Wellposedness of the sixth order Boussinesq equation with non-homogeneous boundary values on a bounded domain

Abstract

This article is concerned with an initial–boundary-value problem (IBVP) for the sixth order Boussinesq equation on a bounded domain with non-homogeneous boundary conditions, (1)utt−uxx+βuxxxx−uxxxxxx+(u2)xx=0,x∈(0,1),t>0,u(x,0)=φ(x),ut(x,0)=ψ(x),ux(0,t)=h1(t),uxxx(0,t)=h2(t),uxxxxx(0,t)=h3(t),ux(1,t)=h4(t),uxxx(1,t)=h5(t),uxxxxx(1,t)=h6(t),where β=±1. For 0≤s≤6, it is shown that the IBVP is locally well-posed if the initial data lie in the product of L2-based Sobolev spaces, Hs(0,1)×Hs−3(0,1), provided the boundary data (h1,h2,h3) and(h4,h5,h6) lie in the product space, Hloc(s+2)∕3(R+)×Hlocs∕3(R+)×Hloc(s−2)∕3(R+).