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

TL;DR

This paper classifies the long-term behavior of solutions to the Sobolev critical Fujita type heat equation in six dimensions, showing they either converge to ground states, decay to zero, or blow up in finite time.

Contribution

It extends the classification of solution dynamics near ground states to the critical six-dimensional case, previously known for higher dimensions.

Findings

Solutions near ground states either converge, decay, or blow up.

The classification applies specifically to initial data close to ground states in H^1.

The result bridges the gap in understanding the 6D case, previously studied for n≥7.

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 $\|u_0-{\sf Q}\|_{\dot H^1(\mathbb{R}^6)}\ll1$, 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-Rapha\"el, to the borderline case $n=6$.