Local decay and asymptotic profile for the damped wave equation in the asymptotically Euclidean setting

TL;DR

This paper establishes local decay estimates for the damped wave equation in asymptotically Euclidean spaces, revealing detailed asymptotic profiles and improved decay rates across different dimensions.

Contribution

It provides new decay estimates and asymptotic profiles for the damped wave equation, extending understanding in asymptotically Euclidean settings and for the damped case.

Findings

Established local decay estimates in asymptotically Euclidean spaces.

Derived asymptotic profiles for solutions in even dimensions.

Improved decay rates in odd dimensions beyond previous results.

Abstract

We prove local decay estimates for the wave equation in the asymptotically Euclidean setting. In even dimensions we go beyond the optimal decay by providing the large time asymptotic profile, given by a solution of the free wave equation. In odd dimensions, we improve the best known estimates. In particular, we get a decay rate that is better than what would be the optimal decay in even dimensions. The analysis mainly relies on a comparison of the corresponding resolvent with the resolvent of the free problem for low frequencies. Moreover, all the results hold for the damped wave equation with short range absorption index.