Global boundedness and asymptotic behavior of the solutions to an attraction–repulsion chemotaxis-growth system. Issue 17 (22nd November 2022) - Liu, Yong; Li, Zhongping
Abstract : This paper deals with the following attraction–repulsion chemotaxis system with logistic source 1 { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) + f ( u ), x ∈ Ω, t > 0, 0 = Δ v + α u − β v, x ∈ Ω, t > 0, w t = Δ w + γ u − δ w, x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n ( n ≥ 2 ) with smooth boundary, where χ, ξ, α, β, γ, δ are assumed to be positive constants and f ( s ) = μ s ( 1 − s θ ) with μ > 0 and θ ≥ 1 . It is shown that the system admits a unique globally bounded classical solution provided that space dimension n  = 2, or n ≥ 3 and θ > 1, or n ≥ 3, θ = 1 and μ > C ( n ) ξ γ + χ α with some C ( n ) > 0 . Furthermore, under the additional assumption μ is suitably large, we show that the global classical solution will converge to the constant steady state ( 1, α β, γ δ ) exponentially as t → ∞ . Our results imply that the logistic source plays an important role on the behavior of the solutions in this model.