Sharp higher-order $L^2$-asymptotic expansion of solutions to σ-evolution equations with different damping types

In this paper, our main goal is to achieve the high-order asymptotic expansion of solutions to $\sigma$-evolution equations with different damping types in the $L^2$ framework. Throughout this, we observe the influence of parabolic like models corresponding to ${\sigma }_1\in [0,\sigma \mathrm{/}2)$ and $\sigma$-evolution like models corresponding to ${\sigma }_2\in (\sigma \mathrm{/}2,\sigma ]$ on the asymptotic behavior of solutions.