Lower regularity solutions of non-homogeneous boundary value problems of the sixth order Boussinesq equation in a quarter plane

Abstract

In this article, we study an initial-boundary-value problem of the sixth order Boussinesq equation on a half line with nonhomogeneous boundary conditions: \[ u_{tt}-u_{xx}+\beta u_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad x>0\mbox{, }t>0,\] \[u(x,0)=\varphi (x), u_t(x,0)=\psi ''(x),\] \[ u(0,t)=h_1(t), u_{xx}(0,t)=h_2(t), u_{xxxx}(0,t)=h_3(t),\] where $\beta=\pm1$. It is shown that the problem is locally well-posed in $H^s(\mathbb{R}^+)$ for $-\frac12<s\leq 0$ with initial condition $(\varphi,\psi)\in H^s(\mathbb{R}^+)\times H^{s-1}(\mathbb{R}^+)$ and boundary condition $(h_1,h_2,h_3) $ in the product space $H^{\frac{s+1}{3}}(\mathbb{R}^+)\times H^{\frac{s-1}{3}}(\mathbb{R}^+)\times H^{\frac{s-3}{3}}(\mathbb{R}^+)$.