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

This paper is a continuation of authors' previous work [6]. We extend the argument [6] 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 $ [8,17]. 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.