On the decay of solutions for the negative fractional KdV equation

TL;DR

This paper investigates how solutions to the negative fractional KdV equation behave in weighted spaces, revealing conditions under which polynomial weights are preserved despite negative dispersion effects.

Contribution

It provides new insights into the propagation of polynomial weights for the fractional KdV equation with negative dispersion, highlighting differences from positive dispersion cases.

Findings

Weighted solutions persist under certain initial conditions

Negative dispersion allows for propagation of specific polynomial weights

Results differ from positive dispersion scenarios

Abstract

We explore the limits of fractional dispersive effects and their incidence in the propagation of polynomial weights. More precisely, we consider the fractional KdV equation when a differential operator of negative order determines the dispersion. We investigate what magnitude of weights and conditions on the initial data that allow solutions of the equation to persist in weighted spaces. As a consequence of our results, it follows that even in the presence of negative dispersion, it is still possible to propagate weights whose maximum magnitude is related to the dispersion of the equation. We also observe that our results in weighted spaces do not follow specific properties and limits that their counterparts with positive dispersion.