  In this paper, we would like to consider the Cauchy problem for a
multi-component weakly coupled system of semi-linear $\sigma$-evolution
equations with double dissipation for any $\sigma\ge 1$. The first main purpose
is to obtain the global (in time) existence of small data solutions in the
supercritical condition by assuming additional $L^1$ regularity for the initial
data and using multi-loss of decay wisely. For the second main one, we are
interested in establishing the blow-up results together with sharp estimates
for lifespan of solutions in the subcritical case. The proof is based on a
contradiction argument with the help of modified test functions to derive the
upper bound estimates. Finally, we succeed in catching the lower bound estimate
by constructing Sobolev spaces with the time-dependent weighted functions of
polynomial type in their corresponding norms.
