Two stability results for the Kawahara equation with a time-delayed boundary control

TL;DR

This paper establishes exponential stability for the Kawahara equation with a boundary delay, using Lyapunov and compactness-uniqueness methods, and characterizes the critical length phenomenon related to Möbius transformations.

Contribution

It introduces two different approaches to prove stability of the Kawahara equation with boundary delay and characterizes the critical length set via Möbius transformations.

Findings

Exponential stability under certain domain lengths.

Two distinct proof methods: Lyapunov and compactness-uniqueness.

Characterization of the critical length set related to Möbius transformations.

Abstract

In this paper, we consider the Kawahara equation in a bounded interval and with a delay term in one of the boundary conditions. Using two different approaches, we prove that this system is exponentially stable under a condition on the length of the spatial domain. Specifically, the first result is obtained by introducing a suitable energy and using the Lyapunov approach, to ensure that the unique solution of the Kawahara system exponentially decays. The second result is achieved by means of a compactness-uniqueness argument, which reduces our study to prove an observability inequality. Furthermore, the main novelty of this work is to characterize the critical set phenomenon for this equation by showing that the stability results hold whenever the spatial length is related to the M\"obius transformations.