Global dynamics for the energy-critical nonlinear heat equation
Masahiro Ikeda, César J. Niche, Gabriela Planas
TL;DR
This work analyzes the energy-critical nonlinear heat equation on $\mathbb{R}^d$ for $d\ge 3$ in the energy space $\dot{H}^1(\mathbb{R}^d)$. It establishes a sharp threshold at the ground state $W$ (the Aubin–Talenti bubble) yielding a dissipativity vs. blow-up dichotomy: if $E(u_0) \le E(W)$ and $\|\nabla u_0\|_{L^2} < \|\nabla W\|_{L^2}$, solutions exist globally with $\|u(t)\|_{\dot{H}^1} \to 0$, whereas if $\|\nabla u_0\|_{L^2} > \|\nabla W\|_{L^2}$ (with $u_0 \in L^2$) they blow up in finite time. The paper extends these insights to all dimensions $d \ge 3$ using a concentration-compactness/rigidity framework to rule out critical elements and prove the dissipative behavior below the ground state, and develops decay estimates via the decay character $r^*$ and Fourier splitting, providing explicit decay rates for the $\dot{H}^1$ norm (including a universal log-decay for all $d$ and sharper polynomial decay for $3 \le d \le 10$). The results rely on a linear profile decomposition, a perturbation lemma, and a Lyapunov structure of the energy, yielding a robust approach to threshold dynamics and long-time decay in the 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