On the equipartition of energy for critical NLW

TL;DR

This paper investigates the energy distribution and asymptotic behavior of global solutions to the critical nonlinear wave equation, establishing identities that generalize Morawetz estimates and demonstrating energy equipartition in the limit.

Contribution

It introduces a family of identities extending Morawetz estimates for critical NLW solutions, leading to a precise energy equipartition result for solutions in natural functional spaces.

Findings

Global solutions satisfy energy identities linking spatial and spacetime integrals.

Energy distribution converges to initial energy as spatial radius tends to infinity.

Results apply to both focusing and defocusing critical NLW solutions.

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