#### Vol. 11, No. 5, 2018

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Nonautonomous maximal $L^p$-regularity under fractional Sobolev regularity in time

### Stephan Fackler

Vol. 11 (2018), No. 5, 1143–1169
##### Abstract

We prove nonautonomous maximal ${L}^{p}$-regularity results on UMD spaces, replacing the common Hölder assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on ${L}^{q}\left(\Omega \right)$ we obtain maximal ${L}^{p}$-regularity for $p\ge 2$ and elliptic operators in divergence form with uniform VMO-modulus in space and ${W}^{\alpha ,p}$-regularity for $\alpha >\frac{1}{2}$ in time.

##### Keywords
nonautonomous maximal regularity, parabolic equations in divergence form, quasilinear parabolic problems
##### Mathematical Subject Classification 2010
Primary: 35B65
Secondary: 35K10, 35B45, 47D06