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In this paper, we would like to consider the Cauchy problem for semi-linear$\sigma$-evolution equations with double structural damping for any $\sigma\ge1$. The main purpose of the present work is to not only study the asymptoticprofiles of solutions to the corresponding linear equations but also describelarge-time behaviors of globally obtained solutions to the semi-linearequations. We want to emphasize that the new contribution is to find out thesharp interplay of ``parabolic like models" corresponding to $\sigma_1 \in[0,\sigma/2)$ and ``$\sigma$-evolution like models" corresponding to $\sigma_2\in (\sigma/2,\sigma]$, which together appear in an equation. In thisconnection, we understand clearly how each damping term influences theasymptotic properties of solutions.

On asymptotic properties of solutions to $\sigma$-evolution equations with general double damping