Dispersive decay bound of small data solutions to higher order scattering-supercritical KdV-type equations

TL;DR

This paper proves that small localized initial data for higher order scattering-supercritical KdV-type equations lead to solutions with linear dispersive decay, but only over a finite time span, using space-time resonance analysis.

Contribution

It introduces a finite-time dispersive decay result for small data solutions to higher order scattering-supercritical KdV equations using novel oscillatory integral analysis.

Findings

Solutions decay linearly over finite time for small data

Utilizes space-time resonance method for analysis

Analyzes oscillatory integrals on the Fourier side

Abstract

In this article, we prove that small localized data yield solutions to Higher order Korteweg-de Vries type equation with scattering-supercritical nonlinearity have linear dispersive decay in only a finite length of time. The proof is done by using space-time resonance method and analyzing the oscillatory integrals on the Fourier side.