  This paper concerns the Cauchy problems for the nonlinear Rayleigh-Stokes
equation and the corresponding system with time-fractional derivative of order
$\alpha\in(0,1)$, which can be used to simulate the anomalous diffusion in
viscoelastic fluids. It is shown that there exists the critical Fujita exponent
which separates systematic blow-up of the solutions from possible global
existence, and the critical exponent is independent of the parameter $\alpha$.
Different from the general scaling argument for parabolic problems, the main
ingredients of our proof are suitable decay estimates of the solution operator
and the construction of the test function.
