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

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\alpha =\frac{2\beta -1}{1-m}$ and $\frac{2}{1-m}<\frac{\alpha }{\beta }<\frac{n-2}{m}$. We give a new direct proof using fixed point method on the existence of singular radially symmetric forward self-similar solution of the form $V(x,t)=t^{-\alpha }f(t^{-\beta }x)$ $\forall x\in {\mathbb{R}}^n\setminus \{0\}$, $t>0$, 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*} Δ(f^m/m) + αf + βx \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 $A_1|x{|}^{-\gamma }\le u_0(x)\le A_2|x{|}^{-\gamma }$ in ${\mathbb{R}}^n\setminus \{0\}$, for some constants $A_2>A_1>0$ and $n\le \gamma <\frac{n-2}{m}$.