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

TL;DR

This paper investigates the asymptotic behavior of solutions to the fast diffusion equation near extinction time, establishing existence, uniqueness, and detailed asymptotics of associated elliptic solutions, and deriving implications for the original PDE.

Contribution

It provides new existence and uniqueness results for singular elliptic solutions and characterizes their asymptotics, leading to a deeper understanding of the extinction behavior of fast diffusion solutions.

Findings

Existence and uniqueness of radially symmetric singular solutions.

Higher order asymptotics of solutions as |x|→∞.

Inversion formula for radially symmetric solutions.

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 $T>0$.