Asymptotic behaviour of solutions of the equation \(u_t=\Delta\log u\) near the extinction time (Q1413007)

Let \(u>0\) be a solution of the Cauchy problem \[ \begin{cases} u_t=\Delta \log u &\text{ in }\mathbb R^2\times(0,T),\\ u(x,0)=u_0(x) &\text{ in }\mathbb R^2 \end{cases} \] where \(0\leq u_0\in L^1({\mathbb R}^2)\cap L^p({\mathbb R}^2),\) \(p>1,\) is a radially symmetric and monotone decreasing function of \(r=|x|\) such that either \(\text{ supp }u_0\) is compact or \(0\leq u_0\leq C \min\{1,|x|^{-2-\delta}\},\) \(C>0,\;\delta>0.\) Suppose further \[ r{{\partial (\log u(x,t))}\over {\partial r}} \to -4\quad \text{ uniformly\;on\;compact\;subsets\;of} (0,T)\quad \text{ as} \;r\to\infty \] and \[ \int_{{\mathbb R}^2} u(x,t) dx= \int_{{\mathbb R}^2} u_0(x) dx -8\pi t\quad \forall 0\leq t < T= {1\over {8\pi}}\int_{{\mathbb R}^2} u_0(x) dx. \] Setting \(s=-\log(T-t),\) the author shows that the function \(v(x,s)=u(x,t)/(T-t)\) converges uniformly on compact subsets of \({\mathbb R}^2\) as \(s\to\infty\) to a solution of the equation \[ \Delta \log v+v=0\quad \text{ in } \;{\mathbb R}^2. \] In other words, \[ u(x,t)\approx {{8\lambda(T-t)}\over {(\lambda+|x|^2)^2}}\qquad \text{ as } t\to T \] for some \(\lambda>0.\)