  In this article, we set up the continuous maximal regularity theory for a
class of linear differential operators on manifolds with singularities. These
operators exhibit degenerate or singular behaviors while approaching the
singular ends. Particular examples of such operators include differential
operators defined on domains, which degenerate fast enough toward the boundary.
Applications of the theory established herein are shown to the Yamabe flow, the
porous medium equation, the parabolic $p$-Laplacian equation and the thin film
equation. Some comments about the boundary blow-up problem, and waiting time
phenomena for singular or degenerate parabolic equations can also be found in
this paper.
