The fractional porous medium equation on manifolds with conical singularities II

Nikolaos Roidos and Yuanzhen Shao

Mathematische Nachrichten, 2023, vol. 296, issue 4, 1616-1650

Abstract: This is the second of a series of two papers that studies the fractional porous medium equation, ∂tu+(−Δ)σ(|u|m−1u)=0$\partial _t u +(-\Delta )^\sigma (|u|^{m-1}u )=0$ with m>0$m>0$ and σ∈(0,1]$\sigma \in (0,1]$, 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–Poincaré, Nash and Super Poincaré 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∞−$L_\infty -$initial data and all m>0$m>0$. We further investigate a number of properties of the solutions, including comparison principle, Lp−$L_p-$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.

Date: 2023
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