Evolution Equations on Manifolds with Conical Singularities

Abstract

This is an introduction to the analysis of nonlinear evolution equations on manifolds with conical singularities via maximal regularity techniques. We address the specific difficulties due to the singularities, in particular the choice of extensions of the conic Laplacian that guarantee the existence of a bounded $H_\infty$-calculus. We introduce the relevant technical tools and survey, as main examples, applications to the porous medium equation, the fractional porous medium equation, the Yamabe flow, and the Cahn-Hilliard equation.