Global well-posedness of the Cauchy problem for a fifth-order KP-I equation in anisotropic Sobolev spaces

Abstract

In this paper, we consider the Cauchy problem for the fifth-order KP-I equation \begin{align*} u_t + \partial_x^5u+\partial_x^{-1}\partial_y^2u + \frac{1}{2}\partial_x(u^2)=0. \end{align*} Firstly, we establish the local well-posedness of the problem in the anisotropic Sobolev spaces $H^{s_1,s_2}({\mathbb{R}}^2)$ with $s_1>-\frac{9}{8}$ and $s_2\ge 0$. Secondly, we establish the global well-posedness of the problem in $H^{s_1,0}({\mathbb{R}}^2)$ with $s_1>-\frac{4}{7}$. Our result improves considerably the results of Saut and Tzvetkov (J. Math.\ Pures Appl.\ 79(2000), 307--338.) and Li and Xiao (J. Math.\ Pures Appl.\ 90(2008), 338--352.) and Guo, Huo and Fang (J. Diff.\ Eqns.\ 263 (2017), 5696--5726).

54pages. arXiv admin note: substantial text overlap with arXiv:1709.01983, arXiv:1709.06077

References in corpus (1)

• Global well-posedness for the KP-I equation on the background of a non localized solution