Well-posedness of the Fractional Porous Medium Equation on Manifolds with Conical Singularities

Nikolaos Roidos, Yuanzhen Shao

Published 2019-08-20Version 1

In this article, we consider the fractional porous medium equation, $\partial_t u +(-\Delta)^\sigma (|u|^{m-1}u )=0 $, posed on a Riemannian manifold with isolated conical singularities, with $m>0$ and $\sigma\in (0,1]$. For $L_\infty-$initial data, we establish existence and uniqueness of a global weak solution for all $m>0$, and we show that this solution is strong for $m \geq 1$. We further investigate a number of properties of the solutions, including comparison principle, $L_p-$contraction and conservation of mass. In particular, it is shown that mass is conserved for all $t\geq 0$ and $m>0$. The method in this paper is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.