Lp asymptotic stability of 1D damped wave equation with nonlinear damping

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 begin by addressing the well-posedness problem, establishing the existence and uniqueness of weak and strong solutions for $p\in [1,\infty )$, under suitable assumptions on the damping function.

Next, 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 compared to similar problems with boundary damping.

Our work is motivated by earlier studies, particularly, those by Chitour, Marx and Prieur (2020), and Haraux (1978). The proofs combine arguments from Kafnemer, Mebkhout and Chitour (2022) for wave equation in the $L^p$ framework with a linear damping, techniques of weighted energy estimates introduced in Martinez (1999), new integral inequalities for $p>2$, and convex analysis tools when $p\in (1,2)$.