On the stabilization of $L^2$ and $H^1$ norms for the Zakharov-Kuznetsov equation with damping

Abstract

In this paper we establish exponential decay results for solutions of the damped $n$-dimensional Zakharov--Kuznetsov equation for $2\le n\le 3$. More precisely, we prove the exponential decay of the $L^2({\mathbb{R}}^n)$ norm when the damping is localized. In addition, when the dissipative mechanism acts on the whole space ${\mathbb{R}}^n$, we prove the exponential decay of the $H^1({\mathbb{R}}^n)$ norm. Our strategy of proof combines a Kato's type smoothing effect, unique continuation and an observability inequality.