The porous medium equation in a two-component domain

We consider a system of two porous medium equations defined on two different components of the real line, which are connected by the nonlinear contact condition ux = vx , v = ψ(u) on the contact line S. First we prove existence and uniqueness of a solution (u, v) on a bounded domain. Furthermore, we are interested in the behaviour of the interface of the porous medium equation when it crosses the contact line S between the two components. To this end we solve the Cauchy problem on unbounded components, consider self similar solutions for special ψ(u) =Muω and derive a formula for the shape of the interface in that case.