Energy decay for linear dissipative wave equations in exterior domains

TL;DR

This paper proves that in exterior domains, the total energy of a damped wave equation decays uniformly at rates similar to heat equations, highlighting the importance of damping at infinity for energy decay.

Contribution

It establishes the uniform decay of both local and global energies for damped wave equations in exterior domains, extending previous results to include damping at infinity.

Findings

Uniform decay rates match those of heat equations.

Effective damping at infinity enhances energy decay.

Both local and global energies decay uniformly.

Abstract

In earlier works, we have shown the uniform decay of the local energy of the damped wave equation in exterior domain when the damper is spatially localized near captive rays. In order to have uniform decay of the total energy, the damper has also to act at space infinity. In this work, we establish uniform decay of both the local and global energies. The rates of decay turns out to be the same as those for the heat equation, which shows that an effective damper at space infinity strengthens the parabolic structure in the equation.