On persistence properties in weighted spaces for solutions of the fractional Korteweg-de Vries equation

TL;DR

This paper investigates how solutions to a fractional Korteweg-de Vries equation maintain their decay properties over time in weighted spaces, revealing the limits imposed by weak dispersion and establishing optimal decay conditions.

Contribution

It extends the analysis of persistence in weighted spaces to a weakly dispersive fractional KdV model, identifying the maximal decay rates compatible with the equation's dispersive nature.

Findings

Polynomial weights are not propagated beyond certain decay rates.

Dispersive effects influence the maximum decay rate of solutions.

Optimal decay conditions are established for the fractional KdV equation.

Abstract

Persistence problems in weighted spaces have been studied for different dispersive models involving non-local operators. Generally, these models do not propagate polynomial weights of arbitrary magnitude, and the maximum decay rate is associated with the dispersive part of the equation. Altogether, this analysis is complemented by unique continuation principles that determine optimal spatial decay. This work is intended to establish the above questions for a weakly dispersive perturbation of the inviscid Burgers equation. More precisely, we consider the fractional Korteweg-de Vries equation, which comprises the Burgers-Hilbert equation and dispersive effects weaker than those of the Benjamin-ono equation.