Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces

TL;DR

This paper investigates the well-posedness and ill-posedness of the regularized Benjamin-Ono equation in weighted Sobolev spaces, establishing conditions for existence, uniqueness, and decay properties of solutions.

Contribution

It provides new results on local and global well-posedness, as well as unique continuation properties, clarifying the behavior of solutions in weighted Sobolev spaces.

Findings

Established local and global well-posedness in weighted Sobolev spaces

Proved a unique continuation property affecting decay preservation

Identified sharp conditions for well-posedness based on decay properties

Abstract

We consider the initial value problem associated to the regularized Benjamin-Ono equation, rBO. Our aim is to establish local and global well-posedness results in weighted Sobolev spaces via contraction principle. We also prove a unique continuation property that implies that arbitrary polinomial type decay is not preserved yielding sharp results regarding well-posedness of the initial value problem in most weighted Sobolev spaces.