The Fractional Porous Medium Equation on noncompact Riemannian manifolds

TL;DR

This paper investigates solutions to the fractional porous medium equation on various noncompact Riemannian manifolds, establishing existence and smoothing estimates for a broad class of initial data, including Euclidean and hyperbolic spaces.

Contribution

It extends the analysis of the fractional porous medium equation to a wider class of noncompact manifolds and larger initial data spaces than previously studied.

Findings

Existence of solutions on noncompact Riemannian manifolds.

Smoothing estimates for solutions with initial data in larger spaces.

Results include Euclidean and hyperbolic spaces.

Abstract

We study nonnegative solutions to the Fractional Porous Medium Equation on a suitable class of connected, noncompact Riemannian manifolds. We provide existence and smoothing estimates for solutions, in an appropriate weak (dual) sense, for data belonging either to the usual $L^1$ space or to a considerably larger weighted space determined in terms of the fractional Green function. The class of manifolds for which the results hold include both the Euclidean and the hyperbolic spaces and even in the Euclidean situation involve a class of data which is larger than previously known one.