NLS blowup

In the $L^{2}\,$-supercritical focussing NLS one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality

$\partial _{t}^{2}\int x^{2}|u|^{2}dx\leq H(u)$;

see e.g. OgTs1991. By scaling this implies that we have instantaneous blowup in $H^{s}\,$ for $s<s_{c}\,$ in the focusing case. In the defocusing case blowup
is not known, but the $H^{s}\,$ norm can still get arbitrarily large arbitrarily quickly for $s<s_{c}\,$ CtCoTa-p2. In addition, the work about sharp criteria of blowup and global existence, see Zhj2002a, Zhj2002b.