The Cauchy problem for dissipative Benjamin-Ono equation in weighted Sobolev spaces

Abstract

We study the well-posedness in weighted Sobolev spaces, for the initial value problem (IVP) associated with the dissipative Benjamin-Ono (dBO) equation. We establish persistence properties of the solution flow in the weighted Sobolev spaces Zs,r=Hs(R)∩L2(|x|2rdx), s≥r>0. We also prove some unique continuation properties in these spaces. In particular, such results of unique continuation show that our results of well posedness are sharp.