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

Abstract: 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 $\sigma_1 \in [0, \sigma/2)$ and $\sigma$-evolution like models corresponding to $\sigma_2 \in (\sigma/2, \sigma]$ on the asymptotic behavior of solutions.