How far does small chemotactic interaction perturb the Fisher-KPP\n dynamics?

2016/10/25 by Johannes Lankeit, Lankeit, Johannes, Masaaki Mizukami +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #35A09 #35B35 #35K51 (Primary) #35Q92 #92C17 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Gene Regulatory Network Analysis #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1610.07981

openalex publication_date 2016/10/25 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

This paper deals with nonnegative solutions of the Neumann initial-boundary\nvalue problem for the fully parabolic chemotaxis-growth system \n(u_\ε)_t =\Δ u_\ε - \ε \∇ \⋅ (ν_\ε \∇ v_\ε) + \μ u_\ε(1 - u_\ε), \n(v_\ε)_t=\Δ v_\ε -v_\ε+u_\ε, with\npositive small parameter \ε>0 in a bounded convex domain\n\Ω\⊂\ℝⁿ (n\≥ 1) with smooth boundary. The solutions\nconverge to the solution u to the Fisher-KPP equation as \ε\→ 0.\nIt is shown that for all \μ>0 and any suitably regular nonnegative initial\ndata (u_init,v_init) there are some constants \ε₀>0 and C>0\nsuch that \
sup_tgt;0
|u_
varepsilon(
cdot,t)-u(
cdot,t)
|_{L^
infty(
Omega)}\n
leq C
varepsilon
quad for
all

varepsilon
in(0,
varepsilon₀). n

Citations

• A user’s guide to PDE models for chemotaxis
• Exponential attractor for a chemotaxis-growth system of equations

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