  We introduce an $R$-sectoriality perturbation technique for non-commuting
operators defined in Bochner spaces. Based on this and on bounded
$H^{\infty}$-functional calculus results for the Laplacian on manifolds with
conical singularities, we show maximal $L^{q}$-regularity for the Laplacian on
manifolds with edge type singularities in appropriate weighted Sobolev spaces.
As an application, we consider the porous medium equation on manifolds with
edges and show short time existence, uniqueness and maximal regularity for the
solution. We also provide space asymptotics near the singularities in terms of
the local geometry.

Analysis of PDEs;Functional Analysis