Collapsing behaviour of a singular diffusion equation

Abstract

Let $0\le u_0(x)\in L^1(\R^2)\cap L^{\infty}(\R^2)$ be such that $u_0(x) =u_0(|x|)$ for all $|x|\ge r_1$ and is monotone decreasing for all $|x|\ge r_1$ for some constant $r_1>0$ and ${ess}\inf_{\2{B}_{r_1}(0)}u_0\ge{ess} \sup_{\R^2\setminus B_{r_2}(0)}u_0$ for some constant $r_2>r_1$. Then under some mild decay conditions at infinity on the initial value $u_0$ we will extend the result of P. Daskalopoulos, M.A. del Pino and N. Sesum \cite{DP2}, \cite{DS}, and prove the collapsing behaviour of the maximal solution of the equation $u_t=\Delta\log u$ in $\R^2\times (0,T)$, $u(x,0)=u_0(x)$ in $\R^2$, near its extinction time $T=\int_{R^2}u_0dx/4\pi$.