LAUSR

Abstract We study the initial-boundary-value problem of the fifth-order KdV equation ∂ t u − ∂ x 5 u = c 1 u ∂ x u + c 2 u 2 ∂ x u + b 1 ∂ x u ∂ x 2 u + b 2 u ∂ x 3 u , b 2 ≠ 0 with initial data u 0 ∈ H s ( 0 , L ) . The main analysis difficulty of this model is caused by the nonlinear dispersive term u ∂ x 3 u since the Kato smoothing effect of the solution to the linear fifth-order KdV equation with free source term can only improve two orders concerning the initial data. The Cauchy problem of this model has recently been proved to be globally well-posed in H − 1 ( R ) by Bringmann-Killip-Visan (2019). Under appropriate non-homogeneous boundary data, we prove that it is locally well-posed in H s ( 0 , L ) with s ∈ [ 0 , 3 2 ) and admits a unique local solution as s ≥ 3 2 . As far as we know, this is the latest well-posedness result of the fifth-order KdV type equation with the nonlinear dispersive term when posed in any finite domain. The main innovation in this paper is that we construct a source term space which makes that the solution to the linear fifth-order KdV equation improves three orders on the source term.