Lower bounds on waiting times for degenerate parabolic equations and systems

Abstract

We extend the method of [19] to obtain quantitative estimates of waiting times for free boundary problems associated with degenerate parabolic equations and systems. Our approach is multi-dimensional, it applies to a large class of equations, including thin-film equations, (doubly) degenerate equations of second and of higher order and also systems of semiconductor equations. For these equations, we obtain lower bounds on waiting times which we expect to be optimal in terms of scaling. This assertion is true for the porous medium equation which seems to be the only PDE for which optimal quantitative estimates of the waiting time have been established so far.