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

2021/12/29 by Roberto de A. Capistrano–Filho, Capistrano-Filho, Roberto de A., Victor H. Gonzalez Martinez +1 · 1 citation
Computer Science · Engineering · Mathematics · #35Q53 #93C20 #93D15 #93D30 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2112.14854

openalex publication_date 2021/12/29 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

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.

Cited by

• Long-Time Dynamics of the Zakharov-Kuznetsov Equation

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