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Blow‐up analysis for a class of nonlinear reaction diffusion equations with Robin boundary conditions
Author(s) -
Ding Juntang,
Shen Xuhui
Publication year - 2017
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.4697
Subject(s) - mathematics , reaction–diffusion system , bounded function , upper and lower bounds , domain (mathematical analysis) , robin boundary condition , nonlinear system , boundary (topology) , mathematical analysis , regular polygon , class (philosophy) , diffusion , boundary value problem , free boundary problem , geometry , physics , quantum mechanics , artificial intelligence , computer science , thermodynamics
This paper is devoted to the study of the blow‐up phenomena of following nonlinear reaction diffusion equations with Robin boundary conditions:g ( u )t = ∇ · ( ρ ( | ∇ u | 2 ) ∇ u ) + k ( t ) f ( u ) in Ω × ( 0 , t ∗ ) ,∂ u ∂ ν + γ u = 0 on ∂ Ω × ( 0 , t ∗ ) ,u ( x , 0 ) = u 0 ( x ) inΩ ¯ .Here, Ω ⊂ R n( n ⩾ 2 ) is a bounded convex domain with smooth boundary. With the aid of a differential inequality technique and maximum principles, we establish a blow‐up or non–blow‐up criterion under some appropriate assumptions on the functions f , g , ρ , k , and u 0 . Moreover, we dedicate an upper bound and a lower bound for the blow‐up time when blowup occurs.