Asymptotic large time behavior of singular entire solutions of the fast diffusion equation

Abstract

Let $n\ge 3$, $0 0$, for the fast diffusion equation $u_t=\Delta (u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, where $f$ satisfies \begin{equation*} \Delta (f^m/m) + \alpha f + \beta x \cdot \nabla f =0 \quad \text{in} \; \mathbb{R}^n\setminus\{0\} \end{equation*} with $\lim_{|x| \to 0} |x|^{ \frac{\alpha}{\beta}}f(x)=A$ and $\lim_{|x| \to \infty}f(x) = D_A$ for some constants $A>0$, $D_A > 0$. We also obtain an asymptotic expansion of such singular radially symmetric solution $f$ near the origin. We will also prove the asymptotic large time behaviour of the singular solutions of the fast diffusion equation $u_t= \Delta (u^m/m)$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$, $u(x,0)=u_0(x)$ in $\mathbb{R}^n\setminus\{0\}$, satisfying the condition $A_1|x|^{-\gamma}\leq u_0(x)\leq A_2|x|^{-\gamma}$ in $\mathbb{R}^n\setminus\{0\}$, for some constants $A_2>A_1>0$ and $n\le\gamma<\frac{n-2}{m}$.