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

TL;DR

This paper develops high-order $L^2$-asymptotic expansions for solutions to $\sigma$-evolution equations with various damping types, revealing how different damping influences long-term behavior.

Contribution

It provides the first high-order asymptotic expansions in $L^2$ for $\sigma$-evolution equations considering multiple damping models.

Findings

Parabolic-like damping models significantly affect solution decay rates.

High-order asymptotic expansions accurately describe long-term solution behavior.

Different damping types lead to distinct asymptotic profiles.

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.