Lp-asymptotic stability analysis of a 1D wave equation with a nonlinear damping

• Published in: Journal of Differential Equations Vol. 269; no. 10; pp. 8107 - 8131
• Main Authors: Chitour, Yacine, Marx, Swann, Prieur, Christophe
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 05.11.2020
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2020.06.007

This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation with Dirichlet boundary conditions subject to a nonlinear distributed damping within an Lp functional framework, p∈[2,∞]. Some well-posedness results are provided together with exponential decay to zero of trajectories, with an estimation of the decay rate. The well-posedness results are proved by considering an appropriate energy functional in the appropriate functional spaces and introduced by Haraux in [A. Haraux, Int. J. Math. Modelling Num. Opt., 2009]. The asymptotic behavior analysis is based on an attractivity result of a trajectory of an infinite-dimensional linear time-varying system with a special structure, which relies on the introduction of a suitable Lyapunov functional. Note that some of the results of this paper apply for a large class of nonmonotone dampings.