Global well-posedness of the generalized KP-II equation in anisotropic Sobolev spaces

Abstract

In this paper, we consider the Cauchy problem for the generalized KP-II equation \begin{eqnarray*} u_{t}-|D_{x}|^αu_{x}+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,α\geq4. \end{eqnarray*} The goal of this paper is two-fold. Firstly, we prove that the problem is locally well-posed in anisotropic Sobolev spaces H^{s_{1},\>s_{2}}(\R^{2}) with s_{1}>\frac{1}{4}-\frac{3}{8}α, s_{2}\geq 0 and α\geq4. Secondly, we prove that the problem is globally well-posed in anisotropic Sobolev spaces H^{s_{1},\>0}(\R^{2}) with -\frac{(3α-4)^{2}}{28α}<s_{1}\leq0. and α\geq4. Thus, our global well-posedness result improves the global well-posedness result of Hadac (Transaction of the American Mathematical Society, 360(2008), 6555-6572.) when 4\leq α\leq6.

We correct some misprints. arXiv admin note: substantial text overlap with arXiv:1709.01983, arXiv:1712.09334