LAUSR

This paper is concerned with the two-species chemotaxis-competition system ut=d1Δu−χ1∇·(u∇w)+μ1u(1−u−a1v)inΩ×(0,∞),vt=d2Δv−χ2∇·(v∇w)+μ2v(1−a2u−v)inΩ×(0,∞),0=d3Δw+αu+βv−γwinΩ×(0,∞), where Ω is a bounded domain in Rn with smooth boundary ∂Ω, n≥2; χi and μi are constants satisfying some conditions. The above system was studied in the cases that a1,a2∈(0,1) and a1>1>a2, and it was proved that global existence and asymptotic stability hold when χiμi are small. However, the conditions in the above 2 cases strongly depend on a1,a2, and have not been obtained in the case that a1,a2≥1. Moreover, convergence rates in the cases that a1,a2∈(0,1) and a1>1>a2 have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all a1,a2>0 which covers the case that a1,a2≥1, and lead to convergence rates for solutions of the above system in the cases that a1,a2∈(0,1) and a1≥1>a2.