Differential operators on conic manifolds: maximal regularity and parabolic equations

We study an elliptic differential operator A on a manifold with conic points. Assuming A to be defined on the smooth functions supported away from the singularities, we first address the question of possible closed extensions of A to L^p Sobolev spaces and then explain how additional ellipticity conditions ensure maximal regularity for the operator A. Investigating the Lipschitz continuity of the maps f(u)=|u|^s, s≥1, and f(u)=u^s, s integer, and using a result of Clément and Li, we finally show unique solvability of a quasilinear equation of the form (d/dt - a(u) \Delta) u = f(u), \Delta the Laplacian, in suitable spaces.