Some well‐posed results on the time‐fractional Rayleigh–Stokes problem with polynomial and gradient nonlinearities

Abstract

In this work, we ponder on a Cauchy problem for the Rayleigh–Stokes equation accompanied by polynomial and gradient nonlinearities. We particularly concern about the behavior of mild solutions for the different instances of the nonlinear source term. In the case of polynomial nonlinearities, we present the local‐in‐time existence and uniqueness of the mild solution. Moreover, we claim that either it is the global‐in‐time or it blows up at a finite time. With reference to the case that the source function is global Lipschitzian, we observe that the solution always uniquely exists for a finite time and is continuously dependent. Eventually, we establish some regularity results for the mild solution.