Interfaces and Free Boundaries


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Volume 8, Issue 1, 2006, pp. 111–129
DOI: 10.4171/IFB/137

Published online: 2006-03-31

Lower bounds on waiting times for degenerate parabolic equations and systems

Lorenzo Giacomelli[1] and Günther Grün[2]

(1) Università di Roma La Sapienza, Italy
(2) Universität Erlangen-Nünberg, Germany

We extend the method of \cite{GuLoRo00} 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 (submitted for publication in {\it Interfaces and Free Boundaries}).

Keywords: Waiting time, nonlinear higher-order PDE, degenerate parabolic equations, degenerate parabolic systems, thin-film equations, porous medium equations, free boundary problems, lubrication theory

Giacomelli Lorenzo, Grün Günther: Lower bounds on waiting times for degenerate parabolic equations and systems. Interfaces Free Bound. 8 (2006), 111-129. doi: 10.4171/IFB/137