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Long-Time Dynamics of the Zakharov-Kuznetsov Equation

Roberto de A. Capistrano Filho, Ailton Nascimento

TL;DR

The paper analyzes exponential stabilization of the two-dimensional Zakharov--Kuznetsov equation on bounded domains with localized damping and internal delay. It develops two complementary frameworks: (i) a damping-delayed formulation leading to local exponential stabilization for a reduced $\mu_i$-system with explicit decay rates, and (ii) a Lyapunov-based local stabilization plus Lions’ compactness-uniqueness method for global stabilization of the $\mu_i$-system, yielding global exponential decay under suitable damping. The results highlight how delay can destabilize energy unless countered by weighted damping, and they provide sharp decay estimates along with conditions under which they are optimal. The methods show potential for extension to other multidimensional dispersive models via unique continuation techniques. These contributions advance understanding of how damping and delay interact in nonlinear dispersive PDEs and offer practitioners explicit criteria for achieving exponential stabilization.

Abstract

This manuscript presents the results of stabilization for the Zakharov--Kuznetsov equation, a two-dimensional Korteweg--de Vries-type equation. We provide rigorous proofs using two different approaches, showing that when a damping mechanism and an internal delay term (anti-damping) are introduced, the solutions of the Zakharov--Kuznetsov equation exhibit both local and global exponential stabilization properties. A significant contribution of our work is the determination of the optimal constant and the minimal time required to ensure exponential decay of the energy associated with this two-dimensional system.

Long-Time Dynamics of the Zakharov-Kuznetsov Equation

TL;DR

The paper analyzes exponential stabilization of the two-dimensional Zakharov--Kuznetsov equation on bounded domains with localized damping and internal delay. It develops two complementary frameworks: (i) a damping-delayed formulation leading to local exponential stabilization for a reduced -system with explicit decay rates, and (ii) a Lyapunov-based local stabilization plus Lions’ compactness-uniqueness method for global stabilization of the -system, yielding global exponential decay under suitable damping. The results highlight how delay can destabilize energy unless countered by weighted damping, and they provide sharp decay estimates along with conditions under which they are optimal. The methods show potential for extension to other multidimensional dispersive models via unique continuation techniques. These contributions advance understanding of how damping and delay interact in nonlinear dispersive PDEs and offer practitioners explicit criteria for achieving exponential stabilization.

Abstract

This manuscript presents the results of stabilization for the Zakharov--Kuznetsov equation, a two-dimensional Korteweg--de Vries-type equation. We provide rigorous proofs using two different approaches, showing that when a damping mechanism and an internal delay term (anti-damping) are introduced, the solutions of the Zakharov--Kuznetsov equation exhibit both local and global exponential stabilization properties. A significant contribution of our work is the determination of the optimal constant and the minimal time required to ensure exponential decay of the energy associated with this two-dimensional system.

Paper Structure

This paper contains 18 sections, 13 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Problem setting
  4. Notations and main results
  5. Outline of the article
  6. Well-Posedness: An overview
  7. Linear system
  8. Linear system with source term
  9. Nonlinear system: Global results.
  10. The damping-delayed system: Optimal local result
  11. Optimal local stabilization: Proof of Theorem \ref{['th:ZKes']}
  12. -system: Stability results
  13. Local stabilization: Lyapunov approach
  14. Global stabilization: Compactness-uniqueness method
  15. Conclusion and novelty of the article
  16. ...and 3 more sections

Key Result

Theorem 1.2

Assume that the functions $a(x,y)$ and $b(x,y)$ satisfy the conditions stated in Assumption A1. Then there exist constants $\delta>0$, $r>0$, $C>0$, and $\gamma>0$, depending on $T_{\min}$, $\xi$, $L$, and $h$, such that if $\|b\|_{\infty} \leq \delta$, such that for every $(\zeta_{0}, z_{0}) \in \m

Theorems & Definitions (23)

  • Theorem 1.2: Optimal local stabilization
  • Theorem 1.3: Local stabilization
  • Theorem 1.4: Global stabilization
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 13 more