# Stability analysis of a 1D wave equation with a nonmonotone distributed damping

## TL;DR

This paper investigates the stability of a 1D wave equation with a nonmonotone distributed damping, providing well-posedness in nonstandard functional spaces and analyzing the asymptotic behavior of solutions.

## Contribution

It introduces a novel stability analysis for a wave equation with nonmonotone damping, including well-posedness in Lp spaces and asymptotic behavior characterization.

## Key findings

- Well-posedness established in Lp spaces for p in [2, ∞]
- Asymptotic stability characterized via attractivity of a linear time-variant system
- Provides a detailed description of the system's long-term behavior

## 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 p $\in$ [2, $\infty$], and relies mostly on some results collected in Haraux (2009). The asymptotic behavior analysis is based on an attractivity result on a specific infinite-dimensional linear time-variant system.