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On non-autonomous and quasilinear parabolic equations

The lecture is concerned with extrapolation principles, both for elliptic and parabolic
system operators, based on the famous Snejberg theorem. First it is proved that

-{\rm div} (\mu \nabla \cdot) + \lambda :\mathbb{W}^{1,q}_D(\Omega) \to \mathbb{W}^{-1,q}_D(\Omega),

being an isomorphism for $q=2$ by Lax-Milgram, extrapolates under minimal assumptions to $q$'s close to $2$ and even to differential indices close, but different from $\pm 1$.
Secondly, we prove that spaces of maximal parabolicity interpolate as the corresponding $L^r(J;X)$ spaces, provided that there exists one \emph{autonomous} reference operator which satisfies maximal parabolic regularity on both Banach spaces $X_\pm$ under consideration.
Having this at hand, one may show that even \emph{non-autonomous} maximal parabolic regularity, when being known for \emph{one} interpolation index $\theta$ in case of the spaces of maximal parabolic regularity, extrapolates to neighbouring ones.
This can substantially improve the knowledge on the functional analytic quality for the solution for non-autonomous parabolic systems.

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08. Dezember 2022, 13:45-15:10

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