On the equipartition of energy for the critical NLW

Abstract

We study some qualitative properties of global solutions to the following focusing and defocusing critical NLW: □ u + λ u | u | 2 ∗ − 2 = 0 , λ ∈ R , u ( 0 ) = f ∈ H ˙ 1 ( R n ) , ∂ t u ( 0 ) = g ∈ L 2 ( R n ) on R × R n for n ⩾ 3 , where 2 ∗ ≡ 2 n n − 2 . We will consider the global solutions of the defocusing NLW whose existence and scattering property is shown in [J. Shatah, M. Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Int. Math. Res. Not. (7) (1994) 303–309 (electronic); H. Bahouri, J. Shatah, Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (6) (1998) 783–789] and [H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1) (1999) 131–175], without any restriction on the initial data ( f , g ) ∈ H ˙ 1 ( R n ) × L 2 ( R n ) . As well as the solutions constructed in [H. Pecher, Nonlinear small data scattering for the wave and Klein–Gordon equation, Math. Z. 185 (2) (1984) 261–270] to the focusing NLW for small initial data and to the ones obtained in [C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation, preprint], where a sharp condition on the smallness of the initial data is given. We prove that the solution u ( t , x ) satisfies a family of identities, that turn out to be a precised version of the classical Morawetz estimates (see [C. Morawetz, Time decay for the nonlinear Klein–Gordon equation, Proc. Roy. Soc. London Ser. A 306 (1968) 291–296]). As a by-product we deduce that any global solution to critical NLW belonging to a natural functional space satisfies: lim R → ∞ 1 R ∫ R ∫ | x | < R | ∇ x u ( t , x ) | 2 d x d t = lim R → ∞ 1 2 R ∫ R ∫ | x | < R ( | ∇ t , x u ( t , x ) | 2 + 2 λ 2 ∗ | u ( t , x ) | 2 ∗ ) d x d t = ∫ R n ( | ∇ t , x u ( 0 , x ) | 2 + 2 λ 2 ∗ | u ( 0 , x ) | 2 ∗ ) d x .