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 $L^p$-spaces ${\mathcal{H}}_p^{0,\gamma }(\mathbb{B})$ 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 ${\mathcal{H}}_p^{s,\gamma }(\mathbb{B})$, $s\ge 0$. As an application we then 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.

21 pages

References in corpus (1)

• The biharmonic heat operator on edge manifolds and non-linear fourth order equations