Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured Euclidean space

Kin Ming Hui, Jinwan Park

Published 2020-07-14Version 1

We study the existence, uniqueness, and asymptotic behaviour of the singular solution of the fast diffusion equation, which blows up at the origin for all time. For $n\ge 3$, $0<m<\frac{n-2}{n}$, $\beta<0$ and $\alpha=\frac{2\beta}{1-m}$, we prove the existence and asymptotic behaviour of singular eternal self-similar solution of the fast diffusion equation. As a consequence, we prove the existence and uniqueness of solution of Cauchy problem for the fast diffusion equation. For $n=3, 4$ and $\frac{n-2}{n+2}\le m<\frac{n-2}{n}$, under appropriate condition on the initial value $u_0$, we prove the asymptotic large time behaviour, the rescaled function $\tilde u(x,t):=e^{\alpha t}u(e^{\beta t}x,t)$ converges uniformly on every compact subset of $\mathbb R^n\setminus\{0\}$ to $f_{\lambda_0}$ as $t\to\infty$, for some $\lambda_0>0$. Furthermore, for the radially symmetric initial value $u_0$, $3 \le n < 8$, $1- \sqrt{\frac{2}{n}} \le m \le \min \left \{\frac{2(n-2)}{3n}, \frac{n-2}{n+2}\right \}$, we also have the asymptotic large time behaviour.