Global solutions versus finite time blow-up for the supercritical fast diffusion equation with inhomogeneous source

Solutions in self-similar form, either global in time or presenting finite time blow-up, to the supercritical fast diffusion equation with spatially inhomogeneous source $${\mathrm{\partial }}_{t}u=\mathrm{\Delta }{u}^m+\mathrm{\mid }x{\mathrm{\mid }}^{\sigma }{u}^p,(x,t)\in {\mathbb{R}}^N\times (0,\mathrm{\infty })$$ with $$m_c=\frac{(N-2{)}_{+}}{N}\le m<1,\sigma \in (\max \{-2,-N\},\mathrm{\infty }),p>\max \{1+\frac{\sigma (1-m)}{2},1\}$$ are considered. It is proved that global self-similar solutions with the specific tail behavior $$u(x,t)\sim C(m)\mathrm{\mid }x{\mathrm{\mid }}^{-2\mathrm{/}(1-m)},\mathrm{a}\mathrm{s}\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$$ exist exactly for $p\in (p_F(\sigma ),p_s(\sigma ))$, where $$p_F(\sigma )=m+\frac{\sigma +2}{N},p_s(\sigma )=\{\begin{array}{ll}\frac{m(N+2\sigma +2)}{N-2},& N\ge 3,\\ \mathrm{\infty },& N\in \{1,2\},\end{array}$$ are the renowned Fujita and Sobolev critical exponents. In contrast, it is shown that self-similar solutions presenting finite time blow-up exist for any $\sigma \in (-2,0)$ and $p$ as above, but do not exist for any $\sigma \ge 0$ and $p\in (p_F(\sigma ),p_s(\sigma ))$. We stress that all these results are \emph{new also in the homogeneous case $\sigma =0$}.