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

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 $b<\frac{1}{2}$, which is almost optimal compared to the standard $X^{s,b}$ nonlinear estimates for $b>\frac{1}{2}$ \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.

60 pages. More expositions are complemented in the introduction section, and minor typos are corrected. arXiv admin note: text overlap with arXiv:1805.05229