ON COLLAPSING RING BLOW-UP SOLUTIONS TO THE MASS SUPERCRITICAL NONLINEAR SCHR?DINGER EQUATION

摘要

We consider the nonlinear Schr?dinger equation i?t_u+?u+ulul~p-1=0 in dimension N≥2 and in the mass supercritical and energy subcritical range 1+ 4/N < min{N+2/N-2,5} For initial data u_0εH~1 with radial symmetry, we prove a universal upper bound on the blow-up speed. We then prove that this bound is sharp and attained on a family of collapsing ring blow-up solutions first formally predicted in Fibich et al.