Septic NLW/NLKG on R3

• Scaling is $s=7/6$.
• LWP for $s\geq 7/6$ by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
  • When $s=7/6$ 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]
• Global existence of large smooth solutions is unknown in the defocussing case; in the focussing case one certainly has blowup by ODE methods.
  • In the energy class $s=1$, one has ill-posedness in the sense that the solution map is not uniformly continuous Leb2000; for higher dimensions see BrKum2000. This is despite an a priori bound on the $H^{1}xL^{2}$ norm in the defocussing case from energy conservation. A variant of this result appears in [CtCoTa-p2].
  • For small data one of course has GWP and scattering LbSo1995
  • It is not known what happens to large smooth solutions in the defocusing case, even in the radial case. This can be viewed as an extremely simplified model problem for the global regularity issue for Navier-Stokes.

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