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# $L^p$-asymptotic stability analysis of a 1D wave equation with a nonlinear damping

3 GIPSA-INFINITY - GIPSA - Infinite Dimensional Dynamics
GIPSA-PAD - GIPSA Pôle Automatique et Diagnostic
Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-02193922
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Submitted on : Friday, October 2, 2020 - 5:54:44 PM
Last modification on : Wednesday, April 27, 2022 - 3:55:56 AM

### Citation

Yacine Chitour, Swann Marx, Christophe Prieur. $L^p$-asymptotic stability analysis of a 1D wave equation with a nonlinear damping. Journal of Differential Equations, Elsevier, 2020, 269 (10), pp.8107-8131. ⟨10.1016/j.jde.2020.06.007⟩. ⟨hal-02193922v2⟩

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