Flux-saturated porous media equations and applications

Abstract

The aim of this paper is to review the main recent results about the dynamics of nonlinear partial differential equations describing flux-saturated transport mechanisms, eventually in combination with porous media flow and/or reactions terms. The result is a system characterized by the presence of wave fronts which move defining an interface. This can be used to model different process in applications in a variety of areas as Developmental Biology or Astrophysics. The concept of solution and its properties (well-posedness in a Bounded Variation scenario, Rankine–Hugoniot and geometric conditions for jumps, regularity results, finite speed of propagation, …), qualitative study of these fronts (traveling waves in particular) and application in morphogenesis cover the panorama of this review.