Dynamics of 3D focusing, energy-critical wave equation with radial data

TL;DR

This paper investigates the long-term behavior of radial solutions to the energy-critical wave equation in 3D, revealing that soliton resolution occurs almost always, with implications for understanding blow-up solutions and multi-soliton interactions.

Contribution

It establishes a link between energy radiation and soliton resolution, providing new insights into the dynamics of solutions and their long-time behavior in the energy-critical wave equation.

Findings

Soliton resolution occurs at all times except short intervals.

A correspondence between energy radiation and solution behavior is identified.

Applications include analysis of blow-up solutions and multi-soliton interactions.

Abstract

In this article we discuss the long-time dynamics of the radial solutions to the energy-critical wave equation in 3-dimensional space. Given a solution defined for all time $t\geq 0$, we show that the soliton resolution phenomenon happens at all times $t>0$ except for a few relatively short time intervals. The main tool is the radiation theory of wave equations and the major observation of this work is a correspondence between the energy radiation and the soliton resolution/collision behaviour of solutions. We also give a few applications of the main observation on the type II blow-up solutions and ``one pass'' theory near pure mutli-solitons.