In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi-linear \(\sigma \) -evolution equations with different damping mechanisms for any \(\sigma > 1\) , “parabolic like damping” and “ \(\sigma \) -evolution like damping”. Motivated strongly by the well-known Nakao’s problem, the main goal of this work is to determine the critical curve between the power exponents p and q of nonlinear terms by not only establishing the global well-posedness property of small data solutions but also indicating blow-up in finite time weak solutions. We want to point out that the application of a modified test function associated with a judicious choice of test functions really plays an essential role to show a blow-up result for solutions and upper bound estimates for lifespan of solutions, where \(\sigma \) is assumed to be any fractional real number. To end this paper, lower bound estimates for lifespan of Sobolev solutions are also shown to verify their sharp results in some spatial dimensions.
