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arxiv: 2606.31643 · v1 · pith:V6BLTXKJnew · submitted 2026-06-30 · 🧮 math.AP

Fujita-type blow-up for inhomogeneous semilinear heat equations with regularly varying forcing

Vishvesh Kumar , Mohamed Majdoub This is my paper

Pith reviewed 2026-07-01 04:22 UTC · model grok-4.3

classification 🧮 math.AP
keywords Fujita exponentsemilinear heat equationregular variationblow-upinhomogeneous forcingtest-function methodcritical exponentglobal existence
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The pith

The critical Fujita exponent for the inhomogeneous semilinear heat equation is set by the variation index of the forcing term's spatial mass.

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

The paper replaces classical integrability conditions on the forcing with quantitative regular variation properties of its spatial mass F(R). Using regular variation theory and the Mitidieri--Pohozaev test-function method, it proves global solutions cannot exist below a critical value of p determined by the variation index of F, and establishes blow-up in the critical case when suitable slowly varying corrections are present. The same framework yields optimality of the mass condition and extends the analysis to sign-changing forcings, space-time dependent terms, Riesz potentials, anisotropic operators, and fractional Laplacians. A sympathetic reader would care because the results give sharp thresholds for blow-up in equations with spatially inhomogeneous sources whose total mass grows in a regularly varying way at infinity.

Core claim

Using techniques from regular variation theory together with the Mitidieri--Pohozaev test-function method, we establish sharp Fujita-type nonexistence results and identify the critical exponent in terms of the variation index of F. We prove that global solutions do not exist in the subcritical range and obtain critical-case blow-up under suitable slowly varying corrections. The regular variation framework further shows the optimality of the underlying mass condition, extends naturally to anisotropic settings through operator regular variation, and yields sufficient blow-up criteria for sign-changing forcings via the Gaussian-Laplace transform.

What carries the argument

Quantitative regular variation properties of the spatial mass F(R) = integral of w(x) over |x| <= R, which enable the Mitidieri--Pohozaev test-function method to produce nonexistence and blow-up thresholds.

If this is right

  • Global solutions do not exist in the subcritical range defined by the variation index of F.
  • Critical-case blow-up occurs under suitable slowly varying corrections to the forcing.
  • The mass condition on F is optimal, as shown by the regular variation framework.
  • The method applies to sign-changing forcings via the Gaussian-Laplace transform and to space-time dependent forcings.
  • The results extend to anisotropic settings, Riesz-potential forcings, and equations with the fractional Laplacian.

Where Pith is reading between the lines

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

  • The framework could be tested numerically by simulating the heat equation with a forcing whose cumulative mass follows a regularly varying function such as R^alpha log R.
  • Similar variation-based thresholds may apply to blow-up questions in other semilinear parabolic systems driven by inhomogeneous sources.
  • The approach suggests a route to handle forcings that are only asymptotically regularly varying rather than exactly so at every scale.

Load-bearing premise

The spatial mass F(R) of the forcing term satisfies quantitative regular variation properties that enable the Mitidieri--Pohozaev test-function method to produce the stated nonexistence and blow-up thresholds.

What would settle it

Construction of a global nonnegative solution for some p strictly below the critical exponent given by the variation index of F, or a global solution in the critical case when the slowly varying correction condition fails.

read the original abstract

We develop a unified framework for Fujita-type blow-up of solutions to the inhomogeneous semilinear heat equation $$\partial_tu-\Delta u=|u|^p+\mathbf{w}(x), \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0, \cdot)=u_0.$$ The classical integrability assumptions on the forcing term are replaced by quantitative regular variation properties of its spatial mass $$F(R)=\int\limits_{|x|\le R}\mathbf{w}(x)\,dx.$$ Using techniques from regular variation theory together with the Mitidieri--Pohozaev test-function method, we establish sharp Fujita-type nonexistence results and identify the critical exponent in terms of the variation index of $F$. We prove that global solutions do not exist in the subcritical range and obtain critical-case blow-up under suitable slowly varying corrections. The regular variation framework further shows the optimality of the underlying mass condition, extends naturally to anisotropic settings through operator regular variation, and yields sufficient blow-up criteria for sign-changing forcings via the Gaussian-Laplace transform. The approach also applies to space-time dependent forcings, Riesz-potential type forcings, and equations involving the fractional Laplacian, providing a unified description of blow-up thresholds beyond the classical Fujita theory.

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

0 major / 2 minor

Summary. The manuscript develops a unified framework for Fujita-type blow-up of solutions to the inhomogeneous semilinear heat equation ∂_t u − Δu = |u|^p + w(x) by replacing classical integrability assumptions on the forcing with quantitative regular variation properties of its spatial mass F(R). Using regular variation theory and the Mitidieri–Pohozaev test-function method, it establishes sharp nonexistence results in the subcritical range (identified via the variation index of F) and obtains blow-up at criticality under suitable slowly varying corrections. The framework is claimed to extend to anisotropic settings, sign-changing forcings, space-time dependent forcings, Riesz potentials, and fractional Laplacians while demonstrating optimality of the mass condition.

Significance. If the derivations hold, the work provides a meaningful extension of classical Fujita theory to inhomogeneous problems with regularly varying forcings, yielding a parameter-free description of critical exponents in terms of the variation index and showing optimality via the regular-variation hypotheses. The combination of established Mitidieri–Pohozaev techniques with regular variation theory supplies a unified approach that applies across multiple generalizations (anisotropic, fractional, etc.), which is a clear strength for the field of blow-up analysis in semilinear parabolic equations.

minor comments (2)
  1. The abstract states that the regular variation framework 'shows the optimality of the underlying mass condition,' but without the full text it is not possible to verify how the lower-bound constructions or counterexamples are carried out to confirm sharpness.
  2. The description of extensions to 'operator regular variation' and the Gaussian-Laplace transform for sign-changing forcings would benefit from explicit statements of the additional hypotheses needed on w to ensure the test-function integrals remain controlled.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary and positive evaluation of the significance of our unified regular-variation framework for Fujita-type results. The recommendation is listed as 'uncertain,' yet the major comments section contains no specific points or requests for clarification. We interpret this as an invitation to confirm that the derivations are complete and correct as presented; we remain available to supply any additional technical details or expansions if the editor or referee desires them.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation applies the standard Mitidieri-Pohozaev test-function method to the spatial mass F(R) under quantitative regular-variation hypotheses drawn from external regular-variation theory. The critical exponent is expressed directly in terms of the variation index of the input function F, which is an independent assumption rather than a fitted or self-defined quantity. No step reduces by construction to the target nonexistence or blow-up statements; the framework extends classical Fujita theory via known techniques without self-citation chains or ansatzes that are load-bearing for the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of regularly varying functions and the applicability of the test-function method; no free parameters or new entities are introduced.

axioms (2)
  • standard math Properties of regularly varying functions (Karamata theory) hold for the spatial mass F(R).
    Invoked throughout to characterize the asymptotic behavior of F(R) and derive the critical exponent.
  • domain assumption The Mitidieri--Pohozaev test-function method applies directly to the inhomogeneous equation with regularly varying forcing.
    Used to establish nonexistence in the subcritical range.

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Reference graph

Works this paper leans on

28 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Bandle, H

    C. Bandle, H. A. Levine, and Q. S. Zhang,Critical exponents of Fujita type for inhomogeneous parabolic equations and systems, J. Math. Anal. Appl.251(2000), 624–648

    2000

  2. [2]

    Ben Belgacem and M

    R. Ben Belgacem and M. Majdoub,On the Fujita Phenomenon for a Forced Spatio-temporal Fractional Diffusion Equation,Mediterr. J. Math.23(2026), no. 3, Paper No. 122

    2026

  3. [3]

    Ben Belgacem and M

    R. Ben Belgacem and M. Majdoub,On the nonexistence of global solutions for nonlocal parabolic equa- tions with forcing terms,Discrete Contin. Dyn. Syst. Ser. S19(2026), 252–270

    2026

  4. [4]

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

    R. Ben Belgacem and M. Majdoub,Fujita Phenomenon for a Mixed Local–Nonlocal Hardy–H´ enon Equation with Regularly Varying Time Weights, preprint, arXiv:2509.07405,https://arxiv.org/abs/ 2509.07405

    work page internal anchor Pith review Pith/arXiv arXiv

  5. [5]

    N. H. Bingham, C. M. Goldie, and J. L. Teugels,Regular Variation, Encyclopedia of Mathematics and its Applications27, Cambridge University Press, Cambridge, 1987

    1987

  6. [6]

    C´ ordoba and D

    A. C´ ordoba and D. C´ ordoba,A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys.249(2004), 511–528

    2004

  7. [7]

    de Haan and A

    L. de Haan and A. Ferreira,Extreme Value Theory: An Introduction, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006

    2006

  8. [8]

    Deng and H

    K. Deng and H. A. Levine,The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl.243(2000), 85–126

    2000

  9. [9]

    A. Z. Fino and M. Kirane,Qualitative properties of solutions to a time-space fractional evolution equa- tion, Quart. Appl. Math.70(2012), 133–157

    2012

  10. [10]

    Fujita,On the blowing up of solutions of the Cauchy problem for∂ tu= ∆u+u 1+α, J

    H. Fujita,On the blowing up of solutions of the Cauchy problem for∂ tu= ∆u+u 1+α, J. Fac. Sci. Univ. Tokyo Sect. I13(1966), 109–124

    1966

  11. [11]

    Hayakawa,On nonexistence of global solutions of some semilinear parabolic differential equations, Proc

    K. Hayakawa,On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad.49(1973), 503–505

    1973

  12. [12]

    Ju,The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm

    N. Ju,The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys.255(2005), 161–181

    2005

  13. [13]

    Karamata,Sur un mode de croissance r´ eguli` ere des fonctions, Mathematica (Cluj)4(1930), 38–53

    J. Karamata,Sur un mode de croissance r´ eguli` ere des fonctions, Mathematica (Cluj)4(1930), 38–53

    1930

  14. [14]

    Kumar and B

    V. Kumar and B. T. Torebek,Fujita-type results for the degenerate parabolic equations on the Heisenberg groups, J. Differential Equations465(2026), Paper No. 114241

    2026

  15. [15]

    H. A. Levine,The role of critical exponents in blowup theorems, SIAM Rev.32(1990), 262–288

    1990

  16. [16]

    Majdoub,Well-posedness and blow-up for an inhomogeneous semilinear parabolic equation, Differ

    M. Majdoub,Well-posedness and blow-up for an inhomogeneous semilinear parabolic equation, Differ. Equ. Appl.13(2021), 85–100

    2021

  17. [17]

    Majdoub,The Fujita exponent for an inhomogeneous semilinear heat equation revisited, La Matem- atica2(2023), 340–361

    M. Majdoub,The Fujita exponent for an inhomogeneous semilinear heat equation revisited, La Matem- atica2(2023), 340–361

    2023

  18. [18]

    M. M. Meerschaert,Regular variation inR k, Proc. Amer. Math. Soc.102(1988), 341–348. FUJITA-TYPE BLOW-UP VIA REGULAR V ARIATION 35

    1988

  19. [19]

    M. M. Meerschaert and H.-P. Scheffler,Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice, Wiley, New York, 2001

    2001

  20. [20]

    Mitidieri and S

    `E. Mitidieri and S. I. Pohozaev,A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova234(2001), 1–384; translation in Proc. Steklov Inst. Math.234(2001), 1–362

    2001

  21. [21]

    Qi,The critical exponents of parabolic equations and blow-up inR n, Proc

    Y.-W. Qi,The critical exponents of parabolic equations and blow-up inR n, Proc. Roy. Soc. Edinburgh Sect. A128(1998), 123–136

    1998

  22. [22]

    Quittner and Ph

    P. Quittner and Ph. Souplet,Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkh¨ auser Advanced Texts, Birkh¨ auser, Basel, 2007

    2007

  23. [23]

    S. I. Resnick,Extreme Values, Regular Variation, and Point Processes, Applied Probability4, Springer, New York, 1987

    1987

  24. [24]

    S. I. Resnick,Hidden regular variation, second order regular variation and asymptotic independence, Extremes5(2002), 303–336

    2002

  25. [25]

    S. I. Resnick,Heavy-Tail Phenomena: Probabilistic and Statistical Modeling, Springer Series in Opera- tions Research and Financial Engineering, Springer, New York, 2007

    2007

  26. [26]

    Seneta,Regularly Varying Functions, Lecture Notes in Mathematics508, Springer, Berlin, 1976

    E. Seneta,Regularly Varying Functions, Lecture Notes in Mathematics508, Springer, Berlin, 1976

    1976

  27. [27]

    Sharpe,Operator-stable probability distributions on vector groups, Trans

    M. Sharpe,Operator-stable probability distributions on vector groups, Trans. Amer. Math. Soc.136 (1969), 51–65

    1969

  28. [28]

    Q. S. Zhang,Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J.97(1998), 515–539. (V. Kumar)Department of Mathematical Sciences, Indian Institute of Technology (BHU), V aranasi, Uttar Pradesh, 221005, India. Email address:vishvesh.mat@iitbhu.ac.in Email address:vishveshmishra@gmail.com (M. Majdoub)Department of Mathematics, Colle...

    1998