Quintic NLW/NLKG on R3
From DispersiveWiki
Description | |
---|---|
Equation | |
Fields | |
Data class | |
Basic characteristics | |
Structure | Hamiltonian |
Nonlinearity | semilinear |
Linear component | wave |
Critical regularity | |
Criticality | energy-critical |
Covariance | Lorentzian |
Theoretical results | |
LWP | for |
GWP | for (+) and sub-ground-state energy (-) |
Related equations | |
Parent class | Quintic NLW/NLKG |
Special cases | - |
Other related | - |
- 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.