Non-autonomous maximal regularity for fractional evolution equations

Abstract

We consider the problem of maximal regularity for the semilinear non-autonomous fractional equations $$\begin{aligned} B^\alpha u(t)+A(t)u(t)=F(t,u),\, t \text {-a.e}. \end{aligned}$$ Here $$B^\alpha $$ denotes the Riemann–Liouville fractional derivative of order $$\alpha \in (0,1)$$ w.r.t. time and the time- dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space $${\mathcal {H}}.$$ We prove maximal $$L^p$$ -regularity results and other regularity properties for the solution of the above equation under minimal regularity assumptions on the forms and the inhomogeneous term F.