Lp-asymptotic stability of 1D damped wave equations with localized and linear damping

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 $p\in (1,\mathrm{\infty })$. 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 $(S_p(t){)}_{t\ge 0}$ associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether $p\ge 2$ or $1<p<2$.