In this post, we talked about a paper concerning the, roughly speaking, blow-up profiles of solutions to the critical heat equation in large dimensions (). In the ArXiv paper of C. Collot, F. Merle and P. Raphael today, they study the same critical nonlinear heat equation
and classify the behaviour of solutions around the ground state solitary wave
in the dimension .
Given the initial data close enough to the ground state , the results show that the solution of the heat equation could fall into one of the three scenarios:
(i) Convergence to the ground state: such that
(ii) Decaying to zero:
(iii) Blow up in Type I: