On the existence of shock propagation in a flow through deformable porous media

We consider a one-dimensional incompressible flow through a porous medium undergoing deformations such that the porosity and the hydraulic conductivity can be considered to be functions of the flux intensity. The medium is initially dry and we neglect capillarity, so that a sharp wetting front proceeds into the medium. We consider the open problem of the continuation of the solution in the case of onset of singularities, which can be interpreted as a local collapse of the medium, in the general case of convex boundary pressure. We study the behaviour of the solution after the development of a singularity and we study the existence of the solution after the time at which the shock line reaches the surface.