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 𝜕𝑡𝑢=𝛥𝑢𝑚+|𝑥|𝜎𝑢𝑝, posed for (𝑥, 𝑡) ∈ R^𝑁× (0,∞),𝑁≥ 3, in the sub-critical range of the fast diffusion equation 0< 𝑚 < 𝑚𝑐= (𝑁− 2)∕𝑁. We consider 𝜎 >0 and max{𝑝𝑐(𝜎),1}< 𝑝 < 𝑝𝐿(𝜎), where 𝑝𝑐(𝜎) =𝑚(𝑁+𝜎) 𝑁− 2 , 𝑝𝐿(𝜎) = 1 + 𝜎(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^𝛼 𝑓(|𝑥|𝑡^𝛽), 𝑓(𝜉) ∼𝐶 𝜉−(𝑁−2)∕𝑚, 𝛼 >0, 𝛽 >0 exist, provided 0< 𝑚 < 𝑚𝑠= (𝑁− 2)∕(𝑁+ 2)and 𝑝𝑠(𝜎) =𝑚(𝑁+ 2𝜎+ 2)∕(𝑁− 2)< 𝑝 < 𝑝𝐿(𝜎). On the other hand, we prove that there exists 𝑝_0(𝜎) ∈ (𝑝𝑐(𝜎), 𝑝𝑠(𝜎)) such that self-similar solutions presenting finite time extinction are established both for 𝑝∈ (𝑝_ 0(𝜎), 𝑝𝑠(𝜎)) and for 𝑝∈ (𝑝𝑠(𝜎), 𝑝𝐿(𝜎)), but with profiles 𝑓(𝜉) having different spatially decreasing tails as |𝑥|→∞. We also prove non-existence of self-similar solutions in complementary ranges of exponents to the ones described above or if 𝑚≥𝑚𝑐.