Local well-posedness of the fifth-order KdV-type equations on the half-line

TL;DR

This paper proves local well-posedness for fifth-order KdV-type equations on half-lines by extending previous methods to handle different nonlinearities and establishing near-optimal nonlinear estimates in $X^{s,b}$ spaces.

Contribution

It extends existing analysis to fifth-order KdV equations with new nonlinearities where scaling arguments fail, establishing near-optimal $X^{s,b}$ estimates and well-posedness.

Findings

Established local well-posedness for fifth-order KdV on half-lines.

Extended $X^{s,b}$ estimates to $b<rac{1}{2}$, near optimal.

Handled nonlinearities where scaling arguments do not apply.

Abstract

This paper is a continuation of authors' previous work \cite{CK2018-1}. We extend the argument \cite{CK2018-1} to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We establish the $X^{s,b}$ nonlinear estimates for $b < \frac12$, which is almost optimal compared to the standard $X^{s,b}$ nonlinear estimates for $b > \frac12$ \cite{CGL2010, JH2009}. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half-line.