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arXiv · 1908.07138

The fractional porous medium equation on manifolds with conical singularities II

Nikolaos Roidos·Yuanzhen Shao
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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.

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Nikolaos Roidos, Yuanzhen Shao. 2019-08-20. The fractional porous medium equation on manifolds with conical singularities II. https://arxiv.org/abs/1908.07138

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