Parametric dependence of exponents and eigenvalues in focusing porous media flows

DON G. ARONSON, JAN BOUWE VAN DEN BERG, JOSEPHUS HULSHOF
2003 European journal of applied mathematics  
We study the hole-filling problem for the porous medium equation u t = 1 m ∆u m with m > 1 in two space dimensions. It is well known that it admits a radially symmetric self-similar focusing solution u = t 2β−1 F(|x|t −β ), and we establish that the self-similarity exponent β 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
more » ... 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.
doi:10.1017/s0956792503005229 fatcat:23i3n253z5d6ne634v4wi4haba