Existence and properties of solutions of the extended play-type hysteresis model
Koondanibha Mitra · Journal of Differential Equations · 2021
This paper analyses the existence and properties of solutions of the extended play-type model which was proposed in [16] to incorporate hysteresis in unsaturated flow through porous media. The model, when regularised, reduces to a nonlinear degenerate parabolic equation coupled to an ordinary differential equation. This PDE–ODE coupled system, also studied in the fields of bio-film and cellular biology, possesses an interesting mathematical structure which, to our knowledge, remains relatively unexplored. The existence of solutions for the non-degenerate version of the model is shown using the Rothe's method. Furthermore, an L∞ bound is obtained for the solutions under certain restrictions. 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 proven that if the solution for the unregularised case exists, then it is contained within physically consistent bounds.