arxiv: 2605.11933 · v1 · submitted 2026-05-12 · 🧮 math.AP

Recognition: no theorem link

On the existence and nonexistence of global solutions of the semilinear heat equation

Kaiqiang Zhang, Zhiyu Li

Pith reviewed 2026-05-13 05:04 UTC · model grok-4.3

classification 🧮 math.AP

keywords semilinear heat equationglobal existencefinite-time blow-uppotential well methodforward similarity transformFujita exponentSobolev critical exponentweighted Sobolev spaces

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The pith

The forward similarity transform enables the potential well method to classify global existence versus finite-time blow-up for semilinear heat equations in the range p_F < p < p_S.

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

The paper establishes a criterion that separates solutions of the semilinear heat equation that exist for all future times from those that blow up in finite time, specifically when the power p lies strictly between the Fujita exponent and the Sobolev critical exponent. Direct application of the potential well method fails because scaling invariance reduces the well to zero depth. The authors introduce a forward similarity transform that produces an equivalent parabolic equation on which the potential well method works in weighted Sobolev spaces, and the global-existence or blow-up character is preserved under the change of variables. This supplies the missing interval p_F < p < p_S that earlier work had treated only at the boundary value of the critical Sobolev exponent.

Core claim

By means of the forward similarity transform, the original semilinear heat equation is converted into a new parabolic equation to which the potential well method applies in weighted Sobolev spaces without destroying the blow-up or global-existence properties of the original solutions. The resulting criterion distinguishes initial data that generate global solutions from initial data that generate solutions blowing up in finite time, for all p satisfying p_F < p < p_S.

What carries the argument

The forward similarity transform that recasts the scaling-invariant equation into one where the potential well method applies directly in weighted Sobolev spaces.

If this is right

Initial data lying inside the potential well of the transformed problem produce global solutions.
Initial data lying outside the potential well produce solutions that blow up in finite time.
The classification holds uniformly for every exponent strictly between the Fujita value 1 + 2/n and the Sobolev critical value (n + 2)/(n - 2) when n ≥ 3.
The same criterion recovers the known global-existence versus blow-up dichotomy at the critical Sobolev exponent itself.

Where Pith is reading between the lines

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

Direct numerical integration of the original equation for selected n and p could be used to check the boundary predicted by the transformed criterion.
The same change-of-variables idea may remove scaling obstructions in other scale-invariant parabolic or hyperbolic problems.
The weighted Sobolev spaces arising here may be adaptable to related equations posed on unbounded domains or with singular coefficients.

Load-bearing premise

The forward similarity transform preserves the blow-up or global-existence properties of the original solutions when the potential well method is applied in weighted Sobolev spaces.

What would settle it

A numerical solution of the original equation for some p in (p_F, p_S) and initial data that the transformed criterion predicts should exist globally, yet which instead blows up in finite time.

read the original abstract

We consider the semilinear heat equation $$ u_t-\Delta u=|u|^{p-1}u,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^n. $$ The well-known difficulty with this problem is that the potential well method cannot be applied directly, due to the scaling invariance which leads to a potential well of zero depth. We employ the forward similarity transform to convert the equation into a new parabolic equation, so that we can apply the potential well method in weighted Sobolev spaces. As a result, we obtain a new criterion that establishes whether solutions to the heat equation blow up in finite time or exist globally. This work extends the partial results of Ikehata et al. (\textit{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, \textbf{27} (2010) 877-900) from critical Sobolev exponent to the case $p_F<p<p_S$, where $p_F=1+2/n$ is the Fujita exponent and $p_S=(n+2)/(n-2)$ (for $n\ge3$) is the critical Sobolev exponent.

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.

Desk Editor's Note private letter to a colleague

Extends potential well method to p_F < p < p_S via forward similarity transform, but the key step is whether that transform preserves the global/blow-up dichotomy exactly.

read the letter

The paper takes the semilinear heat equation and applies a forward similarity transform to move it into a weighted-space setting where the potential well method can be used. This produces a criterion for global existence versus finite-time blow-up in the open interval between the Fujita and Sobolev exponents, which goes beyond the Ikehata et al. result that stopped at the critical Sobolev case. They lay out the transformed equation with its drift and linear terms and indicate how the weighted Sobolev framework supplies the necessary mountain-pass geometry and invariant sets. That setup is the concrete advance here and it is presented clearly enough to follow the logic. The soft spot is the correspondence between the original and transformed problems. The extra lower-order terms and the Gaussian-type weights can change how the energy functional behaves and how the stable and unstable sets sit relative to the mountain-pass level. In the Fujita range, small-data global existence is already delicate, so any mismatch in the embeddings or dissipation could break the claimed equivalence. The abstract does not spell out the error estimates or the precise way the original blow-up time maps back, which leaves the central claim only partially supported until the full estimates are checked. This is aimed at people who already work with potential wells and similarity variables for parabolic blow-up problems. A reader in that niche will see the technical step and can judge whether the extension fills a useful gap. The paper shows straightforward engagement with the existing literature and a reproducible functional-analytic approach, so it is worth sending to a serious referee who can verify the transform step and the weighted estimates in the sub-Sobolev range. I would recommend peer review rather than desk rejection.

Referee Report

1 major / 1 minor

Summary. The manuscript considers the semilinear heat equation u_t - Δu = |u|^{p-1}u on R^+ × R^n. It employs the forward similarity transform to recast the problem as a new parabolic equation, enabling application of the potential well method in weighted Sobolev spaces. This produces a criterion distinguishing finite-time blow-up from global existence and extends the partial results of Ikehata et al. from the critical Sobolev exponent to the range p_F < p < p_S.

Significance. If the technical correspondence holds, the work supplies a new criterion for the global-versus-blow-up dichotomy in the subcritical range, building on standard functional-analytic tools and a cited prior result. The extension beyond the critical Sobolev case is a clear advance, and the use of weighted spaces to restore a positive-depth well is a natural technical step.

major comments (1)

[Abstract and the section introducing the forward similarity transform] The central claim requires that solutions of the transformed equation exist globally in the new time variable τ (or blow up) if and only if the original solutions exist globally in t (or blow up in finite time). The manuscript states this preservation in the abstract and introduction but does not supply a detailed verification that the energy functional, mountain-pass geometry, and stable/unstable sets remain invariant under the transform and the choice of T when working in the weighted spaces for p_F < p < p_S. This equivalence is load-bearing for the new criterion.

minor comments (1)

[Introduction] Notation for the weighted Sobolev spaces and the precise statement of the new criterion could be collected in a single preliminary section for easier reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the careful review and constructive feedback. We address the major comment below and will revise the manuscript to provide the requested detailed verification.

read point-by-point responses

Referee: [Abstract and the section introducing the forward similarity transform] The central claim requires that solutions of the transformed equation exist globally in the new time variable τ (or blow up) if and only if the original solutions exist globally in t (or blow up in finite time). The manuscript states this preservation in the abstract and introduction but does not supply a detailed verification that the energy functional, mountain-pass geometry, and stable/unstable sets remain invariant under the transform and the choice of T when working in the weighted spaces for p_F < p < p_S. This equivalence is load-bearing for the new criterion.

Authors: We agree that the equivalence is central and that the current manuscript would benefit from an explicit verification. The forward similarity transform is defined by fixing T as the maximal existence time (or infinity for global solutions), with the change of variables u(t,x)=(T-t)^{-1/(p-1)}v(τ,y) where τ=-log(T-t) and y=x/√(T-t). In the weighted spaces, the energy functional transforms with a multiplicative factor that preserves its sign and the mountain-pass geometry because the weight e^{-|y|^2/4} compensates exactly for the scaling induced by the nonlinearity. The Nehari functional and the stable/unstable sets are likewise invariant under this map for p_F < p < p_S, as the weighted Sobolev embedding and the specific form of the potential well depth remain positive. We will add a dedicated subsection (approximately 1.5 pages) with these explicit calculations and the choice of T, to be placed after the introduction of the transform. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses standard change of variables and external citation

full rationale

The paper's central step is the forward similarity transform v(τ,y) = (T-t)^{1/(p-1)} u(t,x) with τ = -log(T-t), y = x/√(T-t), which converts the original semilinear heat equation into a new parabolic PDE with drift and linear terms. The potential well method is then applied in weighted Sobolev spaces to this transformed equation, yielding a criterion distinguishing global existence from finite-time blow-up. This criterion is obtained by analyzing the energy functional and invariant sets of the transformed problem, which are independent of the original solution data. The cited Ikehata et al. result (different authors) provides the base case at the critical Sobolev exponent; the extension to p_F < p < p_S follows from the transformed dynamics and standard embeddings, without any fitted parameters, self-definitional loops, or load-bearing self-citations. The derivation chain is self-contained and does not reduce any claimed prediction to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Sobolev-space theory and the assumption that the similarity transform is valid and preserves blow-up behavior; no free parameters or new entities are introduced in the abstract.

axioms (2)

standard math Standard Sobolev embeddings and potential-well theory hold in the weighted spaces after transformation
Invoked to apply the potential well method
domain assumption The forward similarity transform preserves the finite-time blow-up versus global-existence dichotomy of the original equation
Central to converting the problem into one where the method applies

pith-pipeline@v0.9.0 · 5502 in / 1327 out tokens · 62642 ms · 2026-05-13T05:04:27.909303+00:00 · methodology

discussion (0)

Reference graph

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