## [29.4.2016] Dynamics near the ground state for the energy critical nonlinear heat equation in large dimensions

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 ($d\geq 5$). In the ArXiv paper of C. Collot, F. Merle and P. Raphael today, they study the same critical nonlinear heat equation

$u_t = \Delta u + |u|^{\frac{4}{d-2}}u\qquad x\in \mathbb R^d$

and classify the behaviour of solutions around the ground state solitary wave

$Q(x) = \dfrac{1}{\left(1+\dfrac{|x|^2}{d(d-2)}\right)^{(d-2)/2}}$

in the dimension $d\geq 7$.

Given the initial data $u_0$ close enough to the ground state $Q$, the results show that the solution of the heat equation could fall into one of the three scenarios:

(i) Convergence to the ground state: $\exists (\lambda,z)\in\mathbb R_+\times \mathbb R^d$ such that

$\lim\limits_{t\rightarrow +\infty}\left\|u(t,\cdot) - \dfrac{1}{\lambda^{(d-2)/2}}Q\left(\dfrac{\cdot - z}{\lambda}\right)\right\|_{\dot{H}^1} = 0$.

(ii) Decaying to zero:

$\lim\limits_{t\rightarrow+\infty}\|u(t,\cdot)\|_{H^1\cap L^{\infty}} = 0$.

(iii) Blow up in Type I:

$\|u(t,\cdot)\|_{L^{\infty}} \approx (T-t)^{-(d-2)/4}$.