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 $$u_t=\Delta u^m,m>1$$ 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.

Rev. Mat. Iberoam., to appear