On the Cauchy problem for higher-order nonlinear dispersive equations

TL;DR

This paper investigates the well-posedness and ill-posedness of higher-order nonlinear dispersive equations, specifically KdV-type equations, in various functional spaces, providing conditions for stability and instability.

Contribution

It establishes well-posedness in weighted Besov and Sobolev spaces for small initial data and demonstrates ill-posedness in H^s spaces for any real s, advancing understanding of solution behavior.

Findings

Well-posedness in weighted Besov and Sobolev spaces for small data

Ill-posedness in H^s( ) for any real s

Conditions delineating stability and instability regimes

Abstract

We study a class of higher-order KdV equations. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when in H^s(\R), for any real s.