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Global existence and boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant and logistic source
Author(s) -
Baghaei Khadijeh,
Khelghati Ali
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.4264
Subject(s) - bounded function , mathematics , chemotaxis , domain (mathematical analysis) , neumann boundary condition , homogeneous , boundary (topology) , function (biology) , constant (computer programming) , pure mathematics , combinatorics , mathematical analysis , chemistry , computer science , biochemistry , receptor , evolutionary biology , biology , programming language
This paper deals with the following chemotaxis system:u t = ∇ · ( δ ∇ u − χ u ∇ v ) + f ( u ) , x ∈ Ω , t > 0 ,v t = Δ v − u v , x ∈ Ω , t > 0 ,under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ⩾ 1 , with smooth boundary. Here, δ and χ are some positive constants and f is a smooth function that satisfies f ( 0 ) ⩾ 0andf ( s ) ⩽ a s − b s γ , s > 0with some constants a ⩾0, b > 0, and γ > 1. We prove that the classical solutions to the preceding system are global and bounded provided that ∥ v 0∥L ∞ ( Ω ) <1 χδ 2 ( n + 1 )π + 2 arctan( 1 − δ ) 22 ( n + 1 ) δ, if 0 < δ < 1 ,π χ 2 ( n + 1 ), if δ = 1 ,1 χδ 2 ( n + 1 )π − 2 arctan( δ − 1 ) 22 ( n + 1 ) δ, if δ > 1 .Copyright © 2016 John Wiley & Sons, Ltd.