Some remarks on the asymptotic profile of solutions to structurally damped $\sigma$-evolution equations

TL;DR

This paper investigates the long-term behavior of solutions to structurally damped $\sigma$-evolution equations, identifying decay rates and asymptotic profiles under various damping conditions, including mixed damping effects.

Contribution

It provides new insights into the asymptotic profiles and decay rates of solutions for different damping regimes in $\sigma$-evolution equations, including mixed damping scenarios.

Findings

Derived approximation formulas for solutions.

Identified optimal decay rates for different damping types.

Analyzed effects of mixed damping on asymptotic behavior.

Abstract

In this paper, we are interested in analyzing the asymptotic profiles of solutions to the Cauchy problem for linear structurally damped $\sigma$-evolution equations in $L^2$-sense. Depending on the parameters $\sigma$ and $\delta$ we would like to not only indicate approximation formula of solutions but also recognize the optimality of their decay rates as well in the distinct cases of parabolic like damping and $\sigma$-evolution like damping. Moreover, such results are also discussed when we mix these two kinds of damping terms in a $\sigma$-evolution equation to investigate how each of them affects the asymptotic profile of solutions.