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Boundedness in a four‐dimensional attraction‐repulsion chemotaxis system with logistic source
Author(s) -
Li Yan,
Wang Wei
Publication year - 2018
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.4942
Subject(s) - bounded function , mathematics , attraction , domain (mathematical analysis) , homogeneous , neumann boundary condition , boundary (topology) , logistic function , function (biology) , chemotaxis , mathematical analysis , combinatorics , pure mathematics , philosophy , linguistics , statistics , receptor , biochemistry , chemistry , evolutionary biology , biology
In this paper, we study the attraction‐repulsion chemotaxis system with logistic source: u t = Δ u − χ ∇·( u ∇ v )+ ξ ∇·( u ∇ w )+ f ( u ), 0 = Δ v − β v + α u , 0 = Δ w − δ w + γ u , subject to homogeneous Neumann boundary conditions in a bounded and smooth domain Ω ⊂ R 4 , where χ , α , ξ , γ , β , and δ are positive constants, and f : R → R is a smooth function satisfying f ( s ) ≤ a − b s 3/2 for all s ≥ 0 with a ≥ 0 and b > 0. It is proved that when the repulsion cancels the attraction (ie, ξ γ = χ α ), for any nonnegative initial data u 0 ∈ C 0 ( Ω ¯ ) , the solution is globally bounded. This result corresponds to the one in the classical 2‐dimensional Keller‐Segel model with logistic source bearing quadric growth restrictions.