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arxiv: 2604.05101 · v1 · submitted 2026-04-06 · 🧮 math.AP

Blow-up rates and sets for a quasilinear diffusion equation with weighted source

Ra\'ul Ferreira , Razvan Gabriel Iagar , Ariel S\'anchez This is my paper

Pith reviewed 2026-05-10 19:04 UTC · model grok-4.3

classification 🧮 math.AP
keywords blow-up ratesquasilinear diffusionweighted sourceblow-up setscompactly supported solutionsfinite-time blow-upsupport expansion
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The pith

Compactly supported solutions to the quasilinear diffusion equation with weighted source blow up at explicit rates in height and support size.

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

The paper derives matching upper and lower bounds on the blow-up rate of the supremum norm for solutions of ∂t u = Δu^m + |x|^σ u^p that remain compactly supported until a finite blow-up time T. It also establishes an upper bound on the rate at which the support expands as t approaches T. These estimates are given in terms of explicit exponents α = (σ + 2)/L and β = (m - p)/L, where L = σ(m - 1) + 2(p - 1), for the range 1 < p < m and σ > 0. The analysis further shows that, under an additional condition, the blow-up set is either the entire space or occurs only at spatial infinity. A sympathetic reader would care because the results quantify how singularities form and spread in a nonlinear diffusion model with spatially varying forcing.

Core claim

If u is a compactly supported solution with blow-up time T, then C1 (T - t)^{-α} ≤ ||u(·, t)||_∞ ≤ C2 (T - t)^{-α} and sup{|x| : u(x, t) > 0} ≤ C0 (T - t)^{-β} for t ∈ (0, T), with α = (σ + 2)/L, β = (m - p)/L, L = σ(m - 1) + 2(p - 1). Under a suitable condition, either the blow-up set B(u) equals R^N or blow-up occurs only as |x| → ∞.

What carries the argument

The scaling exponents α and β obtained by balancing the diffusion term, the weighted source |x|^σ u^p, and the time derivative, with L serving as the common factor in the denominator.

Load-bearing premise

The existence of compactly supported solutions that blow up in finite time, together with an unspecified suitable condition used to reach the dichotomy on blow-up sets.

What would settle it

A numerical computation for concrete values of m, p, σ that checks whether the observed growth rate of ||u(·, t)||_∞ as t approaches T matches the predicted exponent α = (σ + 2)/L within the derived constants.

read the original abstract

Blow-up rates are established for general solutions to the quasilinear diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ in the range of exponents $1<p<m$, $\sigma>0$. More precisely, if we consider a compactly supported solution $u(x,t)$ with blow-up time $T=T(u)\in(0,\infty)$, we derive the blow-up rate $$ C_1(T-t)^{-\alpha}\leq \|u(x,t)\|_{\infty}\leq C_2(T-t)^{-\alpha}, \quad t\in(0,T), $$ for some positive constants $C_1$, $C_2$, and the upper rate of expansion of the support $$ \sup\{|x|:u(x,t)>0\}\leq C_0(T-t)^{-\beta}, \quad t\in(0,T), $$ for some constant $C_0>0$, where $$ \alpha=\frac{\sigma+2}{L}, \quad \beta=\frac{m-p}{L}, \quad L=\sigma(m-1)+2(p-1). $$ We also analyze the blow-up sets of solutions $u$, showing, under a suitable condition, that either $B(u)=\mathbb{R}^N$ or blow-up takes place only as $|x|\to\infty$.

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

2 major / 2 minor

Summary. The manuscript establishes blow-up rates for compactly supported solutions of the quasilinear equation ∂_t u = Δu^m + |x|^σ u^p (1 < p < m, σ > 0) that blow up at finite time T. It derives the two-sided estimate C_1 (T-t)^{-α} ≤ ||u(·,t)||_∞ ≤ C_2 (T-t)^{-α} together with the support-radius bound sup{|x| : u(x,t) > 0} ≤ C_0 (T-t)^{-β}, where the exponents are given explicitly by α = (σ+2)/L, β = (m-p)/L and L = σ(m-1) + 2(p-1). Under an additional suitable condition the paper claims a dichotomy for the blow-up set: either B(u) = ℝ^N or blow-up occurs only as |x| → ∞.

Significance. If the comparison arguments and scaling limits are fully justified, the explicit parameter-dependent rates supply concrete information on the competition between porous-medium diffusion and a spatially weighted source. Such rates are useful for classifying singularity formation in inhomogeneous media and can serve as benchmarks for numerical schemes or further asymptotic analysis in the literature on quasilinear blow-up.

major comments (2)
  1. [Introduction / Main results] The main theorems are stated under the standing assumption that compactly supported solutions with finite blow-up time T exist. No existence result or sufficient condition on the initial data is supplied, yet this assumption is load-bearing for all subsequent rate statements.
  2. [Blow-up sets section] The blow-up-set dichotomy is asserted only under an unspecified 'suitable condition'. The precise statement of this condition, together with its role in the proof, must be given explicitly; without it the claim cannot be verified.
minor comments (2)
  1. [Abstract] The notation B(u) for the blow-up set is used in the abstract without prior definition; it should be introduced in the introduction or preliminaries.
  2. [Main theorems] The constants C_0, C_1, C_2 are asserted to exist but no dependence on the initial data or on the parameters m, p, σ is indicated; a brief remark on their possible dependence would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will incorporate clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [Introduction / Main results] The main theorems are stated under the standing assumption that compactly supported solutions with finite blow-up time T exist. No existence result or sufficient condition on the initial data is supplied, yet this assumption is load-bearing for all subsequent rate statements.

    Authors: We agree that existence is assumed rather than constructed. The manuscript focuses on the blow-up rates and support dynamics for any compactly supported solution that reaches finite-time blow-up. To address the concern, we will add a short remark in the introduction noting that such solutions exist for sufficiently large, compactly supported initial data (via comparison with ODE blow-up or explicit supersolutions), while a complete existence theory lies outside the paper's scope. This clarification will appear in the revised version. revision: yes

  2. Referee: [Blow-up sets section] The blow-up-set dichotomy is asserted only under an unspecified 'suitable condition'. The precise statement of this condition, together with its role in the proof, must be given explicitly; without it the claim cannot be verified.

    Authors: We apologize for the ambiguity. The suitable condition is a technical hypothesis on the initial data (ensuring controlled growth at infinity to preclude finite-point blow-up in certain regimes) that is stated in Section 4 but not highlighted in the abstract or introduction. In the revision we will state the condition verbatim at the opening of the blow-up-sets section, explain its precise role in the comparison arguments that yield the dichotomy, and verify that it is compatible with the standing assumptions on compact support and finite-time blow-up. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives the explicit exponents α = (σ+2)/L and β = (m-p)/L with L = σ(m-1) + 2(p-1) by balancing the time-derivative, diffusion, and weighted source terms under the standard scaling ansatz for the blow-up profile and support radius; this balancing is an algebraic step that produces the candidate rates but does not presuppose the final inequalities. The subsequent upper and lower bounds on ||u(·,t)||_∞ and the support expansion are then proved via comparison principles and integral estimates that operate on the PDE itself, independent of the scaling calculation. The blow-up-set dichotomy is stated explicitly under an additional “suitable condition” that is not derived from the rates, and no self-citations, fitted parameters, or self-definitional loops appear in the load-bearing steps. The argument is therefore self-contained once the existence of compactly supported finite-time blow-up solutions is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the existence of compactly supported blowing-up solutions and on standard comparison and scaling techniques for quasilinear parabolic equations; no new entities are introduced and no parameters are fitted to data.

axioms (2)
  • domain assumption Existence of compactly supported solutions with finite blow-up time T
    The rate statements are conditioned on the existence of such solutions.
  • standard math Comparison principles and scaling invariance for the quasilinear operator
    Used to obtain upper and lower bounds on the L^∞ norm and support radius.

pith-pipeline@v0.9.0 · 5563 in / 1546 out tokens · 59915 ms · 2026-05-10T19:04:30.059350+00:00 · methodology

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  • Cost/FunctionalEquation washburn_uniqueness_aczel unclear
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    Relation between the paper passage and the cited Recognition theorem.

    Blow-up rates ... α= (σ+2)/L , β= (m-p)/L , L=σ(m-1)+2(p-1). ... self-similar form U(x,t;σ)=(T-t)^{-α}f(ζ), ζ:=|x|(T-t)^β ... profile f solving (f^m)'' + ... + |ζ|^σ f^p =0

  • Foundation/RealityFromDistinction reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We also analyze the blow-up sets ... either B(u)=R^N or blow-up occurs only as |x|→∞ ... under a suitable condition

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