Flat Fronts and Stability for the Porous Medium Equation

Abstract

This work shows stability of general solutions of the porous medium equation close to flat travelling wave fronts in the sense of homogeneous Lipschitz spaces under rather weak conditions on the initial data that are optimal for the techniques used. Consequences include the smoothness of the solutions alongside an exact estimate, analyticity of the solutions in temporal and tangential directions, and analyticity of the interface between empty and occupied regions in time and space. In the course of the argument a Gaussian estimate in an intrinsically arising space of homogeneous type is crucial to obtain linear estimates by means of non-euclidean singular integral theory.