Parametric dependence of exponents and eigenvalues in focusing porous media flows

摘要

We study the hole-filling problem for the porous medium equation u(1) = 1/m Deltau(m) with m > 1 in two space dimensions. It is well known that it admits a radially symmetric self-similar focusing solution u = t(2beta-1) F(x(-beta)), and we establish that the self-similarity exponent beta is a monotone function of the parameter m. We subsequently use this information to examine in detail the stability of the radial self-similar solution. We show that it is unstable for any m > 1 against perturbations with 2-fold symmetry. In addition, we prove that as m is varied there are bifurcations from the radial solution to self-similar solutions with k-fold symmetry for each k = 3,4,5.... These bifurcations are simple and occur at values m(3) > m(4) > m(5) > (...) --> 1. [References: 25]