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

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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 Hˢ(²), s>2, and in the anisotropic spaces Hˢ¹,ˢ²(²), s₂>2, s₁≥ s₂. We also study the persistence properties of the solution and local well-posedness in the weighted Sobolev class Zₛ,ᵣ=Hˢ(²)∩ L²((1+x² +y²)ʳdxdy), where s>2, r≥ 0, and s≥ 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.