Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

By Nicola Bellomo, Abdelghani Bellouquid, Youshan Tao, Michael Winkler. 2015

969 citations recorded by the source corpus.

Subfields: Biomedical Engineering, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/0907198116f476b20945274733a0c616208aad1e

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Citations in the corpus

Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier--Stokes system with competitive kinetics (2017, neutral): “in Ω× (0,∞), uε → u in L(2)loc([0,∞);L(Ω)) for all p ∈ [1, 6) and a.”
On a parabolic–elliptic system with gradient dependent chemotactic coefficient (2018, background): “In the last decades, several mathematical models have been presented to describe the phenomenon, after the pionnering works of Patlak [29] and Keller and Segel [23] (see also the review articles Horstmann [19], [20] and Bellomo et al [3] and references therein for more extensive literature in the su”
Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion (2016, background): “[10, 9, 3]) where v does not stand for a nutrient to be consumed but a signalling substance produced by the bacteria themselves, i.”
Combining effects ensuring boundedness in an attraction–repulsion chemotaxis model with production and consumption (2022, neutral): “(12) Further, uniform boundedness of (u, v, w) is ensured whenever (boundedness criterion) u ∈ L∞((0, Tmax); L(Ω)), with p > 1 arbitrarily large, for m1,m2,m3 ∈ R, and with p > n2 , for m1 = m2 = m3 = 1 (see, for instance, [1,28]).”
Long-term behaviour in a chemotaxis-fluid system with logistic source (2016, background): “This gives rise to a contribution −χ∇ · (n∇c) to the time derivative of n, thus introducing cross-diffusive effects into the model, which lie at the core of the mathematical difficulties accompanying the analysis of chemotaxis systems like the famous Keller-Segel model ([15, 2]).”
Emergence of large population densities despite logistic growth restrictions in fully parabolic chemotaxis systems (2017, neutral): “also the surveys [8], [3]), in the correspondingly modified variant thereof given by { ut = d∆u− χ∇ · (u∇v) + ρu− μu2, vt = ∆v − v + u, (1.”
Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system (2017, result): “Similar to corresponding situations in the minimal two-dimensional Keller-Segel system ([22], [1]), through a variant of the Gagliardo-Nirenberg inequality due to [3] the above slight improvement of the L1 information from (2.”
Analysis of a chemotaxis model with indirect signal absorption (2019, background): “In the past few decades mathematicians have analyzed several chemotaxis models; for a broader introduction we refer to the survey [2].”
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption (2016, background): “.5) that satisﬁes the boundedness condition (5.2), we have u(·,t) → κ µ in C0(Ω), v(·,t) → 0 in C2(Ω). (5.6) 14 as t → ∞. Proof. For j ∈ Nwe deﬁne uj(x,τ) := u(x,j +τ), vj(x,τ) := v(x,j +τ), x ∈ Ω,τ∈ [0,1]. We let (jk)k∈N⊂ Nbe a sequence satisfying jk → ∞ as k → ∞. By Lemma 5.1 there are α∈ (0,1), C”
The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties (2016, background): “the surveys [15], [14] and [2], for instance).”