Energy decay rates for solutions of the wave equation with linear damping in exterior domain

TL;DR

This paper analyzes the decay rates of energy in solutions to the wave equation with localized damping in exterior domains, establishing specific decay behaviors under geometric control conditions.

Contribution

It proves energy decay rates for damped wave equations in exterior domains, extending understanding of damping effects with geometric control assumptions.

Findings

Energy decays like O(1/t) for solutions with general initial data.

Energy and L^2-norm decay faster, like O(1/t^2) and O(1/t), for weighted initial data.

Total energy remains bounded in L^2-norm for solutions with initial data in (H_0^1, L^2).

Abstract

In this paper we study the behavior of the energy of solutions of the wave equation with localized damping in exterior domain. We assume that the damper is positive at infinity. Under the Geometric Control Condition of Bardos et al (1992), we prove that: 1) The total energy decay like O(1/t) and L^2-norm is bounded for the solutions with initial data in (H_{0}^{1},L^{2}). 2) The total energy and the square of the L^2-norm, repectively, decay like O(1/t^{2}) and O(1/t) for a kind of the weighted initial data.