The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in weighted Sobolev spaces

TL;DR

This paper establishes local well-posedness, persistence, and unique continuation properties for the initial-value problem of the Benjamin-Ono-Zakharov-Kuznetsov equation in various Sobolev and weighted Sobolev spaces, extending known results.

Contribution

It proves local well-posedness in Sobolev and anisotropic spaces and analyzes persistence and unique continuation in weighted Sobolev spaces for the equation.

Findings

Well-posedness in Sobolev spaces $H^{s}( ^2)$ for $s>2$

Persistence properties in weighted Sobolev spaces

Unique continuation principles for solutions

Abstract

In this paper we study the initial-value problem associated with the Benjamin-Ono-Zakharov-Kuznetsov equation. We prove that the IVP for such equation is locally well-posed in the usual Sobolev spaces $H^{s}(\R^2),$ $s>2$, and in the anisotropic spaces $H^{s_1,s_2}(\R^2)$, $s_2>2$, $s_1\geq s_2$. We also study the persistence properties of the solution and local well-posedness in the weighted Sobolev class $$ \mathcal{Z}_{s,r}=H^{s}(\R^{2})\cap L^{2}((1+x^{2} +y^{2})^rdxdy), $$ where $s>2$, $r\geq 0$, and $s\geq 2r$. Unique continuation properties of the solution are also established. These continuation principles show that our persistence properties are sharp. Most of our arguments are accomplished taking into account that ones for the Benjamin-Ono equation.