Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity

Alessandro Palmieri; Vladimir Georgiev

REVIEW 2 major objections 3 minor 21 references

Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read On the Heisenberg group, the damped wave equation with power nonlinearity has the Fujita critical exponent 1 + 2/Q.

desk verdict Genuinely new Fujita-type result on the Heisenberg group with a sound blow-up proof, but the global-existence half rests on linear decay estimates deferred to a companion preprint. read the letter →

arxiv 1908.02989 v1 pith:7L2FXHJ5 submitted 2019-08-08 math.AP

Vladimir Georgiev , Alessandro Palmieri This is my paper · ORCID

classification math.AP MSC 35B3335L7135R0335B4435B4543A8058J45

keywords dampedwaveequationHeisenberggroupsub-LaplacianFujitaexponentcriticalblow-uptestfunctionmethodexponentiallyweightedenergyspaces

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 damped wave equation $u_{tt} - \Delta_{\mathbb{H}} u + u_t = |u|^p$ on the Heisenberg group $\mathbb{H}^n$, where $\Delta_{\mathbb{H}}$ is the sub-Laplacian and the group has homogeneous dimension $Q = 2n + 2$. It claims that the critical exponent separating global existence from finite-time blow-up is the Fujita exponent $p_{\mathrm{Fuj}}(Q) = 1 + 2/Q$, exactly as in Euclidean space with $Q$ in place of the spatial dimension. For $p$ above this threshold, small initial data in an exponentially weighted energy space produce global solutions with explicit decay rates; for $1 < p \leq p_{\mathrm{Fuj}}(Q)$, solutions whose data have sign-definite averages blow up in finite time. This matters because it shows the Fujita threshold is governed by the homogeneous dimension of the ambient group rather than by the noncommutative details of its sub-Riemannian geometry.

What carries the argument

The load-bearing objects are the Heisenberg group $\mathbb{H}^n$ with homogeneous dimension $Q = 2n + 2$ and the time-dependent exponential weight $\psi(t,\eta) = (|x|^2 + |y|^2 + 4|\tau|)/(8(1+t))$. The weight converts the sub-Laplacian into a Schr\"odinger-type operator with a potential that yields a weighted energy identity: multiplying the equation by $e^{2\psi} u_t$ produces a divergence-plus-time-derivative structure with controllable remainders, and $\psi_t \leq 0$ supplies the sign needed to bound the nonlinearity. On the existence side, Proposition 6.1 supplies the linear decay estimates $\|u(t)\|_{L^2} \lesssim (1+t)^{-Q/4}$ for $L^1$ data, and Lemma 4.3, the weighted Gagliardo-Nirenberg inequality, translates $L^2$-decay into the $L^{2p}$-estimates needed to close Duhamel's integral; the integrability condition that yields $p > 1 + 2/Q$ is read off these decay exponents. On the blow-up side, the test functions $\phi_R(t,x,y,\tau) = \beta(t/R^2) \alpha(x/R) \alpha(y/R) \beta(\tau/R^2)$ localize in boxes $D_R$ of measure $\approx R^Q$, and H\"older's inequality forces $I_R \to 0$ for $p < p_{\mathrm{Fuj}}(Q)$, while the critical case $p = p_{\mathrm{Fuj}}(Q)$ is handled by refining the support analysis.

What would settle it

Check the linear decay estimate directly for the homogeneous damped wave equation on the Heisenberg group with initial data in $L^1 \cap L^2$: test whether $(1+t)^{Q/4}\|u(t)\|_{L^2}$ remains bounded. If the true decay exponent is $\alpha < Q/4$, then the integrability argument that produces the threshold $p > 1 + 2/Q$ breaks down and the claimed critical exponent would not follow from this method; alternatively, exhibiting a global solution for $p = 1 + 2/Q$ with data satisfying (9) would refute the blow-up result.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is a complete Fujita-type dichotomy for the Heisenberg damped wave equation. Theorem 2.3 states that when $1 < p \leq p_{\mathrm{Fuj}}(Q)$ and the Cauchy data satisfy $\liminf_{R \to \infty} \int_{D_R} (u_0 + u_1) \, d\eta > 0$, every local weak solution blows up in finite time, no matter how small the data are. Theorem 2.2 states the complementary half: for $p_{\mathrm{Fuj}}(Q) < p \leq p_{\mathrm{GN}}(Q) = Q/(Q-2)$, there is $\varepsilon_0 > 0$ such that data of weighted-energy norm at most $\varepsilon_0$ admit a unique global solution $u$ with $\|u(t)\|_{L^2} \lesssim (1+t)^{-Q/4}$, $\|\nabla_{\mathbb{H}} u(t)\|_{L^2} \lesssim (1+t)^{-Q/4 - 1/2}$, and $\|u_t(t)\|_{L^2} \lesssim (1+t)^{-Q/4 - 1}$. The blow-up proof uses the test function method with scaled bump functions; the existence proof uses Duhamel's principle, decay estimates for the linear equation, weighted Gagliardo-Nirenberg inequalities, and a contraction argument in exponentially weighted energy spaces.

Load-bearing premise

The global-existence half depends on the linear decay estimates of Proposition 6.1, which are taken from a companion preprint rather than proved here; if those rates—for instance $\|u(t)\|_{L^2} \lesssim (1+t)^{-Q/4}$ for $L^1$ data—were different, the claimed threshold $p > 1 + 2/Q$ would not be forced by the Duhamel argument.

Editorial extensions

If this is right

Global small-data solutions exist exactly when $p > 1 + 2/Q$, with the additional technical ceiling $p \leq Q/(Q-2)$; the decay rates of the linear equation are preserved by the nonlinear problem.
For every $1 < p \leq 1 + 2/Q$, any weak solution whose data have positive limiting average over the boxes $D_R$ blows up in finite time, including at the critical value $p = 1 + 2/Q$.
The threshold is determined only by the homogeneous dimension $Q = 2n + 2$, not by the noncommutativity or the step of the group.
The weighted energy space $A(\mathbb{H}^n)$ is sufficient to absorb the nonlinearity: the $L^2$, gradient, and time-derivative norms all decay at the same rates as the Euclidean case with dimension $Q$.
No critical-case global existence occurs under the sign assumption (9); the lifespan at $p = 1 + 2/Q$ is always finite for such data.

Reading between the lines

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

The same test-function argument should go through verbatim for other stratified Lie groups whose homogeneous dimension is $Q$, since only the measure of the boxes $D_R$ and the scaling of the sub-Laplacian enter; the blow-up half of the dichotomy is thus likely generic, while the existence half will track whatever linear decay estimates are available.
If the companion preprint's linear rates are optimal, then the critical exponent here is optimal too, and lifespan upper bounds for $p \leq 1 + 2/Q$ would be the natural next project, matching the known Euclidean picture.
The exponential weight $\psi(t,\cdot)$ is tailored to the heat-like scaling of the damped wave equation; a similar weight should work for the corresponding semilinear heat equation on the Heisenberg group, connecting the two Fujita results.
A testable extension of the paper's reasoning: replace $|u|^p$ by $|u|^p$ with an additional time factor $t^\beta$, and check whether the threshold shifts to $1 + (2+\beta)/Q$, as it does in Euclidean models.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This is a solid paper, not a repackaging. It proves that the semilinear damped wave equation on the Heisenberg group has the expected Fujita exponent: global small-data solutions for p_Fuj < p ≤ p_GN(Q), and finite-time blow-up for 1 < p ≤ p_Fuj under the sign condition. The Euclidean analogue was known; the Heisenberg case is genuinely new.

What the paper does well: the weighted energy identity (15), the inequality (21), and the weighted Gagliardo-Nirenberg inequality of Lemma 4.3 are consistent and worked out in detail. The blow-up proof in Section 8 is self-contained and handles the critical case with a refined cut-off argument; I checked that part and it holds up. The critical exponent is not put in by hand—it emerges naturally from the decay rates and integrability conditions.

The real soft spot is Proposition 6.1. The L1–L2 and L2–L2 decay estimates are imported from the companion preprint [17] without proof. These rates control every Duhamel term in Section 7, and the threshold p > 1+2/Q is read off exactly those exponents. If the rates in [17] shift, the global existence result shifts. The paper explicitly states the dependency, which is honest, but it is still an unverified external pillar. For a definitive version, either [17] should be available and refereed, or the key estimates should be sketched in an appendix.

Secondary issues: the abstract overclaims by omitting the upper bound p ≤ p_GN(Q) required in Theorem 2.2. That is a presentation slip, not a mathematical error. There are also minor typos—Theorem 2.3 states the solution space as L^p_loc([0,T) × R^n) but it should be H^n, and the exponent near (58) is garbled in places. All fixable.

Bottom line: the central theorem appears correct, and the extension of the weighted-energy method to the Heisenberg group is nontrivial and well executed. The blow-up half can stand alone; the global-existence half should be accepted only with the companion paper's estimates verified or included. I would send this to a serious referee rather than desk-reject, with that condition attached.

Referee Report

2 major / 3 minor

Summary. The manuscript studies the Cauchy problem for the semilinear damped wave equation u_tt - Δ_H u + u_t = |u|^p on the Heisenberg group H^n, with initial data in a weighted energy space. The main claim is that the critical exponent is the Fujita exponent p_Fuj(Q) = 1 + 2/Q, where Q = 2n + 2 is the homogeneous dimension. Section 2 states a local existence theorem (Theorem 2.1), a global small-data existence theorem for p_Fuj(Q) < p ≤ p_GN(Q) (Theorem 2.2), and a finite-time blow-up theorem for 1 < p ≤ p_Fuj(Q) under a sign condition on the data (Theorem 2.3). The strategy combines weighted energy identities with the weight ψ(t,η) = (|x|^2 + |y|^2 + 4|τ|)/(8(1+t)), a weighted Gagliardo-Nirenberg inequality (Lemma 4.3), Duhamel's principle with linear decay estimates (Proposition 6.1, cited from the companion paper [17]), and the test-function method for blow-up. The proof structure is standard and the estimates in the submitted text appear internally consistent, but the global existence theorem is not self-contained and the abstract overstates the range of p for which it is proved.

Significance. If the missing linear-theory input is supplied, the paper establishes a Fujita-type dichotomy for a semilinear damped wave equation on a non-abelian stratified Lie group, with the threshold emerging from the homogeneous dimension rather than the topological dimension. The test-function proof is self-contained and gives the sharp threshold including the critical case. The weighted Gagliardo-Nirenberg inequality and the bootstrap argument are natural extensions of the Euclidean approach. However, the claim is not fully self-contained: Theorem 2.2 is conditional on Proposition 6.1, whose proof is deferred to [17], and the abstract overstates the range of p for which global existence is proved.

major comments (2)

[Section 6, Proposition 6.1] The linear decay estimates (36)-(41) are not proved in this manuscript; the proof is only a reference to [17, Theorem 1.1]. These estimates are load-bearing: every Duhamel term in the proof of Theorem 2.2 is bounded using (36)-(41), and the integrability condition (49), which yields the threshold p > 1 + 2/Q, is obtained from the decay exponent -Q/4 in (36). Thus the global existence theorem is conditional on a companion preprint. The dependency is stated honestly, but for a journal paper the proof of Proposition 6.1, or at least of the rates (36)-(38), should be included or made available; otherwise the central claim is not independently verifiable.
[Abstract / Theorem 2.2] The abstract and introduction state global existence for all p > p_Fuj(Q), but Theorem 2.2 proves it only for 1 < p ≤ p_GN(Q) = Q/(Q-2). The upper bound is used in the proof: before (48) the authors note that Lemma 4.3 applies "thanks to the upper bound p ≤ p_GN(Q)". Since for n = 1 (Q = 4) the proved super-Fujita range is only (3/2, 2], the abstract overstates the result. The statements should be corrected to match the theorem, or the range of p should be extended and discussed.

minor comments (3)

[Theorem 2.3] The statement says u ∈ L^p_loc([0,T) × R^n), but the spatial domain should be the Heisenberg group H^n.
[Lemma 4.1] The phrase "where where C is a nonnegative constant" contains a duplicated word; also, C should presumably be a positive constant.
[Throughout] The text contains several spacing and OCR-type artifacts (e.g., "Pon tecorvo", "Bonc hev", "lin ear"); the manuscript should be carefully proofread before final submission.

Circularity Check

0 steps flagged · score 1.0 of 10

No definitional circularity: the Fujita threshold is derived from linear decay rates and a self-contained blow-up argument, not put in by hand.

full rationale

The blow-up part (Theorem 2.3, Section 8) is self-contained: the test function method yields the estimate I_R ≲ R^{Q-(Q+2)/p} I_R^{1/p}, whose right-hand side forces I_R→0 for p < p_Fuj(Q), and the critical case p = p_Fuj(Q) is handled by refined support sets without importing any nonlinear input. The global-existence part (Theorem 2.2, Section 7) does depend on the linear decay estimates (36)–(41) stated in Proposition 6.1, whose proof is explicitly omitted: 'Proof. See [17, Theorem 1.1], where the group Fourier transform on Hn is applied to prove this result.' This is a genuine external dependency and a load-bearing self-citation, since every Duhamel term in (48) is controlled by those rates and the integrability condition (49), -Qp/2 + Q/2 + δp(Q/4 + 1/2) < -1, is read off them. However, this is not circular in the sense tracked here: [17] is a distinct linear Cauchy problem (10), the decay exponents are not fitted to the semilinear outcome, and the value 1 + 2/Q is the point where (49) stops being integrable rather than an ansatz inserted into the proof. The abstract's omission of the upper bound p ≤ p_GN(Q) is a presentation issue, not a circular step.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard analytical input (GN and Sobolev embeddings on Hn, Bihari's inequality), on trace theorems for the Heisenberg group, and most importantly on the linear decay estimates from the companion preprint [17]. No free parameters are fitted to data; the auxiliary σ and δ are proof devices. No invented entities.

free parameters (2)

Weight exponent σ = σ = 1 in the solution space; σ ∈ (0,1] in auxiliary estimates
The exponential weight e^{σψ(t,·)} uses a positive exponent σ chosen by hand. It is a proof device, not fitted to data; the threshold p_Fuj(Q) does not depend on σ.
Small positive δ in Lemmas 7.1 and proof of Theorem 2.2 = arbitrarily small, constrained by (49)
Introduced to make certain (1+t) powers integrable. Standard epsilon; no fitted value.

assumptions (5)

standard math Gagliardo-Nirenberg inequality on the Heisenberg group (Lemma 4.1)
Cited to [5,19]; used to derive weighted GN (Lemma 4.3) and to bound nonlinear terms. Standard for the sub-Laplacian.
domain assumption Trace theorem for H1(Hn) on the hypersurface {τ=0}
Used in Lemma 4.2 proof, equation (24), to evaluate the integral of ∆_Hψ against |f|²; the δ0(τ) term is dropped after claiming trace existence. Cited to [18,3,1,2].
domain assumption Linear L1-L2 and L2-L2 decay estimates for the damped wave equation on Hn (Proposition 6.1)
Proof deferred to companion preprint [17]. These decay rates are the quantitative engine for the Duhamel estimates in Theorem 2.2; if the rates differed, the Fujita threshold would change.
standard math Bihari's inequality (Lemma 5.1)
Used as a nonlinear Gronwall tool in the local existence contraction argument; proof cited to [4].
standard math Sobolev embedding H1(Hn) → L^{p+1}(Hn) for p+1 ≤ 2Q/(Q-2)
Used in Lemma 7.1 to bound the initial-data term; follows from Lemma 4.1 with θ=1.

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Cite this review

Pith. "Pith review of Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity." pith.science (2026). https://pith.science/paper/7L2FXHJ5

@misc{pith2026190802989,
  author       = {Pith},
  title        = {Pith review of: Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7L2FXHJ5}},
  note         = {Machine review of arXiv:1908.02989}
}

abstract

In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. We prove that the critical exponent is the Fujita exponent $p_{\mathrm{Fuj}}(\mathscr{Q}) = 1+2 / \mathscr{Q}$, where $\mathscr{Q}$ is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for $p >p_{\mathrm{Fuj}}(\mathscr{Q})$ in an exponential weighted energy space. On the other hand, a blow-up result for $1 < p \leq p_{\mathrm{Fuj}}(\mathscr{Q})$ under certain integral sign assumptions for the Cauchy data by using the test function method.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[17]

Palmieri A, Decay estimates for the linear damped wave e quation on the Heisenberg group, preprint, arxiv:1908.02657 (2019)

work page Pith review arXiv 2019
[1]

Bahouri H, Chemin J-Y, Xu C-J, Trace and trace lifting the orems in weighted Sobolev spaces, Journal of the Inst. of Math. Jussieu 4(4) (2005), 509-552

work page 2005
[2]

Bahouri H, Chemin J-Y, Xu C-J, Trace theorem on the Heisen berg group, Ann. Inst. Fourier, Grenoble 59(2) (2009), 491-514

work page 2009
[3]

Berhanu S, Pesenson I, The trace problem for vector ﬁelds satisfying H¨ ormanderâĂŹs condition, Math. Z. 231 (1999), 103-122

work page 1999
[4]

Bihari I, A generalization of a lemma of Bellman and its ap plication to uniqueness problems of diﬀerential equations , Acta Math. Acad. Sci. Hungar. 7(1) (1956), 81-94

work page 1956
[5]

Theory Adv

Chen J, Rocha E M, A Class of Sub-elliptic Equations on the Heisenberg Group and Related Interpolation Inequalities, in: Advances in Harmonic Analysis and Operator Theory, in: O per. Theory Adv. Appl., vol. 229, Birkh¨ auser/Springer, Basel AG, Basel, 2013, 123-137

work page 2013
[6]

Chen W, Palmieri A, W eakly coupled system of semilinear w ave equations with distinct scale-invariant terms in the li near part, Z. Angew. Math. Phys. 70(2) (2019) 70: 67. https://doi.org/10.1007/s00033-019-1 112-4

work page doi:10.1007/s00033-019-1 2019
[7]

D’Abbicco M, The threshold of eﬀective damping for semil inear wave equation, Math. Meth. Appl. Sci. 38 (2015), 1032-1045

work page 2015

Show all 21 references

[8]

B, 34 (2013), 345-380

D’Abbicco M, Lucente S, Reissig M, Semi-linear wave equa tions with eﬀective damping, Chinese Annals of Mathematics, Ser. B, 34 (2013), 345-380

2013
[9]

Qualitative pr operties of solutions, phase space analysis, semilinear models, Birkh¨ auser, Basel, 2018

Ebert M R, Reissig M, Methods for partial diﬀerential equations. Qualitative pr operties of solutions, phase space analysis, semilinear models, Birkh¨ auser, Basel, 2018

2018
[10]

Georgiev V, Palmieri A, Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisen berg group, preprint, arxiv:1905.05696 (2019)

2019 arXiv
[11]

Georgiev V, Palmieri A, Upper bound estimates for local in time solutions to the semilinear heat equation on stratiﬁ ed Lie groups in the sub-Fujita case, preprint
[12]

61(7) (2005), 1189-1208

Ikehata R, Tanizawa K, Global existence of solutions fo r semilinear damped wave equations in RN with noncompactly supported initial data, Nonlinear Anal. 61(7) (2005), 1189-1208

2005
[13]

Matsumura A, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci. 12(1) (1976/77), 169-189

1976
[14]

Steklov Inst

Mitidieri E, Pohozaev S, A priori estimates and the abse nce of solutions of nonlinear partial diﬀerential equation s and inequalities, Proc. Steklov Inst. Math. 234 (2001), 1-362

2001
[15]

Nunes do Nascimento W, Palmieri A, Reissig M, Semi-line ar wave models with power non-linearity and scale-invarian t time-dependent mass and dissipation, Math. Nachr. 290(11/12) (2017), 1779-1805

2017
[16]

Palmieri A, Global existence of solutions for semi-lin ear wave equation with scale-invariant damping and mass in e xpo- nentially weighted spaces, J. Math. Anal. Appl. 461(2) (2018), 1215-1240. 20

2018
[18]

Partial Diﬀerential Equations , 19(3/4) (1994), 655-676

Pesenson I, The trace problem and Hardy operator for non -isotropic function spaces on the Heisenberg group, Comm. Partial Diﬀerential Equations , 19(3/4) (1994), 655-676

1994
[19]

Diﬀerential Equations 265 (2018), 5212-5236

Ruzhansky M, Tokmagambetov N, Nonlinear damped wave eq uations for the sub-Laplacian on the Heisenberg group and for Rockland operators on graded Lie groups J. Diﬀerential Equations 265 (2018), 5212-5236

2018
[20]

Ruzhansky M, Yessirkegenov N, Existence and non-exist ence of global solutions for semilinear heat equations and i n- equalities on sub-Riemannian manifolds, and Fujita expone nt on unimodular Lie groups, preprint, arxiv:1812.01933v2 (2018)

2018 arXiv
[21]

Diﬀerential Equations 174(2) (2001), 464-489

Todorova G, Yordanov B, Critical exponent for a nonline ar wave equation with damping, J. Diﬀerential Equations 174(2) (2001), 464-489. 21

2001

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [17]

Palmieri A, Decay estimates for the linear damped wave e quation on the Heisenberg group, preprint, arxiv:1908.02657 (2019)

work page Pith review arXiv 2019

[2] [1]

Bahouri H, Chemin J-Y, Xu C-J, Trace and trace lifting the orems in weighted Sobolev spaces, Journal of the Inst. of Math. Jussieu 4(4) (2005), 509-552

work page 2005

[3] [2]

Bahouri H, Chemin J-Y, Xu C-J, Trace theorem on the Heisen berg group, Ann. Inst. Fourier, Grenoble 59(2) (2009), 491-514

work page 2009

[4] [3]

Berhanu S, Pesenson I, The trace problem for vector ﬁelds satisfying H¨ ormanderâĂŹs condition, Math. Z. 231 (1999), 103-122

work page 1999

[5] [4]

Bihari I, A generalization of a lemma of Bellman and its ap plication to uniqueness problems of diﬀerential equations , Acta Math. Acad. Sci. Hungar. 7(1) (1956), 81-94

work page 1956

[6] [5]

Theory Adv

Chen J, Rocha E M, A Class of Sub-elliptic Equations on the Heisenberg Group and Related Interpolation Inequalities, in: Advances in Harmonic Analysis and Operator Theory, in: O per. Theory Adv. Appl., vol. 229, Birkh¨ auser/Springer, Basel AG, Basel, 2013, 123-137

work page 2013

[7] [6]

Chen W, Palmieri A, W eakly coupled system of semilinear w ave equations with distinct scale-invariant terms in the li near part, Z. Angew. Math. Phys. 70(2) (2019) 70: 67. https://doi.org/10.1007/s00033-019-1 112-4

work page doi:10.1007/s00033-019-1 2019

[8] [7]

D’Abbicco M, The threshold of eﬀective damping for semil inear wave equation, Math. Meth. Appl. Sci. 38 (2015), 1032-1045

work page 2015