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Collasping behaviour of a singular diffusion equation

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  • Let $0\le u_0(x)\in L^1(\mathbb{R}^2)\cap L^{\infty}(\mathbb{R}^2)$ 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 [4], [6], and prove the collapsing behaviour of the maximal solution of the equation $u_t=\Delta\log u$ in $\mathbb{R}^2\times (0,T)$, $u(x,0)=u_0(x)$ in $\mathbb{R}^2$, near its extinction time $T=\int_{R^2}u_0dx/4\pi$ by a simplified method without using the Hamilton-Yau Harnack inequality.
    Mathematics Subject Classification: Primary: 35B40; Secondary: 35K57, 35K65.

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