$L^p$ asymptotic stability of 1D damped wave equation with nonlinear distributed damping

In this paper, we study the one-dimensional wave equation with localized nonlinear damping and Dirichlet boundary conditions, in the $L^p$ framework, with $p\in [1,\infty)$. We start by addressing the well-posedness problem. We prove the existence and the uniqueness of weak and strong solutions for $p\in [1,\infty)$, under suitable assumptions on the damping function. Then we study the asymptotic behaviour of the associated energy when $p \in (1,\infty)$, and we provide decay estimates that appear to be almost optimal as compared to a similar problem with boundary damping. Our study is motivated by earlier works, in particular, \cite{Haraux2009, Chitour-Marx-Prieur-2020}. Our proofs combine arguments from \cite{KMJC2022} (wave equation in the $L^p$ framework with a linear damping) with a technique of weighted energy estimates (\cite{PM-COCV}) and new integral inequalities when $p>2$, and with convex analysis tools when $p\in (1,2)$.