On collapsing ring blow up solutions to the mass supercritical NLS

Abstract

We consider the nonlinear Schr\"odinger equation $i\partial_tu+\Delta u+u|u|^{p-1}=0$ in dimension $N\geq 2$ and in the mass super critical and energy subcritical range $1+\frac 4N<p<\min\{\frac{N+2}{N-2},5\}.$ 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 by Gavish, Fibich and Wang.