Nonlinear evolution equations on singular manifolds

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The aim of this project is to study quasilinear parabolic evolution equations on singular spaces in order to obtain a precise understanding of the influence of the singularity on the evolution process. Prototypes of problems we are interested in are the Cahn-Hilliard equation, the porous medium equation and geometric flows. The former two equations have been studied traditionally in domains in Euclidean space, the flows on smooth manifolds. Recently, however, interest in their analysis on singular objects has risen.We will focus on manifolds with conical singularities both with and without boundary, and on manifolds with edges. We are interested in the short and long time existence of solutions, their regularity and asymptotics near the singular set and their long time behavior.Singular analysis has seen a rapid development during the past 30 years. While initially, the analysis mainly focused on linear elliptic problems and applications in index theory, over the past 15 years, tools for parabolic and hyperbolic problems on singular spaces have been developed. Apart from our own contributions we build on important work by Mazzeo and collaborators, Bahuaud and Vertman, and Shao.The principal tools for the linear part of the theory are the pseudodifferential calculi for conically and edge degenerate operators. Here, the basic concepts exist, but new parts will have to be developed for the analysis of the nonlinearities. One has to find suitable closed extensions for the cone Laplacian on manifolds with boundary and for the edge Laplacian, determine the structure of their resolvents and establish maximal $L^p$-regularity; moreover one needs to gain a better understanding of the real interpolation spaces between the base space and the domain of the extensions.In a subsequent step we shall study existence, uniqueness and regularity of short times solutions to the above problems via maximal $L^p$-regularity techniques. As our previous work indicates, singularity effects and asymptotic properties of the solutions near the singular set should already be visible at this point. The next task will be to establish the existence of long time solutions and their asymptotics. We will do this by extending classical techniques like Hölder estimates for quasilinear equations to the singular setting and combining them with maximal $L^p$-regularity theory. Altogether, we hope to obtain a clear view of the behavior of the solutions close to the singularity, in particular we expect to show how the local geometry near the singular set determines the regularity and the asymptotics of the evolution both for short and long times.