Self-similar extinction for a fast diffusion equation with weighted absorption

Finite time extinction of any bounded solution to the fast diffusion equation with spatially inhomogeneous absorption $$\partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty),$$ with $N\geq1$ and exponents $$p>1, \quad m_c=\frac{(N-2)_+}{N}<m<1, \quad \sigma>\sigma_*:=\frac{2(p-1)}{1-m},$$ is established. Moreover, the existence of self-similar solutions of the form $$U(x,t)=(T-t)^{\alpha}f(|x|(T-t)^{\beta}), \quad \alpha=\frac{\sigma+2}{(1-m)(\sigma-\sigma_*)}, \ \beta=\frac{p-m}{(1-m)(\sigma-\sigma_*)},$$ with $f(0)>0$, $f'(0)=0$ and $$\lim\limits_{\xi\to\infty}\xi^{(\sigma+2)/(p-m)}f(\xi)=L\in(0,\infty).$$ is proved, together with some unbounded self-similar solutions as well. The property of finite time extinction is in striking contrast to the standard fast diffusion equation with absorption (that is, $\sigma=0$), where the strict positivity of solutions for any $t\in(0,\infty)$ is well-known.