LAUSR

We consider initial value problems for abstract evolution equations with fractional time derivative. Concerning the Caputo derivative $$\mathbb {D}^\alpha u$$Dαu, we show that certain assumptions, which are known to be sufficient to get a unique solution with a prescribed regularity, are also necessary. So we establish a maximal regularity result. We consider similar problems with the Riemann–Liouville derivative $$\partial ^\alpha u$$∂αu. Here, we give a complete proof (necessity and sufficiency of the assumptions) of the corresponding maximal regularity results.