On Local Energy decay for solution of the Benjamin-Ono equation

TL;DR

This paper investigates the long-term decay behavior of solutions to the Benjamin-Ono equation, identifying regions where solutions decay to zero over time using virial methods, with complete decay established in exterior regions.

Contribution

It introduces virial techniques to analyze spatial decay regions for the Benjamin-Ono equation and proves complete decay in exterior regions for large solutions.

Findings

Solutions decay to zero in certain regions over time

Complete decay is proven in exterior regions

Identifies regions with strong dispersion and soliton behavior

Abstract

We consider the long time dynamics of large solutions to the Benjamin-Ono equation. Using virial techniques, we describe regions of space where every solution in a suitable Sobolev space must decay to zero along sequences of times. Moreover, in the case of exterior regions, we prove complete decay for any sequence of times. The remaining regions not treated here are essentially the strong dispersion and soliton regions.