Characterization of solutions in Besov spaces for fractional Rayleigh–Stokes equations

Abstract

This paper considers fractional Rayleigh–Stokes equations with a power-type nonlinearity. The linear equation can be simulated a non-Newtonian fluid for a generalized second grade fluid and display a nonlocal behavior in time. Because the coexistence of fractional and classical derivatives leads to the lack of semigroup structure of the solution operator, we need to develop a suitable tool to establish some $L^p-L^q$ estimates in the framework of $L^p$ spaces and Besov spaces, respectively. Further, global existence of solutions is showed in spaces of Besov type.