A self-similar solution for the porous medium equation in a two-component domain

We solve a particular system of nonlinear ODEs defined on the two different components of the real line connected by the nonlinear contact condition w′=h′,h=ψ(w)at the point x=0. We show that, for a prescribed power-law nonlinearity ψψ and using the solution (w,h)(w,h), a self-similar solution to the porous medium equation in the two-component domain can be constructed. We provide the rigorous proof of a theorem in which the formula for the free boundary already presented in Filo and Pluschke (2009) [3] is given.