Optimal decay rate for the wave equation on a square with constant damping on a strip

TL;DR

This paper establishes the precise polynomial decay rate of the energy for the damped wave equation on a square with damping on a strip, answering a previously open question in the field.

Contribution

It proves the exact $t^{-4/3}$ decay rate for the energy of solutions, providing a rigorous answer to a question posed by Anantharaman and Léautaud.

Findings

Proves the $t^{-4/3}$ decay rate for the damped wave equation

Confirms the decay rate is optimal for the given damping configuration

Addresses an open problem in the analysis of damped wave equations

Abstract

We consider the damped wave equation with Dirichlet boundary conditions on the unit square. We assume the damping to be a characteristic function of a strip. We prove the exact $t^{-4/3}$-decay rate for the energy of classical solutions. This answers a question of Anantharaman and L\'eautaud (2014).