Improved regularity for the porous medium equation along zero level sets

Abstract

In the present work we establish sharp regularity estimates for the solutions of the porous medium equation, along their zero level-sets. We work under a proximity regime on the exponent governing the nonlinearity of the problem. Then, we prove that solutions are locally of class $\\mathcal{C}^{1-,\\frac{1}{2}-}$ along free boundary points $x_0\\in \\partial\\left\\lbrace u>0\\right\\rbrace$, both in time and space. Our argument consists of importing information from the heat equation, through approximation and localization methods.