Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time

Abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\rho_1>0$, $\beta\ge\frac{m\rho_1}{n-2-nm}$ and $\alpha=\frac{2\beta+\rho_1}{1-m}$. For any $\lambda>0$, we will prove the existence and uniqueness (for $\beta\ge\frac{\rho_1}{n-2-nm}$) of radially symmetric singular solution $g_{\lambda}\in C^{\infty}(R^n\setminus\{0\})$ of the elliptic equation $\Delta v^m+\alpha v+\beta x\cdot\nabla v=0$, $v>0$, in $R^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{\alpha/\beta}g_{\lambda}(x)=\lambda^{-\frac{\rho_1}{(1-m)\beta}}$. When $\beta$ is sufficiently large, we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as $|x|\to\infty$. We also obtain an inversion formula for the radially symmetric solution of the above equation. As a consequence we will prove the extinction behaviour of the solution $u$ of the fast diffusion equation $u_t=\Delta u^m$ in $R^n\times (0,T)$ near the extinction time $T>0$.