The fractional porous medium equation on manifolds with conical singularities I

TL;DR

This paper establishes maximal regularity and analyzes the behavior of solutions to the fractional porous medium equation on manifolds with conical singularities, extending operator theory to non-local fractional operators.

Contribution

It introduces a novel approach to handle non-local fractional operators on singular manifolds, proving R-sectoriality and regularity results for the fractional porous medium equation.

Findings

Proves R-sectoriality for fractional powers of non-invertible operators.

Establishes existence and uniqueness of solutions with maximal L^q-regularity.

Analyzes the asymptotic behavior of solutions near singularities.

Abstract

This is the first of a series of two papers which studies the fractional porous medium equation on a Riemannian manifold with isolated conical singularities. In this article, we show $R$-sectoriality for the fractional powers of possibly non-invertible $R$-sectorial operators. Applications concern existence, uniqueness and maximal $L^{q}$-regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided and its relation to the local geometry is established. Our method extends the freezing-of-coefficients method to the case of non-local operators that are expressed as linear combinations of terms in the form of a product of a function and a fractional power of a local operator.