Extinction profile of the logarithmic diffusion equation

Abstract

Let $u$ be the solution of $u_t=\Delta\log u$ in $\R^N\times (0,T)$, N=3 or $N\ge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)\le u_0\le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $\4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-\log (T-t)$, converges uniformly on $\R^N$ to the rescaled Barenblatt solution $\4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $s\to\infty$. We also obtain convergence of the rescaled solution $\4{u}(x,s)$ as $s\to\infty$ when the initial data satisfies $0\le u_0(x)\le B_{k_0}(x,0)$ in $\R^N$ and $|u_0(x)-B_{k_0}(x,0)|\le f(|x|)\in L^1(\R^N)$ for some constant $k_0>0$ and some radially symmetric function $f$.