Semi-linear fractional $$\varvec{\sigma }$$ -evolution equations with mass or power non-linearity

In this paper we study the global (in time) existence of small data solutions to semi-linear fractional $$\sigma $$ -evolution equations with mass or power non-linearity. Our main goal is to explain on the one hand the influence of the mass term and on the other hand the influence of higher regularity of the data on qualitative properties of solutions. Using modified Bessel functions we prove some polynomial decay in $$L^p-L^q$$ estimates for solutions to the corresponding linear fractional $$\sigma $$ -evolution equations with vanishing right-hand sides. By a fixed point argument the existence of small data solutions is proved for some admissible range of powers p.