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 conditionux=vx,v=ψ(u)on the contact lineS. First we prove existence and uniqueness of a solution (u,v)(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)=Muωψ(u)=Muω and derive a formula for the shape of the interface in that case.