L^p asymptotic stability of 1D damped wave equation with nonlinear distributed damping

2024/06/17 by Chitour, Yacine, Kafnemer, Meryem, Martinez, Patrick +1
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2406.12085

Abstract

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∈ [1,∞). We start by addressing the well-posedness problem. We prove the existence and the uniqueness of weak and strong solutions for p∈ [1,∞), under suitable assumptions on the damping function. Then we study the asymptotic behaviour of the associated energy when p ∈ (1,∞), and we provide decay estimates that appear to be almost optimal as compared to a similar problem with boundary damping. Our study is motivated by earlier works, in particular, \citeHaraux2009, Chitour-Marx-Prieur-2020. Our proofs combine arguments from \citeKMJC2022 (wave equation in the L^p framework with a linear damping) with a technique of weighted energy estimates (\citePM-COCV) and new integral inequalities when p>2, and with convex analysis tools when p∈ (1,2).

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