The theory of the quintic NLS on is as follows.
- Scaling is .
- LWP is known for CaWe1990.
- For the time of existence depends on the profile of the data as well as the norm.
- For we have ill-posedness, indeed the norm can get arbitrarily large arbitrarily quickly CtCoTa-p2. In the focusing case we have instantaneous blowup from the virial identity and scaling.
- GWP and scattering for in the defocusing case CoKeStTkTa-p
- For radial data this is in Bo1999b, Bo1999.
- Blowup can occur in the focusing case from Glassey's virial identity.