We study some qualitative properties of global solutions to the following focusing and defocusing critical $NLW$: \begin{ equation*} \Box u+ \lambda u|u|^{ 2^*-2}=0, \hbox{ } \lambda\in { \mathbf R} \end{ equation*} $$\hspace{ 2cm} u(0)=f\in \dot H^1({ \mathbf R}^n), \partial_t u(0)=g\in L^2({ \mathbf R}^n)$$ on ${ \mathbf R}\times { \mathbf R}^n$ for $n\geq 3$, where $2^*\equiv \frac{ 2n}{ n-2}$. We will consider the global solutions of the defocusing $NLW$ whose existence and scattering property is shown in \cite{ shst}, \cite{ sb} and \cite{ bg}, without any restriction on the initial data $(f,g)\in \dot H^1({ \mathbf R}^n) \times L^2({ \mathbf R}^n)$. As well as the solutions constructed in \cite{ pecher} to the focusing $NLW$ for small initial data and to the ones obtained in \cite{ mk}, 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 \cite{ mor1}). As a by--product we deduce that any global solution to critical $NLW$ belonging to a natural functional space satisfies: $$\lim_{ R\to \infty}\frac 1R \int_{ \mathbf R} \int_{ |x|<R} |\nabla_{ x} u(t,x)|^2 \hbox{ } dxdt $$ $$=\lim_{ R\to \infty} \frac 1{ 2R} \int_{ \mathbf R} \int_{ |x|<R} (|\nabla_{ t,x} u(t,x)|^2 + \frac{ 2 \lambda}{ 2^*} |u(t,x)|^{ 2^*}) \hbox{ } dxdt$$ $$=\int_{ { \mathbf R}^n} (|\nabla_{ t, x} u(0, x)|^2+ \frac{ 2 \lambda}{ 2^*} |u(0, x)|^{ 2^*}) \hbox{ } dx.$$
Categories: math.AP