Vanishing time behavior of solutions to the fast diffusion equation (Q6185115)

The paper focuses on the initial value problem for the fast diffusion equation \[ \begin{cases} u_t=\Delta u^m, &\text{in }\mathbb{R}^n\times (0,T),\\ u(x,0)=u_0(x), &\text{in }\mathbb{R}^n, \end{cases}\tag{1} \] where \(n\ge 3\), \(m\in \left(0,\frac{n-2}{n}\right)\), and \(u_0\) is close to the initial value of some self-similar solution with \[ u_0(x)=\left(\frac{C_*}{|x|^2|}\right)^{\frac{1}{1-m}}(1+o(1)),\quad\text{as }|x|\to\infty, \] and \(C_*=\frac{2m(n-2-nm)}{1-m}\). Under some assumptions on the parameters from the self-similar solution of the diffusion equation which vanishes identically at time \(T\), the authors investigate various asymptotic behaviours of solutions to problem \((1)\) near the extinction time \(T\).