The fractional porous medium equation on manifolds with conical singularities II

TL;DR

This paper investigates the fractional porous medium equation on manifolds with conical singularities, establishing fundamental properties of solutions, including existence, uniqueness, and various qualitative behaviors, using advanced functional analysis tools.

Contribution

It develops a comprehensive framework for analyzing the fractional porous medium equation on manifolds with conical singularities, including properties of Mellin-Sobolev spaces and Markovian extensions of the conical Laplacian.

Findings

Established existence and uniqueness of global strong solutions for all positive m.

Proved comparison principle, Lp-contraction, and conservation of mass for solutions.

Developed a general approach applicable to manifolds with various singularities.

Abstract

This is the second of a series of two papers which studies the fractional porous medium equation, $\partial_t u +(-\Delta)^\sigma (|u|^{m-1}u )=0 $ with $m>0$ and $\sigma\in (0,1]$, posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin-Sobolev spaces including the Rellich-Kondrachov Theorem and Sobolev-Poincar\'e, Nash and Super Poincar\'e type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for $L_\infty-$initial data and all $m>0$. We further investigate a number of properties of the solutions, including comparison principle, $L_p-$contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.