A quasilinear attraction–repulsion chemotaxis system with logistic source

Abstract

This paper is concerned with the quasilinear attraction–repulsion chemotaxis system with logistic source ut=∇⋅(D(u)∇u)−∇⋅(Φ(u)∇v)+∇⋅(Ψ(u)∇w)+f(u),(x,t)∈Ω×(0,∞),0=Δv+αu−βv,(x,t)∈Ω×(0,∞),0=Δw+γu−δw,(x,t)∈Ω×(0,∞)under Neumann boundary conditions in a bounded domain Ω⊂RN(N≥1), where D,Φ,Ψ∈C2[0,∞) are nonnegative functions with D(s)≥(s+1)p for s≥0, Φ(s)≤χsq,ξsr≤Ψ(s)≤ζsr for s>1, and the smooth function f satisfies f(s)≤μs(1−sk) for s>0,f(0)≥0. Tian et al. (2016) proved that when q=max{r,k} and q−p≥2N, if one of the following assumptions holds: (i) q=r=k,μ>(αχ−γξ)(1−2N(q−p))/(1+2(q−1)N(q−p)); (ii) q=r>k,αχ−γξ<0; (iii) q=k>r,μ>αχ(1−2N(p−q))/(1+2(q−1)N(p−q)), then the solution of the equations is globally bounded. The present work further shows that the same conclusion still holds for the critical cases: (a) q=r=k,μ=(αχ−γξ)(1−2N(q−p))/(1+2(q−1)N(q−p)); (b) q=r>k,αχ−γξ=0 with q−p<1Nmin{4(N+1)N+2,N+2}; (c) q=k>r,μ=αχ(1−2N(p−q))/(1+2(q−1)N(p−q)).