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 $$ u\_t= \Delta u^m, \quad m > 1, $$ 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.