Optimal leading term of solutions to wave equations with strong damping terms

TL;DR

This paper investigates the long-term behavior of solutions to wave equations with strong damping, establishing optimal bounds and validating the effectiveness of certain expansion methods for such equations.

Contribution

It provides the first rigorous analysis of the asymptotic behavior and optimal bounds for solutions with strong damping, confirming the sharpness of existing estimates.

Findings

Lower bounds for solution differences are established.

Optimality of expansion methods is demonstrated.

Asymptotic behavior of solutions is characterized.

Abstract

We analyze the asymptotic behavior of solutions to wave equations with strong damping terms. If the initial data belong to suitable weighted $L^1$ spaces, lower bounds for the difference between the solutions and the leading terms in the Fourier space are obtained, which implies the optimality of expanding methods and some estimates proposed in this paper.