 
Summary: Physica D 138 (2000) 344359
Renormalization study of twodimensional convergent solutions of the
porous medium equation
S.I. Betelú a,
, D.G. Aronson a
, S.B. Angenent b
a School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
b Mathematics Department, University of Wisconsin, Madison, WI 53706, USA
Received 16 August 1999; accepted 13 October 1999
Communicated by C.K.R.T. Jones
Abstract
In the focusing problem, we study a solution of the porous medium equation ut = (um) whose initial distribution is
positive in the exterior of a closed noncircular twodimensional region, and zero inside. We implement a numerical scheme
that renormalizes the solution each time that the average size of the empty region reduces by a half. The initial condition
is a function with circular level sets distorted with a small sinusoidal perturbation of wave number k > 3. We find that
for nonlinearity exponents m smaller than a critical value which depends on k, the solution tends to a selfsimilar regime,
characterized by rounded polygonal interfaces and similarity exponents that depend on m and on the discrete rotational
symmetry number k. For m greater than the critical value, the final form of the interface is circular. ©2000 Elsevier Science
B.V. All rights reserved.
PACS: 02.70.c; 05.10.Cc; 05.10.a; 47.55.Mh
