The focusing problem in nonlinear diffusion: a report from the (moving) front

Professor Don Aronson, University of Minnesota, Department of Mathematics

In the focusing problem for the porous medium equation one solves the initial or initial-boundary value problem with initial data supported outside some compact set K. There is a positive time T when the flow first reaches every point of K, and we are interested in the final form of the empty region for times near T. If K is axially symmetric the problem is completely understood, and the final form is given locally by some member of a one-parameter family of self-similar solutions of the second kind. The axially symmetric interfaces are unstable, and there is an infinite sequence of symmetry breaking bifurcations which lead to new families of self-similar solutions. In two space dimensions, the circular interface is always unstable with respect to perturbations with wave number two. Numerical and formal scaling analysis indicates that the final form is no longer self-similar, but that for elongated holes the major and minor axes obey universal power laws.