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Well-posedness for a quasilinear generalization of the matched microstructure model. (English) Zbl 1258.35125

Summary: In this article, we consider a quasilinear matched microstructure model in the sense of R. E. Showalter and N. J. Walkington [J. Math. Anal. Appl. 155, No. 1, 1–20 (1991; Zbl 0711.60078)] for fluid flow in fractured porous media. It consists of two coupled quasilinear parabolic equations on given bounded domains in \(\mathbb R^{n}\), 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.

MSC:

35K59 Quasilinear parabolic equations
35K51 Initial-boundary value problems for second-order parabolic systems

Citations:

Zbl 0711.60078
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References:

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