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Finite time blow-up for an inhomogeneous parabolic equation

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Pith's one-line read A bounded Lipschitz source does not destroy finite-codimension self-similar blow-up for the nonlinear heat equation in three dimensions.

desk verdict Solid, carefully executed extension of Collot–Raphaël–Szeftel to a non-autonomous source; the new analytic content is real and the bootstrap closes. read the letter →

arxiv 2607.11165 v1 pith:POHYBUZE submitted 2026-07-13 math.AP

Kaiqiang Zhang This is my paper · ORCID
classification math.AP MSC 35B4435C0635B3535K5735Q92
keywords finite-timeblow-upinhomogeneousnonlinearheatequationself-similarprofilesfinite-codimensionstabilitymodulationspectralgapbrokenscalinginvariance
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 shows that adding a bounded, Lipschitz space-dependent forcing term to the supercritical nonlinear heat equation does not destroy the finite-codimension self-similar blow-up mechanism known for the homogeneous equation. For every sufficiently large integer n, every prescribed location, and every sufficiently small initial scale, there is a codimension-n Lipschitz manifold of nonradial initial data whose solutions blow up in finite time at a rate controlled by that scale. After rescaling, the profiles converge to a prescribed self-similar profile of the homogeneous problem, and the blow-up point and blow-up time depend Lipschitz-continuously on the data. The point is that exact scaling invariance, which most earlier constructions rely on, is broken by the source; the paper proves the non-autonomous perturbation remains lower order if the initial scale is small enough, so the same spectral-gap and topological-selection argument still closes. A sympathetic reader cares because many realistic models include inhomogeneous terms that break scaling, and this supplies a template for carrying precise singularity constructions into that larger class.

What carries the argument

The finite-codimension stability mechanism of Collot–Raphaël–Szeftel, adapted to a non-autonomous renormalized flow: geometric decomposition around Φ_n, modulation of scale and center, spectral-gap energy estimates on the orthogonal remainder, L^∞ bounds via parabolic comparison, and topological selection of the unstable coefficients by a Brouwer fixed-point argument that keeps the trajectory inside a bootstrap tube.

What would settle it

If, for some large n used in the construction, the linearized operator around Φ_n fails to have the claimed simple unstable eigenvalues and spectral gap, or if numerical or analytic integration of the renormalized equation with a non-zero f shows that the remainder does not stay O(e^{-μs}) when λ_{0} is small, the bootstrap and topological selection collapse.

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Extended reading notes

Core claim

For the equation ∂_t u − Δu = |u|^{p−1}u + f(x) in R^{3} with p > 5 and f bounded and Lipschitz, and for every sufficiently large n, every fixed x_{0} and every sufficiently small λ_{0} > 0, there exists a codimension-n Lipschitz manifold of nonradial initial data such that the corresponding solutions blow up in finite time T ≃ λ_{0}^{2}, the rescaled profile converges in L^∞ to the prescribed homogeneous self-similar profile Φ_n, the modulation center converges to a blow-up point, and the blow-up-time map is Lipschitz.

Load-bearing premise

The whole argument treats as given the existence, smoothness and spectral-gap properties of the countable family of radial self-similar profiles Φ_n and their eigenfunctions, which are known only for sufficiently large n.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper studies the inhomogeneous nonlinear heat equation ∂_t u − Δu = |u|^{p−1}u + f(x) in R^3 for p > 5 and f ∈ L^∞ ∩ C^{0,1}(R^3). Building on the countable family of radial self-similar profiles Φ_n (and their spectral theory) constructed by Collot–Raphaël–Szeftel for the homogeneous equation, it constructs, for every sufficiently large n, every fixed x_0 and every sufficiently small λ_0 > 0, a codimension-n Lipschitz manifold M_{n,λ_0,x_0} ⊂ L^∞(R^3) of (generally non-radial) initial data whose solutions blow up in finite time T ≍ λ_0^{2}. The solutions admit a modulated self-similar decomposition whose remainder vanishes in L^∞ as t → T, the modulation center converges to a blow-up point, and the blow-up-time map is Lipschitz continuous with respect to the L^∞ topology on the manifold. The argument proceeds by geometric decomposition, modulation ODEs, weighted energy estimates exploiting the spectral gap, L^∞ comparison, bootstrap improvement, and a Brouwer fixed-point selection of the unstable coefficients; the non-autonomous forcing generated by f is controlled as a lower-order term by taking the initial renormalized time large.

Significance. If correct, the result shows that the finite-codimensional stability mechanism for type-I self-similar blow-up developed for the homogeneous supercritical heat equation remains robust under a bounded Lipschitz spatial inhomogeneity that completely breaks scaling and translation invariance. No smallness of ∥f∥_∞ is required; the perturbation is rendered lower-order by a sufficiently small initial scale. This is a genuine advance beyond previous constructions that rely on exact scaling, and the framework is expected to apply to other parabolic problems lacking scaling symmetry. The paper correctly treats the existence and spectral properties of Φ_n as black-box input from Collot–Raphaël–Szeftel (2019) and supplies a complete, self-contained bootstrap-plus-topological argument for the inhomogeneous perturbation, including Lipschitz dependence of the manifold and of the blow-up time.

minor comments (5)
  1. In the statement of Theorem 1.1 and the definition of M_{n,λ_0,x_0} the authors note that higher regularity of the graphing functions a_j remains open; a brief remark on whether the present Lipschitz estimates can be bootstrapped to C^{1,α} (or why they cannot) would help the reader assess the sharpness of the result.
  2. Section 3.1, Lemma 3.1: the map F is introduced as an L^∞-valued expression, yet the subsequent implicit-function argument is applied only to the finite-dimensional map G. A one-sentence clarification that G is smooth as a map into R^{n+4} (while F itself need not be viewed as a smooth Banach-space map) would remove a possible source of confusion.
  3. Equation (3.18) and the subsequent estimates for NL: the Taylor remainder is written ≲ v^{2} + v^p; since p > 5 the quadratic term dominates for small v, but it would be cleaner to record the precise range of validity (e.g., |v| ≤ 1) once and for all.
  4. References [14] and [7] appear as arXiv preprints with future dates; updating the bibliographic data (or marking them as preprints) would improve permanence.
  5. Notation: the same symbol λ is used both for the modulation parameter and, earlier, for the scaling family of the homogeneous equation; a typographic distinction (e.g., λ(t) versus λ_scale) would avoid momentary ambiguity in Section 1.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Φ_n and spectral gap are external black-box inputs; the new non-autonomous estimates and topological selection are independent.

full rationale

The paper's central claim (Theorem 1.1) is a finite-codimension Lipschitz manifold of initial data for the inhomogeneous equation whose solutions blow up with rescaled profile converging to a prescribed homogeneous self-similar profile Φ_n. The existence, asymptotics, and spectral-gap properties of Φ_n (and the eigenfunctions of L_n) are imported wholesale from Collot–Raphaël–Szeftel (Mem. AMS 2019, Props. 1.1 & 3.1) and treated as black-box input in Section 2; they are never re-derived. Once those objects are granted, the paper constructs a dynamical rescaling, closes a bootstrap on the modulation parameters, the weighted energy, and the L^∞ remainder, absorbs the non-autonomous forcing term λ^{2p/(p-1)}f(λ y+x(t)) by taking s_0 large, and selects the unstable coefficients by a Brouwer fixed-point argument. None of these steps reduces by construction to a fitted quantity, a tautological definition, or a self-citation chain that itself asserts the target result. The only self-reference is ordinary dependence on prior external work; the genuinely new analytic content (control of the broken-scaling source) is independent. Score 1 reflects a single, non-load-bearing external citation of the spectral theory, not circularity of the derivation.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper is a pure-existence construction that imports the self-similar profiles and their spectral theory from prior work, then builds a bootstrap around them. Free parameters are only the hierarchical bootstrap constants chosen large or small enough; no data fitting occurs. No new physical entities are postulated.

free parameters (2)
  • bootstrap constants (μ, K_0, K, K', s_0) = sufficiently small/large as needed
    Chosen hierarchically (0<μ<c_n/4, K_0≪1, K≫K_0, K'≫C(K,K_0), s_0 large depending on n,p,||f||_∞) so that all error terms close. They are not fitted to data but are free choices that make the argument work.
  • profile index n = n>N≫1
    Must be larger than an absolute N given by the existence theory of Φ_n; the paper treats N as given by the cited reference.
assumptions (4)
  • domain assumption Existence of a countable family of smooth radial self-similar profiles Φ_n (n large) solving -ΔΦ+ΛΦ-Φ^p=0 with exactly n zeros of ΛΦ, together with the stated asymptotics near 0 and infinity (Prop. 2.1).
    Taken verbatim from Collot–Raphaël–Szeftel 2019; used as the building block of every initial datum and of the target profile.
  • domain assumption Spectral structure of the linearized operator L_n=-Δ+Λ-pΦ_n^{p-1}: simple unstable eigenvalues -μ_j (j=1…n+1), translation modes, and the spectral-gap inequality (2.5) on the orthogonal complement (Prop. 2.2).
    Again imported from CRS 2019; the whole modulation and energy analysis rests on these eigenvalues and the gap constant c_n.
  • domain assumption Local well-posedness of the inhomogeneous Cauchy problem in L^∞ for f∈L^∞∩C^{0,1} (cited [17,20]).
    Needed to guarantee that solutions exist long enough for the dynamical rescaling to be defined.
  • standard math Standard parabolic comparison principle and L^∞ regularity estimates for linear parabolic operators with bounded coefficients (used in Lemma 3.5).
    Classical tools; no novelty claimed.

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Pith. "Pith review of Finite time blow-up for an inhomogeneous parabolic equation." pith.science (2026). https://pith.science/paper/POHYBUZE

@misc{pith2026260711165,
  author       = {Pith},
  title        = {Pith review of: Finite time blow-up for an inhomogeneous parabolic equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/POHYBUZE}},
  note         = {Machine review of arXiv:2607.11165}
}
abstract

We consider the inhomogeneous nonlinear heat equation \[ \partial_t u-\Delta u=|u|^{p-1}u+f(x),\qquad x\in\mathbb{R}^3,\quad p>5, \] where \(f\in L^\infty\cap C^{0,1}(\mathbb{R}^3)\). For every sufficiently large integer \(n\), we construct a codimension-\(n\) Lipschitz manifold of non-radial initial data whose corresponding solutions blow up in finite time and whose rescaled profiles converge to the prescribed self-similar profile \(\Phi_n\) of the homogeneous equation. The main novelty is to show that the finite-codimensional stability mechanism for self-similar blow-up, developed in the work of Collot, Rapha\"el and Szeftel [Mem. Amer. Math. Soc. (2019)] for the homogeneous equation, is robust under the addition of a bounded, Lipschitz spatially inhomogeneous source term. In contrast with the homogeneous problem, the equation considered here has no exact scaling invariance, which is a key ingredient in many previous constructions. We expect that the framework developed in this work may also be useful for related problems in which exact scaling invariance is broken.

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