Chemotaxis can prevent thresholds on population density

By Johannes Lankeit. 2014

141 citations recorded by the source corpus.

We define and (for $q>n$) prove uniqueness and an extensibility property of $W^{1,q}$-solutions to $u_t =-\nabla\cdot(u\nabla v)+\kappa u-\mu u^2$ $ 0 =\Delta v-v+u$ $\partial_\nu v|_{\partial\Omega} = \partial_\nu u|_{\partial\Omega}=0,$ $ u(0,\cdot)=u_0 $ in balls in $\mathbb{R}^n$, which we then use to obtain a criterion guaranteeing some kind of structure formation in a corresponding chemotaxis system - thereby extending recent results of Winkler to the higher dimensional (radially symmetric) case. Keywords: chemotaxis, logistic source, blow-up, hyperbolic-elliptic system

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/50aa928306efdced6f09cc613056462c38a8ecbd

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Cell Biology

In Modeling and Simulation

In Molecular Biology

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”
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems (2017, background): “More subtle qualitative facets of chemotaxis-growth interaction could up to now be rigorously detected only in simplified parabolic-elliptic settings and under the essential additional assumption that cell diffusion is suitably weak: In the hyperbolic-elliptic limit case d = 0 of such sytsems, namel”
A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization (2019, background): “also [34] for a precedent concerning a a related parabolicelliptic variant).”
Very weak global solutions to a parabolic–parabolic chemotaxis-system with logistic source (2016, background): “[31] and [11]) for a certain class of initial data, existence and properties of solutions are discussed; in particular such solutions are globally bounded in time for μ ≥ 1 and blow up if μ < 1, in the one-dimensional setting and in a ball of R, with n ≥ 1, respectively.”
How strong a logistic damping can prevent blow-up for the minimal Keller–Segel chemotaxis system? (2016, unknown): “13), verifications can be found in [27, 59].”
Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$ (2016, background): “It is shown in [22], [49] that in presence of suitably weak logistic dampening (that is, small b) certain transient growth phenomena do occur for some initial data.”
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source (2014, background): “On the other hand, the model ut = ε∆u−∇ · (u∇v) + κu− μu(2) (2) 0 = ∆v − v + u has recently been shown to exhibit the following property [12]: If μ ∈ (0, 1) and the (radially symmetric) initial datum u0 is large in a certain L p(Ω)−space, there exists some finite time such that up to this time any g”