Renormalization study of two-dimensional convergent solutions of the porous medium equation

In the focusing problem, we study a solution of the porous medium equation ut=Δ(um) whose initial distribution is positive in the exterior of a closed noncircular two-dimensional 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 self-similar 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.