Repulsion effects on boundedness in the higher dimensional fully parabolic attraction–repulsion chemotaxis system.

This paper deals with an attraction–repulsion chemotaxis system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) , x ∈ Ω , t > 0 , τ 1 v t = Δ v + α u − β v , x ∈ Ω , t > 0 , τ 2 w t = Δ w + γ u − δ w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R N ( N ≥ 2 ), where parameters τ i ( i = 1 , 2 ) , χ , ξ , α , β , γ and δ are positive, and diffusion coefficient D ( u ) ∈ C 2 ( 0 , + ∞ ) satisfies D ( u ) > 0 for u ≥ 0 , D ( u ) ≥ d u m − 1 with d > 0 and m ≥ 1 for all u > 0 . It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution for m > 2 − 2 N . In particular in the case τ 1 = τ 2 and χ α = ξ γ , the solution is globally bounded if m > 2 − 2 N − N + 2 N 2 − N + 2 . Therefore, due to the inhibition of repulsion to the attraction, the range of m > 2 − 2 N of boundedness is enlarged and the results of [21] is thus extended to the higher dimensional attraction–repulsion chemotaxis system with nonlinear diffusion. [ABSTRACT FROM AUTHOR]