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

Abstract

In this short note, we prove that $\alpha$-concavity of the pressure is not preserved for the porous medium equation in dimensions $n=3$ and higher for any $\alpha\in [0,1]\backslash \{\frac{1}{2}\}$. Together with the result of Chau-Weinkove for $n=2$, this fully resolves an open problem posed by V\'asquez on whether pressure concavity is preserved in general for the porous medium equation.