Well-posedness and lower bounds of the growth of weighted norms 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-lines for a wide class of initial–boundary data, including the energy regularity for initial data. Assuming homogeneous boundary conditions, for the problem on the positive half-line, 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, for a certain class of regular initial data, the following result was proved: if the respective solution does not exhibit finite-time blow-up in energy space on the left half-line, then the norm in the virial space blows up at infinity time with super-linear rate; this is obtained by using a satisfactory algebraic manipulation of a new global virial-type identity associated with the system, which does not work in the context of whole real line.