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

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 Hs(R2), s>2, and in the anisotropic spaces Hs1,s2(R2), s2>2, s1⩾s2. We also study the persistence properties of the solution and local well-posedness in the weighted Sobolev classZs,r=Hs(R2)∩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.