PDE Seminar (2015/10/19): Classification of the solutions to the focusing nonlinear Schrödinger equation with a repulsive Dirac delta potential

Date
2015-10-19 16:30 - 2015-10-19 17:30
Place
Faculty of Science Building #3, Room 309
Speaker/Organizer
Takahisa Inui (Kyoto University)
 
We consider the focusing nonlinear Schrödinger equation with a repulsive Dirac delta potential in the \(L^2\) supercritical case for \(H^1\) data. Banica and Visciglia proved that all solutions scatter in the defocusing case. However, in the focusing case, the existence of non-scattering solution is well known. Our aim is to classify the behavior of the solutions by the initial data. In this talk, we prove that the sign of a functional at initial time, which appears naturally from the viewpoint of variational argument, determines whether solutions scatter or blow up if the energy-mass is less than that of the standing wave. This talk is based on a joint work with Masahiro Ikeda in Kyoto university.