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

Abstract

In this paper, we study the Lp-asymptotic stability of the one dimensional linear damped wave equation with Dirichlet boundary conditions in [0, 1], with p ∈ (1, ∞). 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 (Sp(t))t≥0 associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether p ≥ 2 or 1 < p < 2.