An attraction-repulsion chemotaxis system with nonlinear productions

Abstract

This paper studies the semilinear attraction-repulsion chemotaxis system with nonlinear productions and logistic source: ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u), 0=Δv+αuk−βv, 0=Δw+γul−δw, in bounded domain Ω⊂Rn, n≥1, subject to the non-flux boundary conditions, where the nonlinear productions for the attraction and repulsion chemicals are described via αuk and γul respectively, and the logistic source f∈C2[0,∞) satisfying f(u)≤u(a−bus) with s>0, f(0)≥0. It is proved that if one of the random diffusion, logistic source and repulsion mechanisms dominates the attraction with max⁡{l,s,2n}>k, the solutions would be globally bounded. Furthermore, under the three balance situations, namely, k=s>l, k=l>s or k=s=l, the boundedness of solutions depends on the sizes of the coefficients involved. This extends the global boundedness criteria established by Zhang and Li (2016) [20] for the attraction-repulsion chemotaxis system with linear productions and logistic source.