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Extinction of solutions to a class of fast diffusion systems with nonlinear sources
Author(s) -
Han Yuzhu,
Gao Wenjie
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3571
Subject(s) - extinction (optical mineralogy) , mathematics , bounded function , diffusion , domain (mathematical analysis) , nonlinear system , class (philosophy) , reaction–diffusion system , mathematical analysis , diffusion process , pure mathematics , combinatorics , physics , thermodynamics , computer science , innovation diffusion , quantum mechanics , optics , artificial intelligence , knowledge management
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.