The fractional porous medium equation on manifolds with conical singularities I

Optimal regularity in time and space for the porous medium equation

“…Discussion of the literature: In [51], for fractional porous medium equations (PME) the inner Hölder regularity of ∂ t u and (−∆) a/2 Φ(u) is shown for S = 0 under the constraint Φ ∈ C 1,γ (R), γ ∈ (0, 1). The regularity of fractional PMEs on manifolds in the context of L p -spaces has been considered in the works [44,45]. In [45,Theorem 6.2] the authors showed under the condition S ∈ C([0, T ], L p (M)) and u 0 ∈ B 2a−2a/p p,q (M) short time existence and uniqueness of a solution u ∈ L q (0, T ; H 2a (M)) ∩ H 1,q (0, T ; L p (M)), albeit under the additional assumption that the initial condition satisfies u 0 c > 0, in which case the machinery of quasilinear evolution equations by Prüß [43] and Clément-Li [17] based on maximal L p -theory is applicable.…”

The fractional porous medium equation on manifolds with conical singularities II

“…In [56], we initiated the study of (1.1) on conical manifolds, where we proved that operators of the form $$w (-\Delta _g)^\sigma$$ enjoys maximal $$L_p-$$regularity property for some proper strictly positive continuous function w . Then combining with a fixed point theorem argument, we established the existence and uniqueness of classical solution for strictly positive Hölder continuous data.…”