Cubic NLW/NLKG on R3

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