Well-posedness for a quasilinear generalization of the matched microstructure model

In this article, we consider a quasilinear matched microstructure model in the sense of Showalter and Walkington (1991) [21] for fluid flow in fractured porous media. It consists of two coupled quasilinear parabolic equations on given bounded domains in Rn, the so-called macro- and microscale. Two cases are investigated: a quasilinear equation in the macroscale and a semilinear one in the other as well as the case of a quasilinear equation on the macroscopic scale combined with an ansatz for the quasilinear form of the operator in the microscale. The proof of well-posedness in a strong Sobolev setting is based on an approach via maximal regularity.