Dynamic Stability of the Kawahara Equation Under the Effect of a Boundary Finite Memory

• Roberto de A. Capistrano-Filho ^[1] ; Boumediène Chentouf ^[2] ; Isadora Maria de Jesus ^[3]
  1. [1] Universidade Federal de Pernambuco
  2. [2] Kuwait University
  3. [3] Universidade Federal de Pernambuco (UFPE) & Universidade Federal de Alagoas (UFAL)

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 1, 2024
• Idioma: inglés
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  • In this work, we are interested in a detailed qualitative analysis of the Kawahara equation, which models numerous physical phenomena such as magneto-acoustic waves in a cold plasma and gravity waves on the surface of a heavy liquid. First, we design a feedback control law, which combines a damping component and another one of finite memory type. Then, we are capable of proving that the problem is wellposed under a condition involving the feedback gains of the boundary control and the memory kernel. Afterward, it is shown that the energy associated with this system exponentially decays by employing two different methods: the first one utilises the Lyapunov function and the second one uses a compactness–uniqueness argument which reduces the problem to prove an observability inequality.