Asymptotic behavior in an attraction-repulsion chemotaxis system with nonlinear productions

Abstract

This paper studies the semilinear attraction-repulsion chemotaxis system with nonlinear productions and logistic source{ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u),x∈Ω,t>0,0=Δv+αuk1−βv,x∈Ω,t>0,0=Δw+γuk2−δw,x∈Ω,t>0 under the non-flux boundary conditions and initial conditions, where Ω⊂Rn is a bounded domain with smooth boundary, the nonlinear productions for the attraction and repulsion chemicals are described via αuk1 and γuk2 respectively, and the logistic source f∈C2[0,∞) satisfying f(u)≤u(a−bus) with s>0,f(0)≥0. In the previous paper [3], when k1=max⁡{k2,s}≥2n, Hong et al. have proved that if one of the following assumptions holds: k1=k2=s, k1n−2k1n(αχ−γξ)<b; k1=k2>s, αχ−γξ<0; k1=s>k2, k1n−2k1nαχ<b, for any given u0(x)∈C(Ω¯) this system possesses a globally bounded classical solution. The present work further shows that the same conclusion still holds for the critical cases k1=k2=s, k1n−2k1n(αχ−γξ)=b; k1=s>k2, k1n−2k1nαχ=b; k1=k2>s, αχ−γξ=0, nk1(nk1−2)<4, 0<k1=k2≤1 in high dimension (n≥2). Furthermore, we continuously develop the asymptotic behavior of the globally bounded solutions for the logistic source f(u)=u(a−bus).