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Well-Posedness and Long-Time Dynamics of a Water-Waves Model with Time-Varying Boundary Delay

G. Bautista, R. de A. Capistrano--Filho, B. Chentouf, O. Sierra Fonseca

TL;DR

The paper addresses well-posedness and long-time dynamics of a fifth-order Boussinesq-type water-waves model with a time-varying boundary delay. It combines Kato's variable-norm framework and a transposition-based fixed-point approach to prove local well-posedness for small initial data, and constructs a Lyapunov functional to establish exponential decay for the linearized delayed system. Under explicit param and delay-structure conditions, it also derives an optimal decay rate via a careful optimization of the Lyapunov constants. The results extend previous work by incorporating additional high-order nonlinearities and a delay at the boundary, with implications for stabilization of dispersive systems with time-varying delays. The work highlights both the potential for rapid stabilization in delayed dispersive models and the challenges in extending these results to global-in-time nonlinear dynamics.

Abstract

A higher-order nonlinear Boussinesq system with a time-dependent boundary delay is considered. Sufficient conditions are presented to ensure the well-posedness of the problem by utilizing Kato's variable norm technique and the Fixed-Point Theorem. More significantly, the energy decay for the linearized problem is demonstrated using the energy method.

Well-Posedness and Long-Time Dynamics of a Water-Waves Model with Time-Varying Boundary Delay

TL;DR

The paper addresses well-posedness and long-time dynamics of a fifth-order Boussinesq-type water-waves model with a time-varying boundary delay. It combines Kato's variable-norm framework and a transposition-based fixed-point approach to prove local well-posedness for small initial data, and constructs a Lyapunov functional to establish exponential decay for the linearized delayed system. Under explicit param and delay-structure conditions, it also derives an optimal decay rate via a careful optimization of the Lyapunov constants. The results extend previous work by incorporating additional high-order nonlinearities and a delay at the boundary, with implications for stabilization of dispersive systems with time-varying delays. The work highlights both the potential for rapid stabilization in delayed dispersive models and the challenges in extending these results to global-in-time nonlinear dynamics.

Abstract

A higher-order nonlinear Boussinesq system with a time-dependent boundary delay is considered. Sufficient conditions are presented to ensure the well-posedness of the problem by utilizing Kato's variable norm technique and the Fixed-Point Theorem. More significantly, the energy decay for the linearized problem is demonstrated using the energy method.

Paper Structure

This paper contains 11 sections, 11 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notations and main results
  4. Outline
  5. Well-posedness results
  6. Linear problem
  7. Nonlinear problem
  8. Long-time behavior of solutions
  9. Proof of Theorem \ref{['th:Lyapunov0']}
  10. Decay rate: an optimal result
  11. Conclusion

Key Result

Theorem 1.1

Let $T>0$ and the parameters $a, c, a_1, c_1$ verify conditions_para. Then, there exists $\theta=\theta(T)>0$ such that, for every $\left(\eta_0, \omega_0;z_0\right) \in {X}_3\times L^2(0,1)$ satisfying the system eq:KdV-KdV admits a unique solution $(\eta, \omega) \in C\left([0, T] ; X_3 \right)$. Moreover for some positive constant $C=C(T)$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Kato1970
  • Theorem 2.2
  • proof
  • Claim 1
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 14 more