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arxiv: 2604.23764 · v1 · submitted 2026-04-26 · 🧮 math.AP

A new critical exponent for the semilinear damped wave equation with Hartree-type nonlinearity and initial data from homogeneous Besov spaces

Duc An Phan This is my paper

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classification 🧮 math.AP
keywords damped wave equationHartree nonlinearitycritical exponenthomogeneous Besov spacesglobal existencefinite-time blow-upsmall data solutions
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The pith

The critical exponent for global existence versus blow-up of small solutions to the damped wave equation with Hartree nonlinearity is p1 + p2 = 1 + (4 + 2γ)/(n + 2β) when data lie in homogeneous Besov spaces.

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

The paper determines the sharp threshold value for the sum of powers in a nonlocal Hartree nonlinearity that decides the long-time behavior of solutions to a damped wave equation. Using decay estimates for the linear damped wave problem in homogeneous Besov spaces together with harmonic analysis bounds on the nonlinearity, the authors prove that small initial data produce global solutions precisely when the combined exponent meets or exceeds 1 plus (4 plus 2 gamma) divided by (n plus 2 beta). They further show that when the sum lies strictly below this value but above 2, every weak solution blows up in finite time regardless of data size. This sharpens the classical Fujita-type theory by incorporating both damping and the nonlocal Hartree term while working in spaces with negative regularity.

Core claim

We identify a new critical exponent given by p1 + p2 := p_Fuji((n + 2β)/(2 + γ)) := 1 + (4 + 2γ)/(n + 2β) for β ∈ [0, n/2) and γ ∈ [0, n). Global (in time) existence of small-data solutions holds in the supercritical and critical regimes p1 + p2 ≥ p_Fuji((n + 2β)/(2 + γ)). Finite-time blow-up of weak solutions occurs, even for arbitrarily small initial data, in the subcritical range 2 < p1 + p2 < p_Fuji((n + 2β)/(2 + γ)). The proofs rely on decay estimates for the linear damped wave equation in homogeneous Besov spaces dot B_{2,∞}^{-β} combined with refined harmonic analysis tools.

What carries the argument

The critical exponent p_Fuji((n + 2β)/(2 + γ)) = 1 + (4 + 2γ)/(n + 2β), derived from linear decay estimates in dot B_{2,∞}^{-β} that control the Hartree term I_γ(|u|^{p1}) |u|^{p2} via harmonic analysis.

Load-bearing premise

The linear damped wave equation admits decay estimates in homogeneous Besov spaces dot B_{2,∞}^{-β} that are strong enough, when paired with harmonic analysis, to control the nonlocal Hartree nonlinearity.

What would settle it

An explicit weak solution with p1 + p2 slightly below the critical value that remains global for small data, or a solution at the critical value that blows up in finite time, would disprove the claimed sharpness.

read the original abstract

In this paper, we investigate the critical exponent for a semi-linear damped wave equation involving a Hartree-type nonlinearity of the form $\mathcal{I}_\gamma\left(|u|^{p_1}\right)|u|^{p_2}, p_1, p_2>0, \gamma \in[0, n)$, with initial data taken in the homogeneous Besov spaces $\dot{B}_{2, \infty}^{-\beta}$, where $\beta \in\left[0, \frac{n}{2}\right)$. Our approach is based on deriving decay estimates for solutions to the associated linear damped wave equation with initial data belonging to $\dot{B}_{2, \infty}^{-\beta}$, combined with refined tools from Harmonic Analysis. As a consequence, we identify a new critical exponent given by $$ p_1+p_2:=p_{\mathrm{Fuji}}\left(\tfrac{n+2\beta}{2+\gamma}\right):=1+\tfrac{4+2\gamma}{n+2\beta} \quad \text{ for } \beta \in\left[0, \tfrac{n}{2}\right) \text{ and } \gamma \in [0, n). $$ More precisely, we establish the global (in time) existence of small data solutions in the supercritical and critical regimes $p_1+p_2 \geq p_{\mathrm{Fuji}}\left(\frac{n+2 \beta}{2+\gamma}\right)$. In contrast, we prove finite-time blow-up of weak solutions, even for arbitrarily small initial data, in the subcritical range $2<p_1+p_2<p_{\mathrm{Fuji}}\left(\frac{n+2 \beta}{2+\gamma}\right)$.

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 paper studies the critical exponent for the semilinear damped wave equation with Hartree-type nonlinearity of the form I_γ(|u|^{p1})|u|^{p2} and initial data in homogeneous Besov spaces Ḃ_{2,∞}^{-β}. It derives linear decay estimates for the associated linear damped wave equation in these spaces, combines them with harmonic analysis tools to control the nonlocal term, and identifies the critical value p1 + p2 = 1 + (4 + 2γ)/(n + 2β). Global existence of small-data solutions is proved for p1 + p2 ≥ this threshold, while finite-time blow-up of weak solutions (even for small data) is shown for 2 < p1 + p2 < the threshold.

Significance. If the linear decay estimates are sharp, the result extends Fujita-type critical exponents to a nonlocal Hartree nonlinearity in the setting of homogeneous Besov data, with the parameter-dependent threshold providing a precise, falsifiable prediction. The approach of pairing linear decay with refined harmonic analysis for the fractional integral operator is a methodological strength that could apply to related nonlocal problems.

major comments (2)
  1. [Linear estimates section] The linear decay estimates for the damped wave equation in Ḃ_{2,∞}^{-β} (derived in the section on linear estimates) are load-bearing for both the existence and blow-up results. The claimed time-decay rate must be verified explicitly against the Fourier multiplier for the damped-wave propagator to confirm it yields precisely the exponent p_Fuji((n+2β)/(2+γ)) uniformly for β ∈ [0, n/2) and γ ∈ [0, n). A weaker decay would invalidate the sharpness of the critical regime.
  2. [Existence proof section] In the fixed-point argument for global existence (in the supercritical/critical regime), the control of the Hartree term I_γ(|u|^{p1})|u|^{p2} via Besov embeddings and fractional integrals must close the Duhamel integral with the exact decay rate from the linear part. The manuscript should state the admissible range for p1, p2 separately (beyond their sum) to ensure the estimates hold without additional restrictions on the parameters.
minor comments (2)
  1. [Abstract] The abstract introduces the notation p_Fuji without a brief parenthetical explanation of its relation to the classical Fujita exponent; adding this would improve readability for a broad audience.
  2. [Throughout] Notation for the homogeneous Besov space is written inconsistently as Ḃ versus dot{B}; standardize throughout the manuscript.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and indicate the revisions planned.

read point-by-point responses

  1. Referee: [Linear estimates section] The linear decay estimates for the damped wave equation in Ḃ_{2,∞}^{-β} (derived in the section on linear estimates) are load-bearing for both the existence and blow-up results. The claimed time-decay rate must be verified explicitly against the Fourier multiplier for the damped-wave propagator to confirm it yields precisely the exponent p_Fuji((n+2β)/(2+γ)) uniformly for β ∈ [0, n/2) and γ ∈ [0, n). A weaker decay would invalidate the sharpness of the critical regime.

    Authors: We appreciate this observation regarding the centrality of the linear estimates. In the manuscript, the decay rates are derived from the Fourier multiplier of the damped-wave propagator via Littlewood-Paley decomposition in Ḃ_{2,∞}^{-β}, yielding time-decay that produces exactly the stated critical exponent p_Fuji((n+2β)/(2+γ)) uniformly over β ∈ [0, n/2) and γ ∈ [0, n). To address the request for explicit verification, we will revise the linear estimates section to include a more detailed computation of the multiplier estimates confirming this decay rate and its uniformity. revision: yes

  2. Referee: [Existence proof section] In the fixed-point argument for global existence (in the supercritical/critical regime), the control of the Hartree term I_γ(|u|^{p1})|u|^{p2} via Besov embeddings and fractional integrals must close the Duhamel integral with the exact decay rate from the linear part. The manuscript should state the admissible range for p1, p2 separately (beyond their sum) to ensure the estimates hold without additional restrictions on the parameters.

    Authors: We thank the referee for this suggestion to improve clarity. While the critical exponent depends on the sum p1 + p2, the estimates for the nonlocal term via the fractional integral I_γ and Besov embeddings do depend on the individual values to close without further restrictions. In the revised manuscript, we will explicitly state the admissible ranges for p1 and p2 separately in the existence section (in addition to the sum condition), ensuring the fixed-point argument applies under the stated parameter regimes. revision: yes

Circularity Check

0 steps flagged

No circularity; critical exponent follows from linear decay estimates

full rationale

The manuscript derives decay estimates for the linear damped wave equation in Ḃ_{2,∞}^{-β} and applies standard harmonic-analysis estimates to the Hartree term to obtain the threshold p1+p2 = 1 + (4+2γ)/(n+2β). Global existence above the threshold and blow-up below it are closed via Duhamel iteration and test-function arguments that use these decay rates directly. No parameter is fitted to data and then renamed as a prediction, no self-citation chain is load-bearing, and the exponent is not defined in terms of itself. The argument is therefore self-contained against the stated linear estimates and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Besov spaces and linear decay estimates for damped waves; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Decay estimates hold for solutions of the linear damped wave equation with initial data in Ḃ_{2,∞}^{-β}
    This is the key input used to obtain the critical exponent via harmonic analysis.
  • standard math Standard embedding and multiplier theorems in homogeneous Besov spaces apply to the Hartree nonlinearity
    Invoked to close the nonlinear estimates.

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