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arxiv: 2606.18006 · v1 · pith:25FKZZVOnew · submitted 2026-06-16 · 🧮 math.AP

Absence of critical mass phenomena in one-dimensional critical quasilinear Keller-Segel systems

Xinru Cao , Mario Fuest This is my paper

Pith reviewed 2026-06-26 23:49 UTC · model grok-4.3

classification 🧮 math.AP
keywords Keller-Segel systemquasilinear chemotaxiscritical massglobal boundednessone-dimensional domainNeumann boundary conditionsparabolic-elliptic simplification
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The pith

In one-dimensional critical quasilinear Keller-Segel systems with m ≤ -1, all solutions from regular initial data remain globally bounded with no critical mass phenomenon.

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

The paper establishes that the one-dimensional version of this critical quasilinear chemotaxis system exhibits no critical mass threshold when the parameter m is at most -1. Instead, every solution starting from suitably regular initial data stays globally bounded for all time. The argument draws on one-dimensional features such as a lower bound for the energy functional, the embedding of W^{1,1} into L^∞, and the fact that the mass accumulation function obeys a non-degenerate parabolic equation. The same boundedness holds in certain parabolic-elliptic simplifications for wider ranges of m, and even for all real m when the initial datum is monotone. This behavior differs sharply from the critical-mass blow-up known to occur in higher-dimensional analogues of the system.

Core claim

We show that there is no critical mass phenomenon if m ≤ -1; that is, in that case all solutions emanating from suitably regular initial data are globally bounded. For certain parabolic-elliptic simplifications, we obtain the same conclusion for all m ∈ (-∞, -1] ∪ (0, ∞) and even for all m ∈ ℝ if the initial datum is additionally assumed to be monotone. This stands in contrast to critical mass phenomena known to occur for critical quasilinear Keller-Segel systems considered in higher-dimensional domains.

What carries the argument

The mass accumulation function, which satisfies a spatially non-degenerate parabolic equation due to the one-dimensional Neumann setting.

If this is right

  • All solutions remain globally bounded when m ≤ -1.
  • In the parabolic-elliptic simplification, boundedness holds for every m ≤ -1 and every m > 0.
  • When the initial datum is monotone, boundedness holds for every real m in the parabolic-elliptic case.
  • The ratio between diffusion and taxis sensitivity, which grows like u^{2} for large u, does not produce a mass threshold in one dimension.

Where Pith is reading between the lines

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

  • The non-degenerate parabolic structure of the mass accumulation function may be the main reason boundedness persists even at the critical ratio.
  • Similar one-dimensional quasilinear systems with different sensitivity exponents could be checked for boundedness by adapting the same energy and embedding arguments.
  • The transition in behavior as m crosses -1 could be tested by tracking the L^∞ norm of numerical solutions for values of m just above and just below -1.

Load-bearing premise

The system is posed in one dimension on the bounded interval (0,1) with Neumann boundary conditions.

What would settle it

Finding or numerically constructing a solution that becomes unbounded in finite time for some m ≤ -1 and regular initial data would falsify the global boundedness claim.

read the original abstract

We consider the Neumann initial boundary value problem associated to the chemotaxis system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = \big((u+1)^{m-1} u_x - u(u+1)^m v_x\big)_x & \text{in $(0, 1) \times (0, \infty)$}, \\ v_t = v_{xx} - v + u, &\text{in $(0, 1) \times (0, \infty)$}, \end{cases} \end{align} where $m \in \mathbb R$ is a given parameter. The relation between diffusion and taxis sensitivity is critical since the ratio $u(u+1)^m/(u+1)^{m-1}$ grows like $u^{2/n}$ for large $u$ with $n = \dim((0, 1)) = 1$. Nonetheless, we show that there is no critical mass phenomenon if $m \le -1$; that is, in that case all solutions emanating from suitably regular initial data are globally bounded. For certain parabolic-elliptic simplifications of \eqref{prob:abstract}, we obtain the same conclusion for all $m \in (-\infty, -1] \cup (0, \infty)$ and even for all $m \in \mathbb R$ if the initial datum is additionally assumed to be monotone. This stands in contrast to critical mass phenomena known to occur for critical quasilinear Keller-Segel systems considered in higher-dimensional domains. Accordingly, we make use of several special features of the one-dimensional setting such as the boundedness of the energy functional from below, the embedding $W^{1, n} \hookrightarrow L^\infty$, and the fact that the mass accumulation function solves a spatially non-degenerate parabolic equation.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the one-dimensional critical quasilinear Keller-Segel system (⋆) on (0,1) with Neumann boundary conditions. It proves that for m ≤ -1 all solutions with suitably regular initial data remain globally bounded, establishing the absence of a critical mass phenomenon. Analogous global boundedness results are obtained for parabolic-elliptic simplifications when m ∈ (-∞,-1] ∪ (0,∞) and, under the additional assumption of monotone initial data, for all real m.

Significance. The result demonstrates that the critical quasilinear Keller-Segel system behaves qualitatively differently in one dimension than in higher dimensions, where critical-mass blow-up thresholds are known to exist. The proofs exploit standard one-dimensional features (energy bounded from below, the embedding W^{1,1} ↪ L^∞, and the mass accumulation function satisfying a non-degenerate parabolic equation) rather than ad-hoc parameter fitting. If the derivations are complete, the work supplies a clean, dimension-specific positive result that clarifies the range of parameters for which global existence holds without mass restrictions.

minor comments (3)
  1. [Abstract] The abstract refers to “suitably regular initial data” without specifying the precise function space; the main theorem statement (presumably in §2 or §3) should state the exact regularity assumptions explicitly.
  2. [Introduction] The ratio u(u+1)^m / (u+1)^{m-1} is stated to grow like u^{2/n} with n=1; a short calculation confirming the exponent for the given diffusion-taxis relation would help readers verify the criticality claim.
  3. Notation for the mass accumulation function and the energy functional should be introduced once and used consistently; cross-references to their definitions would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript and the recommendation of minor revision. The report does not raise any specific major comments.

Circularity Check

0 steps flagged

No circularity: derivation relies on independent 1D embeddings and energy bounds

full rationale

The paper's central claim—that solutions remain globally bounded for m ≤ -1—rests on the standard one-dimensional facts that the energy is bounded from below, W^{1,1} embeds into L^∞, and the mass accumulation function satisfies a non-degenerate parabolic equation. These properties are external to the specific system and hold independently of the target boundedness result; they are invoked directly rather than derived from fitted parameters, self-citations, or redefinitions within the paper. No load-bearing step reduces by construction to the conclusion itself, and the abstract explicitly contrasts the 1D case with higher-dimensional critical-mass phenomena without importing uniqueness theorems or ansatzes from prior author work. The derivation is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard 1D Sobolev embeddings and energy estimates invoked as known facts rather than derived; no free parameters or new entities are introduced.

axioms (3)
  • standard math The embedding W^{1,1} → L^∞ holds in one dimension on a bounded interval.
    Invoked to obtain L^∞ control from integrability of the derivative.
  • domain assumption The energy functional is bounded from below in the 1D setting.
    Listed as a special feature of one dimension used in the proof.
  • domain assumption The mass accumulation function satisfies a spatially non-degenerate parabolic equation.
    Listed as a special feature of one dimension used in the proof.

pith-pipeline@v0.9.1-grok · 5876 in / 1437 out tokens · 41459 ms · 2026-06-26T23:49:36.121112+00:00 · methodology

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