  In this paper, we study the Cauchy problems for weakly coupled systems of
semi-linear structurally damped $\sigma$-evolution models with different power
nonlinearities. By assuming additional $L^m$ regularity on the initial data,
with $m \in [1,2)$, we use $(L^m \cap L^2)- L^2$ and $L^2- L^2$ estimates for
solutions to the corresponding linear Cauchy problems to prove the global (in
time) existence of small data Sobolev solutions to the weakly coupled systems
of semi-linear models from suitable function spaces.
