In the -supercritical focussing NLS one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality
see e.g. OgTs1991. By scaling this implies that we have instantaneous blowup in for in the focusing case. In the defocusing case blowup
is not known, but the norm can still get arbitrarily large arbitrarily quickly for CtCoTa-p2. In addition, the work about sharp criteria of blowup and global existence, see Zhj2002a, Zhj2002b.