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 $\underset{|x|\to 0}{\lim }|x{|}^{\alpha /\beta }{g}_{\lambda }(x)={\lambda }^{-\frac{{\rho }_1}{(1-m)\beta }}$. 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 $T>0$.

arXiv comment22 pages, the proof of Theorem 1.4 is completely re-written, Lemma 5.2 is added and other typos corrected