Sharp polynomial decay for polynomially singular damping on the torus

TL;DR

This paper investigates energy decay rates for damped wave equations with unbounded, singular damping on the torus, establishing sharp polynomial decay and exploring implications for observability and decay on various manifolds.

Contribution

It provides the first sharp polynomial decay results for polynomially singular damping on the torus and links Schrödinger observability to decay rates for $L^p$-damping.

Findings

Sharp polynomial decay for singular damping on the torus.

Polynomial decay depends on the damping's $L^p$ class.

Exponential decay achieved with polynomially controlled singular damping on the circle.

Abstract

We study energy decay rates for the damped wave equation with unbounded damping, without the geometric control condition. Our main decay result is sharp polynomial energy decay for polynomially controlled singular damping on the torus. We also prove that for normally $L^p$-damping on compact manifolds, the Schr\"odinger observability gives $p$-dependent polynomial decay, and finite time extinction cannot occur. We show that polynomially controlled singular damping on the circle gives exponential decay.