# Quintic NLW/NLKG on R3

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

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