L^p$-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 05.11.2020
• Subjects: Analysis of PDEs; Mathematics
• 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 with an L p 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 functional of the energy in the desired functional spaces introduced by Haraux in [A. Haraux, Int. J. Math. Modelling Num. Opt., 2009]. Asymptotic behavior analysis is based on an attractivity result on 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.