Non-preservation of $\alpha$-concavity for the porous medium equation

We show that the porous medium equation does not in general preserve $\alpha$-concavity of the pressure for $0\le\alpha<1/2$ or $1/2<\alpha\le 1$. In particular, this resolves an open problem of V\'azquez 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 $\alpha>0$. 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 $0\le\alpha<1/2$, we give another set of examples to show that concavity can be broken at a boundary point.