Cubic NLW/NLKG on R2

Scaling is $s c = 0$ .
LWP for by Strichartz estimates (see e.g. LbSo1995)
- This is sharp by concentration examples in the focusing case; the defocusing case is still open.
GWP for for defocussing NLKG Bo1999 and for defocussing NLW Fo-p
- For $s \geq 1$ this is clear from energy conservation (for both NLKG and NLW).
- In the focussing case there is blowup from large data by the ODE method.
Remark: This is a symplectic flow with the symplectic form of $H 1 / 2$ , as in the one-dimensional case.