Quintic NLW/NLKG on R3

Quintic NLW/NLKG on R^3

Description
Equation	$\Box u=m^{2}u\pm u^{5}$
Fields	$u:\mathbb {R} ^{1+3}\to \mathbb {C}$
Data class	$u[0]\in H^{s}\times H^{s-1}(\mathbb {R} ^{3})$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear
Linear component	wave
Critical regularity	${\dot {H}}^{1}\times L^{2}(\mathbb {R} ^{3})$
Criticality	energy-critical
Covariance	Lorentzian
Theoretical results
LWP	$H^{s}\times H^{s-1}(\mathbb {R} )$ for $s\geq 1$
GWP	$H^{s}\times H^{s-1}(\mathbb {R} )$ 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 $s<s_{c}$ one has instantaneous blowup in the focusing case, and unbounded growth of $H^{s}$ 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.