On the initial value problem for the nonlinear fractional Rayleigh-Stokes equation

In this paper, an initial-boundary value problem for the nonlinear fractional Rayleigh-Stokes equation is studied in two cases, namely when the source term is globally Lipschitz or locally Lipschitz. The time-fractional derivative used in this work is the classical Riemann-Liouville derivative. Thanks to the spectral decomposition, a fixed point argument, and some useful function spaces, we establish global well-posed results for our problem. Furthermore, we demonstrate that the mild solution exists globally or blows up in finite time.