Global dynamics for the energy-critical nonlinear heat equation

We examine the energy-critical nonlinear heat equation in the critical space $ \dot{H}^1 ( \mathbb{R} ^d) $ for any dimension $ d \geq 3 $. 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.