Preventing blow-up in a chemotaxis model

By R. Kowalczyk. 2005

196 citations recorded by the source corpus.

Subfields: Modeling and Simulation, Molecular Biology, Oncology

View Paper: https://www.semanticscholar.org/paper/cd2ef17ea4a390523a105510035def1ddb822d52

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Citations in the corpus

Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, methodology): “1) involving nonlinear diffusion which are critical in higher dimensions have also been studied [10,14,50,51,65,66] (see also the related [53]).”
Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior (2010, background): “This leads to a nonlinear porous–medium–like diffusion instead of a linear one, see e. g. [7] [16].”
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system (2011, unknown): “[32], [22], [23], [24], [9], [4], [19], [33] and also the survey [16]).”
Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in $${\mathbb{R}^2}$$R2 with Measure-Valued Initial Data (2012, background): “1) involving nonlinear diffusion which are critical in higher dimensions have also been studied [51, 14, 65, 66, 10, 50] (see also the related [53]).”
Depleting the signal: Analysis of chemotaxis‐consumption models—A survey (2023, unknown): “Various types of nonlinear diffusion mechanisms play important roles in microbial motion ([60], [91], [46]), and corresponding generalizations of (CC1) and (CC2), such as    ut = ∇ · ( D(u)∇u ) u−∇ · ( uS(x, u, v) · ∇v ) , x ∈ Ω, t > 0, vt = ∆v − uv x ∈ Ω, t > 0, ( D(u)∇u ) u− uS(x, u, v) · ∇”
On the global existence of solutions to an aggregation model (2008, methodology): “This simplifled model, called the aggregation model, was proposed originally in [ 9 ], where the author focused on the blow-up problem and boundedness of solutions.”
Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux (2017, background): “On the other hand, for general m ∈ R, the existence of global bounded solutions can be ensured by m + α > 2 − 2 d (see [5,6,16,19,22,27] and references therein).”
Finite-time blow-up in a quasilinear system of chemotaxis (2008, methodology): “In particular we strengthen the two-dimensional result in [14].”
Functional inequalities, thick tails and asymptotics for the critical mass Patlak–Keller–Segel model (2010, background): “It follows that the level sets of D cannot be weakly compact in L1(R2).”
A coupled chemotaxis-fluid model: Global existence (2011, background): “This leads to a nonlinear porous-medium-like diffusion instead of a linear one, see e.g. [4] [11].”
A user’s guide to PDE models for chemotaxis (2009, unknown): “This Lyapunov function has been used by Kowalczyk [50] to prove global existence.”
Infinite time aggregation for the critical Patlak‐Keller‐Segel model in ℝ2 (2007, background): “free energy functional and related functional inequalities played an essential role in improving the range of existence of global in time solutions up to the critical mass in [22, 13] for the PKS system (1), for nonlinear diffusion and chemotactic models in [32, 14] and for related models [15].”
Uniqueness for Keller-Segel-type chemotaxis models (2012, background): “In the classical parabolic-elliptic KS equation ε = α = 0 and d = 2, global in time bounded solutions in the subcritical case m < 8π have been obtained joining the results in [12, 24, 14].”
A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay (2012, background): “The Keller–Segel system in higher space dimensions has been investigated in [13,14], and results for the system with nonlinear diffusion to prevent blow-up can be found in [9,24].”
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion (2012, background): “Since the diffusion of bacteria (or, more generally, of cells) in a viscous fluid is more like movement in a porous medium (see the discussions in [24], [20], [2] and [8], for instance), the authors in [3] extended the model (1.”