  Extending earlier results on the existence of bounded imaginary powers for
cone differential operators on weighted $L^p$-spaces
$\mathcal{H}^{0,\gamma}_p(\mathbb{B})$ 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
$\mathcal{H}^{s,\gamma}_p(\mathbb{B})$, $s\geq0$. 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 Cl\'ement and Li. We also obtain the short time
asymptotics of the solution near the conical point.
