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

Considered in this article is the Cauchy problem of a generalized Korteweg-de Vries equation {u(t) + u(xxx) + uu(x) + vertical bar D(x)vertical bar(2 alpha)u = 0, t is an element of R(+), t is an element of R(+), x is an element of R, u(x, 0) = phi(x) with 0 <= alpha <= 1. The local well-posedness of the Cauchy problem in the homogeneous Sobolev space (H) over dot(s)(R) for s is an element of (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.