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

TL;DR

This paper derives sharp regularity and Lipschitz estimates for solutions of the porous medium equation near the heat equation, enhancing understanding of solution behavior close to linear diffusion.

Contribution

It provides new regularity estimates for solutions near the heat equation, specifically at the free boundary, which was not previously well-understood.

Findings

Sharp regularity estimates near the free boundary

Local Lipschitz continuity established

Enhanced understanding of solution behavior close to 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.