Let $0le u_0(x)in L^1(R^2)cap L^{infty}(R^2)$ 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 ${ess}inf_{2{B}_{r_1}(0)}u_0ge{ess} sup_{R^2setminus B_{r_2}(0)}u_0$ 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 $u_t=Deltalog u$ in $R^2times (0,T)$, $u(x,0)=u_0(x)$ in $R^2$, near its extinction time $T=int_{R^2}u_0dx/4pi$.
تحميل البحث