Fractional porous medium equation on manifolds with nonnegative Ricci curvature: existence of solutions and smoothing effects via potential methods

Abstract

We study the fractional porous medium equation on complete noncompact Riemannian manifolds with nonnegative Ricci curvature for $s\in(0,1]$ and $n>2s$. Assuming that the manifold is $s$-nonparabolic, so that the fractional Laplacian admits a suitable positive minimal Green function, we use this Green function to introduce a natural weighted space of initial data, strictly larger than $L^1$. This leads to a weak dual, or potential, formulation of the equation, for which we prove existence for nonnegative initial data in the weighted space. We then establish quantitative local smoothing estimates for initial data in either $L^1$ or the Green-weighted space. Under additional noncollapsing and uniform volume-growth assumptions, we obtain global smoothing estimates. We also show that, even in $\mathbb{R}^n$, data in the Green-weighted space need not generate bounded solutions unless a suitable uniform weighted integrability condition is imposed. The short- and long-time behaviors predicted by our estimates are shown to be optimal in appropriate senses. When $s\in(0,1/2)$, the results require only $\operatorname{Ric}\geq0$, together with a noncollapsing assumption where needed. Our estimates also cover the case $s=1$, several of them being new in that setting. Finally, we extend the approach to more general filtration equations.