Energy decay for the Klein-Gordon equation with highly oscillating damping

TL;DR

This paper investigates energy decay in the Klein-Gordon equation with periodic damping, demonstrating uniform decay under geometric conditions and improved polynomial decay with highly oscillating damping, using semigroup theory techniques.

Contribution

It provides a parameter-dependent analysis of energy decay for the Klein-Gordon equation with oscillating damping, extending existing results to this specific setting.

Findings

Energy decay is uniform when geometric conditions are met.

Highly oscillating damping can lead to slightly better polynomial decay.

The analysis employs a parameter-dependent version of semigroup theory results.

Abstract

We consider the free Klein-Gordon equation with periodic damping. We show on this simple model that if the usual geometric condition holds then the decay of the energy is uniform with respect to the oscillations of the damping, and in particular the size of the derivatives do not play any role. We also show that without geometric condition the polynomial decay of the energy is even slightly better for a highly oscillating damping. To prove these estimates we provide a parameter dependent version of well known results of semigroup theory.