In this work we shall consider the initial value problem associated to the generalized derivative Schrödinger equations ∂tu = i∂ 2x u + µ |u| α ∂xu, x, t ∈ R, 0 < α ≤ 1 and |µ| = 1, andFollowing the argument introduced by Cazenave and Naumkin [3] we shall establish the local well-posedness for a class of small data in an appropriate weighted Sobolev space. The other main tools in the proof include the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation established by Kenig-Ponce-Vega in [21].

In this note we shall continue our study on the initial value problem associated for the generalized derivative Schrödinger (gDNLS) equationx u + µ |u| α ∂xu, x, t ∈ R, 0 < α ≤ 1 and |µ| = 1. Inspiring by Cazenave-Naumkin's works we shall establish the local wellposedness for a class of data of arbitrary size in an appropriate weighted Sobolev space, thus removing the size restriction on the data required in our previous work. The main new tool in the proof is the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation with lower order variable coefficients established by Kenig-Ponce-Vega.Key words and phrases. Derivative nonlinear Schrödinger, local well-posedness. F.L. was partially supported by CNPq and FAPERJ/Brazil.

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In this paper we prove the exponential decay of the energy for the high-order Kadomtsev-Petviashvili II equation with localized damping. To do that, we use the classical dissipation-observability method and a unique continuation principle introduced by Bourgain in [3] here extended for the highorder Kadomtsev-Petviashvili. A similar result is also obtained for the twodimensional Zakharov-Kuznetsov (ZK)equation. The method of proof works better for the ZK equation, so we were led to make some subtle modifications on it to include KP type equations. In fact, to reach a key estimate we use an anisotropic Gagliardo-Nirenberg inequality to drop the y-derivative of the norm.