  We establish both extinction and non-extinction self-similar profiles for the
following fast diffusion equation with a weighted source term $$
\partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for
$(x,t)\in\real^N\times(0,\infty)$, $N\geq3$, in the sub-critical range of the
fast diffusion equation $0<m<m_c=(N-2)/N$. We consider $\sigma>0$ and
$\max\{p_c(\sigma),1\}<p<p_L(\sigma)$, where $$
p_c(\sigma)=\frac{m(N+\sigma)}{N-2}, \qquad
p_L(\sigma)=1+\frac{\sigma(1-m)}{2}. $$ We show that, on the one hand, positive
self-similar solutions at any time $t>0$, in the form $$
u(x,t)=t^{\alpha}f(|x|t^{\beta}), \qquad f(\xi)\sim C\xi^{-(N-2)/m}, \qquad
\alpha>0, \ \beta>0 $$ exist, provided $0<m<m_s=(N-2)/(N+2)$ and
$p_s(\sigma)=m(N+2\sigma+2)/(N-2)<p<p_L(\sigma)$. On the other hand, we prove
that there exists $p_0(\sigma)\in(p_c(\sigma),p_s(\sigma))$ such that
self-similar solutions presenting finite time extinction are established both
for $p\in(p_0(\sigma),p_s(\sigma))$ and for $p\in(p_s(\sigma),p_L(\sigma))$,
but with profiles $f(\xi)$ having different spatially decreasing tails as
$|x|\to\infty$. We also prove non-existence of self-similar solutions in
complementary ranges of exponents to the ones described above or if $m\geq
m_c$.
