We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+\mu\partial_x(u^{ k+1})=0$, where $k\geq5$ is an integer number and $\mu=\pm1$. In the focusing case ($\mu=1$), we show that if the initial data $u_0$ belongs to $H^1(\R)$ and satisfies $E(u_0)^{ s_k} M(u_0)^{ 1-s_k} < E(Q)^{ s_k} M(Q)^{ 1-s_k}$, $E(u_0)\geq0$, and $\|\partial_x u_0\|_{ L^2}^{ s_k}\|u_0\|_{ L^2}^{ 1-s_k} < \|\partial_x Q\|_{ L^2}^{ s_k}\|Q\|_{ L^2}^{ 1-s_k}$, where $M(u)$ and $E(u)$ are the mass and energy, then the corresponding solution is global in $H^1(\R)$. 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 $H^s(\mathbb{ R})$, $s>\frac{ 4(k-1)}{ 5k}$.
Categories: math.AP