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In this paper we are concerned with $L^p$-maximal parabolic regularity forabstract nonautonomous parabolic systems and their quasilinear counterpart innegative Sobolev spaces incorporating mixed boundary conditions. Our resultsare derived in the setting of nonsmooth domains with mixed boundary conditionsby an extrapolation technique which also yields uniform estimates for theparabolic solution operators. We require only very mild boundary regularity,not in the Lipschitz-class, and generally only bounded and measurable complexcoefficients. The nonlinear functions in the quasilinear formulation can benonlocal-in-time; this allows also to consider certain systems whose stationarycounterpart fails to satisfy the usual ellipticity conditions.

Nonautonomous maximal parabolic regularity for nonsmooth quasilinear parabolic systems