# Cubic NLW/NLKG on R3

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