On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation

Abstract

We consider the nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u+u|u{|}^{p-1}=0$ in dimension $N\ge 2$ and in the mass supercritical and energy subcritical range $1+\frac{4}{N}<p<min\hspace{0.17em}\{\frac{N+2}{N-2},5\}$. For initial data $u_0\in H^1$ 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.