Energy decay for the time dependent damped wave equation

TL;DR

This paper proves that the energy of solutions to the time-dependent damped wave equation on compact manifolds decays exponentially when the damping meets a time-dependent geometric control condition, improving previous inequalities.

Contribution

It introduces a refined time-dependent observability inequality that removes previous technical restrictions, establishing exponential energy decay under broader conditions.

Findings

Energy decays exponentially under the geometric control condition.

Improved observability inequalities without technical initial data assumptions.

Applicable to wave equations on compact Riemannian manifolds.

Abstract

Energy decay is established for the damped wave equation on compact Riemannian manifolds where the damping coefficient is allowed to depend on time. Using a time dependent observability inequality, it is shown that the energy of solutions decays at an exponential rate if the damping coefficient satisfies a time dependent analogue of the classical geometric control condition. Existing time dependent observability inequalities are improved by removing technical assumptions on the permitted initial data.