Stability of a critical nonlinear neutral delay differential equation

Stéphane Junca 1, 2 Bruno Lombard 3
2 COFFEE - COmplex Flows For Energy and Environment
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR7351
3 O&I - Ondes et Imagerie
LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]
Abstract : This work deals with a scalar nonlinear neutral delay differential equation issued from the study of wave propagation. A critical value of the coefficients is considered, where only few results are known. The difficulty follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. An analysis based on the energy method provides new results about the asymptotic stability of the constant and periodic solutions. A complete analysis of the stability diagram is given. Lastly, existence of periodic solutions is discussed, involving a Diophantine condition on the period.
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Stéphane Junca, Bruno Lombard. Stability of a critical nonlinear neutral delay differential equation. Journal of Differential Equations, Elsevier, 2014, 26, pp.2368-2391. ⟨10.1016/j.jde.2014.01.004⟩. ⟨hal-00743970⟩

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