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The focusing problem for the Eikonal equation

In: Nonlinear Evolution Equations and Related Topics

Author

Listed:
  • S. B. Angenent

    (UW Madison, Mathematics Department)

  • D. G. Aronson

    (University of Minnesota, School of Mathematics)

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.

Suggested Citation

  • S. B. Angenent & D. G. Aronson, 2003. "The focusing problem for the Eikonal equation," Springer Books, in: Wolfgang Arendt & Haïm Brézis & Michel Pierre (ed.), Nonlinear Evolution Equations and Related Topics, pages 137-151, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7924-8_7
    DOI: 10.1007/978-3-0348-7924-8_7
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