Qualitative and numerical study of the stability of a nonlinear time-delayed dispersive equation

TL;DR

This paper analyzes the stability of a nonlinear time-delayed dispersive PDE of order four, proving well-posedness, exponential decay under certain conditions, and supporting findings with numerical simulations.

Contribution

It provides the first comprehensive stability analysis of this specific nonlinear time-delayed dispersive equation, including well-posedness and numerical validation.

Findings

Zero solution exponentially converges to zero with small delay

Well-posedness and regularity of the system established

Numerical illustrations support theoretical stability results

Abstract

This paper deals with the stability analysis of a nonlinear time-delayed dispersive equation of order four. First, we prove the well-posedness of the system and give some regularity results. Then, we show that the zero solution of the system exponentially converges to zero when the time tends to infinity provided that the time-delay is small and the damping term satisfies reasonable conditions. Lastly, an intensive numerical study is put forward and numerical illustrations of the stability result are provided.