A free boundary problem in an absorbing porous material with saturation dependent permeability

The authors consider a free boundary describing the one-dimensional propagation of a liquid into a porous medium containing absorbing granules (diapers are the specific application). The situation here studied is the first stage of penetration, when the flow is unsaturated. The ultimate form of the model is a Stefan-like problem for a parabolic equation in which the main coefficient depends in a nonlinear way on the free boundary and on the history of the problem. Indeed, the absorption process, occurring in the wet region, modifies the medium porosity, influencing the rheological parameters of the system. Here the authors prove an existence theorem, allowing k to be a twice continuously differentiable function of the saturation