Existence and properties of solutions of extended play-type hysteresis model

Abstract: This paper analyses the well-posedness and properties of the extended play-type model which was proposed in [van Duijn & Mitra (2018)] to incorporate hysteresis in unsaturated flow through porous media. The model, when regularised, reduces to a nonlinear degenerate parabolic equation coupled with an ordinary differential equation. This has an interesting mathematical structure which, to our knowledge, still remains unexplored. The existence of solutions for the non-degenerate version of the model is shown using the Rothe's method. An equivalent to maximum principle is proven for the solutions. Existence of solutions for the degenerate case is then shown assuming that the initial condition is bounded away from the degenerate points. Finally, it is shown that if the solution for the unregularised case exists, then it is contained within physically consistent bounds.