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 < ¹ ₂ , which is almost optimal compared to the standard X ^s,b nonlinear estimates for b > ₂ ¹ [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.