On the equipartition of energy for critical NLW

Abstract

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: $$\lim_{R\to \infty}\frac 1R \int_{\mathbf R} \int_{|x|<R} |\nabla_{x} u(t,x)|^2 \hbox{} dxdt $$ $$=\lim_{R\to \infty} \frac 1{2R} \int_{\mathbf R} \int_{|x|<R} (|\nabla_{t,x} u(t,x)|^2 + \frac{2 \lambda}{2^*} |u(t,x)|^{2^*}) \hbox{} dxdt$$ $$=\int_{{\mathbf R}^n} (|\nabla_{t, x} u(0, x)|^2+ \frac{2 \lambda}{2^*} |u(0, x)|^{2^*}) \hbox{} dx.$$