  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|^\alpha, with real \alpha \ge 1, and
f(u)=u^\alpha, with \alpha a natural number, and using a result of Cl\'ement
and Li, we finally show unique solvability of a quasilinear equation of the
form \dot{u} - a(u) \Delta u = f(u) in suitable spaces.
