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+ λu|u|^{2^*-2}=0, \hbox{} λ\in {\mathbf R} \end{equation*} $$u(0)=f\in {\dot{H}}^1({\mathbf{R}}^n),{\partial }_{t}u(0)=g\in L^2({\mathbf{R}}^n)$$ on $\mathbf{R}\times {\mathbf{R}}^n$ for $n\ge 3$, where $2^{\ast }\equiv \frac{2n}{n-2}$. 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: $$\underset{R\to \infty }{\lim }\frac{1}{R}{\int }_{\mathbf{R}}{\int }_{|x|<R}|{\nabla }_{x}u(t,x){|}^{2}dxdt$$ $$=\underset{R\to \infty }{\lim }\frac{1}{2R}{\int }_{\mathbf{R}}{\int }_{|x|<R}(|{\nabla }_{t,x}u(t,x){|}^2+\frac{2\hat{I}»}{2^{\ast }}|u(t,x){|}^{2^{\ast }})dxdt$$ $$={\int }_{{\mathbf{R}}^n}(|{\nabla }_{t,x}u(0,x){|}^2+\frac{2\hat{I}»}{2^{\ast }}|u(0,x){|}^{2^{\ast }})dx.$$

to appear on Journal of Functional Analysis