On Decay Properties of Solutions to the IVP for the Benjamin–Ono Equation

Abstract

In this work we investigate unique continuation properties of solutions to the initial value problem associated to the Benjamin–Ono equation in weighted Sobolev spaces $$Z_{s,r}=H^s(\mathbb R )\cap L^2(|x|^{2r}dx)$$ for $$s\in \mathbb R $$ , and $$s\ge 1$$ , $$s\ge r$$ . More precisely, we prove that the uniqueness property based on a decay requirement at three times can not be lowered to two times even by imposing stronger decay on the initial data.