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

2018/10/03 by Cavalcante, Márcio, Corcho, Adán
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.01905

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 H¹(\R^±)× H¹(\R^±) 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 H¹(\R^-)× H¹(\R^-), then the norm of the weighted space L²(\R^-, |x|dx)× L²(\R^-, |x|dx) 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 to the system .

Related

• The initial-boundary value problem for Schrödinger-Korteweg-de Vries system on the half-line
• Sharp local well-posedness for the Schrödinger-Korteweg-de Vries system
• On long time behavior of solutions of the Schrödinger-KdV system with and without resonant interactions
• Well-posedness for the Schrodinger-KdV system on the half-line
• On long time behavior of solutions of the Schrödinger-Korteweg-de Vries system