A new result for asymptotic stability in a two-species chemotaxis model with signal-dependent sensitivity

This paper deals with a two-species chemotaxis model with signal-dependent sensitivity ut=Δu−∇⋅(uχ1(w)∇w)+μ1u(1−u),x∈Ω,t>0,vt=Δv−∇⋅(vχ2(w)∇w)+μ2v(1−v),x∈Ω,t>0,wt=dΔw+h(u,v,w),x∈Ω,t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn ( n≥1 ), where the parameters μ1>0, μ2>0 and d>0, and the chemotactic function χi(w) ( i=1,2 ) are smooth and satisfy some conditions. The global existence and asymptotic behavior of solutions to this system have been studied under some conditions (Mizukami and Yokota, 2016; Mizukami, 2016). The purpose of this work is to improve the conditions assumed in (Mizukami and Yokota, 2016; Mizukami, 2016) for asymptotic behavior.