We consider the problem of maximal regularity for semilinear non-autonomous fractional equations \(\begin{aligned} \sum _{i=1}^{n} \lambda _i \partial ^{\alpha _i} (u-u_0)(t)+{\mathscr {A}}(t)u(t)=F(t,u(t))\quad t {\text {-a.e.}}, \lambda _i\in {\mathbb {C}}, n\in {\mathbb {N}}. \end{aligned}\) Here, \(\partial ^{\alpha _i} \) denotes the Riemann–Liouville fractional derivative of order \(\alpha _i \in (0,1)\) w.r.t. time and each operator \( {\mathscr {A}}(t)\) arises from a time depending sesquilinear form \(\mathfrak {a}(t)\) on a Hilbert space \({\mathscr {H}}\) with constant domain \({\mathscr {V}},\) such that \({\mathscr {V}}\) is continuously and densely embedded into \({\mathscr {H}}.\) We prove non-autonomous maximal \(L^p\) -regularity results on \({\mathscr {V}}'\) and other regularity properties for the solutions of the above equation under minimal regularity assumptions on the forms, the initial data \(u_0\) and the inhomogeneous term F.
