We consider the Cauchy problem
\[
\left\{ \begin{array}{ll}
u_\tau = \Delta (u^m),
\qquad & x\in\mathbb{R}^n, \ \tau\in (0,T), \\[2mm]
u(x,0)=u_0(x)\ge 0, \qquad & x\in\mathbb{R}^n,
\end{array} \right.
\]
where \(n\ge 3\), \(T>0\) and \(0<m<1\).
It is known that for \(m<m_c:=(n-2)/n\) all
solutions with initial data satisfying
\[
u_0(x)=O\left(|x|^{-\frac{2}{1-m}}\right)\qquad\mbox{as }|x|\to\infty,
\]
extinguish in finite time. We shall
consider solutions which vanish at \(\tau=T\)
and study their behaviour near \(\tau=T\).
The function
\begin{equation}\label{sep}
u(x,\tau):=\big((1-m)(T-\tau)\big)^{\frac{1}{1-m}}\varphi^{\frac{1}{m}}(x)
\end{equation}
is a solution of the fast diffusion equation \(u_\tau = \Delta (u^m)\) if
\(\varphi\) satisfies
\[
\Delta\varphi +\varphi^p=0, \qquad x\in\mathbb{R}^n, \qquad p:=\frac{1}{m} \, .
\]
We call a nontrivial solution of the form (\ref{sep}) separable. We shall
show that separable solutions are stable in a suitable sense if
\[
n>10,\qquad 0<m<\frac{(n-2)(n-10)}{(n-2)^2-4n+8\sqrt{n-1}} \, .
\]
We also find optimal rates of convergence to separable solutions.
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