A note on the quartic generalized Korteweg-de Vries equation in weighted Sobolev spaces

TL;DR

This paper proves the persistence of solutions for the quartic generalized Korteweg-de Vries equation within weighted Sobolev spaces, extending understanding of solution behavior with specific initial data.

Contribution

It establishes the persistence property for solutions in weighted Sobolev spaces for the first time for this equation with low regularity initial data.

Findings

Solutions persist in weighted Sobolev spaces under specified conditions.

Persistence holds for initial data with regularity s=1/12+ε.

Results extend the theory of solution behavior for the quartic gKdV equation.

Abstract

In this paper we establish the persistence property for solutions of the quartic generalized Korteweg-de Vries equation with initial data in weighted Sobolev spaces $H^{s}(\mathbb{R})\cap L^2(|x|^{2r}dx)$ for $s =1/12 + \varepsilon$ and any $r\in (0,R)$, for some $0 < \varepsilon < 1/4$ and $0<R<s/2$.