Local existence and nonexistence of fractional Rayleigh–Stokes equations with a superlinear source term

Abstract Fractional Rayleigh–Stokes equations can be described as the viscoelasticity of non‐Newtonian fluids behavior for a generalized second grade fluid. In this paper, we present the monotone iteration method to investigate the nonlinear fractional Rayleigh–Stokes equations from the perspective of the supersolution. More precisely, we analyze the local existence, boundedness, and convergence of nonnegative mild solutions under the superlinear growth conditions. Further, the local nonexistence results of mild solutions are also given.