The focusing problem for the Eikonal equation

S. B. Angenent () and D. G. Aronson ()
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S. B. Angenent: UW Madison, Mathematics Department
D. G. Aronson: University of Minnesota, School of Mathematics

A chapter in Nonlinear Evolution Equations and Related Topics, 2003, pp 137-151 from Springer

Abstract: Abstract We study the focusing problem for the eikonal quation $$ {\partial _{t}}u = |\nabla u{|^{2}}, $$ i.e., the initial value problem in which the support of the initial datum is outside some compact set in R d . The hole in the support will be filled in finite time and we are interested in the asymptotics of the hole as it closes. We show that in the radially symmetric case there are self-similar asymptotics, while in the absence of radial symmetry essentially any convex final shape is possible. However in R 2 , for generic initial data the asymptotic shape will be either a vanishing triangle or the region between two parabolas moving in opposite directions (a closing eye). We compare these results with the known results for the porous medium pressure equation which approaches the eikonal equation in the limit as m → 1.

Keywords: Free Boundary; Viscosity Solution; Filling Time; Contact Element; Radial Solution (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-7924-8_7

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DOI: 10.1007/978-3-0348-7924-8_7

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