  In this paper, we study the $L^p$-asymptotic stability with $p\in (1,\infty)$
of the one-dimensional nonlinear damped wave equation with a localized damping
and Dirichlet boundary conditions in a bounded domain $(0,1)$. We start by
addressing the well-posedness problem. We prove the existence and the
uniqueness of weak solutions for $p\in [2,\infty)$ and the existence and the
uniqueness of strong solutions for all $p\in [1,\infty)$. The proofs rely on
the well-posedness already proved in the $L^\infty$ framework by [4] combined
with a density argument. Then we prove that the energy of strong solutions
decays exponentially to zero. The proof relies on the multiplier method
combined with the work that has been done in the linear case in [8].
