On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative

TL;DR

This paper establishes maximal regularity results for a mixed linear parabolic problem with a fractional Caputo time derivative, extending known results to a broader range of fractional orders.

Contribution

It provides new maximal regularity results for fractional parabolic problems with derivatives of order between 0 and 2, including the classical case when alpha equals 1.

Findings

Maximal regularity in spaces of continuous functions.

Maximal regularity in H"older continuous spaces.

Extension of known results to fractional derivatives with alpha in (0, 2).

Abstract

We prove two maximal regularity results in spaces of continuous and H\"older continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^\alpha$. This derivative is intended in the sense of Caputo and $\alpha$ is taken in $(0, 2)$. In case $\alpha = 1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.