Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on ℝ^N

2021/07/04 by Shen, Wenxian, Xue, Shuwen
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.01551

Abstract

The current paper is concerned with the spreading speeds of the following parabolic-parabolic chemotaxis model with logistic source on ℝ^N, \begincases u_t=Δu-χ∇⋅ ( u∇ v) + u(a-bu), x∈ℝ^N,\cr v_t=Δv -λv+μu, x∈ ℝ^N. \endcases(1) where χ, a, b, λ, μ are positive constants. Assume b>(Nμχ)/(4). Among others, it is proved that 2√(a) is the spreading speed of the global classical solutions of (1) with nonempty compactly supported initial functions, that is, lim_t→∞sup_{|x|≥ ct}u(x,t;u₀,v₀)=0 ∀ cgt;2√(a) and \liminf_t→∞inf_{|x|≤ ct}u(x,t;u₀,v₀)>0 ∀ 0(Nμχ)/(4), the chemotaxis neither speeds up nor slows down the spatial spreading in the Fisher-KPP equation.

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