On the equipartition of energy for the critical NLW

We study some qualitative properties of global solutions to the following focusing and defocusing critical NLW: u+ λu|u|2∗−2 = 0, λ∈ R, u(0) = f ∈ H˙ 1 Rn , ∂t u(0) = g ∈ L2 Rn on R×Rn for n 3, where 2∗ ≡ 2n 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(Rn)×L2(Rn). 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:L. Vega, N. Visciglia / Journal of Functional Analysis 255 (2008) 726–754 727 lim R→∞ 1 R R |x|