Dispersive Decay Bound of Small Data Solutions to Kawahara Equation in a Finite Time Scale

TL;DR

This paper establishes that small localized initial data lead to solutions of the Kawahara equation exhibiting linear dispersive decay within a finite time, extending understanding of decay bounds for fifth-order dispersive PDEs.

Contribution

It provides the first small data global bounds for the Kawahara equation with quadratic nonlinearity, using a simplified method similar to that for KdV.

Findings

Solutions decay linearly over finite time for small localized data

Method simplifies previous approaches for dispersive decay bounds

First known small data global bounds for fifth-order dispersive equations

Abstract

In this article, we prove that small localized data yield solutions to Kawahara type equation which have linear dispersive decay on a finite time. We use the similar method used to derive the dispersive decay bound of the solutions to the KdV equation, with some steps being simpler. This result is expected to be the first result of the small data global bounds of the fifth-order dispersive equations with quadratic nonlinearity.