The Cauchy problem for two dimensional generalized Kadomtsev-Petviashvili-I equation in anisotropic Sobolev spaces

Abstract

The goal of this paper is three-fold. Firstly, we prove that the Cauchy problem for generalized KP-I equation \begin{eqnarray*} u_{t}+|D_{x}|^{\alpha}\partial_{x}u+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,\alpha\geq4 \end{eqnarray*} is locally well-posed in the anisotropic Sobolev spaces$ H^{s_{1},\>s_{2}}(\R^{2})$ with $s_{1}>-\frac{\alpha-1}{4}$ and $s_{2}\geq 0$. Secondly, we prove that the problem is globally well-posed in $H^{s_{1},\>0}(\R^{2})$ with $s_{1}>-\frac{(\alpha-1)(3\alpha-4)}{4(5\alpha+3)}$ if $4\leq \alpha \leq5$. Finally, we prove that the problem is globally well-posed in $H^{s_{1},\>0}(\R^{2})$ with $s_{1}>-\frac{\alpha(3\alpha-4)}{4(5\alpha+4)}$ if $\alpha>5$. Our result improves the result of Saut and Tzvetkov (J. Math. Pures Appl. 79(2000), 307-338.) and Li and Xiao (J. Math. Pures Appl. 90(2008), 338-352.).