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

Shenghao Li; Min Chen; Bing-Yu Zhang

doi:10.3934/dcds.2018104

Article Contents

2018, Volume 38, Issue 5: 2505-2525. Doi: 10.3934/dcds.2018104

This issue Previous Article Topological stability and spectral decomposition for homeomorphisms on noncompact spaces Next Article Stability of the distribution function for piecewise monotonic maps on the interval

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
^*Corresponding author: Bing-Yu Zhang

Received: October 31, 2016

Revised: November 30, 2017

Published: June 2018

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Abstract

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:

$\begin{equation}\label{0}\begin{cases}u_{tt}-u_{xx}+β u_{xxxx}-u_{xxxxxx}+(u^2)_{xx} = 0, \, \, \, \, \,\,\,\,\, \mbox{for }x>0\mbox{, }t>0, \\u(x, 0) = \varphi (x), u_t(x, 0) = ψ "(x), \\u(0, t) = h_1(t), u_{xx}(0, t) = h_2(t), u_{xxxx}(0, t) = h_3(t), \end{cases}\, \, \, \, \, \, \, \, \, \, (1)\end{equation}$

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.

Keywords:

Mathematics Subject Classification: Primary: 35Q53, 35Q55; Secondary: 35Q35.

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Figure . Sketch of the half line case

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