Cubic NLW/NLKG on R3

• Scaling is $s_{c}=1/2$.
• LWP for $s\geq 1/2$ 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 $s\geq 1$ 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 $s>5$.