Cubic NLW/NLKG on R4

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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 has strong uniqueness in the energy class Pl-p5, FurPlTer2001. This argument extends to other energy-critical and sub-critical powers in dimensions 4 and higher.
    • For one has instantaneous blowup in the focusing case, and unbounded growth of norms in the defocusing case (CtCoTa-p2).
  • GWP for in the defocussing case SaSw1994 (see also Gl1990, Gl1992, Sw1988, Sw1992, BaSa1998, BaGd1997).
    • In the focussing case there is blowup from large data by the ODE method.