Non-homogeneous initial-boundary-value problem of the fifth-order Korteweg-de Vries equation with a nonlinear dispersive term

Abstract

We study the initial-boundary-value problem of the fifth-order KdV equation∂tu−∂x5u=c1u∂xu+c2u2∂xu+b1∂xu∂x2u+b2u∂x3u,b2≠0 with initial data u0∈Hs(0,L). The main analysis difficulty of this model is caused by the nonlinear dispersive term u∂x3u 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 Hs(0,L) with s∈[0,32) and admits a unique local solution as s≥32. 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.