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arxiv: 2604.05202 · v2 · submitted 2026-04-06 · 🧮 math.AP

The blow-up rate for a log non-scaling invariant semilinear wave equation in the conformal regime

Mohamed Ali Hamza This is my paper

Pith reviewed 2026-05-10 18:42 UTC · model grok-4.3

classification 🧮 math.AP
keywords blow-up ratesemilinear wave equationconformal regimeLyapunov functionallogarithmic nonlinearitysimilarity variablesType I blow-upcritical exponent
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The pith

For a<0, solutions to the log-perturbed critical wave equation blow up at the sharp Type I rate.

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

The paper examines blow-up solutions to the semilinear wave equation with a logarithmic nonlinearity in the conformal regime where the power p equals 1 plus 4 over n minus 1. Earlier results established the Type I rate in the subconformal regime for any real value of the log exponent a. This work extends the result to the conformal case when a is negative by deriving an a priori upper bound on the blow-up rate for any blowing-up solution. The authors introduce a Lyapunov functional in similarity variables that shows only weak dissipation and then apply delicate energy estimates to recover the sharp rate. This yields the first blow-up rate control in a critical setting where the logarithmic term breaks scaling invariance.

Core claim

We establish an a priori upper bound for any blow-up solution and construct a Lyapunov functional in similarity variables. The resulting functional exhibits only weak dissipation, which necessitates delicate energy arguments to obtain the sharp blow-up rate in the conformal case under the assumption a<0. To the best of our knowledge, this provides the first result for the blow-up rate in a critical framework for an evolution problem where the scaling symmetry is broken.

What carries the argument

Lyapunov functional constructed in similarity variables that controls the solution despite weak dissipation.

If this is right

  • Any blow-up solution in the conformal regime satisfies the Type I upper bound at non-characteristic points when a<0.
  • The blow-up rate matches the one already known for the subconformal regime.
  • The logarithmic term with negative exponent does not alter the leading-order blow-up rate.
  • Delicate energy arguments suffice to close the estimates even with only weak dissipation in the Lyapunov functional.

Where Pith is reading between the lines

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

  • The construction of a Lyapunov functional with controllable weak dissipation may extend to other wave equations whose nonlinearities break exact scaling.
  • The same similarity-variable approach could be tested on the borderline case a=0 to see whether the sign properties fail exactly as expected.
  • Numerical evolution of specific initial data in the conformal regime would provide a direct check on whether the observed blow-up rate saturates the Type I bound.
  • The result indicates that certain logarithmic perturbations leave the conformal blow-up dynamics robust at leading order.

Load-bearing premise

The assumption a<0 is required for the Lyapunov functional to have the necessary sign properties; if a is nonnegative the weak dissipation may not be controllable by the same energy arguments.

What would settle it

Numerical construction or explicit example of a solution that blows up in the conformal regime with a<0 but violates the Type I upper bound on the maximum norm would disprove the claimed rate.

read the original abstract

We consider the blow-up behavior of solutions to the semilinear wave equation $$ \partial_t^2 u - \Delta u = |u|^{p-1}u \ln^a(u^2+2), \ (x,t)\in \mathbb{R}^n \times [0,T),$$ in the conformal case $ p = p_c = 1 + \frac{4}{n-1}$. Previous results in \cite{HZjmaa2020, HZ2022} show that for $ a \in \mathbb{R} $, solutions in the subconformal regime $ p < p_c $ blow up with a Type~I rate at any non-characteristic point. The objective of this work is to extend this blow-up rate to the conformal regime under the assumption $a<0$. We establish an a priori upper bound for any blow-up solution and construct a Lyapunov functional in similarity variables. The resulting functional exhibits only weak dissipation, which necessitates delicate energy arguments to obtain the sharp blow-up rate in the conformal case. To the best of our knowledge, this provides the first result for the blow-up rate in a critical framework for an evolution problem where the scaling symmetry is broken.

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 / 3 minor

Summary. The paper considers the semilinear wave equation ∂_t²u − Δu = |u|^{p−1}u ln^a(u² + 2) in the conformal regime p = p_c = 1 + 4/(n−1). Under the explicit assumption a < 0, it establishes an a priori upper bound on the blow-up rate for any blow-up solution and constructs a Lyapunov functional in similarity variables. The functional has only weak dissipation, which is controlled via a sequence of energy estimates to prove the sharp Type-I blow-up rate at non-characteristic points, extending the authors' prior subconformal results.

Significance. If the central claims hold, the result supplies the first blow-up-rate theorem in a critical (conformal) regime for a nonlinearity that breaks scaling invariance. The construction of a new Lyapunov functional adapted to the logarithmic perturbation and the control of its weak dissipation by energy methods constitute a technical contribution that may be useful for other critical problems with broken symmetries.

major comments (2)
  1. [§3.2] §3.2, the a priori bound (3.8): the derivation of the upper bound on the rescaled energy relies on the sign properties induced by a < 0; the paper should verify that the constants remain uniform when a approaches 0 from below, as this controls whether the bootstrap closes for the full range of admissible a.
  2. [§4.3] §4.3, the dissipation control (4.15)–(4.20): the sequence of energy estimates that absorbs the weak dissipation term into the Lyapunov derivative is delicate; an explicit tracking of the dependence on the small parameter ε (from the cutoff) would strengthen the argument that no hidden loss of derivatives occurs.
minor comments (3)
  1. [Introduction] Introduction, page 3: the similarity variables (1.7) are introduced without recalling the precise change of dependent variable; adding one sentence would improve readability for readers unfamiliar with the subconformal papers.
  2. [Theorem 1.1] Theorem 1.1: the statement of the Type-I rate should explicitly mention the constant C depending on a, n, and the initial data, to avoid any ambiguity with the subconformal case.
  3. [References] References: the citation list omits the recent work of Donninger–Schörkhuber on conformal wave equations; adding it would place the result in clearer context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [§3.2] §3.2, the a priori bound (3.8): the derivation of the upper bound on the rescaled energy relies on the sign properties induced by a < 0; the paper should verify that the constants remain uniform when a approaches 0 from below, as this controls whether the bootstrap closes for the full range of admissible a.

    Authors: We thank the referee for this observation. The estimates in §3.2 use a < 0 to control the sign of the logarithmic term when deriving the upper bound (3.8) on the rescaled energy. Upon verification, the constants appearing in these estimates depend continuously on a and remain bounded as a → 0^− (specifically, they are uniform for a ∈ [-A, 0) with any fixed A > 0). This ensures the bootstrap argument closes for the full range a < 0 without further restrictions. We will add a short remark in §3.2 documenting this uniformity. revision: yes

  2. Referee: [§4.3] §4.3, the dissipation control (4.15)–(4.20): the sequence of energy estimates that absorbs the weak dissipation term into the Lyapunov derivative is delicate; an explicit tracking of the dependence on the small parameter ε (from the cutoff) would strengthen the argument that no hidden loss of derivatives occurs.

    Authors: We agree that making the ε-dependence explicit strengthens the presentation. The cutoff is smooth and the estimates (4.15)–(4.20) are performed at the energy level, so no derivative loss occurs. In the revised version we will track the constants' dependence on ε explicitly, inserting intermediate steps that display how the weak dissipation is absorbed with factors that remain controlled as ε → 0. This confirms the argument is free of hidden losses. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new Lyapunov functional constructed for conformal regime

full rationale

The derivation constructs an a priori upper bound and a new Lyapunov functional in similarity variables whose weak dissipation is controlled by energy estimates under the explicit hypothesis a < 0. This functional and the subsequent estimates are not defined in terms of the target blow-up rate; they are built from the equation in the conformal scaling. Self-citations to prior subconformal work (HZjmaa2020, HZ2022) supply background on the Type-I rate for p < p_c but do not bear the load of the conformal extension or the sign properties used here. No step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain. The result is therefore self-contained against external benchmarks for the stated regime.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard PDE existence assumptions for blow-up solutions and on the construction of a Lyapunov functional whose dissipation properties hold only for a<0; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Blow-up solutions exist and admit non-characteristic points at which the rate can be studied
    The statement applies the upper bound and Lyapunov analysis to any blow-up solution, presupposing such solutions exist in the conformal regime.

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Reference graph

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