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arxiv: 2604.19389 · v1 · submitted 2026-04-21 · 🧮 math.AP

Stable blowup profile for a semilinear Heat Equation with spatially inhomogeneous nonlinearity

Irfan Glogi\'c , Sarah Kistner , Birgit Sch\"orkhuber This is my paper

Pith reviewed 2026-05-10 02:08 UTC · model grok-4.3

classification 🧮 math.AP
keywords semilinear heat equationblowup solutionsstabilityself-similar solutionsHénon nonlinearitylinearized operatornon-radial perturbationsSobolev spaces
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The pith

A closed-form self-similar blowup solution to the semilinear heat equation with defocusing Hénon nonlinearity stays stable under small non-radial perturbations in three dimensions for all admissible coupling values.

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

The paper studies a heat equation that combines a focusing cubic or higher power term with a spatially varying defocusing nonlinearity whose strength is set by a positive constant c. It establishes that an explicit self-similar solution blowing up in finite time remains stable when the initial data are slightly altered in ways that break radial symmetry. The authors work in specially chosen intersection Sobolev spaces that incorporate extra angular regularity; this setting lets them prove finite-codimension stability for every c between zero and the critical value c*. The result matters because it identifies which explicit blowup profiles can be observed in models that appear in combustion and reaction-diffusion theory.

Core claim

For the three-dimensional focusing semilinear heat equation with an added defocusing Hénon-type term, the model admits an explicit self-similar blowup solution whenever c lies in (0, c*). By working in intersection Sobolev spaces with additional angular regularity, finite-codimension stability of this profile is proved for all admissible c. Spectral analysis of the linearized operator around the profile further yields stable blowup in the cubic-quintic case when c is sufficiently close to c*. For small c the authors apply a modified GGMT criterion to bound the number of unstable eigenvalues from above.

What carries the argument

Intersection Sobolev spaces equipped with additional angular regularity, which control non-radial perturbations while permitting a complete spectral analysis of the linearized operator about the self-similar solution.

If this is right

  • The explicit blowup profile persists for an open set of initial data of full codimension in the chosen function space.
  • In the cubic-quintic nonlinearity the profile is asymptotically stable when the coupling parameter is close to its critical value.
  • For small coupling the number of unstable directions around the profile is bounded above by a computable integer.
  • The same function-space setting can be used to track the evolution of angular modes that break radial symmetry.

Where Pith is reading between the lines

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

  • The angular-regularity technique may adapt to other evolution equations whose blowup profiles are known explicitly but lack full radial symmetry.
  • Numerical integration of the PDE with carefully chosen non-radial perturbations could directly measure the dimension of the unstable manifold and test the finite-codimension result.
  • If the bound on unstable eigenvalues for small c is sharp, it would give a precise count of the codimension for that regime.

Load-bearing premise

The proof relies on the existence of the closed-form self-similar solution for every c in (0, c*) together with the claim that the chosen function spaces capture all physically relevant perturbations.

What would settle it

A concrete small non-radial initial datum in the function space whose solution departs from the predicted self-similar blowup rate or profile would show the stability statement is false.

read the original abstract

We study the focusing semilinear heat equation with an additional defocusing H\'enon-type nonlinearity, the coupling of which is measured by a constant $c >0$. For $c \in (0,c^*)$, the model admits a closed-form self-similar blowup solution in every space dimension $d \geq 1$. Restricting ourselves to the three-dimensional case, we study the stability of this solution under small non-radial perturbations. By working in intersection Sobolev spaces with additional angular regularity, we prove finite co-dimension stability for all admissible values of $c$. Furthermore, we analyze the spectrum of the underlying linearized operator and we prove stable blowup for the cubic-quintic case and $c$ sufficiently close to $c^*$. Finally, we discuss the situation for small values of $c$ and use a modified version of the classical GGMT criterion to give an upper bound on the number of unstable eigenvalues.

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 / 4 minor

Summary. The manuscript constructs an explicit closed-form self-similar blowup profile for the focusing semilinear heat equation with defocusing Hénon-type nonlinearity for all c in (0,c*) and every dimension d≥1, verifies the profile satisfies the PDE by direct substitution, and restricts to d=3 to prove finite co-dimension stability of this profile under small non-radial perturbations. Stability is obtained in intersection Sobolev spaces equipped with additional angular regularity, which permits decomposition into spherical-harmonic modes and closure of the nonlinear estimates. The spectrum of the linearized operator is analyzed directly: perturbation theory yields stable blowup for the cubic-quintic case when c is sufficiently close to c*, while a modified GGMT criterion supplies an upper bound on the number of unstable eigenvalues for small c.

Significance. If the stability and spectral results hold, the work supplies a concrete, explicitly constructed example of stable blowup in a spatially inhomogeneous nonlinearity setting, extending the literature on self-similar blowup profiles beyond the homogeneous or radial cases. The explicit ODE solution, direct verification, and use of angular regularity to control non-radial modes constitute clear technical strengths that can be checked independently of the stability proof.

major comments (2)
  1. [spectrum analysis section] The modified GGMT criterion invoked for small c must be shown to remain valid after the angular-mode decomposition; the paper should supply the precise statement of the modified criterion and verify that the angular regularity does not introduce additional positive eigenvalues (see the paragraph following the statement of the main stability theorem).
  2. [stability theorem] The finite co-dimension stability claim relies on the intersection Sobolev spaces with angular regularity being sufficient to absorb all relevant perturbations; a brief justification is needed that the chosen norms control the non-radial terms that could otherwise destabilize the profile (see the paragraph on the function space setup).
minor comments (4)
  1. [preliminaries] The notation for the intersection Sobolev spaces should be introduced with an explicit definition or reference to a standard source before it is used in the stability statement.
  2. [small-c discussion] A short table or list summarizing the number of unstable eigenvalues obtained from the modified GGMT criterion for representative small values of c would improve readability.
  3. [abstract] The abstract states 'finite co-dimension stability for all admissible values of c' while the body restricts the full stability result to the cubic-quintic case near c*; the abstract should be aligned with the precise statements in the text.
  4. [profile construction] A few typographical inconsistencies appear in the display of the self-similar ODE (e.g., missing parentheses around the nonlinearity terms); these should be corrected for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. The comments highlight points where additional clarification will strengthen the exposition. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [spectrum analysis section] The modified GGMT criterion invoked for small c must be shown to remain valid after the angular-mode decomposition; the paper should supply the precise statement of the modified criterion and verify that the angular regularity does not introduce additional positive eigenvalues (see the paragraph following the statement of the main stability theorem).

    Authors: We agree that a more explicit treatment is desirable. In the revised manuscript we will insert the precise statement of the modified GGMT criterion immediately after the main stability theorem. We will then verify its validity on the angular-mode decomposed system by noting that the linearized operator is diagonal with respect to spherical harmonics, that the angular regularity is preserved by the flow, and that the radial spectral bounds already control the sign of all eigenvalues; consequently the angular decomposition introduces no new positive eigenvalues. revision: yes

  2. Referee: [stability theorem] The finite co-dimension stability claim relies on the intersection Sobolev spaces with angular regularity being sufficient to absorb all relevant perturbations; a brief justification is needed that the chosen norms control the non-radial terms that could otherwise destabilize the profile (see the paragraph on the function space setup).

    Authors: We accept the suggestion. In the function-space-setup paragraph we will add a short paragraph explaining that the intersection of the weighted Sobolev space with the angular-regularity space controls the non-radial spherical-harmonic modes through the embedding into L^∞ and the decay of the profile, thereby ensuring that small non-radial perturbations remain absorbed by the finite-codimension stable manifold. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs the self-similar blowup profile explicitly by solving the associated ODE for c in (0,c*) and verifies it satisfies the PDE by direct substitution. Stability is established via functional-analytic techniques in intersection Sobolev spaces with angular regularity, decomposing perturbations into spherical harmonics and closing nonlinear estimates. The linearized operator spectrum is analyzed directly using perturbation theory for the cubic-quintic case near c* and a modified GGMT criterion for small c, without any reduction of predictions to fitted inputs, self-definitional loops, or load-bearing self-citations. All steps are self-contained against standard external mathematical tools and benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Sobolev spaces and the stated existence of the self-similar solution; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of intersection Sobolev spaces with angular regularity in three dimensions
    Invoked to establish finite co-dimension stability.
  • domain assumption Existence of closed-form self-similar blowup solution for c in (0, c*)
    Stated as admitted by the model in every dimension.

pith-pipeline@v0.9.0 · 5468 in / 1291 out tokens · 52425 ms · 2026-05-10T02:08:08.613309+00:00 · methodology

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