 
Summary: The Focusing Problem for the Eikonal Equation
S. B.Angenent and D. G.Aronson
Dedicated to the Memory of Philippe B´enilan
ABSTRACT. We study the focusing problem for the eikonal equation
tu = u2
,
i.e., the initial value problem in which the support of the initial datum is outside some
compact set in Rd. 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 selfsimilar asymptotics, while in the absence of radial symmetry essentially any
convex final shape is possible. However, 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.
1. Introduction
In this paper we compare the focusing problem for the eikonal equation
(EE) tu = u2
with the corresponding problem for the porous medium (pressure) equation
(PME) tu = (m1)uu+u2
,
