Quintic NLW/NLKG on R3

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Quintic NLW/NLKG on R^3
Description
Equation \Box u = m^2 u \pm u^5
Fields u: \R^{1+3} \to \mathbb{C}
Data class u[0] \in H^s \times H^{s-1}(\R^3)
Basic characteristics
Structure Hamiltonian
Nonlinearity semilinear
Linear component wave
Critical regularity \dot H^1 \times L^2(\R^3)
Criticality energy-critical
Covariance Lorentzian
Theoretical results
LWP H^s \times H^{s-1}(\R) for s \geq 1
GWP H^s \times H^{s-1}(\R) for s \geq 1 (+)
s \geq 1 and sub-ground-state energy (-)
Related equations
Parent class Quintic NLW/NLKG
Special cases -
Other related -


  • Scaling is s = 1. Thus this equation is energy-critical.
  • LWP for s \geq 1 by Strichartz estimates (see e.g. LbSo1995; earlier references exist)
    • When s = 1 the time of existence depends on the profile of the data and not just on the norm.
    • For s < sc one has instantaneous blowup in the focusing case, and unbounded growth of Hs norms in the defocusing case (CtCoTa-p2).
  • GWP for s = 1 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.
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