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

On the Cauchy problem for a weakly coupled system of semi-linear $\sigma$-evolution equations with double dissipation