Extinction of solutions to a class of fast diffusion systems with nonlinear sources

Y. Wang In this paper, the finite time extinction of solutions to the fast diffusion system u t =div(|∇ u | p − 2 ∇ u ) + v m , v t =div(|∇ v | q − 2 ∇ v ) + u n is investigated, where 1 < p , q < 2, m , n > 0 and Ω ⊂ R N( N ≥ 1 ) is a bounded smooth domain. After establishing the local existence of weak solutions, the authors show that if m n > ( p − 1)( q − 1), then any solution vanishes in finite time provided that the initial data are ‘comparable’; if m n = ( p − 1)( q − 1) and Ω is suitably small, then the existence of extinction solutions for small initial data is proved by using the De Giorgi iteration process and comparison method. On the other hand, for 1 < p = q < 2 and m n < ( p − 1) 2 , the existence of at least one non‐extinction solution for any positive smooth initial data is proved. Copyright © 2015 John Wiley & Sons, Ltd.