Sharp conditions for exponential and non-exponential uniform stabilization of the time-dependent damped wave equation

TL;DR

This paper establishes sharp conditions under which the damped wave equation with time-dependent damping achieves exponential or non-exponential uniform stabilization, extending classical geometric control results.

Contribution

It generalizes the geometric control condition to time-dependent damping and provides bounds on stabilization rates using observability and Gaussian beam techniques.

Findings

Exponential stabilization occurs under a generalized geometric control condition.

Bounds on non-exponential decay rates are derived for damping not satisfying the control condition.

A new uniform time-dependent observability inequality is introduced.

Abstract

It is classical that uniform stabilization of solutions to the autonomous damped wave equation is equivalent to every geodesic meeting the positive set of the damping, which is called the geometric control condition. In this paper, it is shown that for time-dependent damping a generalization of the geometric control condition is equivalent to uniform stabilization at an exponential rate. Additionally, upper and lower bounds on non-exponential uniform stabilization rates are computed for damping which do not satisfy this geometric control condition. Decay rates are guaranteed via an observability argument, including a new uniform time-dependent observability inequality. Bounds on decay rates are proved via a Gaussian beam construction.