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

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.