Collasping behaviour of a singular diffusion equation
Kin Ming Hui · Discrete and Continuous Dynamical Systems · 2012
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 decreasingfor 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 onthe initial value $u_0$ we will extend the result of P. Daskalopoulos,M.A. del Pino and N. Sesum [4], [6], and prove thecollapsing behaviour of the maximal solution of theequation $u_t=\Delta\log u$ 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.