Smoothing effects and extinction in finite time for fractional fast diffusions on Riemannian manifolds

TL;DR

This paper investigates the behavior of solutions to a fractional fast diffusion equation on Riemannian manifolds, demonstrating smoothing effects, infinite propagation speed, and finite-time extinction with sharp rates.

Contribution

It introduces a framework for weak dual solutions on manifolds, establishes smoothing estimates, and analyzes finite-time extinction for fractional fast diffusion equations.

Findings

Solutions exhibit smoothing effects in weighted spaces

Solutions propagate infinitely fast

Finite-time extinction occurs with sharp rates

Abstract

We study nonnegative solutions to the Cauchy problem for the Fractional Fast Diffusion Equation on a suitable class of connected, noncompact Riemannian manifolds. This parabolic equation is both singular and nonlocal: the diffusion is driven by the (spectral) fractional Laplacian on the manifold, while the nonlinearity is a concave power that makes the diffusion singular, so that solutions lose mass and may extinguish in finite time. Existence of mild solutions follows by nowadays standard nonlinear semigroups techniques, and we use these solutions as the building blocks for a more general class of so-called weak dual solutions, which allow for data both in the usual $L^1$ space and in a larger weighted space, determined in terms of the fractional Green function. We focus in particular on a priori smoothing estimates (also in weighted $L^p$ spaces) for a quite large class of weak dual solutions. We also show pointwise lower bounds for solutions, showing in particular that solutions have infinite speed of propagation. Finally, we start the study of how solutions extinguish in finite time, providing suitable sharp extinction rates.