Asymptotic behaviour of solutions of $u_t=\Delta$ log u$ in a bounded domain

Abstract

We will show that if $u$ is the solution of $u_t=\Delta$ 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 $\Omega\subset R^n$ where $\Omega$ is a smooth convex bounded domain, then for $c_1=\infty$ the rescaled function $w=$log $(u/t)$ will converge uniformly on every compact subset of $\Omega$ to the unique solution $\psi$ of the equation $\Delta\psi-e^{\psi}=0$, $\psi >0$, in $\Omega$ with $\psi=\infty$ on $\partial\Omega$ as $t\to\infty$. When $0 <c_1 <\infty$, $n=1,2$, or $3$, and $u_0\ge c_1$ on $\Omega$, then the function $w=e^{(\lambda_1/c_1)t}$log $(u/c_1)$ will converge uniformly on $\overline{\Omega}$ to $A\phi_1$ as $t\to\infty$ where $\lambda_1>0$ and $\phi_1$ are the first positive eigenvalue and positive eigenfunction of the Laplace operator $-\Delta$ on $\Omega$ with $\|\phi_1\|_{L^2(\Omega)}=1$ respectively and $A=\lim_{t\to\infty}\|w(\cdot ,t)\|_{L^2(\Omega )}$.