Periodic solutions of a quasilinear parabolic boundary value problem arising in unsaturated flow through a porous medium

Abstract

We prove the existence and uniqueness of periodic solutions of the equation ut = φ(u)xx + b(u)x with Dirichlet and N~umann boundary periodic data. The equation arises, for instance, in the mathematical modelling of the flow of groundwater in a homogeneous isotropic unsaturated porous medium. Those kinds of boundary data represent different periodic behaviours on the top and bottom of the medium due, for instance, to the seasons in the course of the years. The equation arises in many other contexts. We improve precedent results in the literature limited to the non degenerate case ( φ'(0) > 0) and uniformly bounded convection terms.