Transformations of Self-Similar Solutions for porous medium equations of fractional type

TL;DR

This paper explores self-similar solutions of fractional porous medium equations, introducing transformations that connect different models and reveal properties like finite propagation, advancing the understanding of nonlinear fractional diffusion.

Contribution

It introduces transformations linking solutions of various fractional porous medium equations, providing new solutions and insights into their properties.

Findings

Transformations map solutions between different fractional models.

New classes of self-similar solutions are constructed.

Finite propagation property is analyzed in these models.

Abstract

We consider four different models of nonlinear diffusion equations involving fractional Laplacians and study the existence and properties of classes of self-similar solutions. Such solutions are an important tool in developing the general theory. We introduce a number of transformations that allow us to map complete classes of solutions of one equation into those of another one, thus providing us with a number of new solutions, as well as interesting connections. Special attention is paid to the property of finite propagation.