# American Institute of Mathematical Sciences

May  2018, 38(5): 2505-2525. doi: 10.3934/dcds.2018104

## A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane

 1 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA 2 Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, USA 3 Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China

*Corresponding author: Bing-Yu Zhang

Received  November 2016 Revised  December 2017 Published  March 2018

The paper is concerned with an initial-boundary-value problem of the sixth order Boussinesq equation posed on a quarter plane with non-homogeneous boundary conditions:
 $$$\label{0}\begin{cases}u_{tt}-u_{xx}+β u_{xxxx}-u_{xxxxxx}+(u^2)_{xx} = 0, \, \, \, \, \,\,\,\,\, \mbox{for }x>0\mbox{, }t>0, \\u(x, 0) = \varphi (x), u_t(x, 0) = ψ "(x), \\u(0, t) = h_1(t), u_{xx}(0, t) = h_2(t), u_{xxxx}(0, t) = h_3(t), \end{cases}\, \, \, \, \, \, \, \, \, \, (1)$$$
where
 $β = ± 1$
. It is shown that the problem is locally well-posed in the space $H^s(\mathbb{R}^+)$ for any 0≤s<
 $\frac{13}{2}$
with the initial data
 $(\varphi, ψ)$
in the space
 $H^s(\mathbb{R}^+)× H^{s-1}(\mathbb{R}^+)$
and the naturally compatible boundary data
 $\mbox{$h_1∈ H_{loc}^{\frac{s+1}{3}}(\mathbb{R}^+)$,$h_2∈ H_{loc}^{\frac{s-1}{3}}(\mathbb{R}^+) \text{and}\,\,\, h_3∈ H_{loc}^{\frac{s-3}{3}}(\mathbb{R}^+)$}$
with optimal regularity.
Citation: Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104
##### References:

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##### References:
Sketch of the half line case
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