Non-existence of radially symmetric singular self-similar solutions of the fast diffusion equation

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\gamma >0$ and $\eta >0$. Suppose either (i) $\alpha \ne 0$ and $\beta =0$ or (ii) $\alpha \in \mathbb{R}$ and $\beta \ne 0$ holds. We will study the elliptic equation $\Delta (f^m/m)+\alpha f+\beta x\cdot \nabla f=0$, $f>0$, in ${\mathbb{R}}^n\setminus \{0\}$ with $\underset{\begin{array}{c}r\to 0\end{array}}{\lim }{r}^{\gamma }f(r)=\eta$. This equation arises from the study of the singular self-similar solutions of the fast diffusion equation which blow up at the origin. We will prove that if there exists a radially symmetric singular solution of the above elliptic equation, then either $\gamma =\frac{2}{1-m}$ and $\alpha >\frac{2\beta }{1-m}$ or $\gamma >\frac{2}{1-m}$, $\beta \ne 0$ and $\gamma =\alpha /\beta$. As a consequence we obtain the non-existence of radially symmetric self-similar solution of the fast diffusion equation $u_t=\Delta (u^m/m)$, $u>0$, which blows up at the origin with rate $|x{|}^{-\gamma }$ when either $0<\gamma \ne \frac{2}{1-m}$ and $\gamma \ne \alpha /\beta$, $\alpha \in \mathbb{R}$ and $\beta \ne 0$ or $\gamma =\frac{2}{1-m}$ and $\left(\alpha -\frac{2\beta }{1-m}\right)\eta \ne \frac{2(n-2-nm)}{(1-m{)}^2}$ holds.

11 pages