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

TL;DR

This paper establishes global well-posedness results for the supercritical generalized KdV equation in the energy space and below, using variational methods and Sobolev space analysis for both focusing and defocusing cases.

Contribution

It proves global existence and well-posedness of the gKdV equation for high nonlinearity powers in the energy space and below, extending previous results to supercritical regimes.

Findings

Global well-posedness in energy space for focusing case with initial data below ground state.

Global well-posedness in Sobolev spaces for even k in the defocusing case.

Conditions involving mass and energy that guarantee global solutions.

Abstract

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 $H^s(\mathbb{R})$, $s>\frac{4(k-1)}{5k}$.