On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping

Abstract

<p style='text-indent:20px;'>In this paper we prove the exponential decay of the energy for the high-order Kadomtsev-Petviashvili II equation with localized damping. To do that, we use the classical dissipation-observability method and a unique continuation principle introduced by Bourgain in [<xref ref-type="bibr" rid="b3">3</xref>] here extended for the high-order Kadomtsev-Petviashvili. A similar result is also obtained for the two-dimensional Zakharov-Kuznetsov (ZK)equation. The method of proof works better for the ZK equation, so we were led to make some subtle modifications on it to include KP type equations. In fact, to reach a key estimate we use an anisotropic Gagliardo-Nirenberg inequality to drop the <inline-formula><tex-math id="M1">\begin{document}$ y $\end{document}</tex-math></inline-formula>-derivative of the norm.</p>