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

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.