Decay rates for the damped wave equation with finite regularity damping

TL;DR

This paper investigates how the energy of solutions to the damped wave equation on a torus decays over time, showing that decay rates depend on the regularity of the damping function, with smoother damping leading to faster decay.

Contribution

It establishes decay rates for the damped wave equation with finite regularity damping and introduces a pseudodifferential calculus approach for low regularity symbols.

Findings

Energy decays at rate 1/t^{2/3} with invariant damping.

Additional regularity improves decay to 1/t^{4/5- extdelta}.

Method uses positive commutator argument and pseudodifferential calculus.

Abstract

Decay rates for the energy of solutions of the damped wave equation on the torus are studied. In particular, damping invariant in one direction and equal to a sum of squares of nonnegative functions with a particular number of derivatives of regularity is considered. For such damping energy decays at rate $1/t^{2/3}$. If additional regularity is assumed the decay rate improves. When such a damping is smooth the energy decays at $1/t^{4/5-\delta}$. The proof uses a positive commutator argument and relies on a pseudodifferential calculus for low regularity symbols.