Well-posedness for the supercritical gKdV equation

Abstract

In this paper we consider the supercritical generalized Korteweg-de Vries equation $\partial_tψ+ \partial_{xxx}ψ+ \partial_x(|ψ|^{p-1}ψ) = 0$, where $5\leq p\in\R$. We prove a local well-posedness result in the homogeneous Besov space $\dot B^{s_p,2}_{\infty}(\mathbb{R})$, where $s_p=\frac12-\frac{2}{p-1}$ is the scaling critical index. In particular local well-posedness in the smaller inhomogeneous Sobolev space $H^{s_p}(\mathbb{R})$ can be proved similarly. As a byproduct a global well-posedness result for small initial data is also obtained.