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In this paper, we would like to study the linear Cauchy problems forsemi-linear $\sigma$-evolution models with mixing a parabolic like damping termcorresponding to $\sigma_1 \in [0,\sigma/2)$ and a $\sigma$-evolution likedamping corresponding to $\sigma_2 \in (\sigma/2,\sigma]$. The main goals areon the one hand to conclude some estimates for solutions and their derivativesin $L^q$ setting, with any $q\in [1,\infty]$, by developing the theory ofmodified Bessel functions effectively to control oscillating integralsappearing the solution representation formula in a competition between thesetwo kinds of damping. On the other hand, we are going to prove the global (intime) existence of small data Sobolev solutions in the treatment of thecorresponding semi-linear equations by applying $(L^{m}\cap L^{q})- L^{q}$ and$L^{q}- L^{q}$ estimates, with $q\in (1,\infty)$ and $m\in [1,q)$, from thelinear models. Finally, some further generalizations will be discussed in theend of this paper.

Applications of $L^p-L^q$ estimates for solutions to semi-linear $\sigma$-evolution equations with general double damping