Well-posedness and long time behavior for the Schrödinger-Korteweg-de Vries interactions on the half-Line

Abstract

The initial-boundary value problem for the Schrödinger-Korteweg-de Vries system is considered on the left and right half-line for a wide class of initial-boundary data, including the energy regularity $H^1({\mathbb{R}}^{\pm })\times H^1({\mathbb{R}}^{\pm })$ for initial data. Assuming homogeneous boundary conditions it is shown for positive coupling interactions that local solutions can be extended globally in time for initial data in the energy space; furthermore, for negative coupling interactions it was proved, for a certain class of regular initial data, the following result: if the respective solution does not exhibits finite time blow-up in $H^1({\mathbb{R}}^{-})\times H^1({\mathbb{R}}^{-})$, then the norm of the weighted space $L^2({\mathbb{R}}^{-},|x|dx)\times L^2({\mathbb{R}}^{-},|x|dx)$ blows-up at infinity time with \textit{super-linear rate}, this is obtained by using a satisfactory algebraic manipulation of a new global virial type identity associated to the system .

26 pages