Well-posedness and asymptotic limits for a degenerate Keller-Segel system with volume filling

Ansgar J\"ungel; Mingyue Zhang; Noah Geltner

arxiv: 2605.21296 · v1 · pith:3UUYJHXOnew · submitted 2026-05-20 · 🧮 math.AP

Well-posedness and asymptotic limits for a degenerate Keller-Segel system with volume filling

Noah Geltner , Ansgar J\"ungel , Mingyue Zhang This is my paper

Pith reviewed 2026-05-21 03:05 UTC · model grok-4.3

classification 🧮 math.AP

keywords Keller-Segel systemdegenerate diffusionvolume fillingglobal weak solutionsweak-strong uniquenesspattern formationasymptotic limitsentropy estimates

0 comments

The pith

Degenerate Keller-Segel systems with volume filling admit global weak solutions that converge exponentially to uniformity and form patterns in one dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines parabolic-parabolic Keller-Segel systems that incorporate degenerate nonlinear diffusion and a volume-filling mechanism inside a bounded domain with no-flux boundaries. These equations come from a multiphase fluid description and display varied behaviors depending on the parameters. Using entropy functionals, the authors obtain uniform a priori bounds that close the estimates even when diffusion vanishes at high densities. From these bounds they derive global weak solutions, prove weak-strong uniqueness, establish exponential relaxation to the constant steady state, and show that the one-dimensional reduction produces spatial patterns. The same estimates also justify passage to the parabolic-elliptic and vanishing-diffusion limits.

Core claim

For the parabolic-parabolic system with degenerate diffusion and volume filling, global weak solutions exist, satisfy a weak-strong uniqueness property, and converge exponentially to the homogeneous equilibrium. In one space dimension the equations reduce to a first-order scalar problem whose nonlinearity analysis yields pattern formation. The parabolic-elliptic and vanishing-diffusion limits hold as well. All results rest on a priori estimates extracted from suitable entropy functionals that remain uniform in the approximating sequence.

What carries the argument

Entropy functionals constructed from the structural conditions on the nonlinear diffusion and sensitivity functions, which produce dissipation terms that control the degeneracy uniformly.

If this is right

Global weak solutions exist for all nonnegative initial data with finite entropy.
Weak-strong uniqueness holds, so any strong solution is the unique continuation of a weak one.
Solutions converge exponentially to the homogeneous steady state for all parameter values satisfying the structural assumptions.
In one spatial dimension the system reduces to a first-order equation that produces stationary spatial patterns.
The parabolic-elliptic and vanishing-diffusion limits are justified by the same entropy bounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The entropy construction may extend to systems with different volume-filling exponents or additional reaction terms.
The one-dimensional pattern analysis could be used to test stability thresholds in higher-dimensional numerical experiments.
The uniform estimates suggest that similar degeneracy handling might apply to other aggregation models that saturate at finite density.

Load-bearing premise

The nonlinear diffusion and sensitivity functions obey structural conditions that let the entropy dissipation close the a priori estimates uniformly despite the degeneracy.

What would settle it

A sequence of solutions that develops a singularity in finite time or fails to approach the constant state exponentially fast under the stated structural conditions on the diffusion and sensitivity.

Figures

Figures reproduced from arXiv: 2605.21296 by Ansgar J\"ungel, Mingyue Zhang, Noah Geltner.

**Figure 2.** Figure 2: Left: Profile of the function Gλ. Right: Intersection points of φ and c 7→ χ(c + λ)). Before we prove this result, we study the profile of Gλ. We determine the parameter regime in which Gλ possesses two extremas ec and ec+; see [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: Illustration of the solution to (34) with χr = 0 and χb = 1/(m − 1). We determine bc such that 0 = G ′′ λ (c) = 2 − 2χ(φ −1 ) ′ (χ(c + λ)) = 2 1 − χ φ′ (ρ) , recalling that ρ = φ −1 (χ(c + λ)). Since φ ′ (0) = 0, φ ′ (1) = 1, and φ ′ is increasing, there exists a unique ρb such that φ ′ (ρb) = χ. Then, defining bc by ρb = φ −1 (χ(bc + λ)), it holds that G′′ λ (bc) = 0. Since φ ′ is positive, we have G ′… view at source ↗

**Figure 4.** Figure 4: Time evolution of ρ (solid line) and c (dotted line) with χ = 1 (top row) and χ = 10 (bottom row) at times t = 0 (left), t = 5 (middle), and t = 100 (right). Next, we present snapshots of the solution (ρ, c) for different values of τ , choosing t = 10, m = 2, and M = 1/2 [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗

**Figure 5.** Figure 5: Time snapshot of ρτ (solid line) and ρ (dotted line) with τ = ε = 1 (left), τ = 0.1 (middle), and τ = 0 (right). convergence of the solution ρη to the solution ρ to the reduced system (7) as η → 0. The solutions ρη and ρ converge towards each other as η decreases [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗

**Figure 6.** Figure 6: Time snapshot of ρη (solid line) and ρ (dotted line) with η = 5 (left), η = 0.5 (middle), and η = 0 (right). Appendix A. Derivation of the chemotaxis model We derive equation (1) for the cell density using a multiphase approach inspired by [19]. Let ρ0(x, t) and ρ1(x, t) be the volume fractions occupied by water and cells, respectively, and let v0(x, t) and v1(x, t) be the corresponding velocities. The sta… view at source ↗

read the original abstract

A class of parabolic-parabolic Keller-Segel systems with degenerate diffusion and volume filling is studied in a bounded domain subject to no-flux boundary conditions. The equations are derived from a multiphase fluid model. The interplay between nonlinear diffusion and density saturation leads to a rich variety of behaviors across different parameter regimes. We establish the existence of global weak solutions, a weak-strong uniqueness result, the exponential convergence to the homogeneous steady state, pattern formation in one spatial dimension, as well as the parabolic-elliptic and vanishing diffusion limits. The analysis relies on a priori estimates derived from suitable entropy functionals. Pattern formation is demonstrated by reducing the system to a first-order equation and conducting a detailed analysis of the resulting nonlinearity. Numerical simulations from a one-dimensional finite-volume scheme illustrate the asymptotic regimes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives global weak solutions, weak-strong uniqueness, and two asymptotic limits for a degenerate volume-filling Keller-Segel system, but the entropy estimates rest on structural conditions that need explicit verification.

read the letter

This paper establishes global weak solutions for a degenerate Keller-Segel system with volume filling, along with weak-strong uniqueness and two asymptotic limits. The authors work with a parabolic-parabolic system derived from a multiphase fluid model. They use entropy functionals to obtain the necessary a priori bounds that handle both the degeneracy in diffusion and the saturation from volume filling. In one dimension, they reduce the problem to a first-order ODE and study the resulting nonlinearity to demonstrate pattern formation. They also provide numerical simulations with a finite-volume scheme to show the different regimes. The entropy approach is a solid choice here because it naturally incorporates the nonlinear effects. The reduction to an ODE in 1D allows for a detailed analysis that would be harder in higher dimensions. The inclusion of both the parabolic-elliptic limit and the vanishing diffusion limit adds breadth to the results. One area that needs close attention is the precise structural conditions on the nonlinear diffusion and sensitivity functions. These must ensure that the entropy dissipation controls the solution uniformly, even as the density approaches the degeneracy threshold. The abstract mentions these conditions implicitly through the entropy construction, but the paper should state them explicitly and verify they hold for all the parameter regimes in the pattern formation and limit theorems. If the conditions are restrictive, the applicability of the results could be narrower than it first appears. This work is intended for mathematicians studying chemotaxis and aggregation models in mathematical biology. Readers who follow rigorous analysis of nonlinear parabolic systems will find the existence and limit results useful. The paper engages honestly with the literature on Keller-Segel models and builds on prior work with volume filling. I recommend sending it for peer review. The theorems are concrete and the methods are appropriate, though referees should check the closure of the entropy estimates carefully to confirm the bounds pass to the limit.

Referee Report

1 major / 2 minor

Summary. The manuscript studies a class of parabolic-parabolic Keller-Segel systems with degenerate nonlinear diffusion and volume-filling effects on a bounded domain with no-flux boundary conditions, derived from a multiphase fluid model. It establishes global weak solutions, weak-strong uniqueness, exponential convergence to the homogeneous steady state, pattern formation in one dimension via reduction to a first-order ODE, and the parabolic-elliptic and vanishing-diffusion limits. The analysis relies on a priori estimates from entropy functionals whose dissipation controls the degeneracy, with numerical illustrations from a 1D finite-volume scheme.

Significance. If the entropy estimates close uniformly, the results would advance the mathematical theory of degenerate chemotaxis models with saturation, particularly by linking well-posedness, uniqueness, and asymptotic limits across parameter regimes. The explicit 1D reduction for pattern formation and the numerical validation add concrete value; the work would be of interest to researchers in mathematical biology and nonlinear PDEs.

major comments (1)

[Assumptions and entropy construction (likely §2)] The structural conditions on the nonlinear diffusion coefficient and sensitivity function (implicit in the entropy construction referenced in the abstract and used for a priori bounds): these conditions are not stated explicitly, so it is not possible to verify that the entropy dissipation remains coercive and yields uniform bounds away from the degeneracy threshold for all parameter values appearing in the pattern-formation and asymptotic-limit statements. This is load-bearing for passing to the limit in the approximating sequence and for justifying the parabolic-elliptic and vanishing-diffusion limits.

minor comments (2)

[Introduction and main results] The statement of the main theorems would benefit from an explicit list of the structural assumptions (A1)–(A3) or equivalent, placed before the theorems rather than referenced only indirectly.
[Section on 1D reduction] In the 1D pattern-formation analysis, the precise monotonicity or convexity properties required of the reduced nonlinearity should be stated explicitly and shown to follow from the general assumptions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive overall assessment and for highlighting the need for greater clarity on the structural assumptions. We address the major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: [Assumptions and entropy construction (likely §2)] The structural conditions on the nonlinear diffusion coefficient and sensitivity function (implicit in the entropy construction referenced in the abstract and used for a priori bounds): these conditions are not stated explicitly, so it is not possible to verify that the entropy dissipation remains coercive and yields uniform bounds away from the degeneracy threshold for all parameter values appearing in the pattern-formation and asymptotic-limit statements. This is load-bearing for passing to the limit in the approximating sequence and for justifying the parabolic-elliptic and vanishing-diffusion limits.

Authors: We agree that the structural conditions should be stated more explicitly to facilitate verification. In the original manuscript the assumptions on the diffusion coefficient D(ρ) (degenerate at the volume-filling threshold ρ=1) and the sensitivity function χ(ρ) are introduced in the model formulation and used to construct the entropy functional in Section 3, but they are not collected in a single, self-contained list. In the revised version we have added a dedicated subsection 2.1 entitled 'Structural Assumptions' that explicitly lists: (i) D(ρ) ≥ 0 with D(ρ)=0 for ρ≥1 and D(ρ)∼(1-ρ)^α near ρ=1 for some α>0; (ii) χ(ρ) bounded and compatible with the entropy dissipation; and (iii) the resulting coercivity estimate that yields uniform L^∞ bounds away from the degeneracy threshold for all parameter values appearing in the pattern-formation and limit statements. These explicit hypotheses directly justify the a-priori estimates, the passage to the limit in the approximating sequence, and the parabolic-elliptic and vanishing-diffusion limits. The mathematical arguments themselves remain unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity: existence, uniqueness, and limit results derived from entropy dissipation and ODE analysis without reduction to inputs or self-referential definitions

full rationale

The paper establishes mathematical theorems on global weak solutions, weak-strong uniqueness, exponential convergence to equilibrium, pattern formation via reduction to a first-order ODE, and asymptotic limits. These follow from a priori estimates obtained from entropy functionals whose dissipation controls the degeneracy, combined with standard compactness arguments and ODE analysis in one dimension. No step reduces a claimed result to a fitted parameter, a self-defined quantity, or a load-bearing self-citation whose validity depends on the present work. The structural conditions on diffusion and sensitivity are part of the hypotheses that close the estimates; they are not smuggled in as outputs. The derivation chain is self-contained against external mathematical benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of suitable entropy functionals whose dissipation yields uniform bounds, on structural assumptions on the nonlinear diffusion and sensitivity that close those estimates, and on the reduction of the 1D system to a first-order scalar equation. No explicit free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Suitable entropy functionals exist whose time derivative produces a priori bounds that control the degeneracy and the volume-filling constraint.
Invoked to obtain global weak solutions and to pass to the limit in the approximation scheme.
domain assumption The nonlinearities satisfy structural conditions that allow the entropy dissipation to remain non-negative and coercive.
Required for the estimates to close uniformly; location implicit in the statement that analysis relies on a priori estimates from entropy functionals.

pith-pipeline@v0.9.0 · 5667 in / 1607 out tokens · 36418 ms · 2026-05-21T03:05:02.292337+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The analysis relies on a priori estimates derived from suitable entropy functionals... F_τ(ρ,c)=∫[ρ(logρ−1)+(1−ρ)(log(1−ρ)−1)+χτρc]dx
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We establish the existence of global weak solutions, a weak-strong uniqueness result, the exponential convergence to the homogeneous steady state, pattern formation in one spatial dimension, as well as the parabolic-elliptic and vanishing diffusion limits.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

[1]

Abdo and F.-N

E. Abdo and F.-N. Lee. Logarithmic Sobolev inequalities for bounded domains and applications to drift-diffusion equations.J. Funct. Anal.288 (2025), no. 110716, 13 pages

work page 2025
[2]

Arumugam and J

G. Arumugam and J. Tyagi. Keller–Segel chemotaxis models: a review.Acta Appl. Math.171 (2021), 1–82

work page 2021
[3]

Bellomo, A

N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler. Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues.Math. Models Meth. Appl. Sci.25 (2015), 1663–1763

work page 2015
[4]

Bendahmane, K

M. Bendahmane, K. Karlsen, and J. Urbano. On a two-sidedly degenerate chemotaxis model with volume-filling effect.Math. Models Meth. Appl. Sci.17 (2007), 783–804

work page 2007
[5]

Byrne and M

H. Byrne and M. Owen. A new interpretation of the Keller–Segel model based on multiphase modelling. J. Math. Biol.49 (2004), 604–626

work page 2004
[6]

Burger, M

M. Burger, M. Di Francesco, and Y. Dulak-Struss. The Keller–Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion.SIAM J. Math. Anal.38 (2006), 1288– 1315

work page 2006
[7]

Calvez and J

V. Calvez and J. A. Carrillo. Volume effects in the Keller–Segel model: energy estimates preventing blow-up.J. Math. Pures Appl.86 (2006), 155–175

work page 2006
[8]

J. A. Carrillo, X. Chen, Q. Wang, Z. Wang, and L. Zhang. Phase transitions and bump solutions of the Keller–Segel model with volume exclusion.SIAM J. Appl. Math.80 (2020), 232–261

work page 2020
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Cie´ slak and C

T. Cie´ slak and C. Stinner. Finite-time blowup and global-in-time unbounded solutions to a para- bolic–parabolic quasi-linear Keller–Segel system in higher dimensions.J. Differ. Eqs.252 (2012), 5832–5851

work page 2012
[10]

D´ ıaz, G

I. D´ ıaz, G. Galiano, and A. J¨ ungel. On a quasilinear degenerate system arising in semiconductors theory. Part I: Existence and uniqueness of solutions.Nonlin. Anal. Real World Appl.2 (2001), 305– 336

work page 2001
[11]

Di Francesco and J

M. Di Francesco and J. Rosado. Fully parabolic Keller–Segel model for chemotaxis with prevention of overcrowding.Nonlinearity21 (2008), 2715–2730

work page 2008
[12]

Hillen and K

T. Hillen and K. Painter. Global existence for a parabolic chemotaxis model with prevention of overcrowding.Adv. Appl. Math.26 (2001), 280–301

work page 2001
[13]

L. Hong, W. Wang, and S. Zheng. Uniqueness of weak solutions to a high dimensional Keller–Segel equation with degenerate diffusion and nonlocal aggregation.Nonlin. Anal.134 (2016), 204–214

work page 2016
[14]

J. Jiang. Convergence to equilibria of global solutions to a degenerate quasilinear Keller–Segel system. Z. Angew. Math. Phys.69 (2018), no. 130, 20 pages

work page 2018
[15]

Kawakami and Y

T. Kawakami and Y. Sugiyama. Uniqueness theorem on weak solutions to the Keller–Segel system of degenerate and singular types.J. Differ. Eqs.257 (2014), 4064–4086

work page 2014
[16]

Keller and L

E. Keller and L. Segel. Initiation of slime mold aggregation viewed as an instability.J. Theor. Biol. 26 (1970), 399–415. A DEGENERATE KELLER–SEGEL SYSTEM WITH VOLUME FILLING 33

work page 1970
[17]

Kowalczyk and Z

R. Kowalczyk and Z. Szyma´ nska. On the global existence of solutions to an aggregation model.J. Math. Anal. Appl.343 (2008), 379–398

work page 2008
[18]

Lauren¸ cot and D

P. Lauren¸ cot and D. Wrzosek. A chemotaxis model with threshold density and degenerate diffusion. In: H. Brezis, M. Chipot, and J. Escher (eds.),Nonlinear Elliptic and Parabolic Problems, pp. 273–290. Birkh¨ auser, Basel, 2005

work page 2005
[19]

Lemon, J

G. Lemon, J. King, H. Byrne, O. Jensen, and K. Shakesheff. Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory.J. Math. Biol.52 (2006), 571–594

work page 2006
[20]

A. Moussa. Some variants of the classical Aubin–Lions lemma.J. Evol. Eqs.16 (2016), 65–93

work page 2016
[21]

Painter and T

K. Painter and T. Hillen. Volume-filling and quorum-sensing in models for chemosensitive movement. Canad. Appl. Math. Quart.10 (2002), 501–543

work page 2002
[22]

C. Patlak. Random walk with persistence and external bias.Bull. Math. Biophys.15 (1953), 311–338

work page 1953
[23]

Perthame and M

B. Perthame and M. Zhang. Patterns in the Keller–Segel system with density cut-off. To appear in EMS Series of Congress Reports, 2026. hal-04825133

work page 2026
[24]

Tao and M

Y. Tao and M. Winkler. Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity.J. Differ. Eqs.252 (2012), 692–715

work page 2012
[25]

Wang and T

Z. Wang and T. Hillen. Classical solutions and pattern formation for a volume filling chemotaxis model.Chaos17 (2007), no. 037108, 13 pages

work page 2007
[26]

Z.-A. Wang, M. Winkler, and D. Wrzosek. Global regularity versus infinite-time singularity formation in a chemotaxis model with volume-filling and degenerate diffusion.SIAM J. Math. Anal.44 (2012), 3502–3525

work page 2012
[27]

D. Wrzosek. Volume filling effect in modelling chemotaxis.Math. Model. Nat. Phenom.5 (2010), 123–147. Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Email address:noah.geltner@tuwien.ac.at Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Email addre...

work page 2010

[1] [1]

Abdo and F.-N

E. Abdo and F.-N. Lee. Logarithmic Sobolev inequalities for bounded domains and applications to drift-diffusion equations.J. Funct. Anal.288 (2025), no. 110716, 13 pages

work page 2025

[2] [2]

Arumugam and J

G. Arumugam and J. Tyagi. Keller–Segel chemotaxis models: a review.Acta Appl. Math.171 (2021), 1–82

work page 2021

[3] [3]

Bellomo, A

N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler. Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues.Math. Models Meth. Appl. Sci.25 (2015), 1663–1763

work page 2015

[4] [4]

Bendahmane, K

M. Bendahmane, K. Karlsen, and J. Urbano. On a two-sidedly degenerate chemotaxis model with volume-filling effect.Math. Models Meth. Appl. Sci.17 (2007), 783–804

work page 2007

[5] [5]

Byrne and M

H. Byrne and M. Owen. A new interpretation of the Keller–Segel model based on multiphase modelling. J. Math. Biol.49 (2004), 604–626

work page 2004

[6] [6]

Burger, M

M. Burger, M. Di Francesco, and Y. Dulak-Struss. The Keller–Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion.SIAM J. Math. Anal.38 (2006), 1288– 1315

work page 2006

[7] [7]

Calvez and J

V. Calvez and J. A. Carrillo. Volume effects in the Keller–Segel model: energy estimates preventing blow-up.J. Math. Pures Appl.86 (2006), 155–175

work page 2006

[8] [8]

J. A. Carrillo, X. Chen, Q. Wang, Z. Wang, and L. Zhang. Phase transitions and bump solutions of the Keller–Segel model with volume exclusion.SIAM J. Appl. Math.80 (2020), 232–261

work page 2020

[9] [9]

Cie´ slak and C

T. Cie´ slak and C. Stinner. Finite-time blowup and global-in-time unbounded solutions to a para- bolic–parabolic quasi-linear Keller–Segel system in higher dimensions.J. Differ. Eqs.252 (2012), 5832–5851

work page 2012

[10] [10]

D´ ıaz, G

I. D´ ıaz, G. Galiano, and A. J¨ ungel. On a quasilinear degenerate system arising in semiconductors theory. Part I: Existence and uniqueness of solutions.Nonlin. Anal. Real World Appl.2 (2001), 305– 336

work page 2001

[11] [11]

Di Francesco and J

M. Di Francesco and J. Rosado. Fully parabolic Keller–Segel model for chemotaxis with prevention of overcrowding.Nonlinearity21 (2008), 2715–2730

work page 2008

[12] [12]

Hillen and K

T. Hillen and K. Painter. Global existence for a parabolic chemotaxis model with prevention of overcrowding.Adv. Appl. Math.26 (2001), 280–301

work page 2001

[13] [13]

L. Hong, W. Wang, and S. Zheng. Uniqueness of weak solutions to a high dimensional Keller–Segel equation with degenerate diffusion and nonlocal aggregation.Nonlin. Anal.134 (2016), 204–214

work page 2016

[14] [14]

J. Jiang. Convergence to equilibria of global solutions to a degenerate quasilinear Keller–Segel system. Z. Angew. Math. Phys.69 (2018), no. 130, 20 pages

work page 2018

[15] [15]

Kawakami and Y

T. Kawakami and Y. Sugiyama. Uniqueness theorem on weak solutions to the Keller–Segel system of degenerate and singular types.J. Differ. Eqs.257 (2014), 4064–4086

work page 2014

[16] [16]

Keller and L

E. Keller and L. Segel. Initiation of slime mold aggregation viewed as an instability.J. Theor. Biol. 26 (1970), 399–415. A DEGENERATE KELLER–SEGEL SYSTEM WITH VOLUME FILLING 33

work page 1970

[17] [17]

Kowalczyk and Z

R. Kowalczyk and Z. Szyma´ nska. On the global existence of solutions to an aggregation model.J. Math. Anal. Appl.343 (2008), 379–398

work page 2008

[18] [18]

Lauren¸ cot and D

P. Lauren¸ cot and D. Wrzosek. A chemotaxis model with threshold density and degenerate diffusion. In: H. Brezis, M. Chipot, and J. Escher (eds.),Nonlinear Elliptic and Parabolic Problems, pp. 273–290. Birkh¨ auser, Basel, 2005

work page 2005

[19] [19]

Lemon, J

G. Lemon, J. King, H. Byrne, O. Jensen, and K. Shakesheff. Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory.J. Math. Biol.52 (2006), 571–594

work page 2006

[20] [20]

A. Moussa. Some variants of the classical Aubin–Lions lemma.J. Evol. Eqs.16 (2016), 65–93

work page 2016

[21] [21]

Painter and T

K. Painter and T. Hillen. Volume-filling and quorum-sensing in models for chemosensitive movement. Canad. Appl. Math. Quart.10 (2002), 501–543

work page 2002

[22] [22]

C. Patlak. Random walk with persistence and external bias.Bull. Math. Biophys.15 (1953), 311–338

work page 1953

[23] [23]

Perthame and M

B. Perthame and M. Zhang. Patterns in the Keller–Segel system with density cut-off. To appear in EMS Series of Congress Reports, 2026. hal-04825133

work page 2026

[24] [24]

Tao and M

Y. Tao and M. Winkler. Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity.J. Differ. Eqs.252 (2012), 692–715

work page 2012

[25] [25]

Wang and T

Z. Wang and T. Hillen. Classical solutions and pattern formation for a volume filling chemotaxis model.Chaos17 (2007), no. 037108, 13 pages

work page 2007

[26] [26]

Z.-A. Wang, M. Winkler, and D. Wrzosek. Global regularity versus infinite-time singularity formation in a chemotaxis model with volume-filling and degenerate diffusion.SIAM J. Math. Anal.44 (2012), 3502–3525

work page 2012

[27] [27]

D. Wrzosek. Volume filling effect in modelling chemotaxis.Math. Model. Nat. Phenom.5 (2010), 123–147. Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Email address:noah.geltner@tuwien.ac.at Institute of Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Email addre...

work page 2010