Asymptotic behaviour of solutions of the equation $u_t=\Delta log u$ near the extinction time

Abstract

Suppose $0\le u_0\in L^1(R^2)\cap L^p(R^2)$ for some constant $p>1$ is a radially symmetric and monotone decreasing function of $r=|x|\ge 0$ such that either supp $u_0$ is compact or $0\le u_0(x)\le C\min (1,|x|^{-(2+\delta)})$ in $R^2$ for some constants $C>0$, $\delta>0$. We will show that if $u$ is the solution of the equation $u_t=\Delta\text{ log }u$, $u>0$, in $R^2\times (0,T)$, $u(x,0)=u_0(x)$ in $R^2$, which satisfies the conditions $r\partial ($log $u(x,t)) /\partial r\to -4$ uniformly on any compact subset of $(0,T)$ as $r=|x|\to\infty$ and $\int_{R^2}u(x,t)dx =\int_{R^2}u_0(x)dx-8\pi t$ $\forall 0\le t < T$, where $T=\int_{R^2}u_0dx/8\pi$, then the function $v(x,s)=u(x,t)/(T-t)$ where $s=-\text{ log }(T-t)$ will converge uniformly on every compact subsets of $R^2$ to a solution of the equation $\Delta\text{ log }v+v=0$ in $R^2$ as $s\to\infty$. In other words $u(x,t)\approx 8 \lambda (T-t)/(\lambda +|x|^2)^2$ as $t$ tends to $T$ for some $\lambda >0$.