Stability Analysis of a 1D Wave Equation with a Nonmonotone Distributed Damping

Abstract

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation subject to a nonmonotone distributed damping. A well-posedness result is provided together with a precise characterization of the asymptotic behavior of the trajectories of the system under consideration. The well-posedness is proved in the nonstandard $L^p$ functional spaces, with $pin [2,infty]$, and relies mostly on some results collected in cite{haraux1D}. The asymptotic behavior analysis is based on an attractivity result on a specific infinite-dimensional linear time-variant system.