Non-preservation of α-concavity for the porous medium equation

Abstract

We show that the porous medium equation does not in general preserve α-concavity of the pressure for 0≤α<1/2 or 1/2<α≤1. In particular, this resolves an open problem of Vázquez on whether concavity of pressure is preserved by the porous medium equation. Our results strengthen an earlier work of Ishige-Salani, who considered the case of small α>0. Since Daskalopoulos-Hamilton-Lee showed that 1/2-concavity is preserved, our result is sharp.Our explicit examples show that concavity can be instantaneously broken at an interior point of the support of the initial data. For 0≤α<1/2, we give another set of examples to show that concavity can be broken at a boundary point.