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

TL;DR

This paper establishes the existence, uniqueness, and regularity of solutions for quasilinear parabolic PDEs with mixed boundary conditions in a Bessel dual scale framework, using advanced maximal regularity techniques.

Contribution

It introduces new nonautonomous maximal parabolic regularity results on Bessel-potential spaces for PDEs with nonsmooth data and mixed boundary conditions, extending previous theories.

Findings

Proves global-in-time existence and uniqueness of solutions.

Develops improved regularity results via bootstrapping.

Derives new maximal regularity results for nonautonomous operators.

Abstract

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 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 $H^{\theta-1,p}_D = [W^{-1,p}_D,L^p]_\theta$ 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 H\"older-continuous coefficients on Bessel-potential spaces. The upper limit for $\theta$ is derived from the maximum degree of H\"older continuity for solutions to an elliptic mixed boundary value problem in $L^p$.