Critical exponent for a weakly coupled system of semi-linear \(\sigma\)-evolution equations with frictional damping

We are interested in studying the Cauchy problem for a weakly coupled system of semi-linear \(\sigma\)-evolution equations with frictional damping. The main purpose of this paper is two-fold. We would like to not only prove the global (in time) existence of small data energy solutions but also indicate the blow-up result for Sobolev solutions when \(\sigma\) is assumed to be any fractional number.