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

In this paper, we study the $L^p$-asymptotic stability with $p\in (1,\mathrm{\infty })$ 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 $p\in [2,\mathrm{\infty })$ and the existence and the uniqueness of strong solutions for all $p\in [1,\mathrm{\infty })$. The proofs rely on the well-posedness already proved in the $L^{\mathrm{\infty }}$ 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].