Traveling wave solutions for fully parabolic Keller-Segel chemotaxis systems with a logistic source
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This article concerns traveling wave solutions of the fully parabolic Keller-Segel chemotaxis system with logistic source,
ut = Δu - X∇ ⋅ (u∇v) + u(α - bu), x ∈ ℝN,
τvt = Δv - λv + μu, x ∈ ℝN,
where X, μ, λ, α, b are positive numbers, and τ ≥ 0. Among others, it is proved that if b > 2Xμ and τ ≥ 1/2(1 - λ/α)+, then for every c ≥ 2√α, this system has a traveling wave solution (u, v)(t, x) = (Uτ,c(x ⋅ ξ - ct), Vτ,c(x ⋅ ξ - ct)) (for all ξ ∈ ℝN) connecting the two constant steady states (0, 0) and (α/b, μ/λ α/b), and there is no such solutions with speed c less than 2√α, which improves the results established in , and shows that this system has a minimal wave speed c*0 = 2√α, which is independent of the chemotaxis.
CitationSalako, R. B., & Shen, W. (2020). Traveling wave solutions for fully parabolic Keller-Segel chemotaxis systems with a logistic source. Electronic Journal of Differential Equations, 2020(53), pp. 1-18.
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