Preprint series: 06-08, Reports on Analysis

MSC:

35K60 Nonlinear boundary value problems for linear parabolic PDE; boundary value problems for nonlinear parabolic PDE

35B05 General behavior of solutions of PDE (comparison theorems; oscillation, zeros and growth of solutions; mean value theorems)

35K45 Initial value problems for pararabolic systems

35K50 Boundary value problems for parabolic systems

Abstract: 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 $u_x=v_x$, $v=\psi(u)$ 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 $\psi(u)=Mu^\omega$ and derive a formula for the shape of the interface in that case.

Keywords: porous medium equation, contact conditions, existence, uniqueness, Cauchy problem, qualitative behaviour

Upload: 2006-07-07

Update: 2006