Dynamics near the ground state for the Sobolev critical Fujita type heat equation in 6D

Abstract

This paper investigates the asymptotic behavior of solutions to $u_t=\Delta u+|u{|}^{p-1}u$ 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 $\Vert u_0-\mathsf{Q}{\Vert }_{{\dot{H}}^1({\mathbb{R}}^6)}\ll 1$, 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 $n\ge 7$, previously obtained by Collot-Merle-Raphaël, to the borderline case $n=6$.