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 Zs,r = H s (R 2 ) ∩ L 2 ((1 + x 2 + y 2 ) r dxdy), 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.2010 Mathematics Subject Classification. Primary 35A01, 35Q53 ; Secondary 35Q35.

In this work, we study the initial-value problem associated with the Kuramoto-Sivashinsky equation. We show that the associated initial value problem is locally and globally well-posed in Sobolev spaces H s (R), where s > 1/2. We also show that our result is sharp, in the sense that the flow-map data-solution is not C 2 at origin, for s < 1/2. Furthermore, we study the behavior of the solutions when µ ↓ 0.

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We study the well-posedness in weighted Sobolev spaces, for the initial value problem (IVP) associated with the dissipative Benjamin-Ono (dBO) equation. We establish persistence properties of the solution flow in the weighted Sobolev spaces Zs,r = H s (R) ∩ L 2 (|x| 2r dx), s ≥ r > 0. We also prove some unique continuation properties in these spaces. In particular, such results of unique continuation show that our results of well posedness are sharp.