Long-Time Dynamics of the Zakharov-Kuznetsov Equation

TL;DR

This paper investigates the stabilization of the two-dimensional Zakharov--Kuznetsov equation, demonstrating exponential decay of solutions through damping and delay mechanisms, and determining optimal constants and minimal stabilization times.

Contribution

It provides rigorous proofs of stabilization using two approaches and identifies optimal constants and minimal times for exponential decay in the system.

Findings

Solutions exhibit local and global exponential stabilization.

Optimal constants for decay are determined.

Minimal time for stabilization is established.

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.