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

TL;DR

This paper investigates the well-posedness and unique continuation properties of the dissipative Benjamin-Ono equation within weighted Sobolev spaces, establishing sharp results for the initial value problem.

Contribution

It provides new well-posedness and unique continuation results for the dissipative Benjamin-Ono equation in weighted Sobolev spaces, demonstrating the sharpness of these results.

Findings

Persistence of solutions in weighted Sobolev spaces

Unique continuation properties established

Results are shown to be sharp

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 $\z_{s,r}=H^s(\R)\cap L^2(|x|^{2r}dx)$, $s\geq r>0$. 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.