Quintic NLW/NLKG on R3
- Scaling is . Thus this equation is energy-critical.
- 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.
- 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 (Gl1990, Gl1992). The main new ingredient is energy non-concentration (Sw1988, Sw1992).
- Further decay estimates and scattering were obtained in BaSa1998, Na1999d, Ta2006; global Lipschitz dependence was obtained in BaGd1997.
- For smooth data GWP and scattering was shown in Gl1992; see also SaSw1994
- For radial data GWP and scattering was shown in Sw1988
- For data with small energy this was shown for general quintic non-linearities (and for either NLW or NLKG) in Ra1981.
- Global weak solutions can be constructed by general methods (e.g. Sr1989, Sw1992); uniqueness was shown in Kt1992
- In the focussing case there is blowup from large data by the ODE method.
- When there is a convex obstacle GWP for smooth data is known SmhSo1995.