# Cubic NLW/NLKG on R3

From DispersiveWiki

- Scaling is .
- LWP for by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
- GWP for (KnPoVe-p2) for defocussing NLKG.(An alternate proof is in GalPl2003).
- For this is clear from energy conservation (for both NLKG and NLW).
- One also has GWP and scattering for data with small norm for general cubic non-linearities (and for either NLKG or NLW).
- In the defocussing case one has scattering for large data (BaeSgZz1990), see also (Hi-p3).
- Improvement is certainly possible, both in lowering the s index and in replacing NLKG with NLW.
- In the focussing case there is blowup from large data by the ODE method.

- For periodic defocussing NLKG there is a weak turbulence effect in for (low frequencies decay in time) but a symplectic non-squeezing effect in H^{1/2} Kuk1995b.In particular cannot be a symplectic phase space for .