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

Abstract

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 $\underset{\begin{array}{c}r\to 0\end{array}}{\lim }\frac{r^{2}f(r{)}^{1-m}}{\log r^{-1}}=\frac{2(n-1)(n-2-nm)}{|\beta |(1-m)}$ and $\underset{\begin{array}{c}r\to \infty \end{array}}{\lim }{r}^{\frac{n-2}{m}}f(r)={\lambda }^{\frac{2}{1-m}-\frac{n-2}{m}}$, 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 $f_{{\lambda }_1}(x)\le u_0(x)\le f_{{\lambda }_2}(x)$, $\forall x\in {\mathbb{R}}^n\setminus \{0\}$, which 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 behaviour of such singular solution $u$ of the fast diffusion equation as $t\to \infty$ when $n=3,4$ and $\frac{n-2}{n+2}\le m<\frac{n-2}{n}$ holds. Asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to \infty$ 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 $u_0$.

34 pages