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

Abstract

We prove existence and uniqueness of global-in-time solutions in the $W_D^{-1,p}$-$W_D^{1,p}$-setting for abstract quasilinear parabolic PDEs with nonsmooth data and mixed boundary conditions, including a nonlinear source term with at most linear growth. Subsequently, we use a bootstrapping argument to achieve improved regularity of these global-in-time solutions within the functional-analytic setting of the interpolation scale of Bessel-potential dual spaces $H_D^{\theta -1,p}=[W_D^{-1,p},L^p{]}_{\theta }$ with $\theta \in [0,1]$ for the abstract equation under suitable additional assumptions. This is done by means of new nonautonomous maximal parabolic regularity results for nonautonomous differential operators operators with Hölder-continuous coefficients on Bessel-potential spaces. The upper limit for $\theta$ is derived from the maximum degree of Hölder continuity for solutions to an elliptic mixed boundary value problem in $L^p$.