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 $\underset{\substack{r\to 0}}{\lim}\,r^{\gamma}f(r)=\eta$. 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.