# Quintic NLS on R3

The theory of the quintic NLS on ${\displaystyle \mathbb {R} ^{3}}$ is as follows.

• Scaling is ${\displaystyle s_{c}=1\,}$.
• LWP is known for ${\displaystyle s\geq 1\,}$ CaWe1990.
• For ${\displaystyle s=1\,}$ the time of existence depends on the profile of the data as well as the norm.
• For ${\displaystyle s we have ill-posedness, indeed the ${\displaystyle H^{s}\,}$ 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 ${\displaystyle s\geq 1\,}$ 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.