  In this paper, we study the $L^p$-asymptotic stability of the one-dimensional
linear damped wave equation with Dirichlet boundary conditions in $[0,1]$, with
$p\in (1,\infty)$. The damping term is assumed to be linear and localized to an
arbitrary open sub-interval of $[0,1]$. We prove that the semi-group
$(S_p(t))_{t\geq 0}$ associated with the previous equation is well-posed and
exponentially stable. The proof relies on the multiplier method and depends on
whether $p\geq 2$ or $1<p<2$.
