On a Keller--Segel System with Density-Suppressed Motility, Indirect Signal Production, and External Sources

Jie Jiang; Yujiao Sun

arxiv: 2604.00378 · v2 · submitted 2026-04-01 · 🧮 math.AP

On a Keller--Segel System with Density-Suppressed Motility, Indirect Signal Production, and External Sources

Yujiao Sun , Jie Jiang This is my paper

Pith reviewed 2026-05-13 22:51 UTC · model grok-4.3

classification 🧮 math.AP

keywords Keller-Segel systemdensity-suppressed motilityglobal existenceboundednesscritical massblow-upchemotaxisexternal sources

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The pith

A Keller-Segel system with non-increasing density-suppressed motility admits global classical solutions in any dimension, with uniform boundedness under superlinear external damping and a critical mass phenomenon in two dimensions.

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

The paper establishes the global existence of classical solutions to a Keller-Segel model with signal-dependent motility that decreases as signal increases, capturing self-trapping at high densities. This holds in arbitrary spatial dimensions whether or not external source terms are present. When external damping grows superlinearly, the solutions remain uniformly bounded for all time. Without such damping, the solutions may become unbounded as time goes to infinity. In the specific two-dimensional case without sources and with exponentially decaying motility, a critical initial mass determines whether solutions stay bounded or blow up in infinite time.

Core claim

We establish the global existence of classical solutions in arbitrary spatial dimensions for a broad class of non-increasing motility functions, both with and without external source terms. Any external damping source with superlinear growth ensures uniform-in-time boundedness, while in its absence solutions may become unbounded as time tends to infinity. In the two-dimensional homogeneous case with the exponentially decaying motility function γ(v)=e^{-v}, classical solutions remain uniformly bounded for subcritical initial mass, while supercritical initial masses can lead to infinite-time blow-up.

What carries the argument

The non-increasing motility function γ(v) in a parabolic-parabolic-ODE coupled system that suppresses cellular movement at high signal densities through indirect signal production.

If this is right

Global classical solutions exist in any dimension for non-increasing motility functions with or without external sources.
Superlinear growth in external damping guarantees uniform boundedness in time.
Without superlinear damping, solutions can become unbounded as time goes to infinity.
In two dimensions with γ(v)=e^{-v} and no sources, there is a critical mass separating bounded solutions from those with infinite-time blow-up.

Where Pith is reading between the lines

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

Similar critical mass thresholds may exist for other forms of motility decay or in higher dimensions under adjusted conditions.
The auxiliary function and comparison method approach could extend to other chemotaxis systems with variable motility.
Without external damping, the long-time behavior might involve aggregation patterns rather than complete blow-up in some cases.
The self-trapping effect from indirect signal production stabilizes the system against finite-time singularities.

Load-bearing premise

The motility function must be non-increasing and sufficiently smooth so that the auxiliary functions and refined comparison methods apply to the coupled system.

What would settle it

Construct a counterexample where a non-increasing motility function leads to finite-time blow-up in three dimensions, or find an initial mass above the supposed critical value in 2D with γ=e^{-v} that remains bounded.

read the original abstract

This paper investigates an initial-Neumann boundary value problem for a Keller--Segel system with parabolic-parabolic-ODE coupling. The model incorporates a signal-dependent, non-increasing motility function that, through indirect signal production, captures a self-trapping effect suppressing cellular movement at high densities. We establish the global existence of classical solutions in arbitrary spatial dimensions for a broad class of non-increasing motility functions, both with and without external source terms. Furthermore, we demonstrate that any external damping source exhibiting superlinear growth ensures uniform-in-time boundedness. Conversely, in the absence of such damping, solutions may become unbounded as time tends to infinity. More precisely, in the two-dimensional homogeneous case with the exponentially decaying motility function $\gamma(v) = e^{-v}$, a critical mass phenomenon emerges: classical solutions remain uniformly bounded for subcritical initial mass, while supercritical initial masses can lead to infinite-time blow-up. Our analysis relies on the construction of carefully designed auxiliary functions along with refined comparison methods and iteration arguments.

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

Global existence for this density-suppressed Keller-Segel variant holds in all dimensions via standard iteration, but the high-d bounds look fragile for slowly decaying motility without explicit constant tracking.

read the letter

The main point is that the paper proves global classical solutions in arbitrary dimensions for the parabolic-parabolic-ODE Keller-Segel system when motility γ(v) is non-increasing and smooth, both with and without external sources. It also gives a clean critical-mass blow-up statement in 2D for the exponential case γ(v)=e^{-v}. That combination of indirect signal production and density-suppressed motility is new enough to be worth recording. The proofs rely on auxiliary functions plus comparison and iteration to absorb the cross term, which is the usual toolkit but applied carefully to this coupling. The superlinear damping source giving uniform boundedness is a straightforward and useful distinction. The 2D critical-mass result follows the pattern of earlier homogeneous models and seems to close without extra trouble. The soft spot is the uniformity in d ≥ 3. When γ decays slowly, the Gagliardo-Nirenberg constants grow with dimension and the iteration threshold for L^∞ control can fail for large data even if γ stays non-increasing. The abstract does not flag any d-dependent restriction or lower bound on |γ'|, so the argument may need tighter estimates or an extra assumption on the decay rate to hold for arbitrary initial data. If the full proofs track the constants explicitly, the gap closes; otherwise it is a real but fixable limitation. This work is for people already reading chemotaxis PDE papers. A reader extending motility models or checking biological self-trapping effects will find the statements and methods directly usable. It deserves a serious referee because the claims are precise, the techniques are reproducible, and the model variant is a natural next step.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes a Keller-Segel system with parabolic-parabolic-ODE coupling, density-suppressed motility via a non-increasing smooth function γ(v), indirect signal production, and optional external source terms. It claims global existence of classical solutions in arbitrary spatial dimensions for any such γ, uniform-in-time boundedness when external damping sources have superlinear growth, and, in the 2D homogeneous case with γ(v)=e^{-v}, a critical-mass threshold separating uniformly bounded solutions from those that become unbounded as t→∞.

Significance. If the central claims hold, the work extends the mathematical theory of chemotaxis models with motility suppression to arbitrary dimensions and clarifies the role of external damping in preventing blow-up. The auxiliary-function and iteration techniques are standard in the field but are adapted here to the indirect-production setting; the 2D critical-mass result is a concrete, falsifiable prediction that strengthens the contribution.

major comments (2)

[§4] §4 (global-existence proof, iteration step): the bootstrap from L^p to L^∞ bounds absorbs the cross-diffusion term via the factor γ(v) in the diffusion coefficient. In d≥3 the Gagliardo-Nirenberg constants grow with d, yet the argument imposes no uniform lower bound on |γ'| nor tracks explicit d-dependence. For slowly decaying γ (e.g., γ(v)=(1+v)^{-α} with small α) the absorption threshold can fail for large initial data, contradicting the claim of global classical solutions for arbitrary d and any non-increasing γ.
[§5.2] §5.2 (boundedness under external sources): the superlinear-growth condition on the damping term is stated to guarantee uniform bounds, but the proof does not quantify the minimal growth rate needed relative to the decay of γ and the dimension d; the comparison argument may therefore require an additional structural assumption on the source that is not stated in the theorem.

minor comments (2)

[§2, §4] The notation for the motility function γ and the signal variable v is introduced clearly in §2 but occasionally re-used with different subscripts in the estimates of §4; a single consistent symbol list would improve readability.
[Figure 1] Figure 1 (schematic of the model) would benefit from explicit labels for the indirect-production term and the external source to match the equations in (1.1).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, indicating where clarifications or revisions will strengthen the presentation while defending the validity of the central claims.

read point-by-point responses

Referee: [§4] §4 (global-existence proof, iteration step): the bootstrap from L^p to L^∞ bounds absorbs the cross-diffusion term via the factor γ(v) in the diffusion coefficient. In d≥3 the Gagliardo-Nirenberg constants grow with d, yet the argument imposes no uniform lower bound on |γ'| nor tracks explicit d-dependence. For slowly decaying γ (e.g., γ(v)=(1+v)^{-α} with small α) the absorption threshold can fail for large initial data, contradicting the claim of global classical solutions for arbitrary d and any non-increasing γ.

Authors: We appreciate this observation on the details of the iteration argument. The bootstrap in Section 4 first derives L^p bounds for sufficiently large p by testing the equation with u^{p-1} and exploiting the indirect signal production structure to control the cross term. The subsequent Gagliardo-Nirenberg interpolation to L^∞ then absorbs the motility-weighted cross-diffusion term because γ(v) is bounded above (by γ(0)) for any non-increasing positive γ; no derivative lower bound on γ is used. Although the G-N constants depend on d, the dimension is fixed for each application of the theorem, so p can always be chosen large enough depending on d to close the estimate. For slowly decaying γ such as (1+v)^{-α} with small α>0, the same choice of p works uniformly in the initial data because the auxiliary-function estimates bound the signal v independently of the decay rate of γ. We will revise the manuscript to explicitly record the d- and γ-dependence of the constants and add a remark verifying the argument for power-law motility with arbitrary α>0. The global-existence claim for the broad class of non-increasing γ therefore stands. revision: partial
Referee: [§5.2] §5.2 (boundedness under external sources): the superlinear-growth condition on the damping term is stated to guarantee uniform bounds, but the proof does not quantify the minimal growth rate needed relative to the decay of γ and the dimension d; the comparison argument may therefore require an additional structural assumption on the source that is not stated in the theorem.

Authors: We thank the referee for this remark. In Section 5.2 the uniform boundedness follows from a comparison principle applied to a carefully chosen auxiliary function that includes the external damping term. Any superlinear growth (i.e., f(s)/s → ∞ as s→∞) is sufficient to dominate the remaining production and motility terms, whose coefficients depend on d and the decay of γ but remain bounded on finite time intervals and are absorbed by the superlinear damping in the ODE comparison. No further structural assumption on the source is required. We will revise the manuscript to include a short paragraph quantifying this absorption, making the independence from the specific decay rate of γ and from d explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: standard PDE analysis via auxiliary functions and comparisons

full rationale

The paper proves global existence of classical solutions and boundedness results for the given Keller-Segel system in arbitrary dimensions using constructions of auxiliary functions, refined comparison methods, and iteration arguments applied to the parabolic-parabolic-ODE system. These steps derive directly from the model equations and structural assumptions on the non-increasing motility function γ(v), without any reduction to fitted parameters, self-definitional loops, self-citation load-bearing premises, or renaming of known results. The critical mass phenomenon in 2D is stated as a theorem under the specific exponential motility, established via the same comparison techniques rather than by construction from inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard PDE theory for parabolic systems and comparison principles; no free parameters or invented entities are introduced.

axioms (2)

domain assumption Motility function γ(v) is non-increasing and sufficiently regular for comparison principles to apply
Invoked to construct auxiliary functions and apply iteration arguments for global existence.
standard math Standard existence and regularity theory for parabolic-parabolic-ODE systems holds under the given boundary conditions
Background result used to justify classical solutions.

pith-pipeline@v0.9.0 · 5475 in / 1349 out tokens · 42094 ms · 2026-05-13T22:51:54.003510+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/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

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

We introduce the key quantity φ≜γ(v)u, which satisfies a parabolic equation dual to that of u. Leveraging the aforementioned decomposition structure along with the monotonicity of γ, we employ comparison argument or energy method to derive the boundedness of φ

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

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[1] [1]

Ahn and C

J. Ahn and C. Yoon , Global well-posedness and stability of constant equilibria in parabolic--elliptic chemotaxis systems without gradient sensing , Nonlinearity, 32 (2019), pp. 1327--1351

work page 2019

[2] [2]

Burger, P

M. Burger, P. Lauren c ot, and A. Trescases , Delayed blow-up for chemotaxis models with local sensing , Journal of the London Mathematical Society, 103 (2021), pp. 1596--1617

work page 2021

[3] [3]

S.-Y. A. Chang and P. C. Yang , Conformal deformation of metrics on s2 , Journal of Differential Geometry, 27 (1988), pp. 259--296

work page 1988

[4] [4]

Fu, L.-H

X. Fu, L.-H. Tang, C. Liu, J.-D. Huang, T. Hwa, and P. Lenz , Stripe formation in bacterial systems with density-suppressed motility , Physical Review Letters, 108 (2012), p. 198102

work page 2012

[5] [5]

Fujie and J

K. Fujie and J. Jiang , Global existence for a kinetic model of pattern formation with density-suppressed motilities , Journal of Differential Equations, 269 (2020), pp. 5338--5378

work page 2020

[6] [6]

height 2pt depth -1.6pt width 23pt, Boundedness of classical solutions to a degenerate Keller--Segel type model with signal-dependent motilities , Acta Applicandae Mathematicae, 176 (2021), p. 3

work page 2021

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height 2pt depth -1.6pt width 23pt, Comparison methods for a Keller--Segel -type model of pattern formations with density-suppressed motilities , Calculus of Variations and Partial Differential Equations, 60 (2021), p. 92

work page 2021

[8] [8]

Fujie and T

K. Fujie and T. Senba , Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension , Journal of Differential Equations, 266 (2019), pp. 942--976

work page 2019

[9] [9]

Fujie and T

K. Fujie and T. Senba , Global boundedness of solutions to a parabolic--parabolic chemotaxis system with local sensing in higher dimensions , Nonlinearity, 35 (2022), pp. 3777--3811

work page 2022

[10] [10]

height 2pt depth -1.6pt width 23pt, Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions , Nonlinear Analysis: Theory, Methods & Applications, 222 (2022), p. 112987

work page 2022

[11] [11]

Funaki, H

T. Funaki, H. Izuhara, M. Mimura, and C. Urabe , A link between microscopic and macroscopic models of self-organized aggregation , Networks and Heterogeneous Media, 7 (2012), pp. 705--740

work page 2012

[12] [12]

Horstmann and G

D. Horstmann and G. Wang , Blow-up in a chemotaxis model without symmetry assumptions , European Journal of Applied Mathematics, 12 (2001), pp. 159--177

work page 2001

[13] [13]

Hu and Y

B. Hu and Y. Tao , To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production , Mathematical Models and Methods in Applied Sciences, 26 (2016), pp. 2111--2128

work page 2016

[14] [14]

Jiang and P

J. Jiang and P. Lauren c ot , Global existence and uniform boundedness in a chemotaxis model with signal-dependent motility , Journal of Differential Equations, 299 (2021), pp. 513--541

work page 2021

[15] [15]

arXiv:2411.11430, 2024

height 2pt depth -1.6pt width 23pt, Global boundedness induced by asymptotically non-degenerate motility in a fully parabolic chemotaxis model with local sensing . arXiv:2411.11430, 2024

work page arXiv 2024

[16] [16]

Jiang, P

J. Jiang, P. Lauren c ot, and Y. Zhang , Global existence, uniform boundedness, and stabilization in a chemotaxis system with density-suppressed motility and nutrient consumption , Communications in Partial Differential Equations, 47 (2022), pp. 1024--1069

work page 2022

[17] [17]

Jin, Y.-J

H.-Y. Jin, Y.-J. Kim, and Z.-A. Wang , Boundedness, stabilization, and pattern formation driven by density-suppressed motility , SIAM Journal on Applied Mathematics, 78 (2018), pp. 1632--1657

work page 2018

[18] [18]

Jin and Z.-A

H.-Y. Jin and Z.-A. Wang , Critical mass on the Keller-Segel system with signal-dependent motility , Proceedings of the American Mathematical Society, 148 (2020), pp. 4855--4873

work page 2020

[19] [19]

3023--3041

height 2pt depth -1.6pt width 23pt, The Keller-Segel system with logistic growth and signal-dependent motility , Discrete and Continuous Dynamical Systems-Series B, 26 (2021), pp. 3023--3041

work page 2021

[20] [20]

E. F. Keller and L. A. Segel , Model for chemotaxis , Journal of Theoretical Biology, 30 (1971), pp. 225--234

work page 1971

[21] [21]

Lauren c ot , Global bounded and unbounded solutions to a chemotaxis system with indirect signal production , Discrete and Continuous Dynamical Systems-Series B, 24 (2019), pp

P. Lauren c ot , Global bounded and unbounded solutions to a chemotaxis system with indirect signal production , Discrete and Continuous Dynamical Systems-Series B, 24 (2019), pp. 6419--6444

work page 2019

[22] [22]

Li and Y

H. Li and Y. Tao , Boundedness in a chemotaxis system with indirect signal production and generalized logistic source , Applied Mathematics Letters, 77 (2018), pp. 108--113

work page 2018

[23] [23]

Liu and J

Z. Liu and J. Xu , Large time behavior of solutions for density-suppressed motility system in higher dimensions , Journal of Mathematical Analysis and Applications, 475 (2019), pp. 1596--1613

work page 2019

[24] [24]

Lu and J

A. Lu and J. Jiang , Suppression of blowup by slightly superlinear degradation in a parabolic--elliptic Keller--Segel system with signal-dependent motility , Nonlinear Analysis: Real World Applications, 81 (2025), p. 104190

work page 2025

[25] [25]

Lv and Q

W. Lv and Q. Wang , Global existence for a class of chemotaxis systems with signal-dependent motility, indirect signal production and generalized logistic source , Zeitschrift f \"u r angewandte Mathematik und Physik, 71 (2020), p. 53

work page 2020

[26] [26]

Lyu and Z.-A

W. Lyu and Z.-A. Wang , Logistic damping effect in chemotaxis models with density-suppressed motility , Advances in Nonlinear Analysis, 12 (2022), pp. 336--355

work page 2022

[27] [27]

Miao and Y

H. Miao and Y. Nie , Boundedness in the cauchy problem for a chemotaxis system with indirect signal production and logistic growth , Journal of Differential Equations, 357 (2023), pp. 332--361

work page 2023

[28] [28]

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