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

In this paper we investigate regularity aspects for solutions of the nonlinear parabolic equation $$u_t= \Delta u^m, \quad m > 1$$ 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.