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Extending earlier results on the existence of bounded imaginary powers forcone differential operators on weighted $L^p$-spaces$\mathcal{H}^{0,\gamma}_p(\mathbb{B})$ over a manifold with conicalsingularities, we show how the same assumptions also yield the existence ofbounded imaginary powers on higher order Mellin-Sobolev spaces$\mathcal{H}^{s,\gamma}_p(\mathbb{B})$, $s\geq0$. As an application we thenconsider the Cahn-Hilliard equation on a manifold with (possibly warped)conical singularities. Relying on our work for the case of straight cones, wefirst establish $R$-sectoriality (and thus maximal regularity) for thelinearized equation and then deduce the existence of a short time solution withthe help of a theorem by Cl\'ement and Li. We also obtain the short timeasymptotics of the solution near the conical point.

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