Exponential decay for the damped wave equation in unbounded domains

TL;DR

This paper investigates the decay rates of solutions to the damped wave equation in unbounded domains, establishing conditions for exponential and logarithmic decay, and extending previous stabilization and control results.

Contribution

It provides new decay estimates under geometric control conditions and weaker assumptions, broadening the understanding of wave stabilization in unbounded domains.

Findings

Exponential decay under geometric control condition

Logarithmic decay under weaker conditions with smooth initial data

Extensions of stabilization and control results

Abstract

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilisation and control.