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REVIEW 2 major objections 5 minor 65 references

The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation

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Pith's one-line read Fast diffusion on weighted graphs extinguishes in finite time when m < 2/ν and smooths when m > 2/ν; under stochastic completeness at infinity, mass loss is charged exactly to the killing term.

desk verdict Solid, honest extension of nonlinear diffusion theory to graphs, but the main theorems lean on an unpublished companion preprint for the definition of the solution object. read the letter →

arxiv 2607.23091 v1 pith:SLY7OHQI submitted 2026-07-25 math.AP

Davide Bianchi , Bobo Hua , Alberto G. Setti , Rados{l}aw K. Wojciechowski This is my paper · ORCID
classification math.AP MSC 35K5535R0247H0605C63
keywords generalizedporousmediumequationfiltrationgraphLaplaciansfinite-timeextinctionfastdiffusionSobolevinequalityongraphsstochasticcompletenessatinfinitymassconservation
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

This paper studies the generalized porous medium equation — ∂t u + Δ(φ(u)) = f — on infinite weighted graphs, where the Laplacian may carry arbitrary edge weights, a killing term, and vertices with infinitely many neighbors. Its central aim is to show that two qualitative features of nonlinear diffusion on Euclidean space survive in this discrete setting with the same critical exponents: finite-time extinction for fast diffusion, and exact accounting of mass. The authors prove, under a ν-Sobolev inequality (a bound on the ℓν norm of compactly supported functions by their Dirichlet energy), that ℓ1 solutions with porous-medium nonlinearity φ(s) = s|s|^{m−1} vanish in finite time when m < 2/ν, with an explicit extinction time, and become instantly smooth for m > 2/ν — recovering the Euclidean threshold mc = (N−2)/N on Cayley graphs of polynomial growth N. They also prove an exact generalized mass balance under stochastic completeness at infinity (the only bounded λ-harmonic functions are trivial, so no heat can leak to infinity): the only mass removed is the mass absorbed by the killing term, and when there is no killing, mass is conserved. Along the way, every graph admits minimal and maximal pointwise solutions for arbitrary bounded initial data, with no assumptions beyond the standing ones. A sympathetic reader would care because these are quantitative, checkable statements — explicit extinction times and an exact conservation law — for a class of graphs that earlier discrete theory did not reach.

What carries the argument

The load-bearing mechanism is a single differential energy inequality (4.8), proved at the implicit-Euler level: ∥u(t)∥q_q + K_{m,q,ν} M^{−γq} ∫ₛᵗ ∥u(τ)∥q^{qδq} dτ ≤ ∥u(s)∥q_q. Its exponents obey δq < 1 exactly when m < 2/ν and δq > 1 exactly when m > 2/ν, so feeding (4.8) into a scalar Gronwall-type comparison (Lemma A.2) yields extinction in the first regime and smoothing in the second — one identity, two theorems. The proof substitutes for the missing chain rule the signed-power inequality (A.1), (b^σ−a^σ)(b^τ−a^τ) ≥ c_{σ,τ} |b^{(σ+τ)/2} − a^{(σ+τ)/2}|², and interpolates between ℓ1 and the Sobolev exponent. The second mechanism is the no-flux identity (Lemma 5.3): under (SC∞), v, Δv ∈ ℓ1∩

What would settle it

Run the implicit Euler scheme (2.4) on a large finite box in ℤN (N ≥ 3) with φ(s)=s|s|^{m−1}, m just below 2/ν, and check whether the measured ℓq norm ever stays strictly above the truncated power [∥u0∥q^{q(1−δq)} − K(1−δq)M^{−γq}t]_{+}^{1/(1−δq)} on the right of (4.15); a violation would disprove the energy inequality (4.8) or the constant K_{m,q,ν}. A second, qualitative check: on the stochastically incomplete birth–death graph of Remark 5.6 with φ(s)=s|s|, the paper predicts exact mass conservation for every nonnegative ℓ1 mild solution — observing any ℓ1 deficit there would delimit the cla

Watch

Extended reading notes

Core claim

Central to the paper is a dichotomy governed by one ratio. For ∂t u + Δ(u|u|^{m−1}) = 0 with ℓ1 data on a graph satisfying a ν-Sobolev inequality (ν>2): if m < 2/ν, every mild solution extinguishes in finite time, via bound (4.15); if m > 2/ν, every ℓ1 solution instantly enters every ℓq, with ∥u(t)∥q ≤ C t^{−θ} M^σ. On Cayley graphs of polynomial growth of order N ≥ 3 the threshold is mc = (N−2)/N, the Euclidean exponent. The second pillar is exact mass accounting: on graphs stochastically complete at infinity, every nonnegative ℓ1-mild solution obeys ∥u(t)∥1 + ∫₀ᵗ Σx κ(x)φ(u(s,x))ds = ∥u0∥1, so every lost unit is charged to the killing term; the same law holds for classical and bounded poin

Load-bearing premise

The load-bearing premise is that the ℓ1 solution theory used by the main theorems is already in place — existence, uniqueness, order-preserving resolvents, and convergence of finite-graph resolvents are all imported from the companion preprint [10] (Theorem 4.1, Lemma 4.4(b), Remarks 4.2 and 5.4), not proved here — so if any of those unpublished results fails on non-locally finite graphs or with unbounded killing, the extinction, smoothing, and mass-balance theorems lose thei

Editorial extensions

If this is right

  • On every graph with a ν-Sobolev inequality (ν>2), the fast-diffusion range 0 < m < 2/ν gives finite-time extinction of ℓ1 mild solutions with an explicit extinction-time bound; at the critical choice q = α the bound is ∥u(t)∥α_α ≤ [∥u0∥α^{1−m} − K(1 − 2/ν)t]_{+}^{ν/(ν−2)}.
  • In the complementary range m > 2/ν, every ℓ1 initial datum instantly produces an ℓq solution for every q > 1, with the explicit smoothing bound ∥u(t)∥q ≤ C t^{−θ} M^σ.
  • On Cayley graphs of polynomial volume growth of order N ≥ 3, the threshold is exactly mc = (N−2)/N — the Euclidean critical exponent (1/2 on the Heisenberg-type group, N = 4) — and the dichotomy is stable under adding any killing term.
  • Under stochastic completeness at infinity, every nonnegative ℓ1 mild solution satisfies the exact balance ∥u(t)∥1 + ∫₀ᵗ Σx κ(x)φ(u(s,x))ds = ∥u0∥1 with arbitrary killing; when κ = 0 this is conservation of mass, and on polynomial-growth Cayley graphs it holds as pure mass conservation throughout mc < m < 1.
  • For arbitrary bounded initial data, minimal and maximal global pointwise solutions exist on every graph (no local finiteness or bounded degree required); under bounded degree and locally Lipschitz nonlinearity the ℓ∞ Cauchy problem is globally well-posed, and the minimal nonnegative pointwise solution also extinguishes for m < 2/ν, even for data not in ℓ1.

Reading between the lines

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

  • The open borderline m = 2/ν should interpolate between the two regimes — likely power-law decay with logarithmic corrections, as on the Euclidean critical line — and pinning that rate on ℤN would complete the dichotomy; the paper stops at m ≠ 2/ν.
  • Because the proofs never use local finiteness, the same Sobolev-energy machinery should transfer to the nonlocal operators the paper cites as motivation (fractional Laplacians, Dirichlet-to-Neumann maps); the authors flag but do not carry out that transfer.
  • The birth–death example in Remark 5.6 shows the mass balance can hold without stochastic completeness, so (SC∞) is sufficient but not necessary; characterizing, for a fixed φ, exactly which graphs satisfy the balance would sharpen the scope of Theorem 5.5.
  • For signed classical solutions the accounting law requires the killing contribution to be absolutely integrable in time; finding natural hypotheses that force that integrability, rather than assuming it, would extend the balance to genuinely sign-changing evolutions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the generalized porous medium equation (GPME) on infinite weighted graphs, with emphasis on the case where the nonlinearity is the signed power φ(s)=s|s|^{m-1}. In Section 3, for arbitrary bounded initial data and continuous ℓ∞-valued forcing, the authors use finite-subgraph comparison principles and a monotone exhaustion argument to construct lower and upper extremal global pointwise solutions; under bounded degree and local Lipschitz nonlinearity they obtain uniqueness in ℓ∞. In Section 4, assuming a ν-Sobolev inequality, they derive energy estimates for ℓ1-mild solutions and obtain quantitative finite-time extinction for 0<m<2/ν and ℓ1-ℓq smoothing for m>2/ν, recovering the Euclidean critical exponent on Cayley graphs of polynomial growth. In Section 5, under stochastic completeness at infinity, they prove an exact generalized mass balance with arbitrary killing term, reducing to conservation of mass when κ=0, for mild solutions and for suitable classical and bounded pointwise solutions. The main results are Theorems 4.6, 4.9, and 5.5.

Significance. If the underlying mild-solution theory is sound, the paper delivers genuinely quantitative extinction and smoothing estimates with explicit constants, and an exact mass-balance law allowing arbitrary killing and non-locally finite graphs. The energy estimate in Lemma 4.5 is carefully derived, the exponent algebra is consistent, and the limit passages in Theorem 5.5 are handled with appropriate care. The recovery of the Euclidean critical exponent m_c=(N-2)/N on Cayley graphs of polynomial growth is a strong and appealing feature. No parameters are fitted: the critical exponent emerges from the Sobolev exponent and the estimates are explicit. However, the existence, uniqueness, and approximation of the ℓ1-mild solution — the very object on which Theorems 4.6, 4.9, and 5.5 act — are imported entirely from the unpublished companion preprint [10]. The in-paper derivations are internally consistent conditional on that import, but the contribution is not self-contained and the central theorems inherit any potential defect in [10].

major comments (2)
  1. [§4.1, Theorem 4.1 and Lemma 4.4; Remarks 4.2 and 5.4] The ℓ1-mild solution used throughout Sections 4 and 5 is not constructed in this paper. Theorem 4.1 is imported from [10, Theorems 3.11 and 4.6], Lemma 4.4(b) from [10, Lemma 3.8], and Remark 5.4 again selects the operator A via [10, Theorem 3.11]. Consequently Theorems 4.6, 4.9, 5.5 and Corollary 5.7 are statements about an object whose existence and approximation properties are assumed from an unpublished companion preprint by overlapping authors. This is load-bearing. Please either include the necessary statements and proofs in an appendix, update to a published reference, or explicitly reformulate the theorems as conditional on the construction in [10].
  2. [Definition 2.9; Theorem 4.1; Remark 5.4] The uniqueness statement in Theorem 4.1 is only uniqueness for the selected m-accretive restriction A, not uniqueness for the formal Cauchy problem as defined in Definition 2.9. Since A⊆L, an ε-approximate solution for A is also an ε-approximate solution for L, so the theorem does not rule out different limits arising from different choices of A. Remark 5.4 acknowledges that A need not coincide with L and that the resolvent limit is independent of the exhaustion only for nonnegative data. Theorems 4.6 and 4.9 allow signed initial data, so 'the mild solution' is not shown to be independent of the choice of A. Please clarify how A is selected and state/prove the needed independence, or restrict the theorems accordingly.
minor comments (5)
  1. [Corollary 4.8] There is a notation clash: u0 is used both for the initial datum and for the minimal positive global pointwise solution. Please use different symbols, e.g. u̲ for the solution.
  2. [Theorem 5.10(ii)] The parenthetical proof of (SC∞) from (BD) only establishes h=0 for 0<λ<(2D)^{-1}. The citation [43, Corollary 27] covers all λ, but the inline argument should say so, or add the standard reduction from small λ to all λ.
  3. [Lemma 4.5, Step 1] The convexity inequality |b|^q-|a|^q ≥ q a^{q-1}(b-a) uses the signed-power convention; this is correct but might be unfamiliar. A one-line reference to the convention in Section 4.2 would help.
  4. [Throughout Section 4] The notation δ_q, γ_q is introduced in (4.5), but the reader would benefit from a brief statement that δ_q>0 for all q≥α and δ_q<1 iff m<2/ν, which is already in the text. No substantive issue.
  5. [Remark 3.6(2)] In the stochastically incomplete example, the identity u^A(t)=P_t1-1 is clear, but it rests on the minimal heat semigroup monotone convergence; please state the cited [64] and [44, Chapter 7] near the display for completeness.

Circularity Check

2 steps flagged · score 4.0 of 10

Central extinction/smoothing/mass-balance theorems are conditional on the ℓ1-mild solution imported from companion [10]; no fitted-input or definitional circularity, but the solution object itself is a load-bearing self-citation chain.

  1. self citation load bearing [Theorem 4.1, Section 4.1; see also Remark 4.2]
    "By [10, Theorem 3.11 and its proof, and Theorem 4.6], there exists an m-accretive operator A on ℓ p(X, µ) that is a restriction of L(p) and whose domain is dense in ℓ p(X, µ), that is, A ⊆ L(p) and dom(A) = ℓ p(X, µ). Since A is m-accretive, ... the abstract theory of evolution equations governed by accretive operators applies ... It yields a unique mild solution u ∈ C([0, T]; ℓ p(X, µ)) of (∂ t + A)u = f with u(0) = u 0, realized as the uniform limit of ε-approximate solutions."

    The paper's central object, the ℓ1-mild solution, is generated by the m-accretive restriction A, but Theorem 4.1 does not construct A in this paper. Its existence, density of domain, order-preserving resolvents, and convergence properties are all imported from [10], an unpublished companion preprint with overlapping authorship (three of the four authors of [10] are authors of the present paper). Theorems 4.6, 4.9, and 5.5 are statements about this externally supplied object. If [10, Theorems 3.11 and 4.6] fail on non-locally finite graphs with unbounded killing, the solution concept used in Sections 4–5 has no foundation within this paper. The manuscript explicitly routes the proofs back to [10] in Remark 4.2, so the load-bearing premise reduces to a self-citation that is not independently

  2. uniqueness imported from authors [Lemma 4.4 and Remark 5.4, Sections 4.2 and 5.1]
    "The finite-domain approximation facts needed below are established in [10, Lemma 3.8 and Theorem 3.11 and its proof]. ... Indeed, [10, Theorem 3.11] provides a nested finite exhaustion (Y n) and a dense set Ω ⊆ dom(L) such that A := L| Ω is m-accretive on ℓ 1(X, µ). ... For nonnegative data, the resolvent limit is independent of the exhaustion, see Step 1 in the proof of [10, Theorem 3.11]."

    Lemma 4.4(b) is the bridge that transfers the finite-graph energy estimate to the whole graph in Step 2 of Lemma 4.5: the zero extensions of finite zero-Dirichlet resolvent solutions are asserted to converge in ℓ1 to Jλ. This convergence is not proved here but cited from [10, Lemma 3.8]. Remark 5.4 additionally imports from [10] the exhaustion-selected restriction A and the uniqueness-type statement that the resolvent limit for nonnegative data is independent of the exhaustion. Thus the energy, extinction, smoothing, and mass-balance arguments all inherit an unverified choice of the underlying operator and resolvent approximation from a companion paper by overlapping authors, rather than deriving that object independently.

full rationale

No fitted parameter is relabelled as a prediction, and no definition makes one theorem equal to its input by construction: the critical exponent 2/ν emerges algebraically from the Sobolev exponent in Lemma 4.5, the extinction time in Theorem 4.6 comes from the scalar inequality in Lemma A.2, and the mass balance in Theorem 5.5 follows from the no-flux identity under (SC∞) plus Green-formula arguments. Section 3's pointwise solutions are constructed in-paper via finite exhausions and comparison principles. The circularity, in the narrow sense used by this pass, is concentrated in the definition of the solution object: Theorem 4.1 imports existence, uniqueness, positivity, and the approximation theory of the ℓ1-mild solution from [10], an unpublished companion preprint by overlapping authors; Lemma 4.4 imports the finite-resolvent convergence needed to pass to the whole graph; and Remark 5.4 explicitly routes the choice of the m-accretive realization A back to [10]. Since no part of the present paper re-proves or independently verifies these companion results for non-locally finite graphs with unbounded killing, Theorems 4.6, 4.9, and 5.5 are conditional on a load-bearing self-citation chain. If [10] were machine-checked, code-reproduced, or otherwise independently verified, the score would be 0–2; as written, the central claim still has substantial independent content (the energy/Sobolev dichotomy and the balance law), so the appropriate finding is score 4 rather than higher.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters anywhere: the only constants (C_ν, K_{m,q,ν}) come from the Sobolev inequality and from definitions. The genuine axiom load is structural: (Sν) for the quantitative dichotomy, (SC∞) and (LG)/finite-measure for the mass balance, and — most fragile — the m-accretive restriction machinery of [10] which defines the ℓ1-mild solution itself. The paper is honest about the borderline m = 2/ν being untreated and about (SC∞) being sufficient but not necessary (Remark 5.6).

assumptions (6)
  • domain assumption (Sν): ∥ψ∥_ν^2 ≤ C_ν Q(ψ) for all ψ ∈ C_c(X), some ν > 2
    Section 2.5; the engine of Lemma 4.5 and Theorems 4.6/4.9. Verified for Cayley graphs with polynomial volume growth of order N ≥ 3 (Corollary 4.11), but assumed in general; the borderline m = 2/ν is excluded.
  • ad hoc to paper m-accretive restriction A ⊆ L(p) with dense domain, order-preserving ℓ1-contractive resolvents; finite zero-Dirichlet resolvents converge to Jλ along a fixed exhaustion
    Imported from companion preprint [10, Thms 3.11, 4.6, Lemma 3.8], invoked in Theorem 4.1, Lemma 4.4, Remarks 4.2 and 5.4. Load-bearing for the very definition of the ℓ1-mild solution; not proved or independently verified in this manuscript.
  • domain assumption (SC∞): the only h ∈ ℓ∞(X) with h + λ∆h = 0, λ > 0, is h = 0
    Section 2.5; used in Lemma 5.3 and Theorem 5.5 to rule out mass loss at infinity and to convert the pointwise identity (5.13) into the mass balance.
  • domain assumption No-flux identity: under (SC∞), if v, ∆v ∈ ℓ1 ∩ ℓ∞ then ∑_x ∆v(x)µ(x) = ∑_x κ(x)v(x) (Lemma 5.3)
    Lemma 5.3 cites [44, Lemma 7.22, Cor. 7.27], [32], and [23]; this identity is the engine of the mass balance (5.7). Author overlap on [44].
  • domain assumption (LG): limsup_{r↓0} φ(r)/r < ∞, or alternatively µ(X) < ∞
    Assumption (5.6) in Theorem 5.5, needed for the uniform ℓ1-bound on H_ε = ∫_0^t Φu_ε(s)ds in the implicit Euler scheme; fails for fast diffusion φ(s) = s^m with 0 < m < 1 on infinite-measure graphs.
  • domain assumption (UM) and (C), or (BD) and (FM), for the classical/pointwise mass balances (Theorem 5.10)
    Section 5.2: (UM)+(C) puts classical solutions' Φu and ∆Φu in ℓ1 ∩ ℓ∞; (BD)+(FM) makes bounded pointwise solutions ℓ1-classical and implies (SC∞).

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Pith. "Pith review of The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation." pith.science (2026). https://pith.science/paper/SLY7OHQI

@misc{pith2026260723091,
  author       = {Pith},
  title        = {Pith review of: The generalized porous medium equation on graphs: well-posedness, extinction, and mass conservation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SLY7OHQI}},
  note         = {Machine review of arXiv:2607.23091}
}
abstract

We study the Cauchy problem for the generalized porous medium equation on infinite weighted graphs. For a general nonlinearity, we establish Dirichlet comparison and weak maximum principles on finite subgraphs and, through an exhaustion argument, construct minimal and maximal pointwise solutions for arbitrary $\ell^\infty$ initial data, controlled by explicit, possibly time-dependent, barriers. For the porous nonlinearity $\phi(s)=s|s|^{m-1}$, assuming a $\nu$-Sobolev inequality with $\nu>2$, we derive quantitative energy estimates for $\ell^1$-mild solutions. These yield finite-time extinction in the fast diffusion range $0<m<2/\nu$ and $\ell^1$-$\ell^q$ smoothing in the range $m>2/\nu$. Interestingly, we recover the Euclidean critical exponent for several model graphs. Finally, we prove an exact generalized mass balance for nonnegative $\ell^1$-mild solutions on graphs that are stochastically complete at infinity, allowing for an arbitrary killing term. The same balance is established for suitable classical and bounded pointwise solutions. In the absence of killing, these reduce to conservation of mass.

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Reference graph

Works this paper leans on

65 extracted references · 36 canonical work pages

  1. [10]

    On the accretivity and m-accretivity of Laplacians and porous medium-type operators on graphs

    D. Bianchi, M. Keller, A. G. Setti, and R. K. Wojciechowski.On the accretivity and m-accretivity of Laplacians and porous medium-type operators on graphs. Preprint. 2026.DOI:10.48550/arXiv.2607.09625. arXiv: 2607.09625 [math.FA]

    work page Pith review arXiv doi:10.48550/arxiv.2607.09625 2026

  2. [32]

    Global properties of Dirichlet forms in terms of Green’s formula

    S. Haeseler, M. Keller, D. Lenz, J. Masamune, and M. Schmidt. “Global properties of Dirichlet forms in terms of Green’s formula”. In:Calc. Var. Partial Differential Equations56.5 (2017), Paper No. 124, 43.ISSN: 0944- 2669,1432-0835.DOI:10.1007/s00526-017-1216-7.URL:https://doi.org/10.1007/s00526-017- 1216-7

    work page doi:10.1007/s00526-017-1216-7.url:https://doi.org/10.1007/s00526-017- 2017

  3. [23]

    Parabolicity and stochastic completeness of manifolds in terms of the Green formula

    A. Grigor’yan and J. Masamune. “Parabolicity and stochastic completeness of manifolds in terms of the Green formula”. In:Journal de Math ´ematiques Pures et Appliqu ´ees100.5 (2013), pp. 607–632.DOI:10.1016/j. matpur.2013.01.015.URL:https://doi.org/10.1016/j.matpur.2013.01.015

    doi:10.1016/j 2013

  4. [1]

    Rates of convergence to non-degenerate asymptotic profiles for fast diffusion via energy methods

    G. Akagi. “Rates of convergence to non-degenerate asymptotic profiles for fast diffusion via energy methods”. In:Archive for Rational Mechanics and Analysis247.2, 23 (2023).DOI:10.1007/s00205-023-01843-2

    work page doi:10.1007/s00205-023-01843-2 2023

  5. [2]

    The porous medium equation

    D. G. Aronson. “The porous medium equation”. In:Nonlinear Diffusion Problems: Lectures given at the 2nd 1985 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held at Montecatini Terme, Italy June 10 – June 18, 1985. Ed. by A. Fasano and M. Primicerio. V ol. 1224. Lecture Notes in Mathematics. Berlin, Hei- delberg: Springer Berlin Heidelbe...

    work page doi:10.1007/bfb0072687 1985

  6. [3]

    A comparison principle for the porous medium equation and its consequences

    B. Avelin and T. Lukkari. “A comparison principle for the porous medium equation and its consequences”. In: Revista matem´atica iberoamericana33.2 (2017), pp. 573–594.DOI:10.4171/RMI/950

    work page doi:10.4171/rmi/950 2017

  7. [4]

    Barbu.Nonlinear differential equations of monotone types in Banach spaces

    V . Barbu.Nonlinear differential equations of monotone types in Banach spaces. Springer Monographs in Math- ematics. Springer, New York, 2010, pp. x+272.ISBN: 978-1-4419-5541-8; 978-1-4419-5542-5.DOI:10.1007/ 978-1-4419-5542-5.URL:https://doi.org/10.1007/978-1-4419-5542-5

    doi:10.1007/978-1-4419-5542-5 2010

  8. [5]

    The Degree of Polynomial Growth of Finitely Generated Nilpotent Groups

    H. Bass. “The Degree of Polynomial Growth of Finitely Generated Nilpotent Groups”. In:Proceedings of the London Mathematical Society. 3rd ser. 25.4 (1972), pp. 603–614.DOI:10.1112/plms/s3-25.4.603

    doi:10.1112/plms/s3-25.4.603 1972

Show all 65 references
  1. [6]

    Nonlinear evolution equations in Banach spaces

    P. B ´enilan, M. G. Crandall, and A. Pazy. “Nonlinear evolution equations in Banach spaces”. Preprint. 1988

    1988

  2. [7]

    Semilinear diffusion equations on infinite graphs: the dissipative and Lipschitz cases

    E. Berchio, D. Bianchi, A. G. Setti, and M. Vallarino. “Semilinear diffusion equations on infinite graphs: the dissipative and Lipschitz cases”. In:Journal of Differential Equations476, 114485 (2026).DOI:10.1016/j. jde.2026.114485

    2026

  3. [8]

    Berchio, F

    E. Berchio, F. Santagati, and M. Vallarino.The fractional Porous Medium Equation on graphs. 2026.DOI: 10.48550/arXiv.2606.23360. arXiv:2606.23360 [math.AP].URL:https://arxiv.org/abs/2606. 23360. 43 Bianchi · Hua · Setti · Wojciechowski GPME on graphs: well-posedness, extinctio...

    2026 doi arXiv

  4. [9]

    Biagi, G

    S. Biagi, G. Meglioli, and F. Punzo.Phragm `en–Lindel¨of type theorems for parabolic equations on infinite graphs. 2025.DOI:10 . 48550 / arXiv . 2504 . 08407. arXiv:2504 . 08407 [math.AP].URL:https : / / arxiv.org/abs/2504.08407

    2025 doi arXiv

  5. [11]

    Laplacian cut-offs, porous and fast diffusion on manifolds and other applications

    D. Bianchi and A. G. Setti. “Laplacian cut-offs, porous and fast diffusion on manifolds and other applications”. In:Calc. Var. Partial Differential Equations57.1 (2018), Paper No. 4, 33.ISSN: 0944-2669.DOI:10.1007/ s00526-017-1267-9.URL:https://doi.org/10.1007/s00526-017-1267-9

    2018 doi

  6. [12]

    The generalized porous medium equation on graphs: exis- tence and uniqueness of solutions withℓ 1 data

    D. Bianchi, A. G. Setti, and R. K. Wojciechowski. “The generalized porous medium equation on graphs: exis- tence and uniqueness of solutions withℓ 1 data”. In:Calc. Var. Partial Differential Equations61.5 (2022), Paper No. 171, 42.ISSN: 0944-2669.DOI:10.1007/s00526-022-02249-w...

    2022 doi

  7. [13]

    The Cauchy-Dirichlet problem for the fast diffusion equation on bounded do- mains

    M. Bonforte and A. Figalli. “The Cauchy-Dirichlet problem for the fast diffusion equation on bounded do- mains”. In:Nonlinear Analysis239, 113394 (2024).DOI:10.1016/j.na.2023.113394

    2024

  8. [14]

    Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations

    M. Bonforte and J. L. V ´azquez. “Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations”. In:Adv. Math.223.2 (2010), pp. 529–578.DOI:10.1016/j.aim.2009.08.021

    2010 doi

  9. [15]

    Cauchy-Dirichlet problems for the porous medium equation

    M. Bowen, J. R. King, and T. P. Witelski. “Cauchy-Dirichlet problems for the porous medium equation”. In: Discrete and Continuous Dynamical Systems43.3–4 (2023), pp. 1143–1174.DOI:10.3934/dcds.2022182

    2023 doi

  10. [16]

    Aronson–B ´enilan estimates for the fast diffusion equation under the Ricci flow

    H.-D. Cao and M. Zhu. “Aronson–B ´enilan estimates for the fast diffusion equation under the Ricci flow”. In: Nonlinear Analysis170 (2018), pp. 258–281.DOI:10.1016/j.na.2018.01.006

    2018 doi

  11. [17]

    Asymptotics near extinction for nonlinear fast diffusion on a bounded domain

    B. Choi, R. J. McCann, and C. Seis. “Asymptotics near extinction for nonlinear fast diffusion on a bounded domain”. In:Archive for Rational Mechanics and Analysis247.2, 16 (2023).DOI:10.1007/s00205- 023- 01850-3

    2023 doi

  12. [18]

    Near field asymptotic behavior for the porous medium equation on the half-line

    C. Cort ´azar, F. Quir´os, and N. Wolanski. “Near field asymptotic behavior for the porous medium equation on the half-line”. In:Advanced Nonlinear Studies17.2 (2017), pp. 245–254.DOI:10.1515/ans-2017-0006

    2017 doi

  13. [19]

    Near-field asymptotics for the porous medium equation in exterior do- mains. The critical two-dimensional case

    C. Cort ´azar, F. Quir´os, and N. Wolanski. “Near-field asymptotics for the porous medium equation in exterior do- mains. The critical two-dimensional case”. In:SIAM Journal on Mathematical Analysis50.3 (2018), pp. 2664– 2680.DOI:10.1137/16M110191X

    2018 doi

  14. [20]

    Generation of semi-groups of nonlinear transformations on general Banach spaces

    M. G. Crandall and T. M. Liggett. “Generation of semi-groups of nonlinear transformations on general Banach spaces”. In:American Journal of Mathematics93.2 (1971), pp. 265–298.DOI:10.2307/2373376

    1971 doi

  15. [21]

    Maximum principle for parabolic inequalities and the heat flow on open manifolds

    J. Dodziuk. “Maximum principle for parabolic inequalities and the heat flow on open manifolds”. In:Indiana Univ. Math. J.32.5 (1983), pp. 703–716.ISSN: 0022-2518,1943-5258.DOI:10.1512/iumj.1983.32.32046. URL:https://doi.org/10.1512/iumj.1983.32.32046

    1983 doi

  16. [22]

    Aronson–B ´enilan gradient estimates for porous medium equations under lower bounds ofN- weighted Ricci curvature withN <0

    Y . Fujitani. “Aronson–B ´enilan gradient estimates for porous medium equations under lower bounds ofN- weighted Ricci curvature withN <0”. In:Kyushu Journal of Mathematics78.1 (2024), pp. 209–223.DOI: 10.2206/kyushujm.78.209

    2024 doi

  17. [24]

    The porous medium equation on noncompact manifolds with non- negative Ricci curvature: A Green function approach

    G. Grillo, D. D. Monticelli, and F. Punzo. “The porous medium equation on noncompact manifolds with non- negative Ricci curvature: A Green function approach”. In:Journal of Differential Equations430, 113191 (2025). DOI:10.1016/j.jde.2025.02.062

    2025 doi

  18. [25]

    Smoothing effects for the porous medium equation on Cartan–Hadamard mani- folds

    G. Grillo and M. Muratori. “Smoothing effects for the porous medium equation on Cartan–Hadamard mani- folds”. In:Nonlinear Analysis131 (2016), pp. 346–362.DOI:10.1016/j.na.2015.07.029

    2016 doi

  19. [26]

    A porous medium equation with rough weights: sharp Widder theory

    G. Grillo, M. Muratori, T. Petitt, and N. Simonov. “A porous medium equation with rough weights: sharp Widder theory”. In:Transactions of the American Mathematical Society(2026).DOI:10.1090/tran/9787

    2026 doi

  20. [27]

    The porous medium equation with large initial data on negatively curved Riemannian manifolds

    G. Grillo, M. Muratori, and F. Punzo. “The porous medium equation with large initial data on negatively curved Riemannian manifolds”. In:Journal de Math ´ematiques Pures et Appliqu ´ees113 (2018), pp. 195–226.DOI: 10.1016/j.matpur.2017.07.021

    2018 doi

  21. [28]

    The porous medium equation with measure data on negatively curved Riemannian manifolds

    G. Grillo, M. Muratori, and F. Punzo. “The porous medium equation with measure data on negatively curved Riemannian manifolds”. In:Journal of the European Mathematical Society20.11 (2018), pp. 2769–2812.DOI: 10.4171/JEMS/824

    2018 doi

  22. [29]

    The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour

    G. Grillo, M. Muratori, and J. L. V ´azquez. “The porous medium equation on Riemannian manifolds with negative curvature. The large-time behaviour”. In:Advances in Mathematics314 (2017), pp. 328–377.DOI: 10.1016/j.aim.2017.04.023. 44 Bianchi · Hua · Setti · Wojciechowski GPME ...

    2017 doi

  23. [30]

    The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case

    G. Grillo, M. Muratori, and J. L. V ´azquez. “The porous medium equation on Riemannian manifolds with negative curvature: the superquadratic case”. In:Mathematische Annalen373.1–2 (2019), pp. 119–153.DOI: 10.1007/s00208-018-1680-1

    2019 doi

  24. [31]

    Croissance polynomiale et p ´eriodes des fonctions harmoniques

    Y . Guivarc’h. “Croissance polynomiale et p ´eriodes des fonctions harmoniques”. In:Bulletin de la Soci ´et´e Math´ematique de France101 (1973), pp. 333–379.DOI:10.24033/bsmf.1764

    1973 doi

  25. [33]

    Cheeger estimates of Dirichlet-to-Neumann operators on infinite subgraphs of graphs

    B. Hua, Y . Huang, and Z. Wang. “Cheeger estimates of Dirichlet-to-Neumann operators on infinite subgraphs of graphs”. In:J. Spectr. Theory12.3 (2022), pp. 1079–1108.ISSN: 1664-039X,1664-0403.DOI:10.4171/ jst/427.URL:https://doi.org/10.4171/jst/427

    2022 doi

  26. [34]

    The existence of extremal functions for discrete Sobolev inequalities on lattice graphs

    B. Hua and R. Li. “The existence of extremal functions for discrete Sobolev inequalities on lattice graphs”. In: Journal of Differential Equations305 (2021), pp. 224–241.DOI:10.1016/j.jde.2021.10.016

    2021 doi

  27. [35]

    Stochastic completeness for graphs with curvature dimension conditions

    B. Hua and Y . Lin. “Stochastic completeness for graphs with curvature dimension conditions”. In:Adv. Math. 306 (2017), pp. 279–302.ISSN: 0001-8708.DOI:10.1016/j.aim.2016.10.022.URL:https://doi.org/ 10.1016/j.aim.2016.10.022

    2017 doi

  28. [36]

    Time regularity and long-time behavior of parabolicp-Laplace equations on infinite graphs

    B. Hua and D. Mugnolo. “Time regularity and long-time behavior of parabolicp-Laplace equations on infinite graphs”. In:J. Differential Equations259.11 (2015), pp. 6162–6190.ISSN: 0022-0396,1090-2732.DOI:10. 1016/j.jde.2015.07.018.URL:https://doi.org/10.1016/j.jde.2015.07.018

    2015 doi

  29. [37]

    Bubbling and extinction for some fast diffusion equations in bounded domains

    T. Jin and J. Xiong. “Bubbling and extinction for some fast diffusion equations in bounded domains”. In: Transactions of the American Mathematical Society, Series B10.36 (2023), pp. 1287–1332.DOI:10.1090/ btran/165

    2023

  30. [38]

    Extinction profiles for the Sobolev critical fast diffusion equation in bounded domains. I. One bubble dynamics

    T. Jin and J. Xiong. “Extinction profiles for the Sobolev critical fast diffusion equation in bounded domains. I. One bubble dynamics”. In:Archive for Rational Mechanics and Analysis250.2, 14 (2026).DOI:10.1007/ s00205-026-02177-5

    2026

  31. [39]

    Optimal boundary regularity for fast diffusion equations in bounded domains

    T. Jin and J. Xiong. “Optimal boundary regularity for fast diffusion equations in bounded domains”. In:Ameri- can Journal of Mathematics145.1 (2023), pp. 151–219.DOI:10.1353/ajm.2023.0003

    2023

  32. [40]

    Regularity of solutions to the Dirichlet problem for fast diffusion equations

    T. Jin and J. Xiong. “Regularity of solutions to the Dirichlet problem for fast diffusion equations”. In:Advances in Mathematics478, 110390 (2025).DOI:10.1016/j.aim.2025.110390

    2025

  33. [41]

    Singular extinction profiles of solutions to some fast diffusion equations

    T. Jin and J. Xiong. “Singular extinction profiles of solutions to some fast diffusion equations”. In:Journal of Functional Analysis283.7, 109595 (2022).DOI:10.1016/j.jfa.2022.109595

    2022

  34. [42]

    Dirichlet forms and stochastic completeness of graphs and subgraphs

    M. Keller and D. Lenz. “Dirichlet forms and stochastic completeness of graphs and subgraphs”. In:J. Reine Angew. Math.666 (2012), pp. 189–223.ISSN: 0075-4102,1435-5345.DOI:10.1515/crelle.2011.122.URL: https://doi.org/10.1515/crelle.2011.122

    2012 doi

  35. [43]

    Unbounded Laplacians on graphs: basic spectral properties and the heat equation

    M. Keller and D. Lenz. “Unbounded Laplacians on graphs: basic spectral properties and the heat equation”. In: Math. Model. Nat. Phenom.5.4 (2010), pp. 198–224.ISSN: 0973-5348.DOI:10.1051/mmnp/20105409.URL: https://doi.org/10.1051/mmnp/20105409

    2010

  36. [44]

    Keller, D

    M. Keller, D. Lenz, and R. K. Wojciechowski.Graphs and discrete Dirichlet spaces. V ol. 358. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Cham, 2021, pp. xv+668.ISBN: 978-3-030-81458-8; 978-3-030-81459-5.DOI:10 . 10...

    2021 doi

  37. [45]

    Gaussian upper bounds, volume doubling and Sobolev inequalities on graphs

    M. Keller and C. Rose. “Gaussian upper bounds, volume doubling and Sobolev inequalities on graphs”. In: Journal de l’ ´Ecole polytechnique — Math´ematiques13 (2026), pp. 965–1005.DOI:10.5802/jep.340.URL: https://jep.centre-mersenne.org/articles/10.5802/jep.340/

    2026 doi

  38. [46]

    Flat fronts and stability for the porous medium equation

    C. Kienzler. “Flat fronts and stability for the porous medium equation”. In:Communications in Partial Differ- ential Equations41.12 (2016), pp. 1793–1838.DOI:10.1080/03605302.2016.1244207

    2016

  39. [47]

    Perron’s method for the porous medium equation

    J. Kinnunen, P. Lindqvist, and T. Lukkari. “Perron’s method for the porous medium equation”. In:Journal of the European Mathematical Society18.12 (2016), pp. 2953–2969.DOI:10.4171/JEMS/658

    2016 doi

  40. [48]

    Porous medium equations with reactions on a metric graph

    R. Kou, P. Lai, and B. Lou. “Porous medium equations with reactions on a metric graph”. In:Discrete and Continuous Dynamical Systems - B38 (2026), pp. 448–462.DOI:10.3934/dcdsb.2026079

    2026 doi

  41. [49]

    Lakshmikantham and S

    V . Lakshmikantham and S. Leela.Differential and Integral Inequalities: Theory and Applications. Volume I: Ordinary Differential Equations. V ol. 55. Mathematics in Science and Engineering. New York and London: Academic Press, 1969. 45 Bianchi · Hua · Setti · Wojciechowski GPM...

    1969

  42. [50]

    Gradient flows of generalized relative entropy and functional inequalities on graphs

    K. Li and X. Xue. “Gradient flows of generalized relative entropy and functional inequalities on graphs”. In: Journal of Mathematical Analysis and Applications543.1, 128862 (2025).DOI:10.1016/j.jmaa.2024. 128862

    2025 doi

  43. [51]

    Porous media equation on locally finite graphs

    L. Ma. “Porous media equation on locally finite graphs”. In:Archivum Mathematicum58.3 (2022), pp. 177– 187.DOI:10.5817/AM2022-3-177

    2022 doi

  44. [52]

    A generalized conservation property for the heat semigroup on weighted man- ifolds

    J. Masamune and M. Schmidt. “A generalized conservation property for the heat semigroup on weighted man- ifolds”. In:Math. Ann.377.3–4 (2020), pp. 1673–1710.ISSN: 0025-5831,1432-1807.DOI:10.1007/s00208- 019-01888-3.URL:https://doi.org/10.1007/s00208-019-01888-3

    2020 doi

  45. [53]

    Ollivier Ricci curvature for general graph Laplacians: heat equation, Laplacian comparison, non-explosion and diameter bounds

    F. M ¨unch and R. K. Wojciechowski. “Ollivier Ricci curvature for general graph Laplacians: heat equation, Laplacian comparison, non-explosion and diameter bounds”. In:Adv. Math.356, 106759 (2019).ISSN: 0001- 8708.DOI:10.1016/j.aim.2019.106759.URL:https://doi.org/10.1016/j.aim...

    2019

  46. [54]

    Nonlinear Integral Inequalities I

    B. G. Pachpatte. “Nonlinear Integral Inequalities I”. In:Inequalities for Differential and Integral Equations. V ol. 197. Mathematics in Science and Engineering. Academic Press, 1998, pp. 99–220.DOI:10.1016/S0076- 5392(98)80004-0

    1998 doi

  47. [55]

    Gradient estimates for porous medium equations under the Ricci flow

    L.-J. Shen, S. Yao, G.-Y . Zhang, and X.-A. Ren. “Gradient estimates for porous medium equations under the Ricci flow”. In:Applied Mathematics-A Journal of Chinese Universities31.4 (2016), pp. 481–490.DOI:10. 1007/s11766-016-3368-1

    2016

  48. [56]

    R. E. Showalter.Monotone operators in Banach space and nonlinear partial differential equations. V ol. 49. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1997, pp. xiii+278. ISBN: 0-8218-0500-2.DOI:10.1090/surv/049.URL:https://doi.org/10.10...

    1997 doi

  49. [57]

    J. L. V ´azquez.A survey on mass conservation, self-similarity and related topics in nonlinear diffusion. 2025. DOI:10.48550/arXiv.2311.18357. arXiv:2311.18357 [math.AP].URL:https://arxiv.org/abs/ 2311.18357

    2025 doi

  50. [58]

    J. L. V ´azquez.Smoothing and decay estimates for nonlinear diffusion equations: equations of porous medium type. V ol. 33. Oxford Lecture Series in Mathematics and Its Applications. Oxford: Oxford University Press, 2006, pp. xiv+234.ISBN: 978-0-19-920297-3.DOI:10 . 1093 / acp...

    2006

  51. [59]

    J. L. V ´azquez.The porous medium equation. Oxford Mathematical Monographs. Mathematical theory. The Clarendon Press, Oxford University Press, Oxford, 2007, pp. xxii+624.ISBN: 978-0-19-856903-9; 0-19- 856903-3.DOI:10 . 1093 / acprof : oso / 9780198569039 . 001 . 0001.URL:https...

    2007

  52. [60]

    Wang.Eigenvalue estimates for the fractional Laplacian on lattice subgraphs

    J. Wang.Eigenvalue estimates for the fractional Laplacian on lattice subgraphs. 2023.DOI:10.48550/arXiv. 2303.15766. arXiv:2303.15766 [math.AP].URL:https://arxiv.org/abs/2303.15766

    2023 doi arXiv

  53. [61]

    Some gradient estimates and Liouville properties of the fast diffusion equation on Riemannian manifolds

    W. Wang, R. Xie, and P. Zhang. “Some gradient estimates and Liouville properties of the fast diffusion equation on Riemannian manifolds”. In:Chinese Annals of Mathematics, Series B42.4 (2021), pp. 529–550.DOI:10. 1007/s11401-021-0276-3

    2021

  54. [62]

    Woess.Random walks on infinite graphs and groups

    W. Woess.Random walks on infinite graphs and groups. V ol. 138. Cambridge Tracts in Mathemat- ics. Cambridge University Press, Cambridge, 2000, pp. xii+334.ISBN: 0-521-55292-3.DOI:10 . 1017 / CBO9780511470967.URL:https://doi.org/10.1017/CBO9780511470967

    doi

  55. [63]

    Heat kernel and essential spectrum of infinite graphs

    R. K. Wojciechowski. “Heat kernel and essential spectrum of infinite graphs”. In:Indiana Univ. Math. J.58.3 (2009), pp. 1419–1442.ISSN: 0022-2518,1943-5258.DOI:10.1512/iumj.2009.58.3575.URL:https: //doi.org/10.1512/iumj.2009.58.3575

    2009 doi

  56. [64]

    Stochastic completeness of graphs

    R. K. Wojciechowski. “Stochastic completeness of graphs”. Ph.D. thesis. New York: The Graduate Center, City University of New York, 2008, p. 87.ISBN: 978-0549-58579-4.URL:https://arxiv.org/abs/0712.1570

    2008 arXiv

  57. [65]

    Fractional Laplace operator and related Schr¨odinger equations on locally finite graphs

    M. Zhang, Y . Lin, and Y . Yang. “Fractional Laplace operator and related Schr¨odinger equations on locally finite graphs”. In:Calc. Var. Partial Differential Equations64.7 (2025), Paper No. 227, 27.ISSN: 0944-2669,1432- 0835.DOI:10.1007/s00526-025-03074-7.URL:https://doi.org/...

    2025 doi

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