Global existence of solutions for weakly coupled systems of semi-linear structurally dampedσ-evolution models

In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped sigma-evolution models with different power nonlinearities. By assuming additional Lm regularity on the initial data, with m is an element of [1, 2), we use (L-m boolean AND (L)2) - L-2 and L-2 - L-2 estimates for solutions to the corresponding linear Cauchy problems to prove the global (in time) existence of small data Sobolev solutions to the weakly coupled systems of semi-linear models from suitable function spaces for any sigma >= 1 and delta is an element of (0, sigma). Moreover, not only the nonexistence of global solution is indicated by using test function method, but also a lifespan estimate is discussed when s and delta is an element of (0, s) are integers.