Dynamics near the ground state for the Sobolev critical Fujita type heat equation in 6D
Junichi Harada
TL;DR
This work analyzes the Sobolev-critical Fujita-type heat equation $u_t=\Delta u+|u|^{p-1}u$ in $\mathbb{R}^6$ (with $p=\frac{n+2}{n-2}$) near the Aubin–Talenti ground state ${\sf Q}$. The authors develop a modulation framework around the ground-state manifold ${\mathcal M}$ and derive a coupled dynamical system for the scaling $\lambda(t)$, translation $z(t)$, and the unstable mode amplitude $a(t)$, together with the remainder $\epsilon$. They establish a complete trichotomy for initial data close to ${\sf Q}$: global convergence to a translated/scaled ground state, global dissipation to zero, or finite-time Type I blowup, extending the previously known $n\ge7$ classification to the borderline case $n=6$ by incorporating an $L^2$-based control of the remainder to close the modulation dynamics. The results provide a sharp, dimension-6 analogue of Collot–Merle–Raphaël’s classification and deepen the understanding of ground-state dynamics at the Sobolev critical threshold. The analysis contributes a rigorous description of the delicate interplay between scale, position, and unstable mode in the most challenging critical dimension.
Abstract
This paper investigates the asymptotic behavior of solutions to $u_t=Δu+|u|^{p-1}u$ in the Sobolev critical case. Our main result is a classification of the dynamics near the ground states in the six dimensional case. It is shown that if the initial data $u_0\in H^1(\mathbb{R}^6)$ satisfies $\|u_0-{\sf Q}\|_{\dot H^1(\mathbb{R}^6)}\ll1$, then the solution falls into one of the following three scenarios: 1) It is globally defined and converge to one of the ground states as $t\to\infty$. 2) It is globally defined and converge to $0$ in $\dot H^1(\mathbb{R}^6)$ as $t\to\infty$. 3) It exhibits finite time blowup with a type I rate. This paper extends the classification result in the case $n\geq7$, previously obtained by Collot-Merle-Raphaël, to the borderline case $n=6$.