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

In this article, we study an initial-boundary-value problem of the sixth order Boussinesq equation on a half line with nonhomogeneous boundary conditions: $$u_{tt}-u_{xx}+\beta u_{xxxx}-u_{xxxxxx}+(u^2{)}_{xx}=0,x>0,t>0,$$ $$u(x,0)=\phi (x),u_t(x,0)={\psi }^{\mathrm{\prime }\mathrm{\prime }}(x),$$ $$u(0,t)=h_1(t),u_{xx}(0,t)=h_2(t),u_{xxxx}(0,t)=h_3(t),$$ 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 $(\varphi,\psi)\in H^s($\mathbb{R}$^+)\times H^{s-1}($\mathbb{R}$^+)$andboundarycondition$(h_1,h_2,h_3) $intheproductspace$H^{\frac{s+1}{3}}($\mathbb{R}$^+)\times H^{\frac{s-1}{3}}($\mathbb{R}$^+)\times H^{\frac{s-3}{3}}($\mathbb{R}$^+)$.