Large time behavior of solutions to a quasilinear attraction–repulsion chemotaxis model with nonlinear secretion

In this paper, we study the large time behavior of a quasilinear attraction–repulsion chemotaxis model with nonlinear secretion: ut = ∇ · (D(u)∇u − χΦ(u)∇v + ξΨ(u)∇w) + λu − μuϵ; 0=Δv−α1v+β1uγ1; 0=Δw−α2w+β2uγ2, x ∈ Ω, t > 0. We show that the global-in-time bounded smooth solution of the system converges exponentially/algebraically to steady state in the large time limit. Those results generalize some of our previous results [G. Ren and B. Liu, Math. Models Methods Appl. Sci. 30(13), 2619–2689 (2020) and G. Ren and B. Liu, J. Differ. Equations 268(8), 4320–4373 (2020)].