The IVP for a nonlocal perturbation of the Benjamin-Ono equation in classical and weighted Sobolev spaces

Abstract

We prove that the initial value problem associated to a nonlocal perturbation of the Benjamin-Ono equation is locally and globally well-posed in Sobolev spaces Hs(R) for any s>−3/2 and we establish that our result is sharp in the sense that the flow map of this equation fails to be C2 in Hs(R) for s<−3/2. Finally, we study persistence properties of the solution flow in the weighted Sobolev spaces Zs,r=Hs(R)∩L2(|x|2rdx) for s≥r>0. We also prove some unique continuation properties of the solution flow in these spaces.