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In this article, we prove the global (in time) existence of small datasolutions from energy spaces basing on $L^q$ spaces, with $q \in (1,\infty)$,to the Cauchy problems for a weakly coupled system of semi-linear visco-elasticdamped $\sigma$-evolution models. Here we consider different powernonlinearities and different $\sigma$ values in the comparison between twosingle equations. To do this, we use $(L^m \cap L^q)- L^q$ and $L^q- L^q$estimates, i.e., by mixing additional $L^m$ regularity for the data on thebasis of $L^q- L^q$ estimates for solutions, with $m \in [1,q)$, to thecorresponding linear Cauchy problems. In addition, allowing loss of decay andthe flexible choice of parameters $\sigma$, $m$ and $q$ bring some benefits torelax the restrictions to the admissible exponents $p$.

A note on a weakly coupled system of semi-linear visco-elastic damped  $\sigma$-evolution models with different power nonlinearities and different  $\sigma$ values