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

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 ⩾ s 2 . We also study the persistence properties of the solution and local well-posedness in the weighted Sobolev class Z s , r = H s ( R 2 ) ∩ L 2 ( ( 1 + x 2 + y 2 ) r d x d y ) , where s > 2 , r ⩾ 0 , and s ⩾ 2 r . Unique continuation properties of the solution are also established. These continuation principles show that our persistence properties are sharp.