Continuous in time bubbling and Soliton Resolution for Non-negative Solutions of the Energy-Critical Heat Flow

TL;DR

This paper proves that finite energy, non-negative solutions to the energy-critical nonlinear heat flow in dimensions three and higher decompose into solitons, radiation, and negligible errors over time, confirming the Soliton Resolution Conjecture.

Contribution

It establishes the Soliton Resolution Conjecture for all dimensions d≥3 for non-negative solutions of the energy-critical heat flow, a significant advancement in understanding long-term behavior.

Findings

Solutions decompose into solitons, radiation, and vanishing errors

Confirmation of the Soliton Resolution Conjecture in dimensions d≥3

Asymptotic stability of soliton solutions

Abstract

We show that any finite energy solution of the energy-critical nonlinear heat flow in dimensions $d\geq 3$ asymptotically resolves into a sum of possibly time-dependent solitons, a radiation term, and an error term that vanishes in the energy space. As a consequence, when the initial data has finite energy and is non-negative, we settle the Soliton Resolution Conjecture for all dimensions $d\geq 3.$