This paper deals with the semilinear Rayleigh–Stokes equation with the fractional derivative in time of order \(\alpha \in (0,1)\) , which can be used to model anomalous diffusion in viscoelastic fluids. An operator family related to this problem is defined, and its regularity properties are investigated. We firstly give the concept of the mild solutions in terms of the operator family and then obtain the existence of global mild solutions by means of fixed point technique. Moreover, the existence and regularity of classical solutions are given.
