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

Abstract

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