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 $$\partial_tu=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty)$$ with $$m_c=\frac{(N-2)_+}{N}\leq m<1, \quad \sigma\in(\max\{-2,-N\},\infty), \quad p>\max\left\{1+\frac{\sigma(1-m)}{2},1\right\}$$ are considered. It is proved that global self-similar solutions with the specific tail behavior $$u(x,t)\sim C(m)|x|^{-2/(1-m)}, \qquad {\rm as} \ |x|\to\infty$$ exist exactly for $p\in(p_F(\sigma),p_s(\sigma))$, where $$p_F(\sigma)=m+\frac{\sigma+2}{N}, \qquad p_s(\sigma)=\left\{\begin{array}{ll}\frac{m(N+2\sigma+2)}{N-2}, & N\geq3,\\\infty, & N\in\{1,2\}, \end{array}\right.$$ 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 $\sigma\geq0$ and $p\in(p_F(\sigma),p_s(\sigma))$. We stress that all these results are \emph{new also in the homogeneous case $\sigma=0$}.