Cubic NLW/NLKG on R3

Scaling is $s c = 1 / 2$ .
LWP for by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
- When $s = 1 / 2$ the time of existence depends on the profile of the data and not just on the norm.
- One can improve the critical space $H 1 / 2$ to a slightly weaker Besov space (Pl-p2).
- For $s < 1 / 2$ one has instantaneous blowup in the focusing case, and unbounded growth of H^s norms in the defocusing case (CtCoTa-p2).
GWP for s > 3 / 4 (KnPoVe-p2) for defocussing NLKG.(An alternate proof is in GalPl2003).
- For $s\geq1$ this is clear from energy conservation (for both NLKG and NLW).
- One also has GWP and scattering for data with small $H 1 / 2$ norm for general cubic non-linearities (and for either NLKG or NLW).
- In the defocussing case one has scattering for large $H 1$ 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 $H s$ for $s > 5$ (low frequencies decay in time) but a symplectic non-squeezing effect in H^{1/2} Kuk1995b.In particular $H s$ cannot be a symplectic phase space for $s > 5$ .