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

Abstract

We establish both extinction and non-extinction self-similar profiles for the following fast diffusion equation with a weighted source term $${\partial }_{t}u=\Delta u^m+|x{|}^{\sigma }{u}^p,$$ posed for $(x,t)\in {\Re }^N\times (0,\infty )$, $N\ge 3$, in the sub-critical range of the fast diffusion equation $0<m<m_c=(N-2)/N$. We consider $\sigma >0$ and $\max \{p_c(\sigma ),1\}<p<p_L(\sigma )$, where $$p_c(\sigma )=\frac{m(N+\sigma )}{N-2},p_L(\sigma )=1+\frac{\sigma (1-m)}{2}.$$ We show that, on the one hand, positive self-similar solutions at any time $t>0$, in the form $$u(x,t)=t^{\alpha }f(|x|t^{\beta }),f(\xi )\sim C{\xi }^{-(N-2)/m},\alpha >0,\beta >0$$ 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 $m\ge m_c$.