Improved regularity for the porous medium equation along zero level-sets

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 $\mathcal{C}^{1-}$ along $\left\lbrace u=0\right\rbrace$, both in time and space. Our argument consists of importing information from the heat equation, through approximation and localization methods.