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: \begin{equation*} \begin{cases} u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad x,t\in \mathbb{R}^+,\\ 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), \end{cases} \end{equation*} where $\beta =\pm 1$. We show that the problem is locally analytically well-posed in the space $H^s({\mathbb{R}}^{+})$ for any $s>-\frac{3}{4}$ with the initial-value data $$(\phi ,\psi )\in H^s({\mathbb{R}}^{+})\times H^{s-1}({\mathbb{R}}^{+})$$ and the boundary-value data $$(h_1,h_2,h_3)\in H^{\frac{s+1}{3}}({\mathbb{R}}^{+})\times H^{\frac{s-1}{3}}({\mathbb{R}}^{+})\times H^{\frac{s-3}{3}}({\mathbb{R}}^{+}).$$

28 pages

References in corpus (1)

• Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation