Dynamics of solutions of a singular diffusion equation

Abstract

In this paper we will obtain the asymptotic profile of solutions of the equation $u_t=\Delta$ log $u$, $u>0$, in $\Omega\times (0,\infty)$, $u=c_1$ on $\partial\Omega\times (0,\infty)$, $u(x,0)=u_0(x)\ge 0$ on $\Omega\subset R^n$ for all $n\in\mathcal {Z}^+$ and $0 <c_1 <\infty$ where $\Omega$ is a smooth, convex, bounded domain.