weakly coupled systems of semi-linear fractional σ evolution equations with mass and different power nonlinearities

In this paper, we are interested to study the global (in time) existence of small data Sobolev solutions to the Cauchy problem for weakly coupled systems of semi-linear fractional sigma-evolution equations with mass and different power nonlinearities. Using L (R)-L-q estimates of Sobolev solutions to related linear models with vanishing right-hand side, we explain connections between regularity assumptions for the data and the admissible range of exponents (p(1),p(2)) in (1.2) which allow to prove the global (in time) existence of small data Sobolev solutions.