[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}.

About baotangquoc

Lecturer School of Applied Mathematics and Informatics Hanoi University of Science and Technology No 1, Dai Co Viet Street, Hanoi
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