On the blow up scenario for a class of parabolic moving boundary problems

We consider maximally continued classical solutions of a large class of parabolic moving boundary problems. If the maximal existence time is finite, we describe the blow up mechanism: either a suitable norm of the bulk density blows up or the geometry of the interface collapses. This can also be seen as a sufficient condition for global in time existence of classical solutions. Moreover, we prove a representation theorem saying, that any closed compact connected hypersurface of Hölder regularity class ck,αck,α can be regarded as a graph over an analytic hypersurface, provided k≥2k≥2.