Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source.

In this paper, we study traveling wave solutions of the chemotaxis system {ut Δ u − χ1∇(u∇v1) + χ2∇(u∇v2) + u(a−bu), x ∈ R τ∂tv1 = (Δ − λ1I)v1+μ1u, x ∈ R, τ∂v2 = (Δ − λ2I)v2 + μ2u, x ∈ R,(0.1) where τ > 0, χi > 0,λi > 0, μi > 0 (i = 1,2) and a > 0, b > 0 are constants. Under some appropriate conditions on the parameters, we show that there exist two positive constant c∗(τ,χ1,μ1,λ1,χ2,μ2,λ2) < c∗∗ (τ,χ1,μ1,λ1,χ2,μ2,λ2) such that for every c∗(τ,χ1,μ1,λ1,χ2,μ2,λ2) ≤ c < c∗∗ (τ,χ1,μ1,λ1,χ2,μ2,λ2), (0.1) has a traveling wave solution (u,v1,v2)(x,t) = (U,V1,V2) (x − ct) connecting (ab,aμ1bλ1,aμ2bλ2) and (0,0,0) satisfying limz→∞U(z)e−μz=1, where μ∈(0,√a) is such that c = cμ: = μ + aμ . Moreover, lim(χ1,χ2) → (0+,0+))c∗∗(τ,χ1,μ1,λ1,χ2,μ2,λ2) = ∞ and lim(χ1,χ2) → (0+,0+))c∗(τ,χ1,μ1,λ1,χ2,μ2,λ2) = c~μ∗, where ~μ∗=min{√a,√λ1+τa(1−τ)+,√λ2+τa(1−τ)+} . We also show that (1) has no traveling wave solution connecting (ab,aμ1bλ1,aμ2bλ2) and (0,0,0) with speed c<2√a. [ABSTRACT FROM AUTHOR]