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

We study the asymptotic large time behavior of singular solutions of the fast diffusion equation $u_t=Δ u^m$ in $({\mathbb R}^n\setminus\{0\})×(0, ∞)$ in the subcritical case $0<m<\frac{n-2}{n}$, $n≥3$. Firstly, we prove the existence of the singular solution $u$ of the above equation that is trapped in between self-similar solutions of the form of $t^{-α} f_i(t^{-β}x)$, $i=1, 2$, with the initial value $u_0$ satisfying $A_1|x|^{-γ}≤ u_0≤ A_2|x|^{-γ}$ for some constants $A_2>A_1>0$ and $\frac{2}{1-m}<γ<\frac{n-2}{m}$, where $β:=\frac{1}{2-γ(1-m)}$, $α:=\frac{2\beta-1}{1-m}, $ and the self-similar profile $f_i$ satisfies the elliptic equation

$Δ f^m+α f+β x· \nabla f=0 \,\,\,\,\,\,\mbox{ in ${\mathbb R}^n\setminus\{0\}$}$

with $\lim_{|x|\to0}|x|^{\frac{ α}{ β}}f_i(x)=A_i$ and $\lim_{|x|\to∞}|x|^{\frac{n-2}{m}}{f_i}(x)= D_{A_i} $ for some constants $D_{A_i}>0$. When $\frac{2}{1-m} < γ < n$, under an integrability condition on the initial value $u_0$ of the singular solution $u$, we prove that the rescaled function

$\tilde u(y, τ):= t^{\, α} u(t^{\, β} y, t),\,\,\,\,\,\, { τ:=\log t}, $

converges to some self-similar profile $f$ as $τ\to∞$.