Bounded imaginary powers of cone differential operators on higher order Mellin–Sobolev spaces and applications to the Cahn–Hilliard equation

Abstract

Extending earlier results on the existence of bounded imaginary powers for cone differential operators on weighted Lp-spaces Hp0,γ(B) over a manifold with conical singularities, we show how the same assumptions also yield the existence of bounded imaginary powers on higher order Mellin–Sobolev spaces Hps,γ(B), s≥0.As an application we consider the Cahn–Hilliard equation on a manifold with (possibly warped) conical singularities. Relying on our work for the case of straight cones, we first establish R-sectoriality (and thus maximal regularity) for the linearized equation and then deduce the existence of a short time solution with the help of a theorem by Clément and Li. We also obtain the short time asymptotics of the solution near the conical point.