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The focusing problem for the Eikonal equation

The focusing problem for the Eikonal equation We study the focusing problem for the eikonal equation¶¶ $ \partial _{t}u=\left| \nabla u\right| ^{2}, $ ¶¶i.e., the initial value problem in which the support of the initial datum is outside some compact set in $ \mathbf{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 $ \mathbf{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\rightarrow 1 $ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

The focusing problem for the Eikonal equation

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Publisher
Springer Journals
Copyright
Copyright © 2003 by Birkhäuser Verlag Basel,
Subject
Mathematics; Analysis
ISSN
1424-3199
DOI
10.1007/s000280300006
Publisher site
See Article on Publisher Site

Abstract

We study the focusing problem for the eikonal equation¶¶ $ \partial _{t}u=\left| \nabla u\right| ^{2}, $ ¶¶i.e., the initial value problem in which the support of the initial datum is outside some compact set in $ \mathbf{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 $ \mathbf{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\rightarrow 1 $ .

Journal

Journal of Evolution Equations – Springer Journals

Published: Feb 1, 2003

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