Extinction and non-extinction profiles for the sub-critical fast diffusion equation with weighted source

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 $m\geq m_c$.