Aronson–Bénilan and Harnack estimates for the discrete porous medium equation

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–Bénilan estimate holds true for positive solutions of the PME on finite graphs. 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.