Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems

By Michael Winkler. 2017

74 citations recorded by the source corpus.

We consider the no-flux initial-boundary value problem for Keller-Segel-type chemotaxis growth systems of the form \begin{document}$\begin{eqnarray*} ≤\left\{ \begin{array}{ll} u_t=Δ u -χ \nabla · (u\nabla v) + ρ u -μ u^2, & x∈Ω, \ t>0, \\ v_t=Δ v -v+u, & x∈Ω, \ t>0, \end{array} \right. \end{eqnarray*}$ \end{document} in a ball \begin{document} $Ω\subset\mathbb{R}^n$ \end{document} , \begin{document} $n≥ 3$ \end{document} , with parameters \begin{document} $χ>0, ρ≥ 0$ \end{document} and \begin{document} $μ>0$ \end{document} . By means of an argument based on a conditional quasi-energy inequality, it is firstly shown that if \begin{document} $χ=1$ \end{document} is fixed, then for any given \begin{document} $K>0$ \end{document} and \begin{document} $T>0$ \end{document} one can find radially symmetric initial data, possibly depending on \begin{document} $K$ \end{document} and \begin{document} $T$ \end{document} , such that for arbitrary \begin{document} $μ∈ (0, 1)$ \end{document} the corresponding local-in-time classical solution \begin{document} $(u, v)$ \end{document} satisfies \begin{document} $\begin{eqnarray*} u(x, t) > \frac{K}{μ} \end{eqnarray*}$ \end{document} with some \begin{document} $x∈Ω$ \end{document} and \begin{document} $t∈ (0, T)$ \end{document} ; in fact, this growth phenomenon is actually identified as being generic in the sense that the set of all initial data having this property is dense in the set of all suitably regular radial initial data in a certain topology. Secondly, turning a focus on possible effects of large chemotactic sensitivities, on the basis of the above it is shown that when \begin{document} $ρ≥ 0$ \end{document} and \begin{document} $μ>0$ \end{document} are fixed, then for all \begin{document} $L>0, T>0$ \end{document} and \begin{document} $χ>μ$ \end{document} one can fix radial initial data \begin{document} $(u_{0, χ}, v_{0, χ})$ \end{document} which decay in \begin{document} $L^∞(Ω)× W^{1, ∞}(Ω)$ \end{document} as \begin{document} $χ\to∞$ \end{document} , and which are such that for the respective solution \begin{document} $(u_χ, v_χ)$ \end{document} there exist \begin{document} $x∈Ω$ \end{document} and \begin{document} $t∈ (0, T)$ \end{document} fulfilling \begin{document} $\begin{eqnarray*} u_χ(x, t) > L. \end{eqnarray*}$ \end{document}

Subfields: Biomedical Engineering, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/8d8d9b9d48dfdaa07a88ca714215c613cc8da427

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Citations in the corpus

Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation (2018, background): “The few available exceptions address the taxis-driven spontaneous emergence of arbitrarily large population densities, possibly in the form of transient phenomena, in presence of suitably small diffusion and death parameters even in spatially one-dimensional, but also in higher-dimensional Keller-Se”
Diffusive Spatial Movement with Memory (2020, background): “7), it is known that the solution is bounded under some conditions [26], or even converges to a constant equilibrium [28], but it is also known that solutions can blow up in finite time despite logistic growth [27], or have other interesting dynamics [29,30].”
Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions (2020, background): “Besides providing numerical evidence that shows remarkably colorful facets in logistic Keller–Segel systems, up to even chaotic behavior [28], previous studies have revealed quite drastic phenomena related to the spontaneous emergence of large population densities, possibly at intermediate timescale”