Non-homogeneous boundary value problems of the fifth-order KdV equations on a bounded interval

Abstract

We investigate well-posedness of initial boundary value problem for the fifth-order KdV equation (or Kawahara equation) posed on a finite interval ∂ t u − ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 with general non-homogeneous boundary conditions. Firstly, all possible boundary conditions are found while searching enough dissipative effects to the initial boundary value problem. Then, boundary smoothing effect is established through estimating explicit solution established by using Laplacian transformation. Finally, based on smoothing effect, local well-posedness of the initial boundary value problem was proved by applying Banach's fixed point theorem.