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The Cauchy problem for two dimensional generalized Kadomtsev-Petviashvili-I equation in anisotropic Sobolev spaces (1709.01983v2)

Published 6 Sep 2017 in math.AP

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.).

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