Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

This paper is concerned with the two‐species chemotaxis‐competition system urn:x-wiley:mma:media:mma4607:mma4607-math-0001 where Ω is a bounded domain in with smooth boundary ∂ Ω, n ≥2; χ i and μ i are constants satisfying some conditions. The above system was studied in the cases that a 1 , a 2 ∈(0,1) and a 1 >1> a 2 , and it was proved that global existence and asymptotic stability hold when are small. However, the conditions in the above 2 cases strongly depend on a 1 , a 2 , and have not been obtained in the case that a 1 , a 2 ≥1. Moreover, convergence rates in the cases that a 1 , a 2 ∈(0,1) and a 1 >1> a 2 have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all a 1 , a 2 >0 which covers the case that a 1 , a 2 ≥1, and lead to convergence rates for solutions of the above system in the cases that a 1 , a 2 ∈(0,1) and a 1 ≥1> a 2 .