Local Well‐Posedness for a Class of KdV‐Type Equations With Second‐Order Derivative Nonlinearities

ABSTRACT We prove that the generalized Korteweg‐de Vries (KdV) equation with second‐order derivative nonlinearities is locally well‐posed in for , and ill‐posed in for . The local well‐posedness of this equation has been studied for sufficiently large by Kenig–Ponce–Vega. The proof in this paper relies on the Fourier restriction norm method, dispersive estimates, and bilinear estimates. Due to the existence of logarithmic divergence, the nonlinear terms or cannot be controlled in Bourgain spaces. However, by reducing the integrability index, this difficulty in can be solved in the Besov spaces for , which implies the local well‐posedness for initial data with small ‐norm in for and for .