  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 $p\in (1,2)$.
