EQUILIBRATION IN A TWO-SPECIES–TWO-CHEMICALS CHEMOTAXIS-COMPETITION SYSTEM.

This paper is concerned { u_t = ∆u− χ₁∇·(u∇v) + µ₁u(1−u−a1w) in Ω×(0,∞), 0 = ∆v−v+w in Ω×(0,∞), w_t = ∆w− χ₂∇·(w∇_z) + µ₂w(1−a₂u−w) in Ω×(0,∞), 0 = ∆z−z+u in Ω×(0,∞), with stabilization in the two-species–two chemicals chemotaxis-competition system where Ω is a bounded domain in R ⁿ (n ≥ 2) with smooth boundary, X1,X2 and µ1,µ2 are constants satisfying some conditions. About this system Tu–Mu–Zheng–Lin (Discrete Contin. Dyn. Syst.;2018;38;3617– 3636) showed global existence and stabilization of solutions under some smallness conditions for χ1 and χ2. Here energy arguments for seeing stabilization in the previous work were based on ideas in Bai–Winkler (Indiana Univ. Math. J.;2016;65;553–583); however, these ideas were recently improved by the first author (Discrete Contin. Dyn. Syst. Ser. S;2020;13;269–278), which implies that the result about stabilization in the previous work seems not to be the best. This paper gives an improvement of conditions for stabilization in the previous work. The feature of the proof is to use the Sylvester criterion in deriving energy estimates. [ABSTRACT FROM AUTHOR]