Global boundedness in quasilinear attraction–repulsion chemotaxis system with logistic source. (August 2016) - Tian, Miaoqing; He, Xiao; Zheng, Sining
Abstract: This paper studies the quasilinear attraction–repulsion chemotaxis system with logistic source u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( Φ ( u ) ∇ v ) + ξ ∇ ⋅ ( Ψ ( u ) ∇ w ) + f ( u ), τ v t = Δ v + α u − β v, τ ∈ 0, 1 , 0 = Δ w + γ u − δ w, in bounded domain Ω ⊂ R N, N ≥ 1, subject to the homogeneous Neumann boundary conditions, D, Φ, Ψ ∈ C 2 [ 0, + ∞ ) nonnegative, with D ( s ) ≥ ( s + 1 ) p for s ≥ 0, Φ ( s ) ≤ χ s q, ξ s r ≤ Ψ ( s ) ≤ ζ s r for s > 1, and f smooth satisfying f ( s ) ≤ μ s ( 1 − s k ) for s > 0, f ( 0 ) ≥ 0 . It is proved that if the attraction is dominated by one of the other three mechanisms with max r, k, p + 2 N > q, then the solutions are globally bounded. Under more interesting balance situations, the behavior of solutions depends on the coefficients involved, i.e., the upper bound coefficient χ for the attraction, the lower bound coefficient ξ for the repulsion, the logistic source coefficient μ, as well as the constants α and γ describing the capacity of the cells u to produce chemoattractant and chemorepellent respectively. Three balance situations (attraction–repulsion balance, attraction–logistic source balance, and attraction–repulsion–logistic source balance) are considered to establish the boundedness of solutions for the parabolic–elliptic–elliptic case (with τ = 0 ) and the parabolic–parabolic–elliptic case (with τ = 1 ) respectively.