一类奇异扩散方程的坍缩性
Collapsing behaviour of a singular diffusion equation
摘要：
设$0\le u_0(x)\in L^1(\R^2)\cap L^{\infty}(\R^2)$是这样的，对于所有$|x|\ge r_1$，$u_0(x) =u_0(|x|)$是单调递减的，对于某个常数$r_1>0$，对于所有$|x|\ge r_1$，$u_0(x) =u_0(|x|)$是单调递减的，对于某个常数$r_2>r_1$，${ess}\inf_{\2{B}_{r_1}(0)}u_0\ge{ess} \sup_{\R^2\setminus B_{r_2}(0)}u_0$是单调递减的。然后在初始值$u_0$的无穷远处的一些温和的衰减条件下，我们推广了P.Daskalopoulos，M.A. del Pino和N. Sesum \cite{DP2}，\cite{DS}，并证明了方程$u_t=Δ\log u$ in $\R^2\times (0,T)$，$u(x,0)=u_0(x)$ in $\R^2$的极大解在其灭绝时刻$T=\int_{R^2}u_0dx/4π$附近的坍缩行为.
英文摘要：
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 $u_t=Δ\log u$ in $\R^2\times (0,T)$, $u(x,0)=u_0(x)$ in $\R^2$, near its extinction time $T=\int_{R^2}u_0dx/4π$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
