We investigate the global existence of small data solutions to the
Cauchy problem for the semi-linear time fractional σ-evolution equation
models with nonlinearity of derivative type and memory. Based on the Lr
− Lq estimates obtained in linear problem, and combined with the global
iteration method, the global existence of small data solutions is proved
under certain conditions of power p. Furthermore, we find that in the
low-dimensional case, the limit of our conclusion can match the critical
exponent in classical results.