Unsaturated incompressible flows in adsorbing porous media

We study a free boundary problem modelling the penetration of a liquid through a porous material in the presence of absorbing granules. The geometry is one dimensional. The early stage of penetration is considered, when the flow is unsaturated. Since the hydraulic conductivity depends both on saturation and on porosity and the latter change due to the absorption, the main coefficient in the flow equation depends on the free boundary and on the history of the process. Some results have been obtained in Fasano (Math. Meth. Appl. Sci. 1999; 22:605) for a simplified version of the model. Here existence and uniqueness are proved in a class of weighted Hölder spaces in a more general situation. A basic tool are the estimates on a non-standard linear boundary value problem for the heat equation in an initially degenerate domain (Rend. Mat. Acc. Lincei 2002; 13:23).