Spreading speed in a fractional attraction–repulsion chemotaxis system with logistic source. (May 2023) - Jiang, Chao; Lei, Yuzhu; Liu, Zuhan; Zhang, Weiyi
Abstract: This paper studies an attraction-repulsion chemotaxis system with logistic source and a fractional diffusion of order α ∈ ( 0, 1 ) on R N : (0.1) u t = − ( − Δ ) α u − χ 1 ∇ ⋅ ( u ∇ v 1 ) + χ 2 ∇ ⋅ ( u ∇ v 2 ) + u ( a − b u ), x ∈ R N, t > 0, 0 = ( Δ − λ 1 I ) v 1 + μ 1 u, x ∈ R N, t > 0, 0 = ( Δ − λ 2 I ) v 2 + μ 2 u, x ∈ R N, t > 0, where α, χ i, μ i, λ i ( 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 u 0, we prove that (0.2) lim inf t → ∞ inf | x | ≤ e c t u ( x, t ; u 0 ) > 0, for all 0 < c < a N + 2 α and (0.3) lim t → ∞ sup | x | ≥ e c t u ( x, t ; u 0 ) = 0, for all c > a N + 2 α . 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).