Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Extinction and propagation phenomena for semilinear parabolic equations on metric trees

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A spectral threshold $f'(0)=E_0$ decides whether KPP-type waves invade or die out on regular metric trees.

desk verdict New threshold result for KPP on metric trees, with a clean spectral constant and explicit speed bounds; two fixable issues—an unpublished comparison-principle citation and a matching typo in Lemma 5.1—need attention before the main theorems are fully supported. read the letter →

arxiv 2505.10712 v1 pith:BJ2ORZCZ submitted 2025-05-15 math.AP

Fabio Punzo , Alberto Tesei This is my paper · ORCID
classification math.AP MSC 35K5535C0735R02
keywords metrictreessemilinearparabolicequationsKPP-typereactiontermspropagationversusextinctionspectralthresholdfrontspeedcomparisonprinciplesNeumannLaplacian
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the semilinear heat equation $u_t=\Delta u+f(u)$ on an infinite regular metric tree, with a Neumann condition at the root and initial data between 0 and 1, for KPP-type sources $f$. Its central claim is that the long-time behaviour is controlled by the sign of $f'(0)-E_0$, where $E_0$ is the bottom of the spectrum of the Neumann Laplacian on the tree. Under the tree's structural assumptions $E_0$ is positive; if $f'(0)>E_0$, every nontrivial solution converges to 1 uniformly on compact subsets, while on homogeneous trees with $f'(0)

What carries the argument

The central objects are the spectral bottom $E_0=\min\sigma(-\Delta)$ of the Neumann Laplacian on the regular tree and symmetric stationary sub- and supersolutions built on a weighted half-line reduction. Symmetric functions on the tree reduce to functions on $\mathbb R^+$ with the branching weight $\beta(\rho)$; the restricted Laplacian becomes an operator $A$ on $L^2(\mathbb R^+)$ with jump conditions at the radii $\rho_n$, and for homogeneous trees $E_0=\theta^2/r^2$ with $\theta=\arccos(2\sqrt b/(b+1))$. The proof of propagation uses eigenfunctions on exhausting domains to build small stationary subsolutions; the proof of extinction uses explicit radial profiles $g$ satisfying $g''+\lambda g=0$ with $\lambda<E_0$ as stationary supersolutions. Comparison principles then trap the solution between these barriers. The speed bounds come from traveling-wave-like symmetric profiles $m(\rho-ct)$ adapted to the Kirchhoff jump conditions at the vertices.

What would settle it

Take the homogeneous tree with $b=2$ and edge length $r=1$, source $f(u)=\lambda u(1-u)$, and let $\lambda$ exceed $E_0=(\arccos(2\sqrt{2}/3))^2$. Simulate the Cauchy-Neumann problem on exhausting finite truncated trees with a small compact initial bump near the root; if the solution does not converge to 1 uniformly on compact subsets, Theorem 2.1 is refuted. Alternatively, a counterexample to the order-preserving comparison principle for sub- and supersolutions on a non-compact tree would invalidate both threshold theorems.

Watch

Extended reading notes

Core claim

The paper establishes a sharp dichotomy for the Cauchy-Neumann problem on an infinite regular metric tree. With a KPP source (so $f(0)=f(1)=0$, $f>0$ in $(0,1)$, and $f(u)/u\le f'(0)$), the bottom of the $L^2$ spectrum $E_0=\min\sigma(-\Delta)$ is positive under the structural assumptions on the tree. If $f'(0)>E_0$, Theorem 2.1 asserts that every solution with $u_0\not\equiv 0$ satisfies $u(x,t)\to 1$ uniformly on compact subsets of $T$; if $T$ is homogeneous and $f'(0)<E_0$, Theorem 2.2 asserts that every suitably small solution satisfies $u(x,t)\to 0$ uniformly on $T$. The propagation speed is bracketed: no front moves faster than $\hat c=M\rho_1 b_1/(b_1-1)$, and when propagation occurs no front moves slower than $\check c=2\sqrt{f'(0)-E_0}$. The mechanism is that the positivity of $E_0$ strengthens diffusion on trees relative to Euclidean space, so a weak source can be suppressed by spreading while a strong source overwhelms it.

Load-bearing premise

The whole argument leans on a set of comparison results that are cited to a not-yet-published companion paper rather than proved here; if any of those results needs extra hypotheses or fails on non-compact trees, the extinction-propagation dichotomy and both speed bounds would lose their proof.

Editorial extensions

If this is right

  • If $f'(0)>E_0$, any nonzero initial datum, however small in $L^2(T)$, leads to full invasion of every compact subset of the tree.
  • On a homogeneous tree with $f'(0)<E_0$, all initial data lying below an explicit radial profile are driven to zero uniformly in the whole tree.
  • The asymptotic front speed is pinned between $2\sqrt{f'(0)-E_0}$ and $M\rho_1 b_1/(b_1-1)$, so the Euclidean KPP speed $2\sqrt{f'(0)}$ is not the tree speed.
  • Lengthening the first edges or lowering the branching number raises the upper speed bound, so the geometry of the tree directly changes invasion rates.
  • A polynomial growth condition near zero, in the spirit of the Fujita condition, also forces extinction on homogeneous trees, extending the threshold phenomenon to a broader class of nonlinearities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the driving mechanism is really the positivity of $E_0$ rather than exponential volume growth, then regular trees whose branching and edge lengths give polynomial volume growth, where the paper conjectures $E_0=0$, should behave like Euclidean space: every nonzero KPP solution propagates to 1, with speed near $2\sqrt{f'(0)}$. This is the paper's conjecture, not one of its theorems.
  • A testable extension is to compute the first Dirichlet eigenvalue on truncations of a homogeneous tree as the truncation grows; the paper's Proposition A.6 says it decreases to $E_0$, so finite-network simulations can empirically probe the threshold $f'(0)=E_0$.
  • A natural sharper target left open is the exact front speed $c_0$; one could seek it as the minimal speed of a traveling-wave-like solution of the half-line problem with Kirchhoff jump conditions, analogous to the Euclidean ODE $q''+cq'+f(q)=0$.
  • Should the deferred comparison principles, cited to an unpublished companion paper, turn out to require additional hypotheses on the tree or the nonlinearity, the dichotomy and speed bounds would inherit those hypotheses; until that companion proof is public, the theorem's full range remains conditional.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the semilinear heat equation u_t = Δu + f(u) on regular metric trees with homogeneous Neumann condition at the root, under geometric assumptions (H0) and hypotheses (H1)-(H5) on the initial datum and on the KPP-type nonlinearity f. The main claims are: if f'(0) > E0 := min σ(-Δ), then every nonzero solution converges to 1 uniformly on compact subsets of the tree (Theorem 2.1); on homogeneous trees, if f'(0) < E0 and the initial datum is suitably small, the solution converges to 0 uniformly (Theorem 2.2), with an analogous extinction result under the sub-Fujita-type assumption (H5) (Theorem 2.3); and the asymptotic speed of propagation is bounded above by ĉ = Mρ1 b1/(b1-1) (Theorem 2.4) and below by č = 2√(f'(0)-E0) (Theorem 2.5). The proofs rely on spectral analysis of the Neumann Laplacian on regular trees (including the explicit formula for homogeneous trees in Theorem 3.3), on symmetric super- and subsolutions, and on several comparison principles.

Significance. If the results are correct, the paper establishes a nontrivial threshold dichotomy for KPP-type equations on metric trees, parallel to the hyperbolic-space theory, and it provides explicit, parameter-free bounds on the propagation speed that depend only on the spectral threshold and on the geometry of the tree. The spectral computation in Theorem 3.3 is self-contained and convincing, and Theorem 4.5 is proved in detail. The constructions of super- and subsolutions are explicit and give falsifiable predictions. However, the central results are not fully established within the manuscript because the comparison principles on which they rely are deferred to an unpublished preprint, and because the matching condition in Lemma 5.1 fails as written.

major comments (3)
  1. [Section 4.2, Theorems 4.2-4.4] The comparison principles stated as Theorems 4.2, 4.3 and 4.4 are not proved in this paper; the text refers only to the unpublished preprint [29]. These results are load-bearing: Theorem 4.2 is used in the proofs of Theorems 2.2 and 2.5, Theorem 4.3 in the proof of Theorem 2.1, and Theorem 4.4 (via Remark 4.4) in the proofs of Theorems 2.3 and 2.4. Since [29] is not publicly available, the central dichotomy and the speed bounds are not verifiable from the manuscript alone. The authors should either include the proofs of these comparison theorems or clearly state that the main results are conditional on [29] and make that preprint accessible.
  2. [Section 5, proof of Theorem 2.1] The application of Theorem 4.3 after (5.2) uses the constant function 1 as the upper stationary supersolution. However, q ≡ 1 is not a stationary supersolution in the sense of Definition 4.4, because it does not belong to H1(T); it is only a weak stationary solution (Remark 4.2). The statement of Theorem 4.3 does not cover weak stationary supersolutions, so the proof as written needs either an extended version of Theorem 4.3 or a separate limiting argument to justify the upper bound in (5.2).
  3. [Lemma 5.1, equation (5.5)] The claimed matching condition (5.4d) fails as written. Substituting (5.5) gives g_n(ρ_n) = α^n b^{n/2} sin(√λ r) and g_{n+1}(ρ_n) = α^n b^{n/2+1} sin(√λ r), which are unequal for b ≥ 2. Consequently the function z(ρ,t) := g(ρ)e^{-[λ-f'(0)]t} used in the proof of Theorem 2.2 is not continuous at the vertices and is not a supersolution of problem (1.1) as defined. Replacing b^{n/2} by b^{-n/2} in (5.5) restores (5.4d); this correction and the resulting changes in the estimates (5.6)-(5.7) should be made explicit before the extinction conclusion of Theorem 2.2 is accepted.
minor comments (4)
  1. [Assumption (H0), Section 1.2] The phrase "T is is a regular metric tree" contains a duplicated "is" and should read "T is a regular metric tree".
  2. [Lemma 6.4, after (6.17)] The sentence "Moreover, there holds ψϵ ≤ 0 in R+" should state ψϵ′ ≤ 0 in R+, since the intended assertion is the nonpositivity of the derivative used later in the proof of Theorem 2.5.
  3. [Proposition 2.6, proof] The notation "θ(0) = π/2" should be "θ(b) → π/2 as b → ∞"; also the numerical value θ(2) ≈ π/12 is inaccurate, since arccos(2√2/3) ≈ 0.34 rad ≈ π/9.
  4. [Definition 4.4] The sentence "A stationary subsolution q is defined by reversing inequalities in (4.13b)-(4.13c)" refers to equations that are introduced later and concern symmetric stationary supersolutions; it should refer to (4.8b)-(4.8c) of the present definition.

Circularity Check

1 steps flagged · score 4.0 of 10

Main theorems depend on comparison principles whose proofs are deferred to the authors' own unpublished preprint [29]; otherwise the E0 threshold and speed bounds are independently computed, not fitted.

  1. self citation load bearing [Section 4.2, before Theorems 4.2-4.4; these theorems are invoked in the proofs of Theorems 2.1, 2.2, 2.3 and 2.5.]
    "We refer the reader to [29] for the proof of the following results."

    The comparison and monotonicity principles that support the central dichotomy and the lower speed bound are not proved in this paper. Reference [29] is an unpublished 2024 preprint by the same two authors, Fabio Punzo and Alberto Tesei. Thus the load-bearing statement 'there is an ordering between sub- and supersolutions' is not verified by an external, machine-checked, or publicly available proof within the submitted text; it is carried by a self-citation. This is not an equation-for-equation reduction, but it is a self-citation that supports the main results, so the circularity score is raised. The rest of the derivation, by contrast, uses independently computed spectral quantities and explicit supersolution constructions without fitted parameters.

full rationale

No fitted-parameter circularity is present. The threshold E0 = min sigma(-Delta) is computed independently in Section 3 using the spectral analyses of [36] and [38], which are external references, and Theorem 3.3 gives the explicit homogeneous-tree value E0 = theta^2/r^2. The upper speed bound (2.5) is obtained from an explicit supersolution m satisfying m'' + c m' + f(m) <= 0 with the estimate f <= M u and an explicit inequality c >= M(ρ_n - ρ_{n-1}) ...; here M, ρ_1, b_1 are given data, not parameters fitted to the conclusion. The lower speed bound (2.8) arises from the spectral gap f'(0) - E0 through the explicit function ψ of Lemma 6.3; again no parameter is tuned to force the result. The uniqueness of the nontrivial weak stationary solution used in Theorem 2.1 is attributed to [42, Theorem 3], an external source, not to the authors' own prior work. The only circularity-adjacent feature is the delegation of Theorems 4.2-4.4 to the authors' unpublished preprint [29]; this is load-bearing for Theorems 2.1, 2.2, 2.3 and 2.5, but it is a verification gap rather than a derivation of the conclusion from an assumed form of it. I also note as a correctness risk, not as circularity, that in Lemma 5.1 formula (5.5) gives g_{n+1}(ρ_n) = α^n b^{n/2+1} sin(√λ r) while g_n(ρ_n) = α^n b^{n/2} sin(√λ r), so the continuity asserted in (5.4d) appears to fail for b >= 2 unless the b^{n/2} factor is corrected; this affects the proof of Theorem 2.2 but does not change the circularity assessment.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new entities. All constants in the statements are either geometric data (b_n, rho_n), spectral quantities (E0, theta), or assumption constants (M, f'(0)). The only notable external reliance is the unpublished comparison-principle preprint [29].

assumptions (4)
  • domain assumption Assumptions (H0)-(H2), and where indicated (H3)-(H5), on the tree T and the reaction f.
    These hypotheses define the class of regular metric trees and nonlinearities for which the theorems are stated.
  • standard math Spectral characterization of the Neumann Laplacian on regular trees: E0 > 0 under (H0), and the explicit formula E0 = theta^2/r^2 for homogeneous trees.
    Theorem 3.1 and 3.3 rest on known spectral results from [38], [36], [25], [26] and are proven in the paper.
  • domain assumption Comparison principles (Theorems 4.2, 4.3, 4.4) are stated without proof and cited to [29], an unpublished preprint by the same authors.
    These results are used in the proofs of Theorems 2.1, 2.2, 2.3 and 2.5; they are not proven in this preprint.
  • standard math Unitary transformation to the weighted space L2(R+;beta) and the form characterization (Proposition A.8, A.9).
    Standard functional analytic framework from [38], [26] used throughout the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Extinction and propagation phenomena for semilinear parabolic equations on metric trees." pith.science (2026). https://pith.science/paper/BJ2ORZCZ

@misc{pith2026250510712,
  author       = {Pith},
  title        = {Pith review of: Extinction and propagation phenomena for semilinear parabolic equations on metric trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BJ2ORZCZ}},
  note         = {Machine review of arXiv:2505.10712}
}
read the original abstract

We study the Cauchy-Neumann problem on a regular metric tree T for the semilinear heat equation with forcing term of KPP type. Propagation and extinction of solutions, as well as asymptotical speed of propagation are investigated.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Nonexistence results for the semilinear wave equation on graphs

    math.AP 2025-06 reject novelty 6.0 of 10

    On infinite weighted graphs, if the graph Laplacian of the distance grows slowly and suitable weighted volume growth holds, every very weak solution of the semilinear wave inequality is identically zero.

Reference graph

Works this paper leans on

43 extracted references · 39 canonical work pages · cited by 1 Pith paper

  1. [29]

    Punzo & A

    F. Punzo & A. Tesei, Monotonicity results for semilinear parabolic equations on metric graphs , preprint (2024)

    work page 2024

  2. [1]

    D. G. Aronson & H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Lecture Notes in Mathematics 446, pp. 6-49 (Springer, 1975)

    work page 1975

  3. [2]

    D. G. Aronson & H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics , Adv. Math. 30 (1978), 33-76

    work page 1978

  4. [3]

    Baker & X

    M. Baker & X. Faber, Metric graphs, Laplacian operators, and electrical networks , in Quantum graphs and their applications , Contemp. Math. 415, pp. 15-33 (Amer. Math. Soc., 2006)

    work page 2006

  5. [4]

    Bandle, M

    C. Bandle, M. A. Pozio & A. Tesei, The Fujita exponent in the hyperbolic space, J. Differential Equations 251 (2011), 2143-2163

    work page 2011

  6. [5]

    Bandle, F

    C. Bandle, F. Punzo & A. Tesei, Existence and nonexistence of patterns on Riemannian manifolds , J. Math. Anal. Appl. 387 (2012), 33-47

    work page 2012

  7. [6]

    Berestycki & F

    H. Berestycki & F. Hamel, Generalized travelling waves for reaction diffusion equations, in Perspectives in Nonlinear Partial Differential Equations. In Honor of H. Brezis , Contemporary Mathematics 446, pp. 101-123 (Amer. Math. Soc., 2007)

    work page 2007

  8. [7]

    Berkolaiko & P

    G. Berkolaiko & P. Kuchment, Introduction to Quantum Graphs (Amer. Math. Soc., 2013)

    2013

Show all 43 references
  1. [8]

    Carlson, Hill’s equation for a homogeneous tree, Electron

    R. Carlson, Hill’s equation for a homogeneous tree, Electron. J. Differential Equations 1997 (1997), 1-30

    1997

  2. [9]

    Carlson, Linear network models related to blood flow

    R. Carlson, Linear network models related to blood flow . In: Quantum graphs and their applications , Contemp. Math. 415, pp. 65-80 (Amer. Math. Soc., 2006)

    2006

  3. [10]

    E. B. Davies, Heat Kernel and Spectral Theory (Cambridge Univ. Press, 1989)

    1989

  4. [11]

    P. C. Fife & J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal. 65 (1977), 335-361

    1977

  5. [12]

    Frank & H

    R.L. Frank & H. Kovarik, Heat kernels of metric trees and applications , SIAM J. Math. Anal. 45 (2013), 1027-1046

    2013

  6. [13]

    Fujita, On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α, J

    H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = ∆u + u1+α, J. Fac. Sci. Tokyo Sect. IA Math. 13 (1966), 109-124

    1966

  7. [14]

    Grigor’yan & J

    A. Grigor’yan & J. Hu, Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces, Invent. Math. 174 (2008), 81-126

    2008

  8. [15]

    H¨ aseler,Heat kernel estimates and related inequalities on metric graphs , arXiv:1101.3010v1

    S. H¨ aseler,Heat kernel estimates and related inequalities on metric graphs , arXiv:1101.3010v1

    arXiv

  9. [16]

    Hoffman & M

    A. Hoffman & M. Holzer, Invasion fronts on graphs: the Fisher-KPP equation on homogeneous trees and Erdos-R´ eyni graphs, arXiv:1610.06877

    arXiv

  10. [17]

    Jones, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions , Rocky Mount

    C. Jones, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions , Rocky Mount. J. Math. 13 (1983), 355-364

    1983

  11. [18]

    Kato, Perturbation Theory for Linear Operators (Springer, 1980)

    T. Kato, Perturbation Theory for Linear Operators (Springer, 1980)

    1980

  12. [19]

    Keller, D

    M. Keller, D. Lenz, H. Vogt & R. Wojciechowski, Note on basic features of large time behaviour of heat kernels, J. Reine Angew. Math. 708 (2015), 73-95

    2015

  13. [20]

    Lakshmikantham, D

    V. Lakshmikantham, D. D. Bainov & P. S. Simeonov, Theory of Impulsive Differential Equations (World Scientific, 1989). 34 PUNZO AND TESEI

    1989

  14. [21]

    Matano, F

    H. Matano, F. Punzo & A. Tesei, Front propagation for nonlinear diffusion equations on the hyperbolic space, J. European Math. Soc. 17 (2015), 1199-1227

    2015

  15. [22]

    Maury, D

    B. Maury, D. Salort & C. Vannier, Trace theorem for trees and application to the human lungs , Netw. Heterog. Media 4 (2009) , 469-500

    2009

  16. [23]

    Mugnolo, Semigroup Methods for Evolution Equations on Networks (Springer, 2014)

    D. Mugnolo, Semigroup Methods for Evolution Equations on Networks (Springer, 2014)

    2014

  17. [24]

    Mugnolo & J.-F

    D. Mugnolo & J.-F. Rault, Construction of exact travelling waves for the Benjamin-Bona-Mahony equa- tion on networks , Bull. Belg. Math. Soc. Simon Stevin 21 (2014), 415-436

    2014

  18. [25]

    Naimark & M

    K. Naimark & M. Solomyak, Eigenvalue estimates for the weighted Laplacian on metric trees , Proc. London Math. Soc. 80 (2000), 690-724

    2000

  19. [26]

    Naimark & M

    K. Naimark & M. Solomyak, Geometry of Sobolev spaces on regular trees and the Hardy inequalities , Russian J. Math. Phys. 8 (2001), 322-335

    2001

  20. [27]

    Nicaise, Some results on spectral theory over networks applied to nerve impulse transmission

    S. Nicaise, Some results on spectral theory over networks applied to nerve impulse transmission . In: Polynˆ omes orthogonaux et applications, C. Brezinski, A. Draux, A.P. Magnus, P. Maroni & A. Ronveaux Eds., Lecture Notes in Math. 1171, pp 532-541 (Springer, 1985)

    1985

  21. [28]

    Punzo & A

    F. Punzo & A. Tesei, Blow-up on metric graphs and Riemannian manifolds , Discrete Contin. Dyn. Syst. Ser. B 28 (2023), 6362-6392

    2023

  22. [30]

    Rach˘ unkov´ a & J

    I. Rach˘ unkov´ a & J. Tomeˇ cek,Impulsive BVPs with nonlinear boundary conditions for the second order differential equations without growth restrictions , J. Math. Anal. Appl. 292 (2004), 525-539

    2004

  23. [31]

    Rach˘ unkov´ a & J

    I. Rach˘ unkov´ a & J. Tomeˇ cek,Singular Dirichlet problem for ordinary differential equation with impulses , Nonlinear Anal. 65 (2006), 210-229

    2006

  24. [32]

    J. M. Ramirez, Population persistence under advection-diffusion in river networks , J. Math. Biol. 65 (2012), 919-942

    2012

  25. [33]

    Sarhad, R

    J. Sarhad, R. Carlson & K. E. Anderson, Population persistence in river networks , J. Math. Biol. 69 (2014), 401-448

    2014

  26. [34]

    Sarhad, S

    J. Sarhad, S. Manifold & K. E. Anderson, Geometric indicators of population persistence in branching continuous-space networks, J. Math. Biol. 74 (2017), 981-1009

    2017

  27. [35]

    Sherwin, V

    S. Sherwin, V. Franke, J. Peiro & K. Parker, One-dimensional modeling of a vascular network in space- time variables, J. Eng. Math. 47 (2003), 217-250

    2003

  28. [36]

    Sobolev & M

    A.V. Sobolev & M. Solomyak, Schr¨ odinger operator on homogeneous metric trees: spectrum in gaps , Rev. Math. Phys. 14 (2002), 421-467

    2002

  29. [37]

    Solomyak, Laplace and Schr¨ odinger operators on regular metric trees: the discrete spectrum case , arXiv:math/0111023v1

    M. Solomyak, Laplace and Schr¨ odinger operators on regular metric trees: the discrete spectrum case , arXiv:math/0111023v1

    arXiv

  30. [38]

    Solomyak, On the spectrum of the Laplacian on regular metric trees , Waves Random Complex Media 14 (2004), 155-171

    M. Solomyak, On the spectrum of the Laplacian on regular metric trees , Waves Random Complex Media 14 (2004), 155-171. Special section on quantum graphs

    2004

  31. [39]

    C.J. Stam, P. Tewarie, E. Van Dellen, E.C.W. van Straaten, A. Hillebrand & P. Van Mieghem, The Trees and the Forest: Characterization of complex brain networks with minimum spanning trees , Int. J. Psychophysiol. 92 (2014), 129-138

    2014

  32. [40]

    Villacorta, J

    J.A. Villacorta, J. Castro, P. Negredo & C.Avenda˜ no,Mathematical foundations of the dendritic growth models, J. Math. Biol. 55 (2007), 817-859

    2007

  33. [41]

    von Below, A maximum principle for semilinear parabolic network equations

    J. von Below, A maximum principle for semilinear parabolic network equations. In: Differential Equations with Applications in Biology, Physics, and Engineering , J. A. Goldstein, F. Keppel & W. Schappacher Eds, pp. 37-45 (Marcel Dekker, 1991)

    1991

  34. [42]

    von Below, Front propagation in diffusion problems on trees

    J. von Below, Front propagation in diffusion problems on trees . In: Calculus of Variations, Applications and Computations (Pont-` a-Mousson, 1994), Pitman Res. Notes Math. Ser. 326, pp. 254-265 (Longman Sci. Tech., Harlow, 1995)

    1994

  35. [43]

    G. Castelnuovo

    A. Yagi, Abstract Parabolic Equations and their Applications (Springer, 2010). Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, I-20133 Milano, Italy Email address: fabio.punzo@polimi.it Dipartimento di Matematica “G. Castelnuovo”, Universit`a Sapienza di Roma...

    2010

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.