On local energy decay for large solutions of the Zakharov-Kuznetsov equation

Abstract

We consider the Zakharov-Kutznesov (ZK) equation posed in R d , with d " 2 and 3. Both equations are globally well-posed in L 2 pR d q. In this paper, we prove local energy decay of global solutions: if uptq is a solution to ZK with data in L 2 pR d q, then lim inffor suitable regions of space Ω d ptq Ď R d around the origin, growing unbounded in time, not containing the soliton region. We also prove local decay for H 1 pR d q solutions. As a byproduct, our results extend decay properties for KdV and quartic KdV equations proved by Gustavo Ponce and the second author. Sequential rates of decay and other strong decay results are also provided as well.

On long time behavior of solutions of the Schrödinger-Korteweg-de Vries system

“…Remark 2.2. The approach we follows is closer to the one introduced in [27]. This method shows to be independent of the integrability of the equation and does not need size restriction.…”

“…Remark 2.4. The idea of the proof of Corollary 2.2 follows the argument used in [27]. Thus, it is enough to define the functional…”

“…Proof of Theorem 2.1. We follow the argument of proof presented in [27]. Then, we will consider the following weighted functions:…”

“…New techniques were introduced by Muñoz, Ponce and Saut [30] to investigate the long time behavoir issue for solutions to the Intermediate Long Wave (IWL) equation. Extensions to higher dimensional model as the Zakharov-Kusnetsov (ZK) equation were made by Mendez, Muñoz, Poblete and Pozo in [27].…”