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 ${\partial }_{t}u=\Delta u^m+{\left|x\right|}^{\sigma }{u}^p,$ posed for $(x,t)\in R^N\times (0,\infty )$, $N\ge 3$, in the sub-critical range of the fast diffusion equation $0<m<m_c=(N-2)/N$. We consider $\sigma >0$ and $max\{p_c\left(\sigma \right),1\}<p<p_L\left(\sigma \right)$, where $p_c\left(\sigma \right)=\frac{m(N+\sigma )}{N-2},p_L\left(\sigma \right)=1+\frac{\sigma (1-m)}{2}.$ We show that, on the one hand, positive self-similar solutions at any time $t>0$, in the form $u(x,t)=t^{\alpha }f\left(\left|x\right|t^{\beta }\right),f\left(\xi \right)\sim C{\xi }^{-(N-2)/m},\alpha >0,\beta >0$ exist, provided $0<m<m_s=(N-2)/(N+2)$ and $p_s\left(\sigma \right)=m(N+2\sigma +2)/(N-2)<p<p_L\left(\sigma \right)$. On the other hand, we prove that there exists $p_0\left(\sigma \right)\in (p_c\left(\sigma \right),p_s\left(\sigma \right))$ such that self-similar solutions presenting finite time extinction are established both for $p\in (p_0\left(\sigma \right),p_s\left(\sigma \right))$ and for $p\in (p_s\left(\sigma \right),p_L\left(\sigma \right))$, but with profiles $f\left(\xi \right)$ having different spatially decreasing tails as $\left|x\right|\to \infty$. We also prove non-existence of self-similar solutions in complementary ranges of exponents to the ones described above or if $m\ge m_c$.