Cubic NLW/NLKG on R2

• Scaling is $s_{c}=0$.
• LWP for $s\geq 1/4$ 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 $s\geq 1/2$ 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.