  We rigorously construct radial $H^1$ solutions to the 3d cubic focusing NLS
equation $i\partial_t \psi + \Delta \psi + 2 |\psi|^2\psi=0$ that blow-up along
a contracting sphere. With blow-up time set to $t=0$, the solutions concentrate
on a sphere at radius $\sim t^{1/3}$ but focus towards this sphere at the
faster rate $\sim t^{2/3}$. Such dynamics were originally proposed
heuristically by Degtyarev-Zakharov-Rudakov (1975) and independently later in
Holmer-Roudenko (2007), where it was demonstrated to be consistent with all
conservation laws of this equation. In the latter paper, it was proposed as a
solution that would yield divergence of the $L_x^3$ norm within the "wide"
radius $\sim |\nabla u(t)|_{L_x^2}^{-1/2}$ but not within the "tight" radius
$\sim |\nabla u(t)|_{L_x^2}^{-2}$, the second being the rate of contraction of
self-similar blow-up solutions observed numerically and described in detail in
Chapter 7 of Sulem-Sulem (1998).
