In this paper, we would like to study the Cauchy problem for linear σ-evolution equations with mixing the parabolic like damping term corresponding to σ 1 ∈ [ 0 , σ / 2 ) and the σ-evolution like damping corresponding to σ 2 ∈ ( σ / 2 , σ ] . The main goals are on the one hand to conclude some estimates for solutions and their derivatives in the L q setting, with any q∈[1 ,∞], by developing the theory of modified Bessel functions effectively to control Fourier multipliers appearing the solution representation formula in a competition between these two kinds of damping. On the other hand, we are going to prove the global (in time) existence of small data Sobolev solutions in the treatment of the corresponding semi-linear equations by applying ( L m ∩ L q ) - L q and L q - L q estimates, with q∈(1 ,∞) and m∈[1 ,q), from the linear models. Thanks to flexible choices of parameters q,m and even suitably required regularities, one recognizes that not only some restrictions for power exponents can be relaxed, but also they allow us to conclude an existence result for global solutions with arbitrarily small regularity in terms of dealing with the semi-linear equations.