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

In this paper, we would like to consider the Cauchy problem for semi-linear $\sigma$-evolution equations with double structural damping for any $\sigma\ge 1$. The main purpose of the present work is to not only study the asymptotic profiles of solutions to the corresponding linear equations but also describe large-time behaviors of globally obtained solutions to the semi-linear equations. We want to emphasize that the new contribution is to find out the sharp 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 this connection, we understand clearly how each damping term influences the asymptotic properties of solutions.