國立清華大學 數學系

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 $\mbox{ess}\inf_{\2{B}_{r_1}(0)}u_0\ge\mbox{ess} \sup_{R^2\setminus B_{r_2}(0)}u_0$ 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 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 $T=\int_{R^2}u_0dx/4\pi$ without using the Hamilton-Yau Harnack inequality. Tea Time: 3:30pm, R707