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

Continuous maximal regularity on singular manifolds and its applications