.\ complete and chain-like graphs.