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In this work, we study some special properties of smoothness concerning to the initial value problem associated with the Zakharov-Kuznetsov-(ZK) equation in the n´dimensional setting, n ě 2.It is known thatÂăthe solutions of the ZK equation in the 2d and 3d cases verify special regularity properties. More precisely, the regularity of the initial data on a family of half-spaces propagates with infinite speed. Our objective in this work is to extend this analysis to the case in that the regularity of the initial data is measured on a fractional scale. To describe this phenomenon we presentÂă newÂă localization formulas Âăthat allow us to portray the regularity of the solution on a certain class ofÂăsubsets of the euclidean space. ż R n |∇u(x, t)| 2 dx ´1 3 ż R n (u(x, t)) 3 dx = I 3 [u](0).

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We consider the Zakharov-Kutznesov (ZK) equation posed in R d , with d " 2 and 3. Both equations are globally well-posed in L 2 pR d q. In this paper, we prove local energy decay of global solutions: if uptq is a solution to ZK with data in L 2 pR d q, then lim inffor suitable regions of space Ω d ptq Ď R d around the origin, growing unbounded in time, not containing the soliton region. We also prove local decay for H 1 pR d q solutions. As a byproduct, our results extend decay properties for KdV and quartic KdV equations proved by Gustavo Ponce and the second author. Sequential rates of decay and other strong decay results are also provided as well.

We show that for any uniformly bounded in time H 1 ∩ L 1 solution of the dispersive generalized Benjamin-Ono equation, the limit infimum, as time t goes to infinity, converges to zero locally in an increasing-in-time region of space of order t/ log t. This result is in accordance with the one established by Muñoz and Ponce [20] for solutions of the Benjamin-Ono equation. Similar to solutions of the Benjamin-Ono equation, for a solution of the dispersive generalized Benjamin-Ono equation, with a mild L 1 -norm growth in time, its limit infimum must converge to zero, as time goes to infinity, locally in an increasing on time region of space of order depending on the rate of growth of its L 1 -norm. As a consequence, the existence of breathers or any other solution for the dispersive generalized Benjamin-Ono equation moving with a speed "slower" than a soliton is discarded. In our analysis the use of commutators expansions is essential.