A critical blow-up exponent for flux limitation in a Keller-Segel system

2020/10/04 by Winkler, Michael · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2010.01553

Abstract

The parabolic-elliptic cross-diffusion system \ u_t = Δu - ∇ ⋅ (uf(|∇ v|²) ∇ v ),
0 = Δv - μ+ u, ∫_Ωv=0, μ:=(1)/(|Ω|) ∫_Ωu dx, . is considered along with homogeneous Neumann-type boundary conditions in a smoothly bounded domain Ω⊂ Rⁿ, n≥ 1, where f generalizes the prototype given by f(ξ) = (1+ξ)^-α, ξ≥ 0, for all ξ≥ 0, with α∈ R. In this framework, the main results assert that if n≥ 2, Ω is a ball and \[ α(n-2)/(2(n-1)), then any explosion is ruled out in the sense that for arbitrary nonnegative and continuous initial data, a global bounded classical solution exists.

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• Mass threshold for global existence in chemotaxis systems with critical flux limitation
• To what extent does the consideration of positive total flux influence the dynamics of Keller-Segel-type models?

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