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

Rayan Fahs, Julien Royer

TL;DR

This work analyzes local decay for the damped wave equation on $\mathbb{R}^d$ in an asymptotically Euclidean setting. It develops a frequency-splitting approach that combines high-frequency propagation (via classical trajectories) with a detailed low-frequency resolvent analysis, culminating in sharp low-frequency bounds by comparing the perturbed resolvent $R(z)=(-\Delta_G-izaw-z^2w)^{-1}$ to the free resolvent $R_0(z)=(-\Delta-z^2)^{-1}$ and employing Mourre theory. In even dimensions, the authors obtain a precise large-time asymptotic profile for the damped solution that coincides with the free-wave evolution, while in odd dimensions they achieve a substantial improvement of the local decay rate. The methodology hinges on a generalized Hardy inequality to exploit spatial decay, elliptic regularity to control low-frequency perturbations, and a Mourre commutator framework to obtain uniform resolvent estimates near the real axis. The results extend to damped waves with short-range absorption and advance the understanding of how asymptotic geometry and damping influence local energy decay and long-time behavior.

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.

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

TL;DR

This work analyzes local decay for the damped wave equation on in an asymptotically Euclidean setting. It develops a frequency-splitting approach that combines high-frequency propagation (via classical trajectories) with a detailed low-frequency resolvent analysis, culminating in sharp low-frequency bounds by comparing the perturbed resolvent to the free resolvent and employing Mourre theory. In even dimensions, the authors obtain a precise large-time asymptotic profile for the damped solution that coincides with the free-wave evolution, while in odd dimensions they achieve a substantial improvement of the local decay rate. The methodology hinges on a generalized Hardy inequality to exploit spatial decay, elliptic regularity to control low-frequency perturbations, and a Mourre commutator framework to obtain uniform resolvent estimates near the real axis. The results extend to damped waves with short-range absorption and advance the understanding of how asymptotic geometry and damping influence local energy decay and long-time behavior.

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.

Paper Structure

This paper contains 7 sections, 36 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. From resolvent estimates to time decay
  3. Strategy of the proof for low frequencies
  4. The generalized Hardy inequality
  5. Elliptic regularity
  6. Mourre commutators method
  7. Additional proof

Key Result

Theorem 1.1

Assume that the damping condition hyp:damping holds. Let $\rho_1 \in ]0,\rho_0[$ and $\delta > d + \frac{5}{2}$. There exists $C > 0$ such that for $f \in H^{1,\delta}$, $g \in L^{2,d}$ and $t \geqslant 0$, we have where $u$ is the solution of the damped wave equation wave and $u_0$ is the solution eq:u0 of the free wave equation wave-free with initial condition

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 52 more