Stabilization results for delayed fifth order KdV-type equation in a bounded domain

TL;DR

This paper proves exponential stability for a delayed fifth order KdV-type equation (Kawahara equation) in a bounded domain, using Lyapunov methods and observability inequalities under certain conditions.

Contribution

It introduces new stabilization results for the Kawahara equation with time delays, extending previous work to semi-global stability without restrictive assumptions.

Findings

Solutions are exponentially stable under specific delay conditions.

Energy decays exponentially with small initial data.

Stability holds for sufficiently large domain length T > T_min.

Abstract

Studied here is the Kawahara equation, a fifth order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients we prove that solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the Kawahara system is exponentially stable for $T> T_{\min}$. After that, introducing a more general delayed system, and by introducing suitable energies, we show using Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.