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

Roberto de A. Capistrano-Filho, Boumediène Chentouf*, Luan S. de Sousa, Victor H. Gonzalez Martinez

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 functional and using Lyapunov’s approach, to ensure that the energy of the Kawahara system goes to 0 exponentially as t→ ∞. The second result is achieved by employing a compactness–uniqueness argument, which reduces our study to prove an observability inequality. Furthermore, the novelty of this work is to characterize the critical lengths phenomenon for this equation by showing that the stability results hold whenever the spatial length is related to the Möbius transformations.

Original languageEnglish
Article number16
JournalZeitschrift fur Angewandte Mathematik und Physik
Volume74
Issue number1
DOIs
Publication statusPublished - Feb 2023

Keywords

  • Boundary time-delay
  • Critical set
  • exponential stability
  • Nonlinear Kawahara equation

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)
  • Applied Mathematics

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