Abstract
Our aim is to show that several important systems of partial differential equations arising in mathematical biology, fluid dynamics and electrokinetics can be approached within a single model, namely, a Keller-Segel-type system with rotational flux terms. In particular, we establish sharp conditions on the optimal critical mass for having global existence and finite time blow-up of solutions in two spatial dimensions. Our results imply that the rotated chemotactic response can delay or even avoid the blow-up. The key observation is that for any angle of rotation α∈(-π, π], the resulting PDE system preserves a dissipative energy structure. Inspired by this property, we also provide an alternative derivation of the general system via an energetic variational approach. ©2020 International Press.
Highlights
We consider the Cauchy problem of the following generalized Keller–Segel system in R2: ρt = ∆ρ − χdiv(ρA∇c), −∆c = ρ, ρ(x,0) = ρ0(x),(x,t) ∈ R2 × R+, (x,t) ∈ R2 × R+, x ∈ R2, where χ is a positive constant and A denotes a 2 × 2 matrix given by A :=cosα −sinα sinα cosα with α ∈ (−π,π] being a constant. (1.4)The system (1.1)–(1.2) generalizes the well-known Keller–Segel model that describes the oriented movement of bacterial cells in response to certain chemical signals [20,26]
Equation (1.1) indicates that the motion of bacteria is driven by the self-diffusion and the gradient of concentration of the chemoattractant, while the Poisson Equation (1.2) means that the chemoattractant is produced by the cells themselves and it is diffusing into the environment
One feature of our system (1.1)–(1.2) is that the chemotactic sensitivity turns out to be a tensor χA instead of a scalar function like in the classical Keller–Segel model. This tensor-valued sensitivity is motivated by some interesting biased chemotactic response that has been observed for different kinds of bacteria in experiments, see for instance, [9, 12], such that when the bacteria swim close to a surface, they may be subject to a net rotational force and form spiral-type patterns
Summary
University of Nottingham Ningbo China, 199 Taikang East Road, Ningbo, 315100, China. The work is licenced to the University of Nottingham Ningbo China under the Global University Publication Licence: https://www.nottingham.edu.cn/en/library/documents/researchsupport/global-university-publications-licence.pdf. OPTIMAL CRITICAL MASS FOR THE TWO-DIMENSIONAL KELLER–SEGEL MODEL WITH ROTATIONAL FLUX TERMS∗
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