Lower regularity solutions of the non-homogeneous boundary-value problem for a higher order Boussinesq equation in a quarter plane

Abstract

We continue to study the initial–boundary-value problem of the sixth order Boussinesq equation in a quarter plane with non-homogeneous boundary conditions: utt−uxx+βuxxxx−uxxxxxx+(u2)xx=0,x,t∈R+,u(x,0)=φ(x),ut(x,0)=ψ′′(x),u(0,t)=h1(t),uxx(0,t)=h2(t),uxxxx(0,t)=h3(t),where β=±1. We show that the problem is locally analytically well-posed in the space Hs(R+) for any s>−34 with the initial-value data (φ,ψ)∈Hs(R+)×Hs−1(R+)and the boundary-value data (h1,h2,h3)∈Hs+13(R+)×Hs−13(R+)×Hs−33(R+).