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}|^α\partial_{x}u+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,α\geq4 \end{eqnarray*} is locally well-posed in the anisotropic Sobolev spaces$H^{s_1,s_2}({\mathbb{R}}^2)$ with $s_1>-\frac{\alpha -1}{4}$ and $s_2\ge 0$. Secondly, we prove that the problem is globally well-posed in $H^{s_1,0}({\mathbb{R}}^2)$ with $s_1>-\frac{(\alpha -1)(3\alpha -4)}{4(5\alpha +3)}$ if $4\le \alpha \le 5$. Finally, we prove that the problem is globally well-posed in $H^{s_1,0}({\mathbb{R}}^2)$ 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.).

57 pages