  In this article, we prove the global (in time) existence of small data
solutions 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-elastic
damped $\sigma$-evolution models. Here we consider different power
nonlinearities and different $\sigma$ values in the comparison between two
single 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 the
basis of $L^q- L^q$ estimates for solutions, with $m \in [1,q)$, to the
corresponding linear Cauchy problems. In addition, allowing loss of decay and
the flexible choice of parameters $\sigma$, $m$ and $q$ bring some benefits to
relax the restrictions to the admissible exponents $p$.
