Quintic NLW/NLKG on R3

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Quintic NLW/NLKG on R^3
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.