Nonautonomous maximal parabolic regularity for nonsmooth quasilinear parabolic systems

Abstract: In this paper we are concerned with $L^p$-maximal parabolic regularity for abstract nonautonomous parabolic systems and their quasilinear counterpart in negative Sobolev spaces incorporating mixed boundary conditions. Our results are derived in the setting of nonsmooth domains with mixed boundary conditions by an extrapolation technique which also yields uniform estimates for the parabolic solution operators. We require only very mild boundary regularity, not in the Lipschitz-class, and generally only bounded and measurable complex coefficients. The nonlinear functions in the quasilinear formulation can be nonlocal-in-time; this allows also to consider certain systems whose stationary counterpart fails to satisfy the usual ellipticity conditions.