\(L^p\)-asymptotic stability analysis of a 1D wave equation with a boundary nonmonotone damping

Abstract

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation with a nonlinear non-monotone damping acting at a boundary. The study is performed in an \(L^p\)-functional framework, \(p\in [1,\infty]\). Some well-posedness results are provided together with exponential decay to zero of trajectories, with an estimation of the decay rate. The well-posedness results rely mainly on some results collected in [7]. Asymptotic behavior results are obtained by the use of a suitable Lyapunov functional if \(p\) is finite and on a trajectory-based analysis if \(p=\infty\).