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

For $ n\ge 3 $, $ 0<m<\frac{n-2}{n} $, $ \beta<0 $ and $ \alpha = \frac{2\beta}{1-m} $, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in $ (\mathbb{R}^n\setminus\{0\})\times \mathbb{R} $ of the form $ U_{\lambda}(x,t) = e^{-\alpha t}f_{\lambda}(e^{-\beta t}x), x\in \mathbb{R}^n\setminus\{0\}, t\in\mathbb{R}, $ where $ f_{\lambda} $ is a radially symmetric function satisfying

$ \frac{n-1}{m}\Delta f^m+\alpha f+\beta x\cdot\nabla f = 0 \text{ in }\mathbb{R}^n\setminus\{0\}, $

with $ \underset{\substack{r\to 0}}{\lim}\frac{r^2f(r)^{1-m}}{\log r^{-1}} = \frac{2(n-1)(n-2-nm)}{|\beta|(1-m)} $ and $ \underset{\substack{r\to\infty}}{\lim}r^{\frac{n-2}{m}}f(r) = \lambda^{\frac{2}{1-m}-\frac{n-2}{m}} $, for some constant $ \lambda>0 $.

As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation $ u_t = \frac{n-1}{m}\Delta u^m $ in $ (\mathbb{R}^n\setminus\{0\})\times (0,\infty) $ with initial value $ u_0 $ satisfying $ f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x) $, $ \forall x\in\mathbb{R}^n\setminus\{0\} $, such that the solution $ u $ satisfies $ U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t) $, $ \forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0 $, for some constants $ \lambda_1>\lambda_2>0 $.

We also prove the asymptotic large time behaviour of such singular solution $ u $ when $ n = 3,4 $ and $ \frac{n-2}{n+2}\le m<\frac{n-2}{n} $ holds. Asymptotic large time behaviour of such singular solution $ u $ is also obtained when $ 3\le n<8 $, $ 1-\sqrt{2/n}\le m<\min\left(\frac{2(n-2)}{3n},\frac{n-2}{n+2}\right) $, and $ u(x,t) $ is radially symmetric in $ x\in\mathbb{R}^n\setminus\{0\} $ for any $ t>0 $ under appropriate conditions on the initial value $ u_0 $.