Abstract
Local and blowing‐up solutions for the Cauchy problem for a system of space and time fractional evolution equations with time‐nonlocal nonlinearities of exponential growth are considered. The existence and uniqueness of the local mild solution is assured by the Banach fixed point principle. Then, we establish a blow‐up result by Pokhozhaev capacity method. Finally, under some suitable conditions, an estimate of the life span of blowing‐up solutions is established.