Asymptotic analysis of some elliptic-parabolic moving boundary problems

We consider a radial elliptic-parabolic initial boundary value problem with ux conditions at the lateral boundary. This problem arises as a model for saturated-unsaturated ow in a porous medium. Using formal methods we describe the non-uniform behaviour of the solutions near the extinction time, which is the time at which the unsaturated zone disappears. Borderline cases are particularly delicate and we include a physically important example in which the methods still apply. A related one-dimensional problem is also analysed, illustrating the widespread applicability of the methods used. We gratefully acknowledge the support of the Dutch Organisation for Scientiic Research (NWO) and the British Council. We thank Juan Luis Vazquez for stimulating discussions.