Radial equivalence and study of self-similarity for two very fast diffusion equations

Abstract

In this paper we continue the study of the radial equivalence between the porous medium equation and the evolution p-Laplacian equation, begun in a previous work. We treat the cases m < 0 and p < 1 . We perform an exhaustive study of self-similar solutions for both equations, based on a phase-plane analysis and the correspondences we discover. We also obtain special correspondence relations and self-maps for the limit case m = − 1 , p = 0 , which is particularly important in applications in image processing. We also find self-similar solutions for the very fast p-Laplacian equation that have finite mass and, in particular, some of them that conserve mass, while this phenomenon is not true for the very fast diffusion equation.