Classification of radial solutions of energy-critical wave systems

TL;DR

This paper analyzes radial solutions of energy-critical wave systems in 3D Minkowski space, showing they decompose into stationary and radiation parts, with a continuous resolution under certain conditions.

Contribution

It extends the channel of energy method to coupled wave systems and establishes a continuous in time decomposition result for radial solutions.

Findings

Radial solutions decompose into stationary and radiation components.

Sequential resolution is achieved using the channel of energy method.

Continuous resolution is proven under additional structural assumptions.

Abstract

This work concerns a general system of energy-critical wave equations in the Minkowski space of dimension $1+3$. The wave equations are coupled by the nonlinearities, which are homogeneous of degree 5. We prove that any radial solution of the system can be written asymptotically as a sum of rescaled stationary solutions plus a radiation term, along any sequence of times for which the solution is bounded in the energy space. With an additional structural assumption on the nonlinearity, we prove a continuous in time resolution result for radial solutions. The proof of the sequential resolution uses the channel of energy method, as in the scalar case treated by Duyckaerts, Kenig and Merle (Cambridge Journal of Mathematics 2013 and arXiv 1204.0031). The proof of the continous in time resolution is based on new compactness and localization arguments.