Global dynamics for the energy-critical nonlinear heat equation

TL;DR

This paper characterizes the long-term behavior of solutions to the energy-critical nonlinear heat equation in higher dimensions, establishing conditions for global existence or finite-time blow-up and determining decay rates.

Contribution

It provides a necessary and sufficient criterion for initial data that predicts solution outcomes and derives decay rates for global solutions in higher dimensions.

Findings

Dichotomy criterion for solution behavior based on initial data

Decay rates for globally existing solutions

Complete characterization of solution dynamics in energy-critical setting

Abstract

We examine the energy-critical nonlinear heat equation in critical spaces for any dimension greater or equal than three. The aim of this paper is two-fold. First, we establish a necessary and sufficient condition on initial data at or below the ground state that dichotomizes the behavior of solutions. Specifically, this criterion determines whether the solution will either exist globally with energy decaying to zero over time or blow up in finite time. Secondly, we derive the decay rate for solutions that exist globally. These results offer a comprehensive characterization of solution behavior for energy-critical conditions in higher-dimensional settings