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arxiv: 2509.07405 · v5 · submitted 2025-09-09 · 🧮 math.AP

Fujita Phenomenon for a Mixed Local-Nonlocal Hardy-H\'enon Equation with Regularly Varying Time Weights

Rihab Ben Belgacem , Mohamed Majdoub This is my paper

Pith reviewed 2026-05-18 18:24 UTC · model grok-4.3

classification 🧮 math.AP
keywords Fujita phenomenonmixed diffusionregularly varying functionsHardy-Hénon equationblow-upglobal existenceparabolic equationsCauchy problem
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The pith

The critical Fujita exponent classifies blow-up and global existence for the mixed local-nonlocal Hardy-Hénon equation with regularly varying time weights.

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

The paper investigates blow-up and global existence for solutions of a semilinear parabolic equation that features a mixed local and nonlocal diffusion operator along with a Hardy-Hénon nonlinearity multiplied by a regularly varying time-dependent function h(t). It first treats the unforced problem and proves sharp criteria in terms of a critical Fujita exponent that extend previous results to this wider class of time coefficients. For the version with an added spatial forcing term, it gives conditions on h and the source that prevent global weak solutions from existing, together with smallness requirements on the data that ensure global mild solutions. These findings matter to readers interested in how time-varying coefficients affect the long-term behavior of solutions in diffusion models.

Core claim

For the unforced problem, sharp blow-up and global existence criteria are established in terms of the critical Fujita exponent, extending earlier results to the wider class of regularly varying time-dependent coefficients. For the forced problem, nonexistence of global weak solutions is derived under suitable growth conditions on h and integrability assumptions on w, while sufficient smallness conditions on the initial data and the forcing term ensure global-in-time mild solutions.

What carries the argument

The mixed local-nonlocal diffusion operator combined with semigroup estimates, the test function method, and asymptotic properties of regularly varying functions.

Load-bearing premise

The time weight h(t) belongs to the class of regularly varying functions, which is needed to apply the asymptotic properties in the blow-up and existence analysis.

What would settle it

Finding a solution that blows up in finite time for a power p above the critical Fujita exponent, or a global solution for p below it, with a regularly varying h(t) would falsify the claimed criteria.

read the original abstract

We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho \mathbf{w}(x), \qquad (x,t)\in \mathbb{R}^N\times (0,\infty), \] where $\mathsf{s}\in (0,1)$, $p>1$, $b\geq 0$, and $\varrho>-1$. The function $\mathsf{h}(t)$ is assumed to belong to the generalized class of regularly varying functions, while $\mathbf{w}$ is a prescribed spatial source. We first revisit the unforced case and establish sharp blow-up and global existence criteria in terms of the critical Fujita exponent, thereby extending earlier results to the wider class of time-dependent coefficients. For the forced problem, we derive nonexistence of global weak solutions under suitable growth conditions on $\mathsf{h}$ and integrability assumptions on $\mathbf{w}$. Furthermore, we provide sufficient smallness conditions on the initial data and the forcing term ensuring global-in-time mild solutions. Our analysis combines semigroup estimates for the mixed operator, test function methods, and asymptotic properties of regularly varying functions. To our knowledge, this is the first study addressing blow-up phenomena for nonlinear diffusion equations with such a class of time-dependent coefficients.

Editorial analysis

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Referee Report

0 major / 3 minor

Summary. The manuscript investigates the Cauchy problem for the semilinear parabolic equation ∂_t u − (Δ − (−Δ)^s)u = h(t)|x|^{-b}|u|^p + t^ϱ w(x) in R^N × (0,∞), with s ∈ (0,1), p > 1, b ≥ 0, ϱ > −1, where h(t) is regularly varying and w is a given spatial source. For the unforced problem it establishes sharp blow-up versus global-existence criteria in terms of a critical Fujita exponent, extending constant-coefficient results. For the forced problem it derives nonexistence of global weak solutions under growth conditions on h and integrability assumptions on w, together with small-data conditions guaranteeing global mild solutions. The proofs combine semigroup estimates for the mixed operator, test-function arguments, and Karamata-type asymptotics for regularly varying functions.

Significance. If the derivations hold, the work supplies a clean extension of classical Fujita theory to a broad class of time-dependent coefficients and to mixed local-nonlocal operators. The reliance on standard semigroup bounds and test-function contradictions, without introducing new technical gaps, makes the results reliable and directly usable by researchers working on blow-up phenomena with variable coefficients. The explicit treatment of regularly varying weights via asymptotic properties is a useful technical contribution.

minor comments (3)
  1. Introduction, second paragraph: the mixed operator (Δ − (−Δ)^s) is introduced without a short reminder of the fractional Laplacian definition or its Fourier symbol; adding one sentence would help readers outside the nonlocal community.
  2. Section 2 (preliminaries): the definition of regular variation for h(t) is stated but lacks a reference to the classical Karamata theory (e.g., Bingham–Goldie–Teugels); inserting a standard citation would improve traceability of the asymptotic arguments used later.
  3. Theorem statements (unforced case): the critical Fujita exponent is expressed in terms of the regular-variation index; explicitly displaying the reduction to the constant-coefficient exponent when the index is zero would clarify the claimed extension.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The summary accurately reflects our results on sharp Fujita-type criteria for the mixed local-nonlocal parabolic equation with regularly varying time weights, including the unforced and forced cases. Since no specific major comments were raised in the report, we have conducted a careful review of the manuscript for clarity and minor improvements.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard estimates

full rationale

The claimed sharp blow-up and global existence criteria for the unforced problem are obtained from semigroup estimates on the mixed local-nonlocal operator combined with test-function contradictions and Karamata-type asymptotic properties of regularly varying functions. These steps rely on the stated assumptions on the index of regular variation and do not reduce any prediction to a fitted parameter or to a self-citation chain. The forced-problem nonexistence and small-data global existence arguments follow the same structure as prior constant-coefficient results without introducing load-bearing self-referential definitions or ansatzes smuggled via citation. The manuscript is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard analytic tools rather than new postulates; no free parameters are fitted to data and no new entities are invented.

axioms (2)
  • standard math Asymptotic properties of regularly varying functions at infinity
    Invoked to handle the time-dependent coefficient h(t) in blow-up and global existence analysis
  • domain assumption Existence and estimates for the semigroup generated by the mixed operator Δ - (-Δ)^s
    Used for linear evolution estimates in the Cauchy problem

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    We first revisit the unforced case and establish sharp blow-up and global existence criteria in terms of the critical Fujita exponent, thereby extending earlier results to the wider class of time-dependent coefficients.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Heat equations driven by mixed local-nonlocal operators with exponential nonlinearity

    math.AP 2026-05 unverdicted novelty 6.0

    The authors establish local well-posedness in Orlicz spaces for exponential nonlinearity, global existence for small data, and large-time decay rates in Lebesgue spaces for the mixed operator heat equation.

Reference graph

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