Non-radiative solutions and long-time dynamics of 5D focusing energy-critical wave equation in the radial case

TL;DR

This paper studies the long-term behavior of radial solutions to the 5D focusing energy-critical wave equation, proving a soliton resolution conjecture in the radial case without energy boundedness assumptions.

Contribution

It provides a detailed analysis of non-radiative solutions, their asymptotics, and establishes a quantitative soliton resolution theorem for all-time solutions in the radial setting.

Findings

Verification of the soliton resolution conjecture in the radial case

Development of a radiation theory linking wave radiation and soliton interactions

Characterization of the asymptotic behavior of non-radiative solutions

Abstract

In this article we discuss the long-time dynamics of the radial solutions to the focusing energy-critical wave equation in 5-dimensional space. We give some details about the asymptotic behaviour, topological structure and time evolution of the non-radiative solutions to this equation. As an application we prove a quantitative version of soliton resolution theorem for solutions defined for all time $t>0$, which immediately verifies the soliton resolution conjecture in the radial case, without a priori boundedness assumption on the energy norm of solution as time tends to infinity. The main tool of this work is the radiation theory of wave equations and the major observation of this work is a correspondence between the radiation and the soliton collision behaviour of solutions.