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We prove existence and uniqueness of global-in-time solutions in the$W^{-1,p}_D$-$W^{1,p}_D$-setting for abstract quasilinear parabolic PDEs withnonsmooth data and mixed boundary conditions, including a nonlinear source termwith at most linear growth. Subsequently, we use a bootstrapping argument toachieve improved regularity of these global-in-time solutions within thefunctional-analytic setting of the interpolation scale of Bessel-potential dualspaces $H^{\theta-1,p}_D = [W^{-1,p}_D,L^p]_\theta$ with $\theta \in [0,1]$ forthe abstract equation under suitable additional assumptions. This is done bymeans of new nonautonomous maximal parabolic regularity results fornonautonomous differential operators operators with H\"older-continuouscoefficients on Bessel-potential spaces. The upper limit for $\theta$ isderived from the maximum degree of H\"older continuity for solutions to anelliptic mixed boundary value problem in $L^p$.

Global-in-time solutions for quasilinear parabolic PDEs with mixed boundary conditions in the Bessel dual scale