On decay and asymptotic properties of solutions to the Intermediate Long Wave equation

TL;DR

This paper investigates the decay and long-term behavior of solutions to the Intermediate Long Wave (ILW) equation, establishing sharp persistence properties and describing energy decay in various spatial regions over time.

Contribution

It provides new results on the persistence of solutions in weighted Sobolev spaces and characterizes their decay properties using virial techniques.

Findings

Persistence properties are sharp in weighted Sobolev spaces.

Energy decays to zero in certain spatial regions over time.

Complete decay is proved in exterior regions for large solutions.

Abstract

We consider solutions to the initial value problem associated to the intermediate long wave (ILW) equation. We establish persistence properties of the solution flow in weighted Sobolev spaces, and show that they are sharp. We also deal with the long time dynamics of large solutions to the ILW equation. Using virial techniques, we describe regions of space where the energy of the solution 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.