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 $L^{\infty }$ 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.

本文分析了文献[16]中提出的扩展游隙型模型解的存在性和性质，该模型将滞后作用纳入了通过多孔介质的非饱和流中。当模型进行正则化时，可简化为非线性简并抛物线方程与常微分方程的耦合。这种PDE-ODE耦合系统也在生物膜和细胞生物学领域进行了研究，据我们所知，它具有有趣的数学结构，至今仍未得到充分研究。使用Rothe方法显示了模型的非退化版本的解的存在性。此外，${\mathrm{大号}}^{\infty }$在一定限制下获得解的界。然后，假设初始条件与简并点有界，则显示简并情况的解的存在性。最终，证明了如果存在非规则情形的解，那么它就包含在物理上一致的范围内。