Some existence-uniqueness results for a class of one-dimensional nonlinear Biot models Original Research Article

The wave propagation in a poro-elastic medium is generally described by a Biot model. This model couples the displacement in the solid structure with the fluid pressure and the most complete system involves coupled equations which are mixed hyperbolic–parabolic. In this paper, we are interested in including nonlinearities in the displacement equation. We restrict our study to the one-dimensional case and we establish existence and uniqueness results in Sobolev spaces using Galerkin approximants. The quasi-static case is also investigated. The hyperbolic character is then suppressed and we get the well-posedness of the system with data less regular than the complete model. But, we also prove that the complete model may be considered as an approximation of the quasi-static model.