The IVP for the Benjamin-Ono equation in weighted Sobolev spaces

TL;DR

This paper investigates the initial value problem for the Benjamin-Ono equation, establishing persistence and unique continuation properties in weighted Sobolev spaces, demonstrating the sharpness of these properties.

Contribution

It provides new results on the persistence and unique continuation of solutions in weighted Sobolev spaces for the Benjamin-Ono equation.

Findings

Persistence properties in weighted Sobolev spaces established

Unique continuation principles proved

Results demonstrate sharpness of persistence properties

Abstract

We study the initial value problem associated to the Benjamin-Ono equation. The aim is to establish persistence properties of the solution flow in the weighted Sobolev spaces $Z_{s,r}=H^s(\R)\cap L^2(|x|^{2r}dx)$, $s\in\R, \,s\geq 1$ and $s\geq r$. We also prove some unique continuation properties of the solution flow in these spaces. In particular, these continuation principles demostrate that our persistence properties are sharp.