Long-Time Dynamics of the Zakharov-Kuznetsov Equation

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.