Abstract.
We will show that if u is the solution of the equation \(u_t=(\text{log u})_{xx}, u>0\), in \(\Bbb{R}\times(0,T), u(x,0)=u_0(x)\in L^1(\Bbb{R})\) is an even function on \(\Bbb{R}\) and is monotone decreasing in \(x\ge 0, 0\le u_0\not\equiv 0\) on \(\Bbb{R}, \lim_{x\to\infty}\int_a^b(\text{log }u)_x(x,t) dt =-f(t)\), \(\lim_{x\to -\infty}\int_a^b(\text{log }u)_x(x,t) dt=f(t)\), \(\forall 0<a<b<T\) where \(0\le f\in C([0,\infty))\) is a monotone increasing function satisfying \(2\int_0^{\infty}fdt>\int_{\Bbb{R}} u_0 dx\) with \(T>0\) being given by \(\int_{\Bbb{R}}u_0 dx =2\int_0^Tfdt\) and \(f(T)>0\), then the rescaled function \(v(x,s) =u(x,t)/(T-t), s=- \log (T-t)\), will converge uniformly on every compact subset of \(\Bbb{R}\) to \(2/(\lambda\text{ cosh}^2 (x/\sqrt{\lambda}))\) as \(s\to\infty\) where \(\lambda=4/f(T)^2\).
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Received: 25 May 2000 / Revised version: 26 October 2001 / Published online: 28 February 2002
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Hsu, SY. Dynamics near extinction time of a singular diffusion equation. Math Ann 323, 281–318 (2002). https://doi.org/10.1007/s002080100304
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DOI: https://doi.org/10.1007/s002080100304