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 $${\mathrm{\partial }}_{t}u=\mathrm{\Delta }{u}^m-\mathrm{\mid }x{\mathrm{\mid }}^{\sigma }{u}^p,(x,t)\in {\mathbb{R}}^N\times (0,\mathrm{\infty }),$$ with $N\ge 1$ and exponents $$p>1,m_c=\frac{(N-2{)}_{+}}{N}<m<1,\sigma >{\sigma }_{\ast }:=\frac{2(p-1)}{1-m},$$ is established. Moreover, the existence of self-similar solutions of the form $$U(x,t)=(T-t{)}^{\alpha }f(\mathrm{\mid }x\mathrm{\mid }(T-t{)}^{\beta }),\alpha =\frac{\sigma +2}{(1-m)(\sigma -{\sigma }_{\ast })},\beta =\frac{p-m}{(1-m)(\sigma -{\sigma }_{\ast })},$$ with $f(0)>0$, $f^{\mathrm{\prime }}(0)=0$ and $$\underset{\xi \to \mathrm{\infty }}{\lim }{\xi }^{(\sigma +2)\mathrm{/}(p-m)}f(\xi )=L\in (0,\mathrm{\infty })\mathrm{.}$$ 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 $t\in (0,\mathrm{\infty })$ is well-known.