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: \[ u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad x>0\mbox{, }t>0,\] \[u(x,0)=φ(x), u_t(x,0)=ψ''(x),\] \[ u(0,t)=h_1(t), u_{xx}(0,t)=h_2(t), u_{xxxx}(0,t)=h_3(t),\] where $\beta =\pm 1$. It is shown that the problem is locally well-posed in $H^s({\mathbb{R}}^{+})$ for $-\frac{1}{2}<s\le 0$ with initial condition $(\phi ,\psi )\in H^s({\mathbb{R}}^{+})\times H^{s-1}({\mathbb{R}}^{+})$ and boundary condition $(h_1,h_2,h_3)$ in the product space $H^{\frac{s+1}{3}}({\mathbb{R}}^{+})\times H^{\frac{s-1}{3}}({\mathbb{R}}^{+})\times H^{\frac{s-3}{3}}({\mathbb{R}}^{+})$.