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arXiv · 2303.04659

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

Fabian Hoppe·Hannes Meinlschmidt·Ira Neitzel
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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 $L^p$.

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Fabian Hoppe, Hannes Meinlschmidt, Ira Neitzel. 2023-03-08. Global-in-time solutions for quasilinear parabolic PDEs with mixed boundary conditions in the Bessel dual scale. https://arxiv.org/abs/2303.04659

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