Cubic NLW/NLKG on R4

• Scaling is $s_{c}=1$.
• LWP for $s\geq 1$ by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
  • When $s=1$ 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 $s<s_{c}$ one has instantaneous blowup in the focusing case, and unbounded growth of $H^{s}$ norms in the defocusing case [CtCoTa-p2]
• GWP for $s=1$ 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.

---