Low regularity solution of the initial-boundary-value problem for the “good” Boussinesq equation on the half line

we study an initial-boundary-value problem for the “good” Boussinesq equation on the half line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ \begin{gathered} \partial _t^2 u - \partial _x^2 u + \partial _x^4 u + \partial _x^2 u^2 = 0, t > 0, x > 0, \hfill \\ u\left( {0,t} \right) = h_1 \left( t \right), \partial _x^2 u\left( {0,t} \right) = \partial _t h_2 \left( t \right), \hfill \\ u\left( {x,0} \right) = f\left( x \right), \partial _t u\left( {x,0} \right) = \partial _x h\left( x \right) \hfill \\ \end{gathered} \right. $$\end{document}. The existence and uniqueness of low reguality solution to the initial-boundary-value problem is proved when the initial-boundary data (f, h, h1, h2) belong to the product space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H^s \left( {\mathbb{R}^ + } \right) \times H^{s - 1} \left( {\mathbb{R}^ + } \right) \times H^{\tfrac{s} {2} + \tfrac{1} {4}} \left( {\mathbb{R}^ + } \right) \times H^{\tfrac{s} {2} + \tfrac{1} {4}} \left( {\mathbb{R}^ + } \right) $$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0 \leqslant s \leqslant \tfrac{1} {2} $$\end{document} . The analyticity of the solution mapping between the initial-boundary-data and the solution space is also considered.