The supercritical generalized KdV equation: Global well-posedness in the energy space and below

Abstract

We consider the generalized Korteweg-de Vries (gKdV) equation ${\partial }_{t}u+{\partial }_x^{3}u+\mu {\partial }_x(u^{k+1})=0$, where $k\ge 5$ is an integer number and $\mu =\pm 1$. In the focusing case ($\mu =1$), we show that if the initial data $u_0$ belongs to $H^1(\mathbb{R})$ and satisfies $E(u_0{)}^{s_k}M(u_0{)}^{1-s_k}<E(Q{)}^{s_k}M(Q{)}^{1-s_k}$, $E(u_0)\ge 0$, and $\Vert {\partial }_x{u}_0{\Vert }_{L^2}^{s_k}\Vert u_0{\Vert }_{L^2}^{1-s_k}<\Vert {\partial }_{x}Q{\Vert }_{L^2}^{s_k}\Vert Q{\Vert }_{L^2}^{1-s_k}$, where $M(u)$ and $E(u)$ are the mass and energy, then the corresponding solution is global in $H^1(\mathbb{R})$. Here, $s_k=\frac{(k-4)}{2k}$ and $Q$ is the ground state solution corresponding to the gKdV equation. In the defocusing case ($\mu =-1$), if $k$ is even, we prove that the Cauchy problem is globally well-posed in the Sobolev spaces $H^s(\mathbb{R})$, $s>\frac{4(k-1)}{5k}$.