pith. sign in

arxiv: 2606.28248 · v1 · pith:33RAWO2Inew · submitted 2026-06-26 · 🧮 math.AP · math.PR

The Fujita exponent across an interface

Mohamed Majdoub , Ezzedine Mliki This is my paper

Pith reviewed 2026-06-29 02:56 UTC · model grok-4.3

classification 🧮 math.AP math.PR
keywords Fujita exponentsemilinear parabolic equationinterface hyperplaneblow-upglobal existencesingular drifttest-function methodGaussian bounds
0
0 comments X

The pith

The Fujita critical exponent stays 1 + 2/N for the semilinear parabolic equation even with a singular interface drift term.

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

The paper examines the semilinear parabolic equation that includes a delta-supported drift term on a fixed hyperplane interface. It first establishes local well-posedness of mild solutions in Lebesgue spaces by combining Gaussian bounds on the fundamental solution with a contraction mapping argument. The central result is a sharp dichotomy: every nontrivial nonnegative solution blows up in finite time when 1 < p ≤ 1 + 2/N, while for p > 1 + 2/N sufficiently small initial data yield global solutions. This critical value is identical to the classical Fujita exponent for the heat equation without any interface. The analysis relies on an adapted test-function method that accounts for the interface.

Core claim

For the equation ∂_t u = Δu + 2q δ_S ∇u + |u|^{p-1}u with |q| ≤ 1 and S a hyperplane, every nontrivial nonnegative solution blows up in finite time when 1 < p ≤ 1 + 2/N, whereas global solutions exist for sufficiently small initial data when p > 1 + 2/N. The critical exponent coincides exactly with the classical Fujita exponent for the heat equation, showing that the Fujita phenomenon remains stable under discontinuous diffusion effects and interface transmission conditions.

What carries the argument

An adapted test-function method that accounts for the interface and uses Gaussian bounds on the fundamental solution of the linear operator with the singular drift term to establish the sharp blow-up versus global-existence dichotomy.

If this is right

  • The critical power separating blow-up from global existence is unaffected by the interface term for any |q| ≤ 1.
  • Local well-posedness holds in Lebesgue spaces despite the spatial inhomogeneity induced by the interface.
  • The Fujita phenomenon is stable under the addition of singular drift supported on a hypersurface.
  • The same dichotomy applies to nonnegative solutions in the presence of the interface transmission conditions.

Where Pith is reading between the lines

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

  • The stability result may extend to curved or time-dependent interfaces if similar Gaussian bounds can be obtained.
  • The method could be tested on equations with nonlinear transmission conditions across the interface.
  • Analogous invariance of the critical exponent might hold for other singular perturbations of the heat equation that preserve the Gaussian kernel estimates.

Load-bearing premise

The Gaussian bounds for the fundamental solution of the linear operator with the singular drift term continue to hold and allow the adapted test-function method to produce the claimed sharp dichotomy.

What would settle it

A positive global solution for some p = 1 + 2/N with nontrivial initial data, or finite-time blow-up for small initial data when p is slightly larger than 1 + 2/N, would disprove the stated threshold.

read the original abstract

We consider the semilinear parabolic equation \[ \partial_t u = \Delta u + 2\mathfrak{q}\,\delta_{\mathbb{S}}\,\nabla u + |u|^{p-1}u \qquad \text{in } (0,\infty)\times\mathbb{R}^N, \] where $|\mathfrak{q}|\le 1$, $p>1$, and $\mathbb{S}$ is a fixed interface hyperplane. Working in Lebesgue spaces, we first establish local well-posedness of mild solutions. This is achieved by combining Gaussian bounds for the associated fundamental solution with a contraction mapping argument adapted to the lack of spatial homogeneity induced by the interface term. We then prove a sharp Fujita-type dichotomy for nonnegative solutions. Specifically, we show that every nontrivial solution blows up in finite time when $1<p \le 1+\frac{2}{N}$, whereas for $p>1+\frac{2}{N}$ global solutions exist for sufficiently small initial data. The blow-up analysis relies on a suitably adapted test-function method that accounts for the presence of the interface. It is noteworthy that the critical exponent coincides with the classical Fujita exponent for the heat equation, indicating that the Fujita phenomenon remains stable under the presence of discontinuous diffusion effects and interface transmission conditions. To the best of our knowledge, this is the first result of this type for operators involving a singular drift supported on a hypersurface.

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 / 3 minor

Summary. The manuscript considers the semilinear parabolic equation δ_t u = Δu + 2q δ_S ∇u + |u|^{p-1}u in (0,∞)×R^N with |q|≤1 and interface hyperplane S. It establishes local well-posedness of mild solutions in Lebesgue spaces by combining Gaussian bounds for the fundamental solution of the linear operator with a contraction-mapping argument adapted to the interface. It then proves a sharp Fujita dichotomy for nonnegative solutions: every nontrivial solution blows up in finite time when 1<p≤1+2/N, while global solutions exist for sufficiently small initial data when p>1+2/N. The blow-up analysis uses a suitably adapted test-function method that accounts for the interface; the critical exponent is shown to coincide with the classical Fujita exponent 1+2/N for the heat equation.

Significance. If the Gaussian bounds and the adapted test-function argument hold as claimed, the work demonstrates stability of the Fujita phenomenon under singular interface drifts and discontinuous diffusion effects. This is the first such result for operators with drift supported on a hypersurface, extending classical techniques while preserving the exact critical exponent. The combination of explicit kernel bounds for local existence and interface-adjusted test functions for the dichotomy supplies a reusable framework for related transmission problems.

minor comments (3)
  1. [§2] §2 (Gaussian bounds): the comparability constants between the fundamental solution and the heat kernel should be stated explicitly with dependence on q and N, as these enter the contraction-mapping radius and the small-data threshold in the global-existence regime.
  2. [§3.2] §3.2 (test-function construction): the choice of the cut-off function near the interface and the integration-by-parts identities that absorb the singular drift term should be written out in full; the current sketch leaves open whether the resulting ODE for the weighted integral is exactly the same as in the classical case or carries a multiplicative factor depending on q.
  3. [Introduction] Notation: the symbol δ_S is used both for the Dirac measure on the hyperplane and, implicitly, for the surface measure in the weak formulation; a single clarifying sentence distinguishing the two usages would remove ambiguity for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report, so we have no points to address point-by-point. We will handle any minor issues (such as typographical corrections or clarifications) in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The derivation relies on establishing Gaussian bounds for the fundamental solution of the linear operator (with singular drift) to obtain local well-posedness via contraction mapping in Lebesgue spaces, followed by an adapted test-function argument to obtain the blow-up/global-existence dichotomy. Both steps adapt standard techniques (Gaussian kernel estimates and test-function methods) to the interface setting without any reduction of the critical exponent or main claims to fitted parameters, self-definitions, or load-bearing self-citations. The coincidence with the classical Fujita exponent 1 + 2/N emerges from the analysis rather than being presupposed by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on standard PDE existence and comparison tools plus one domain-specific assumption about the linear operator; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The fundamental solution of the linear operator with the interface term satisfies Gaussian bounds that enable the contraction mapping argument for local well-posedness.
    Invoked to obtain local existence of mild solutions in Lebesgue spaces.

pith-pipeline@v0.9.1-grok · 5779 in / 1232 out tokens · 49041 ms · 2026-06-29T02:56:29.239126+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references

  1. [1]

    Al-Shahrani and M

    H. Al-Shahrani and M. Majdoub,Existence and Uniqueness for a Class of Singular Fractional Differential Equations, Fractional Differential Calculus, to appear

  2. [2]

    Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge University Press, 2009

    D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge University Press, 2009

    2009

  3. [3]

    Brezis and T

    H. Brezis and T. Cazenave,A nonlinear heat equation with singular initial data, J. Anal. Math.,68(1996), 277–304

    1996

  4. [4]

    Dłotko and Yejuan Wang,Critical parabolic-type problems, De Gruyter Series in Nonlinear Analysis and Applications,34, De Gruyter, Berlin, 2020

    Tomasz W. Dłotko and Yejuan Wang,Critical parabolic-type problems, De Gruyter Series in Nonlinear Analysis and Applications,34, De Gruyter, Berlin, 2020

    2020

  5. [5]

    S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, 1986

    1986

  6. [6]

    Fino and G

    A. Fino and G. Karch,Decay of mass for nonlinear equation with fractional Laplacian, Monatsh. Math.,160 (2010), 375–384

    2010

  7. [7]

    Fujishima, T

    Y. Fujishima, T. Kawakami and Y. Sire,Critical exponent for the global existence of solutions to a semilinear heat equation with degenerate coefficients, Calc. Var. Partial Differential Equations,58(2019), Paper No. 62, 25 pp

    2019

  8. [8]

    Fujishima, K

    Y. Fujishima, K. Hisa, K. shige and R. Laister, Solvability of superlinear fractional parabolic equations, J. Evol. Equ.,23(2023), Paper No. 4, 38

    2023

  9. [9]

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

    H. Fujita,On the blowing up of solutions of the Cauchy problem forut = ∆u+u 1+α, J. Fac. Sci. Univ. Tokyo Sec. IA Math.,13(1966), 109–124

    1966

  10. [10]

    Fujita,On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Proc

    H. Fujita,On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Proc. Sympos. Pure Math.,18(1969), 105–113

    1969

  11. [11]

    Gaveau, M

    B. Gaveau, M. Okada and T. Okada,Second order differential operators and Dirichlet integrals with singular coefficients. I. Functional calculus of one-dimensional operators, Tohoku Math. J. (2),39(1987), 465–504

    1987

  12. [12]

    Guedda and M

    M. Guedda and M. Kirane,Criticality for some evolution equations, Differ. Uravn.,37(2001), 540–550

    2001

  13. [13]

    Guedda and M

    M. Guedda and M. Kirane,Local and global nonexistence of solutions to semilinear evolution equations, Pro- ceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf.,9(2002), 149–160

    2002

  14. [14]

    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

  15. [15]

    Jleli, T

    M. Jleli, T. Kawakami and B. Samet,Critical behavior for a semilinear parabolic equation with forcing term depending of time and space, J. Math. Anal. Appl.,486(2020), 123931

    2020

  16. [16]

    Kirane, A

    M. Kirane, A. Z. Fino and A. Ayoubb,Decay of mass for a semilinear heat equation with mixed local-nonlocal operators, Fract. Calc. Appl. Anal.,28(2025), 1756–1776

    2025

  17. [17]

    Lejay,On the constructions of the skew Brownian motion, Probability Surveys,3(2006), 413–466

    A. Lejay,On the constructions of the skew Brownian motion, Probability Surveys,3(2006), 413–466

    2006

  18. [18]

    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

  19. [19]

    Majdoub,On the Fujita exponent for a Hardy–Hénon equation with a spatial-temporal forcing term, La Matematica,2(2023), 340–361

    M. Majdoub,On the Fujita exponent for a Hardy–Hénon equation with a spatial-temporal forcing term, La Matematica,2(2023), 340–361

    2023

  20. [20]

    Mastrangelo and M

    M. Mastrangelo and M. Talbi,Mouvements Browniens asymétriques modifiés en dimension finie et opérateurs différentiels à coefficients discontinus, Probab. Math. Stat. Fasc.,11(1) (1990), 49–80. 36 M. MAJDOUB, E. MLIKI

    1990

  21. [21]

    Mitidieri and S

    E. Mitidieri and S. I. Pokhozhaev,A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Trudy Mat. Inst. Steklova,234(2001), 3–383

    2001

  22. [22]

    Miyamoto,A doubly critical semilinear heat equation in theL1 space, J

    Y. Miyamoto,A doubly critical semilinear heat equation in theL1 space, J. Evol. Equ.,21(2021), 151–166

    2021

  23. [23]

    Portenko,Diffusion processes with a generalized drift coefficient, Theory Probab

    N. Portenko,Diffusion processes with a generalized drift coefficient, Theory Probab. Appl.,24(1) (1979), 62–78

    1979

  24. [24]

    Portenko,Stochastic differential equations with generalized drift vector, Theory Probab

    N. Portenko,Stochastic differential equations with generalized drift vector, Theory Probab. Appl.,24(2) (1979), 338–353

    1979

  25. [25]

    N. I. Portenko,Generalized Diffusion Processes, Translations of Mathematical Monographs, Vol.83, AMS, Providence, 1990

    1990

  26. [26]

    J. M. Ramirez,Multi-skewed Brownian motion and diffusion in layered media, Proc. Amer. Math. Soc.,139 (2011), 3739–3752

    2011

  27. [27]

    Revuz and M

    D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer, 1999

    1999

  28. [28]

    C. A. Tudor and M. Zili,SPDE with generalized drift and fractional-type noise, NoDEA Nonlinear Differential Equations Appl.,23(2016), Art. 53, 23 pp

    2016

  29. [29]

    J. R. L. Webb,Weakly singular Gronwall inequalities and applications to fractional differential equations, J. Math. Anal. Appl.,471(2019), 692–711

    2019

  30. [30]

    F. B. Weissler,Local existence and nonexistence for semilinear parabolic equations inLp, Indiana Univ. Math. J.,29(1980), 79–102. Department of Mathematics, College of Science, Imam Abdulrahman Bin F aisal University, P. O. Box 1982, Dammam, Saudi Arabia Basic and Applied Scientific Research Center, Imam Abdulrahman Bin F aisal University, P.O. Box 1982, ...

    1980