The Cauchy Problem for Dissipative Benjamin-Ono equation in Weighted Sobolev spaces

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 ${\text{z}}_{s,r}=H^s(\mathbb{R})\cap L^2(\mathrm{\mid }x{\mathrm{\mid }}^{2r}dx)$, $s\ge r>0$. 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.