Global bounded classical solution for an attraction–repulsion chemotaxis system

Abstract

In this paper, we deal with the global and boundedness of solutions to a quasilinear attraction–repulsion chemotaxis system with logistic source ut=Δu−χ∇⋅(u(u+1)r−1∇v)+ξ∇⋅(u(u+1)r−1∇w)+f(u),0=Δv−βv+αu,0=Δw−δw+γuin a bounded domain Ω⊂RN(N≥1) with smooth boundary, where χ, ξ, α, β, δ, γ, r are positive constants and f(u)≤a−buη for all u≥0 with some a≥0, b>0 and η≥1. Under the Neumann boundary conditions, if χα=ξγ and η>max{r+(N−2)+N,1} or χα<ξγ and η≥1, we prove that there exists a unique global-in-time and bounded classical solution for all appropriately regular nonnegative initial data, which extends the results of Li and Xiang (2016), Xu and Zheng (2018) and Xie and Zheng (2021).