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

Abstract

we study an initial-boundary-value problem for the “good” Boussinesq equation on the half line

$$ \left\{ \begin{gathered} \partial _t^2 u - \partial _x^2 u + \partial _x^4 u + \partial _x^2 u^2 = 0, t > 0, x > 0, \hfill \\ u\left( {0,t} \right) = h_1 \left( t \right), \partial _x^2 u\left( {0,t} \right) = \partial _t h_2 \left( t \right), \hfill \\ u\left( {x,0} \right) = f\left( x \right), \partial _t u\left( {x,0} \right) = \partial _x h\left( x \right) \hfill \\ \end{gathered} \right. $$

. The existence and uniqueness of low reguality solution to the initial-boundary-value problem is proved when the initial-boundary data (f, h, h ₁, h ₂) belong to the product space

$$ H^s \left( {\mathbb{R}^ + } \right) \times H^{s - 1} \left( {\mathbb{R}^ + } \right) \times H^{\tfrac{s} {2} + \tfrac{1} {4}} \left( {\mathbb{R}^ + } \right) \times H^{\tfrac{s} {2} + \tfrac{1} {4}} \left( {\mathbb{R}^ + } \right) $$

with $0 \leqslant s \leqslant \tfrac{1} {2}$ . The analyticity of the solution mapping between the initial-boundary-data and the solution space is also considered.