Higher regularity estimates for the porous medium equation near the heat equation

Abstract

In this paper we investigate regularity aspects for solutions of the nonlinear parabolic equation

usually called the porous medium equation. More precisely, we provide sharp regularity estimates for bounded nonnegative weak solutions along the free boundary $\partial \{u>0\}$, when the equation is universally close to the heat equation. As a consequence, local Lipschitz estimates are also established for this scenario.