Spreading speed in a fractional attraction–repulsion chemotaxis system with logistic source

Abstract

This paper studies an attraction-repulsion chemotaxis system with logistic source and a fractional diffusion of order α∈(0,1) on RN: (0.1)ut=−(−Δ)αu−χ1∇⋅(u∇v1)+χ2∇⋅(u∇v2)+u(a−bu),x∈RN,t>0,0=(Δ−λ1I)v1+μ1u,x∈RN,t>0,0=(Δ−λ2I)v2+μ2u,x∈RN,t>0,where α, χi, μi, λi(i=1,2), a and b are constants. We show that (0.1) has a unique global bounded solution and investigate the asymptotic behavior of the global classical solutions under some assumptions on the parameters. Particularly, we consider the spreading properties of the global classical solutions. Under some conditions for the nonnegative initial function u0, we prove that (0.2)lim inft→∞inf|x|≤ectu(x,t;u0)>0,for all0<c<aN+2αand (0.3)limt→∞sup|x|≥ectu(x,t;u0)=0,for allc>aN+2α.This fact shows that the spreading speed of (0.1) is exponential in time, which is different from the linear spreading speed obtained in Salako and Shen (2019).