{"paper":{"title":"The Cauchy problem for two dimensional generalized Kadomtsev-Petviashvili-I equation in anisotropic Sobolev spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jianhua Huang, Jinqiao Duan, Wei Yan, Yongsheng Li","submitted_at":"2017-09-06T20:22:26Z","abstract_excerpt":"The goal of this paper is three-fold. Firstly, we prove that the Cauchy problem for generalized KP-I equation \\begin{eqnarray*}\n  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\n  \\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 probl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.01983","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}