Nonlocal porous medium equation: Barenblatt profiles and other weak solutions

TL;DR

This paper studies a nonlinear nonlocal porous medium equation, establishing the existence of sign-changing weak solutions and constructing explicit self-similar solutions that extend classical Barenblatt profiles.

Contribution

It introduces a new class of solutions for a nonlocal porous medium equation, including sign-changing weak solutions and generalized Barenblatt profiles.

Findings

Existence of sign-changing weak solutions

Construction of explicit self-similar solutions

Generalization of classical Barenblatt profiles

Abstract

A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles --- the well-known solutions of the classical porous medium equation.