Lower Regularity Solutions of the Non-homogeneous Boundary-Value Problem for a Higher Order Boussinesq Equation in a Quarter Plane

TL;DR

This paper establishes local well-posedness for a sixth order Boussinesq equation with non-homogeneous boundary conditions in a quarter plane for low regularity initial data.

Contribution

It extends well-posedness results to solutions with regularity as low as s > -3/4 for a higher order Boussinesq equation with boundary conditions.

Findings

Well-posedness holds for s > -3/4.

Solutions exist and are unique in specified Sobolev spaces.

The analysis covers non-homogeneous boundary conditions.

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}+\beta u_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad x,t\in \mathbb{R}^+,\\ 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), \end{cases} \end{equation*} where $\beta=\pm1$. We show that the problem is locally analytically well-posed in the space $H^s(\mathbb{R}^+)$ for any $ s> -\frac34 $ with the initial-value data $$(\varphi,\psi)\in H^s(\mathbb{R}^+)\times H^{s-1}(\mathbb{R}^+)$$ and the boundary-value data $$(h_1,h_2,h_3) \in H^{\frac{s+1}{3}}(\mathbb{R}^+)\times H^{\frac{s-1}{3}}(\mathbb{R}^+)\times H^{\frac{s-3}{3}}(\mathbb{R}^+).$$