Collapsing behaviour of a singular diffusion equation

TL;DR

This paper studies the collapse behavior of solutions to a singular diffusion equation in two dimensions, extending previous results to more general initial conditions and analyzing the solution's extinction near the critical time.

Contribution

It extends prior work by proving the collapsing behavior of the maximal solution for a singular diffusion equation under broader initial data conditions.

Findings

Solution exhibits collapsing behavior near extinction time.

Results generalize previous findings to wider initial conditions.

Provides detailed analysis of solution behavior at extinction.

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 ${ess}\inf_{\2{B}_{r_1}(0)}u_0\ge{ess} \sup_{\R^2\setminus B_{r_2}(0)}u_0$ 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 $u_t=\Delta\log u$ in $\R^2\times (0,T)$, $u(x,0)=u_0(x)$ in $\R^2$, near its extinction time $T=\int_{R^2}u_0dx/4\pi$.