Quintic NLW/NLKG on R2

• 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.
  • 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 $s>3/4$ for defocussing NLW/NLKG (Fo-p)
  • For $s\geq 1$ this follows energy conservation.
  • One also has GWP and scattering for data with small $H^{1/2}$ norm for general quintic non-linearities (and for either NLW or NLKG).
  • In the focussing case there is blowup from large data by the ODE method.