Asymptotic expansion for a class of second-order evolution equations in the energy space and its applications

TL;DR

This paper derives the asymptotic expansion of solutions to second-order evolution equations in energy space, improving previous regularity requirements, and applies these results to establish optimal decay rates for damped wave equations in exterior domains.

Contribution

It provides a new asymptotic expansion for solutions in energy space, extending previous results, and applies this to obtain optimal decay estimates for damped wave equations.

Findings

Established asymptotic expansion for solutions in energy space.

Derived optimal decay rates for energy functionals.

Obtained decay estimates for local energy in exterior domains.

Abstract

In this paper, we mainly discuss asymptotic profiles of solutions to a class of abstract second-order evolution equations of the form $u''+Au+u'=0$ in real Hilbert spaces, where $A$ is a nonnegative selfadjoint operator. The main result is the asymptotic expansion for all initial data belonging to the energy space, which is naturally expected. This is an improvement for the previous work (required a sufficient regularity). As an application, we focus our attention to damped wave equations in an exterior domain in $\mathbb{R^N (N \geq 2)$ with the Dirichlet boundary condition. By using the asymptotic expansion in the present paper, we could derive the optimal decay rates of energy functional of solutions to damped wave equations. Moreover, some decay estimates of the local energy (energy functional restricted in a compact subset) can be observed via the asymptotic expansion.