# Sharp polynomial decay rates for the damped wave equation with H\"older-like damping

## TL;DR

This paper establishes sharp polynomial decay rates for the energy of solutions to the damped wave equation on a torus with H"older-like damping, using Morawetz multiplier methods.

## Contribution

It provides the first sharp decay rate results for damped wave equations with boundary-like H"older damping on a torus.

## Key findings

- Decay rate of 1/t^{(eta+2)/(eta+3)} for solutions
- Optimal decay rate when damping vanishes like x^{eta}
- Use of Morawetz multiplier method in the proof

## Abstract

We study decay rates for the energy of solutions of the damped wave equation on the torus. We consider dampings invariant in one direction and bounded above and below by multiples of $x^{\beta}$ near the boundary of the support and show decay at rate $1/t^{\frac{\beta+2}{\beta+3}}$. In the case where $W$ vanishes exactly like $x^{\beta}$ this result is optimal by work of the second author. The proof uses a version of the Morawetz multiplier method.