Stability of a critical nonlinear neutral delay differential equation

TL;DR

This paper investigates the stability of a critical nonlinear neutral delay differential equation related to wave propagation, providing new insights into stability conditions, spectral analysis, and periodic solutions at a critical coefficient value.

Contribution

It offers a comprehensive stability analysis at a critical parameter value, including spectral, energy-based, and periodic solution results, which were previously limited.

Findings

Asymptotic stability of constant solutions established.

Complete stability diagram provided.

Existence of periodic solutions under Diophantine conditions.

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.