THE CAUCHY PROBLEM FOR A GENERALIZED KORTEWEG-DE VRIES EQUATION IN HOMOGENEOUS SOBOLEV SPACES

Abstract

Considered in this article is the Cauchy problem of a generalized Korteweg-de Vries equation $$\left\{\begin{array}{l} u_{t} + u_{xxx} + uu_{x} + |D_{x}|^{2\alpha}u = 0, \quad t \in \mathbb{R}^{+},\, x \in \mathbb{R}, \\ u(x,0) = \varphi(x) \end{array}\right. $$ with $0 \leq \alpha \le 1$. The local well-posedness of the Cauchy problem in the homogeneous Sobolev space $\dot{H}^s(\mathbb{R})$ for $s \in (\frac{\alpha-3}{2(2-\alpha)}, 0]$ is proved. In addition, the mapping that associated to appropriate initial-data the corresponding solution is analytic as a function between appropriate Banach spaces.