  Solutions in self-similar form, either global in time or presenting finite
time blow-up, to the fast diffusion equation with spatially inhomogeneous
source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for
$(x,t)\in\mathbb{R}^N\times(0,\infty)$, $N\geq1$, are classified with respect
to the exponents $m\in(0,1)$, $\sigma\in(-2,\infty)$ and
$p>\max\{1,p_L(\sigma)\}$, where $$ p_L(\sigma)=1+\frac{\sigma(1-m)}{2}. $$ In
the supercritical range $m_c=(N-2)_{+}/N<m<1$, global solutions are classified
with respect to their tail behavior as $|x|\to\infty$, proving that a specific
tail behavior $$ u(x,t)\sim C(m)|x|^{-2/(1-m)}, \qquad {\rm as} \ |x|\to\infty
$$ exists 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
$p>p_L(\sigma)$, but do not exist for any $\sigma\geq0$ and
$p\in(p_F(\sigma),p_s(\sigma))$. In the subcritical range $0<m<m_c$, $N\geq3$,
we introduce a new transformation between radially symmetric solutions to the
equation, which can be understood as a kind of symmetry of the solutions with
respect to the critical exponents $m_s=(N-2)/(N+2)$ and $p_s(\sigma)$, and we
employ this symmetry to classify both global and blow-up self-similar
solutions. We stress that all these results are new also in the homogeneous
case $\sigma=0$.
