Qualitative Analysis of the Dynamic for the Nonlinear Korteweg–de Vries Equation with a Boundary Memory

Boumediène Chentouf*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


This paper addresses the impact of the presence of a boundary memory term in the third-order Korteweg–de Vries equation in a bounded interval [0 , ℓ]. First, an overall literature review is provided. Indeed, a comprehensive discussion on the literature constitutes a survey part of the current paper. Thereafter, it is shown that the system under consideration possesses a unique solution under a smallness assumption on the initial data and an appropriate condition on the parameters and the kernel involved in the memory term. Last but not least, we demonstrate that the zero solution is exponentially stable as long as the length ℓ is small enough by means of Lyapunov method, which permits to provide an estimate of the exponential decay rate. These findings improve and complement those of Zhang (in: Desch W, Kappel F, Kunisch K (eds) Proceedings of international conference on control and estimation of distributed parameter systems: nonlinear phenomena, International Series of Numerical Mathematics, vol 118, Birkhauser, Basel, pp 371–389, 1994) (resp. Baudouin et al. in IEEE Trans Autom Control 64:1403–1414, 2019), where no memory term is present (resp. a delay occurs instead of memory).

Original languageEnglish
Article number36
JournalQualitative Theory of Dynamical Systems
Issue number2
Publication statusPublished - Jul 2021
Externally publishedYes


  • Boundary memory
  • Decay rate estimate
  • Exponential Stability
  • Nonlinear third-order Korteweg–de Vries equation
  • Well-posedness

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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