Semigroup decay for the wave equation with unbounded damping

TL;DR

This paper investigates the decay rates of solutions to the damped wave equation with unbounded damping, providing sharp polynomial decay estimates for certain initial conditions despite spectral challenges.

Contribution

It introduces a detailed spectral analysis for unbounded damping in the wave equation, deriving precise polynomial decay rates for solutions.

Findings

Sharp polynomial decay rates for solutions with specific initial conditions

Spectral analysis reveals zero in the spectrum prevents uniform exponential decay

Resolvent norm estimates at low frequencies are key to decay rate determination

Abstract

We study the damped wave equation with a damping coefficient which is possibly singular and unbounded at infinity. In general, zero belongs to the spectrum of the corresponding generator, which prevents a uniform (exponential) decay for the energy. However, for initial conditions in a suitable subspace, a detailed analysis of the resolvent norm for low frequencies leads to sharp polynomial time-decay rates for the solution and its energy.