European Journal of Applied Mathematics

Research Article

A selfsimilar solution to the focusing problem for the porous medium equation

D. G. Aronsona1 and J. Graveleaua2

a1 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

a2 Centre d'Etudes Nucléaires, Commissariat á l'Energie Atomique, 23700 Pierrelatte, France

Abstract

In the focusing problem we seek a solution to the porous medium equation whose initial distribution is in the exterior of some compact set (e.g. a ball). At a finite time T the gas will reach all points of the initially empty region R. We construct a selfsimilar solution of the radially symmetric focusing problem. This solution is an example of a selfsimilar solution of the second kind, i.e. one in which the similarity variable cannot be determined a priori from dimensional considerations. Our solution also shows that in more than one space dimension, the velocity of the gas is infinite at the centre of R at the focusing time T.

(Received May 05 1992)