Boundedness in a nonlinear chemotaxis-consumption model with gradient terms

Alessandro Columbu; Daniel Acosta Soba; Giuseppe Viglialoro

REVIEW 2 major objections 4 minor 1 cited by

Boundedness in a nonlinear chemotaxis-consumption model with gradient terms

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Nonlinear gradient damping keeps a chemotaxis–consumption system globally bounded.

desk verdict A solid, genuinely new boundedness result for a chemotaxis-consumption model with gradient damping, held back by a too-quick appeal to an external L^p-to-L^infty bootstrap lemma. read the letter →

arxiv 2501.13224 v1 pith:I2URTT5Q submitted 2025-01-22 math.AP

Daniel Acosta Soba , Alessandro Columbu , Giuseppe Viglialoro This is my paper · ORCID

classification math.AP MSC 35K5535B4035Q9292C17

keywords chemotaxis-consumptionnonlineardiffusiongradientnonlinearityglobalboundednessclassicalsolutionlogisticsourceKeller–Segelquasilinearparabolicsystem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies a quasilinear chemotaxis–consumption system in which cells diffuse nonlinearly, are attracted by a chemical signal that they consume, and are damped both by a logistic death term and by an explicit gradient-dependent source $-\rho|\nabla u|^q$. The authors seek conditions on the exponents $a_1,a_2$ and the damping exponent $q$ under which every smooth nonnegative initial datum produces a unique classical solution that exists for all times and stays uniformly bounded. They prove that such global boundedness holds when either $q$ exceeds the threshold $\max\{\frac{2n}{n+1},\frac{n}{n+1}(2a_2-a_1+1)\}$, or, in an intermediate range, when the logistic coefficient $\mu$ is large enough relative to a power of $\kappa\|v_0\|_{L^\infty}$. If correct, the result shows that the gradient damping alone can suppress blow-up even for strong chemotactic attraction, and it unifies earlier boundedness results for the linear-gradient and gradient-free versions of the model.

What carries the argument

The engine of the proof is the energy functional $\Phi(t) := \int_\Omega (u+1)^p + \frac{\kappa^{2p}}{2p}\int_\Omega |\nabla v|^{2p}$, whose time derivative is shown to satisfy $\Phi'(t)+\Phi(t)\le C$ on $(0,T_{\max})$. The chain of estimates combines interpolation inequalities, bounds on $\nabla v$ coming from the parabolic equation for $v$, and a set of carefully chosen interpolation exponents; under (A1) the gradient-damping term controls the nonlinear production with the absorption constant free, while under (A2) the condition on $\mu$ makes the coefficient of $\int_\Omega (u+1)^{p+1}$ nonpositive. The resulting uniform bounds feed Lemma 4, the boundedness criterion, which upgrades the $L^p$ information to $L^\infty$ through the external regularity lemma and extends the bound to all times by contradiction with the local blow-up criterion.

What would settle it

A direct counterexample would be a smooth nonnegative initial datum and a parameter set satisfying (A1) or (A2) for which the classical solution of (6) becomes unbounded in $L^\infty$ at some finite time, contradicting Theorem 1. Short of that, checking whether the coefficient pair $D=(u+1)^{a_1-1}$, $S=-\kappa u(u+1)^{a_2-1}\nabla v$ actually satisfies hypotheses (A2)–(A10) of the regularity lemma in [22] for an admissible choice of $a_1,a_2,p,q$ would settle whether the boundedness criterion has a gap.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: under the hypotheses in (7) and with $\lambda,\mu,\rho,\kappa>0$, $a_1,a_2\in\mathbb{R}$, the zero-flux initial–boundary problem (6) admits a unique uniformly bounded classical solution $(u,v)\in(C^{2,1}(\Omega\times[0,\infty))\cap L^\infty(\Omega\times(0,\infty)))^2$, provided either (A1) $\max\{\frac{2n}{n+1},\frac{n}{n+1}(2a_2-a_1+1)\} < q \le 2$, or (A2) $\frac{n}{n+1}(2a_2-a_1+1) < q \le \frac{2n}{n+1}$ together with $\mu > \frac{4}{p_0} K \|\kappa v_0\|_{L^\infty}^{2p_0}$ for explicit constants $p_0=p_0(a_1,a_2,n,q)$ and $K=K(p_0)$. The proof first produces uniform bounds for $u$ in $L^p$ and $\nabla v$ in $L^{2p}$ from an energy inequality, then invokes a parabolic regularity lemma to pass from $L^p$ to $L^\infty$. The damping exponent $q$ is the deciding mechanism: in regime (A1) the term $-\rho|\nabla u|^q$ keeps the energy inequality closed with no restriction on $\mu$, while in (A2) the size condition on $\mu$ makes the destabilizing production term absorbable.

Load-bearing premise

The whole proof rests on Lemma 4, which concludes $u\in L^\infty((0,\infty);L^\infty(\Omega))$ from a uniform $L^{p_2}$ bound by invoking an external regularity lemma whose hypotheses on the coefficients are asserted to follow from the $L^p$ estimate and parabolic regularity of $v$, rather than verified in detail; if those hypotheses fail for $D=(u+1)^{a_1-1}$ and $S=-\kappa u(u+1)^{a_2-1}\nabla v$, the final bootstrap does not go through.

Editorial extensions

If this is right

Under (A1), boundedness holds for every $\mu>0$; the gradient damping alone prevents unboundedness for any logistic strength.
Under (A2), the required lower bound on $\mu$ grows with a power of $\kappa\|v_0\|_{L^\infty}$, making the initial signal concentration the decisive quantity in this regime.
Setting $a_1=a_2=1$ recovers the boundedness criterion of the linear-gradient-source model [7], and setting $\kappa=0$ recovers the gradient-free condition $a_2<\frac{a_1+1}{2}$ with $\mu$ large from [18].
The obtained solution is classical and global, so neither finite-time nor infinite-time blow-up occurs for data satisfying the stated inequalities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The threshold $\frac{n}{n+1}(2a_2-a_1+1)$ interpolates between the diffusion exponent and the sensitivity combination; the paper does not address whether it is sharp, but I would expect an optimality statement in the spirit of the known critical exponent for production chemotaxis models.
The quantity in (A2) is essentially a power of the same product $\kappa\|v_0\|_{L^\infty}$ that governs the classical small-data condition for the consumption model; a natural testable extension is whether the explicit constant $K(p_0)$ can be optimized or the condition is also necessary in two dimensions.
The same energy-absorption scheme appears transferable to consumption models with singular sensitivity or to chemotaxis–fluid systems, replacing the $\nabla v$ bounds with the corresponding regularity estimates; that extension is not attempted in this paper.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main theorem is real and useful: system (6) combines nonlinear diffusion/sensitivity with a gradient source, and the paper proves uniform boundedness under two regimes, (A1) and (A2). The gradient term genuinely removes the smallness condition on the sensitivity in case (A1), which is a clear qualitative advance over [18] and [7]. The proof is built on a careful energy functional Y(t), and Lemmas 5–9 are algebraically consistent; the constants are tracked and the interpolation/absorption steps check out. Credit where due: this is a competent, substantive extension, not a cosmetic one.

The soft spot is exactly where the reader and the stress-test point: Lemma 4, the bridge from L^p to L^infty, is under-justified in seven lines. The appeal to [22, Lemma A.1] is not fully substantiated. In particular, the paper writes f(x,t) = lambda u - mu u^2 - rho|nabla u|^q and then says the equation is solved with f = F, a constant upper bound. That is not a literal identification: the actual source contains the gradient term, so one needs an explicit comparison argument to justify replacing it by a constant. The stress-test's additional concern about S containing nabla v is less accurate: S(u) is a scalar function of u, and nabla v enters as the gradient of the signal, which matches the structure of [22]. So the gap is narrower than the stress-test suggests, and it is very likely fixable. Still, the proof as written leaves the load-bearing bootstrap to an external lemma without verifying hypotheses (A2)–(A10), and a referee should insist on either a direct proof of Lemma 4 or a detailed check of those conditions, including the required regularity of nabla v.

Minor point: in condition (A2), p0 and K(p0) are existential; the continuity argument in the proof of Theorem 1 is acceptable but terse, and the paper would benefit from a more transparent description of these constants.

Bottom line: this paper deserves serious refereeing. The main result is new, the energy estimates are solid, and the Lemma 4 gap is repairable without changing the theorem. I would send it to a competent reviewer, with a request to focus on Lemma 4 and the comparison argument. For my own work, I would not cite it in the next year, but I would not be surprised to see it cited by others working on gradient-damping chemotaxis.

Referee Report

2 major / 4 minor

Summary. This paper studies the Neumann initial-boundary value problem for a two-component parabolic chemotaxis-consumption system (Eq. (6)) whose first equation combines nonlinear diffusion (u+1)^{a1-1}, nonlinear sensitivity -κu(u+1)^{a2-1}∇v, a logistic source λu-μu^2, and a gradient-damping term -ρ|∇u|^q, while the second equation is v_t = Δv - uv. The main result (Theorem 1) asserts the existence of a unique uniformly bounded classical solution under either condition (A1), a lower bound on q, or condition (A2), a lower bound on μ in terms of ‖κv0‖_∞. The proof proceeds through local existence (Lemma 3), an L^p-based energy functional Y(t) (Eq. (15)), a differential inequality with absorption (Lemmas 5-9), and a final bootstrap lemma (Lemma 4) that converts uniform L^{p2} bounds into L^∞ bounds.

Significance. If the bootstrap is properly justified, the result is a nontrivial extension of [7] and [18]: it quantifies how the gradient term |∇u|^q permits boundedness for strong chemotactic sensitivities in regime (A1) without a smallness condition on μ. The energy-estimate part is carefully structured, constants are tracked, and the parameter conditions in Lemma 2 are explicit enough to be checked. The dependence on ‖κv0‖_∞ in (A2) is consistent with previous chemotaxis-consumption literature (Remark 1). The main uncertainty is the external L^p-to-L^∞ bootstrap, which is not verified.

major comments (2)

[Section 4.1, Lemma 4] The proof of Lemma 4 is the only place where the uniform-in-time L^{p2} bound obtained in Section 5 is converted into the uniform L^∞ bound stated in Theorem 1, but the proof is an unverified appeal to two external results. In the first paragraph the identification with [22, problem (A.1)] sets f ≡ F, while the actual source in (6) is λu - μu^2 - ρ|∇u|^q; it is not shown that [22, Lemma A.1] allows replacing this source by its constant upper bound, nor that the vector-valued coefficient S = -κu(u+1)^{a2-1}∇v satisfies the hypotheses on S in [22]. In the second paragraph the assertion that 'parabolic regularity results' give the needed regularity for ∇v is not quantified: no exponent r, no dependence on p2, and no argument that the chosen p2 yields that r are provided. Because Lemma 4 is invoked verbatim in the proof of Theorem 1 (Section 5.2), this gap is load-bearing. The authors should either state the hypotheses (A2)-(A10) of [22] and verify them line by line for the present D, S, and f, or replace Lemma 4 by a self-contained proof of the L^p-to-L^∞ bootstrap.
[Section 5.2, proof of Theorem 1 (condition (A2))] The passage from Lemma 9's condition (28) to the theorem's condition (A2) is only a one-line continuity argument. Lemma 9 is applied with some p > p0 and η > 0, and the right-hand side of (28) contains ‖κv0‖^{2p}_{L^∞}, not ‖κv0‖^{2p0}_{L^∞}. The map (p,η) ↦ (4/p) E(p,n,η) ‖κv0‖^{2p}_{L^∞} is continuous, so the argument can be made rigorous, but as written it is not shown that a choice p > p0 can be made while preserving the strict inequality against the fixed μ. Please spell out the neighborhood argument, or state a monotonicity condition that makes the choice immediate.

minor comments (4)

[Section 3, Lemma 2] The proof of Lemma 2 consists of the sentence 'straightforward as long as p is chosen sufficiently large'; since inequalities (9a)-(9e) determine the admissible range p > p1 that is later used to define p0, please provide at least the explicit verification of (9b) and (9d), or state this as a separate elementary lemma.
[Section 1.3] The motivating source is written as h(u,∇u) = λu^r - μu^q - ρ|∇u|^q, while the model (6) uses the logistic term -μu^2 and the gradient term -ρ|∇u|^q; please align the notation in this paragraph with the rest of the paper.
[Section 2, Lemma 3 and Theorem 1] Lemma 3 states local regularity in C^{2+α,1+α/2}, while Theorem 1 states the solution class C^{2,1} ∩ L^∞; please clarify how the extensibility criterion in (10) relates to the lower regularity asserted in the theorem.
[References] There are typographical errors in the reference list (e.g., 'J. Diﬀererential Equations' in [13] and [23]); also, [7] is a preprint and its status should be marked as such in the bibliography.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central boundedness theorem rests on new energy estimates and an external parabolic regularity lemma; self-citations supply only technical or companion lemmas, and no claimed result reduces to its own inputs.

full rationale

The proof of Theorem 1 is not circular. The main boundedness mechanism is the energy functional Phi(t) in (15), estimated in Lemmas 5–9 to give u in L^∞((0,Tmax);L^p(Ω)) for all p>p1, with the hypotheses (A1)/(A2) entering through the algebraic conditions in Lemma 2 and the large-μ condition in (A2). The bridge from this L^p information to L^∞ is Lemma 4, which invokes the external regularity result [22, Lemma A.1]. One passage in Section 4.1 is terse and is a genuine completeness risk rather than circularity: the proof says that by choosing an appropriately large p2, parabolic regularity for the v-equation gives the necessary regularity for ∇v, and that '(A2)–(A10) are all satisfied', without displaying the verification for the specific coefficient S=−κu(u+1)^{a2−1}∇v or for the replacement of the actual source λu−μu^2−κ|∇u|^q by the constant upper bound F. This concerns the applicability of an external theorem, not a self-referential reduction. The self-citations in the paper—[10,16] for local existence, [16, Lemma 5.2] for extending local L^∞-boundedness to global existence via the blow-up criterion (10), and [7,18] as motivational prior cases—are technical or companion results with independent statements; they are not the target boundedness claim, and no fitted parameter is renamed as a prediction. No equation is defined in terms of the conclusion, and no uniqueness assertion from prior work by the authors is used to force the choice. The paper therefore receives score 2 for the presence of self-citations in the proof, not for circular reasoning.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard PDE tools (interpolation, parabolic regularity, maximum principle) and on the stated smoothness assumptions on the domain and data. The only non-explicit ingredients are the existential constants p0 and K in condition (A2), which are not fitted to data but are also not computable from the paper.

free parameters (2)

p0
Existential constant in condition (A2), defined as max{p1,p2} from Lemmas 2 and 4. No explicit formula or bound is given, so the largeness condition on μ is not directly checkable.
K(p0) = E(p0,n,0)
Defined in the proof of Theorem 1 as K(p0)=E(p0,n,0), with E as in (17). Its value depends on the unknown p0, so the condition remains implicit.

assumptions (5)

standard math Gagliardo-Nirenberg interpolation inequality (Lemma 1, equation (8))
Used in Lemmas 5 and 6 to control higher-order L^p norms by gradients; quoted from [15, Lemma 2.2].
standard math Young's inequality with arbitrary epsilon
Used throughout Section 5 for absorption of gradient and drift terms into negative contributions.
standard math Parabolic and elliptic regularity theory for linear/quasilinear heat equations
Basis for local existence (Lemma 3) and for the boundedness criterion (Lemma 4) via [22, Appendix A].
standard math Maximum principle for v-equation gives 0 <= v <= ||v0||_infinity
Used in (11) and repeatedly (e.g., Lemma 6) to bound ||v||_infinity by initial data.
domain assumption Domain and initial data assumptions in (7): Omega bounded C^{2+delta}, u0, v0 in C^{2+delta}_nu(Omega), nonnegative
Required for classical solution framework and for the local existence theorem.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Boundedness in a nonlinear chemotaxis-consumption model with gradient terms." pith.science (2026). https://pith.science/paper/I2URTT5Q

@misc{pith2026250113224,
  author       = {Pith},
  title        = {Pith review of: Boundedness in a nonlinear chemotaxis-consumption model with gradient terms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I2URTT5Q}},
  note         = {Machine review of arXiv:2501.13224}
}

read the original abstract

We study a chemotaxis-consumption mechanism, in which some chemical signal and cells density interact each other. In order to control the concentration of such a population, sources involving gradient nonlinearities, which introduce a dampening effect on the model, are considered. Moreover, the system is characterized by nonlinear diffusion and sensitivity terms. We derive conditions on some data of the problem so to ensure the boundedness of related solutions.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Active Matter as a framework for living systems-inspired Robophysics
cond-mat.soft 2025-11 unverdicted novelty 3.0 of 10

Active-matter physics is presented as the organizing framework for robophysics, with robot swarms designed around local interactions, shared purpose, and adaptive feedback.

Reference graph

Works this paper leans on

35 extracted references · 34 canonical work pages · cited by 1 Pith paper

[18]

Marras and G

M. Marras and G. Viglialoro. Boundedness in a fully parab olic chemotaxis-consumption system with nonlinear diﬀusion and sensitivity, and logist ic source. Math. Nachr., 291(14– 15):2318–2333, 2018

work page 2018
[7]

A. Columbu. Boundedness criteria for a chemotaxis consum ption model with gradient non- linearities. Preprint, 2024. https://arxiv.org/abs/2408.14250

work page Pith review arXiv 2024
[22]

Tao and M

Y . Tao and M. Winkler. Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity. J. Diﬀerential Equations , 252(1):692–715, 2012

work page 2012
[1]

H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabo lic boundary value problems. H.J. Schmeisser, H. Triebel (Eds.), Function Spaces, Diﬀer ential Operators and Nonlinear Analysis, in: Teubner-Texte Math., vol. 133, Teubner, Stuttgart, Leipzig, pp. 9–126, 1993

work page 1993
[2]

Baghaei and A

K. Baghaei and A. Khelghati. Global existence and bounded ness of classical solutions for a chemotaxismodel with consumption of chemoattractant and logistic source. Math. Meth. Appl. Sci., 40(10):3799–3807, 2016

work page 2016
[3]

Baghaei and A

K. Baghaei and A. Khelghati. Boundedness of classical sol utions for a chemotaxis model with consumption of chemoattractant. C. R. Math. Acad. Sci. Paris , 355(6):633–639, 2017

work page 2017
[4]

Bellomo, A

N. Bellomo, A. Belloquid, J. Nieto, and J. Soler. Multisca le biological tissue models and ﬂux-limited chemotaxis for multicellular growing systems . Math. Models Methods Appl. Sci. , 20(7):1179–1207, 2010

work page 2010
[5]

Cie ´slak and C

T. Cie ´slak and C. Stinner. Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller–Segel system in higher dimensions. J. Diﬀerential Equations, 252(10):5832–5851, 2012

work page 2012

Show all 35 references

[6]

Cie ´slak and C

T. Cie ´slak and C. Stinner. New critical exponents in a fully parabo lic quasilinear Keller–Segel system and applications to volume ﬁlling models. J. Diﬀerential Equations , 258(6):2080–2113, 2015

2015
[8]

Hillen and K

T. Hillen and K. J. Painter. A user’s guide to PDE models for chemotaxis. J. Math. Biol. , 58(1):183–217, 2009. 16 Daniel Acosta-Soba Alessandro Columbu and Giuseppe Vigl ialoro

2009
[9]

Horstmann and M

D. Horstmann and M. Winkler. Boundedness vs. blow-up in a chemotaxis system. J. Diﬀerential Equations, 215(1):52–107, 2005

2005
[10]

Ishida, J

S. Ishida, J. Lankeit, and G. Viglialoro. A Keller–Segel type taxis model with ecological interpretation and boundedness due to gradient nonlineari ties. Discrete Contin. Dyn. Syst. Ser. B., 29(9):3955–3969, 2024

2024
[11]

E. F. Keller and L. A. Segel. Initiation of slime mold aggr egation viewed as an instability. J. Theoret. Biol., 26(3):399–415, 1970

1970
[12]

E. F. Keller and L. A. Segel. Traveling bands of chemotact ic bacteria: a theoretical analysis. J. Theoret. Biol., 30(2):235–248, 1971

1971
[13]

J. Lankeit. Eventual smoothness and asymptotics in a thr ee-dimensional chemotaxis system with logistic source. J. Diﬀererential Equations , 258(4):1158–1191, 2015

2015
[14]

J. Lankeit. Locally bounded global solutions to a chemot axis consumption model with singular sensitivity and nonlinear diﬀusion. J. Diﬀerential Equations. , 262(7):4052–4084, 2017

2017
[15]

Lankeit and Y

J. Lankeit and Y . Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete Contin. Dyn. Syst., 37(12):6099–6121, 2017

2017
[16]

T. Li, D. Acosta Soba, A. Columbu, and G. Viglialoro. Diss ipative gradient nonlinearities prevent /u1D6FF-formations in local and nonlocal attraction-repulsion ch emotaxis models, 2024. https://arxiv.org/abs/2405.03586

2024 arXiv
[17]

D. Liu. Global classical solution to a chemotaxis consum ption model with singular sensitivity. Nonlinear Anal. Real World Appl., 41:497–508, 2018

2018
[19]

Nirenberg

L. Nirenberg. On elliptic partial diﬀerential equation s. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 2(13):115–162, 1959

1959
[20]

Y . Tao. Boundedness in a chemotaxis model with oxygen con sumption by bacteria. J. Math. Anal. Appl., 381(2):521–529, 2011

2011
[21]

Y . Tao, L. Wang, and Z. Wang. Large-time behavior of a para bolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension. Discrete Contin. Dyn. Syst. Ser. B. , 18(3):821–845, 2013

2013
[23]

Tao and M

Y . Tao and M. Winkler. Eventual smoothness and stabiliza tion of large-data solutions in a three-dimensional chemotaxis system with consumption of c hemoattractant. J. Diﬀererential Equations, 252(3):2520–2543, 2012

2012
[24]

T uval, L

I. T uval, L. Cisneros, C. Dombrowski, C.W. Wolgemuth, J. O. Kessler, and R.E. Goldstein. Bacterial swimming and oxygen transport near contact lines . Proc. Natl. Acad. Sci. USA , 102:2277–2282, 2005

2005
[25]

Viglialoro

G. Viglialoro. Blow-up time of a Keller–Segel-type syst em with Neumann and Robin boundary conditions. Diﬀ. Int. Eqns. , 29(3-4):359–376, 2016

2016
[26]

Viglialoro

G. Viglialoro. Very weak global solutions to a parabolic –parabolic chemotaxis-system with logistic source. J. Math. Anal. Appl. , 439(1):197–212, 2016

2016
[27]

Viglialoro

G. Viglialoro. Boundedness properties of very weak solu tions to a fully parabolic chemotaxis- system with logistic source. Nonlinear Anal. Real World Appl., 34:520–535, 2017

2017
[28]

Viglialoro and T.E

G. Viglialoro and T.E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete Contin. Dyn. Syst. Ser. B. , 22(5), 2017

2017
[29]

M. Winkler. Boundedness in the higher-dimensional para bolic-parabolic chemotaxis system with logistic source. Commun. Partial Diﬀer. Equ. , 35(8):1516–1537, 2010

2010
[30]

M. Winkler. Global large-data solutions in a chemotaxis -(Navier–)Stokes system modeling cellular swimming in ﬂuid drops. Commun. Partial Diﬀer. Equ. , 37(2):319–351, 2012

2012
[31]

M. Winkler. Stabilization in a two-dimensional chemota xis-Navier–Stokes system. Arch. Ration. Mech. Anal., 212(2):455–487, 2014

2014
[32]

M. Winkler. Global weak solutions in a three-dimensiona l chemotaxis-navier–stokes system. Ann. Inst. H. Poincar´e Anal. Non Lin ´eaire, 33(5):1329–1352, 2016. Boundedness in a nonlinear chemotaxis-consumption model w ith gradient terms 17

2016
[33]

M. Winkler. The two-dimensional Keller–Segel system wi th singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties. Math. Mod. Meth. Appl. S., 26(05):987–1024, 2016

2016
[34]

M. Winkler. How far do chemotaxis-driven forces inﬂuenc e regularity in the navier–stokes system? Trans. Amer. Math. Soc., 369(5):3067–3125, 2017

2017
[35]

M. Winkler. Renormalized radial large-data solutions t o the higher-dimensional keller–segel system with singular sensitivity and signal absorption. J. Diﬀerential Equations , 264(3):2310– 2350, 2018

2018

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [18]

Marras and G

M. Marras and G. Viglialoro. Boundedness in a fully parab olic chemotaxis-consumption system with nonlinear diﬀusion and sensitivity, and logist ic source. Math. Nachr., 291(14– 15):2318–2333, 2018

work page 2018

[2] [7]

A. Columbu. Boundedness criteria for a chemotaxis consum ption model with gradient non- linearities. Preprint, 2024. https://arxiv.org/abs/2408.14250

work page Pith review arXiv 2024

[3] [22]

Tao and M

Y . Tao and M. Winkler. Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with subcritical sensitivity. J. Diﬀerential Equations , 252(1):692–715, 2012

work page 2012

[4] [1]

H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabo lic boundary value problems. H.J. Schmeisser, H. Triebel (Eds.), Function Spaces, Diﬀer ential Operators and Nonlinear Analysis, in: Teubner-Texte Math., vol. 133, Teubner, Stuttgart, Leipzig, pp. 9–126, 1993

work page 1993

[5] [2]

Baghaei and A

K. Baghaei and A. Khelghati. Global existence and bounded ness of classical solutions for a chemotaxismodel with consumption of chemoattractant and logistic source. Math. Meth. Appl. Sci., 40(10):3799–3807, 2016

work page 2016

[6] [3]

Baghaei and A

K. Baghaei and A. Khelghati. Boundedness of classical sol utions for a chemotaxis model with consumption of chemoattractant. C. R. Math. Acad. Sci. Paris , 355(6):633–639, 2017

work page 2017

[7] [4]

Bellomo, A

N. Bellomo, A. Belloquid, J. Nieto, and J. Soler. Multisca le biological tissue models and ﬂux-limited chemotaxis for multicellular growing systems . Math. Models Methods Appl. Sci. , 20(7):1179–1207, 2010

work page 2010

[8] [5]

Cie ´slak and C

T. Cie ´slak and C. Stinner. Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller–Segel system in higher dimensions. J. Diﬀerential Equations, 252(10):5832–5851, 2012

work page 2012