Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance

Abstract

We consider the quadratic nonlinear Schrödinger system i∂tu+Δu=vu¯,i∂tv+κΔv=u2,onI×Rd,where 1≤d≤6 and κ>0. In the lower dimensional case d=1,2,3, it is known that the H1-solution is global in time. On the other hand, there are finite time blow-up solutions when d=4,5,6 and κ=1∕2. The condition of κ=1∕2 is called mass-resonance. In this paper, we prove finite time blow-up under radially symmetric assumption when d=5,6 and κ≠1∕2 and we show blow-up or grow-up when d=4.