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REVIEW 2 major objections 5 minor 45 references

Boundedness and decay of waves on spatially flat decelerated FLRW spacetimes

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Pith's one-line read This paper claims that on every spatially flat decelerated FLRW spacetime with scale factor $t^q$ for $0<q<1$, solutions of the linear wave equation satisfy uniform twisted-energy bounds, integrated local energy decay, and an…

desk verdict Solid physical-space proof of the full decelerated FLRW wave-decay package, but the abstract's 'optimal' claim outruns the upper-bound theorems—worth a serious referee, not a desk reject. read the letter →

arxiv 2505.16794 v1 pith:DL2SXKUO submitted 2025-05-22 gr-qc math-phmath.APmath.MP

Mahdi Haghshenas This is my paper · ORCID
classification gr-qcmath-phmath.APmath.MP MSC 35L0535B4083C0583F05
keywords linearwaveequationFLRWspacetimesdeceleratedexpansiontwistedenergyr^p-weightedestimatesintegratedlocaldecaypointwisescalefactort^q
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 is trying to establish that waves on a spatially flat decelerated FLRW expanding universe, whose scale factor grows as $t^q$ with $0

What carries the argument

The central mechanism is the twisted energy-momentum tensor (1.21) with twisting function $\beta=t^{-q}$, which replaces $\partial_t\psi$ by the twisted derivative $\partial_t(t^q\psi)$; its divergence identity (1.22) converts wave solutions into spacetime currents. A potential $V=-q(2q-1)t^{-2}$ appears, and its sign change at $q=1/2$ forces a second twisting function $\beta'=t^{q-1}$ for $q>1/2$. The decay engine is the $r^p$ multiplier: applying $t^{-\mu_q}r^{p-1}\partial_v$ to $\varphi=rt^q\psi$ produces identity (3.16) in which every term is a boundary term or has a favorable sign except the mixed error $2r^{p-2}(\gamma-r^2W)\varphi\,\partial_v\varphi$; this error is absorbed using $\gamma=q|1-2q|/|1-q|^2$ and the pointwise bound $rt^{q-1}\le C$. The requirement $2\sqrt{\gamma}<p<2-2\sqrt{\gamma}$ is what produces the admissible range and the parameter $\sigma_q$.

What would settle it

Evolve the radiation-case equation $q=1/2$, which is equivalent to the flat wave equation, with smooth compactly supported data and test the predicted pointwise bound $|\psi|\lesssim 1/t$ in the interior and the $r^2$ weighted estimate (3.13): any data that violate either bound would falsify Theorem 1.3 and Corollary 1.4 in that range. For $q$ between $1/2$ and $1$, a numerical evolution of compact data with finite $E_q[\psi_0,\psi_1]$ should exhibit energy decay at least $\tau^{-(2-\sigma)}$ for every $\sigma>\sigma_q$; a slower measured rate would disprove Proposition 4.1.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that the three standard ingredients of the $r^p$-method hold for the linear wave equation $\Box_{g_q}\psi=0$ on all spatially flat decelerated FLRW spacetimes. For $0<q<1$ and $\varepsilon\ge0$ small, the twisted flux $E_q^\varepsilon[\psi](\tau)$ defined in (1.12) is non-increasing along the hyperboloidal-null foliation; the integrated local energy decay estimate (1.7) holds with weight $\mu_q^\varepsilon$; and for $\varphi=rt^q\psi$ the wave-zone estimate (1.10) holds for all $0<p<\max\{1,2-\sigma_q\}$, where $\sigma_q=2\sqrt{q|1-2q|}/|1-q|^2$. Given these, Corollary 1.4 concludes that for $q\ne1/3$ the energy decays as $\tau^{-(2-\sigma)}$ with arbitrary $\sigma>\sigma_q$ when $\sigma_q<1$ and as $\tau^{-1}$ with a small power loss when $\sigma_q\ge1$, and that $\psi$, $\partial_t\psi$, and the null derivatives decay at the explicit rates listed in (1.11). In the wave zone the $\psi$ rates are optimal for $q\le1/2$ and almost optimal near $q=1/2$.

Load-bearing premise

The hierarchy stands or falls on the uniform inequality $rt^{q-1}\le C$ in the future domain; that bound is what turns the mixed $r$-$t$ error terms into absorbable ones, and it holds only for $q<1$, so the method as written cannot cross into $q\ge1$.

Editorial extensions

If this is right

  • Uniform energy boundedness holds for the twisted flux in both regimes $q\le1/2$ and $q>1/2$, with no growth along the foliation.
  • Integrated local energy decay controls the spacetime integral of a local energy density across the whole decelerated range, with an extra $t$-weight $\mu_q^\varepsilon$ only for $q>1/2$.
  • The $r^p$ hierarchy holds in the wave zone for $p<\max\{1,2-\sigma_q\}$, and in the radiation case $q=1/2$ it extends to $p\le2$.
  • Energy decay is $\tau^{-(2-\sigma)}$ for $q<(1+2\sqrt2)/7$ and $\tau^{-1}$ up to an $\varepsilon$ loss for $q\ge(1+2\sqrt2)/7$; pointwise decay of $\psi$ and its derivatives follows with the rates in (1.11).
  • The estimates for first derivatives in Corollary 1.4 provide explicit rates for $\partial_t\psi$, $\partial_u\psi$, and $\partial_v\psi$, with a logarithmic factor in the radiation case.

Reading between the lines

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

  • The compact-support assumption is not essential: replacing it by a finite twisted weighted flux on $\Sigma_{\tau_0}$ should give the same decay statements, so the results likely extend to all data with the weighted initial energy (4.4).
  • The $\varepsilon$ loss above the threshold $q=(1+2\sqrt2)/7$ is probably a limitation of this multiplier family rather than a property of the equation; a multiplier tuned to the region $u\le0$ may remove the loss.
  • Because the $r^p$-method is designed for quasilinear problems, the linear estimates here are a natural first input for proving small-data global existence of nonlinear wave equations on these backgrounds under a null condition.
  • The excluded case $q=1/3$ is a technical gap rather than a real obstruction: the $0<p<1$ estimates remain valid, and a modified corollary with an arbitrarily small loss should close it.
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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 linear wave equation □_{g_q}ψ=0 on spatially flat FLRW spacetimes M=(0,∞)_t×R^3 with g_q=-dt²+t^{2q}δ_{ij}dx^i dx^j, where 0<q<1 is the decelerated regime. Using twisted energy-momentum currents imported from Dafermos–Holzegel–Rodnianski–Taylor, it proves three main ingredients: uniform boundedness of a twisted energy flux (Theorem 1.1), integrated local energy decay (Theorem 1.2), and a hierarchy of r^p-weighted estimates in the spirit of the Dafermos–Rodnianski method (Theorem 1.3). From these, Corollary 1.4 derives energy decay and pointwise decay of ψ and its first derivatives, with rates split according to q=1/2, q<1/2, 1/2<q<(1+2√2)/7, and q≥(1+2√2)/7; the case q=1/3 is explicitly excluded from the corollary. The proofs are divided into q≤1/2 and q>1/2, and the key Lemma 2.2 controls mixed r-t error terms through the inequality rt^{q-1}≤C, which is precisely the mechanism that uses q<1.

Significance. If the main estimates are correct, the paper provides a self-contained physical-space proof of energy and pointwise decay for waves on decelerated FLRW spacetimes, avoiding the Bessel-function representation used in earlier work by Natário–Rossetti and Wirth. It also extends the Dafermos–Rodnianski r^p method to a class of non-stationary, expanding, conformally flat backgrounds. The paper's strengths are its explicit twisted divergence identities, its transparent multiplier choices, and the fact that the main estimates are derived from inequalities rather than from fitting parameters. The chain from energy boundedness to ILED to r^p estimates and finally to pointwise decay is internally consistent as far as I have checked, and no circularity is evident.

major comments (2)
  1. [Abstract and Section 1.2, Corollary 1.4] The abstract claims that the wave-zone pointwise decay is "optimal" in the radiation and sub-radiation cases and "almost optimal" around the radiation case. The body of the paper, however, proves only upper bounds: Corollary 1.4 and Propositions 4.4 and 4.7 give estimates of the form |ψ|≲C τ^{σ/2}/t, but no matching lower bounds or asymptotic profiles are established for any q. In particular, the t^{-1} bound for q=1/2 is not shown to be attained by generic solutions. The optimality assertion is therefore not supported by the theorems. I recommend either removing or softening this claim, or adding matching lower-bound statements.
  2. [Remark 1.5 and Corollary 1.4] Corollary 1.4 explicitly assumes q≠1/3, while the abstract advertises results across the entire decelerated regime. Remark 1.5 only sketches an "analogous" modified statement for q=1/3 with an arbitrarily small loss, but no proof is given in the paper. If the all-q claim is to be retained, the q=1/3 decay statement should be proved in the text; otherwise the abstract and the theorem statements should be reworded to state the exclusion prominently. This is a boundary-case gap in an otherwise coherent proof structure.
minor comments (5)
  1. [Lemma 2.4] The statement of Lemma 2.4 says h1(τ)≲C/τ^{1−σ}, but the proof ends with h1(τ)≲C/τ^{2−σ}. If the stronger decay is intended, the statement should be updated; if the weaker decay is intended, the final line of the proof should be corrected.
  2. [Equation (1.8)] The displayed definition of σ_q is typeset ambiguously: it is hard to tell whether the denominator |1−q|^2 is inside or outside the square root. Since the admissible p-range in Theorem 1.3 depends on this quantity, the formula should be typeset with an unambiguous fraction.
  3. [Throughout] There are several typos and spacing inconsistencies: "FLR W" appears with inconsistent spacing, "Lorentizian" should be "Lorentzian", "Grönwall" is spelled with variations, and "Klien–Gordon" in the related-works section should be "Klein–Gordon". In Section 3.3 the region D^{τ2}_{τ2} appears where D^{τ2}_{τ1} is meant.
  4. [Remark 1.8] In Remark 1.8, the Sobolev inequality (2.15) is stated for functions on T^3 but the displayed norm is written as L∞(S2); this should be corrected to L∞(T^3).
  5. [Section 4 and (4.1)] For q≥(1+2√2)/7, all decay statements contain an arbitrary ε>0 loss, and the constants in Corollary 1.4 depend on ε, but the paper does not track the dependence or give even a qualitative range for admissible ε. The phrase "almost optimal around the radiation case" should be made quantitative, or the statement should be limited to the upper-bound nature of the estimate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the energy, ILED, and r^p estimates are derived by explicit multiplier computations, and the decay corollary is a direct application of an abstract hierarchy lemma with no fitted parameters.

full rationale

The paper's derivation chain is self-contained. Theorem 1.1 is proved from the twisted divergence identity (2.4) with explicit vector-field multipliers and twisting functions in Propositions 3.1 and 3.4. Theorem 1.2 follows from the same identity with multipliers such as Y1 = ∂r and Y2 = (1+r)^(-δ)∂r, with all boundary and error terms controlled by energy boundedness and Lemma 2.2. Theorem 1.3 is obtained from the pointwise identities (3.16) and (3.41), where every error term is absorbed either by the favourable left-hand-side terms using the bound rt^(q-1) ≤ C or by Gronwall's inequality in the compact region. The decay results in Corollary 1.4 are then a direct consequence of the abstract hierarchical Lemma 2.4 applied to these proven estimates, together with Sobolev and interpolation inequalities. No parameter is fitted to data and no 'prediction' is a restatement of an input: the energy flux whose decay is proved is a fixed weighted flux, not a quantity tuned to force the advertised rate. The only imported identities are the standard twisted energy-momentum divergence formulas from [10, Appendix A.2], which are cited as background algebraic identities rather than as a substitute for the main argument; [10] is not authored by the present author, and in any case the identities do not assert the paper's conclusions. The abstract's claim that the wave-zone pointwise decay is 'optimal' or 'almost optimal' goes beyond what the upper-bound theorems prove, since no matching lower bound is established, but that is an overstatement of the results, not a circularity in the derivation.

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

No constants are fitted to data; q is the fixed background exponent, and epsilon is an arbitrarily small loss parameter whose value does not affect the qualitative statements. The paper postulates no new physical entities; the twisted energy-momentum tensor is a mathematical construction taken from the prior literature.

assumptions (5)
  • domain assumption FLRW metric g_q = -dt^2 + t^{2q}((dx^1)^2+(dx^2)^2+(dx^3)^2) with 0<q<1 and R^3 spatial topology is the background, and the wave operator is given by (1.2).
    Definition (1.1) and Section 1; the entire paper studies this class.
  • standard math The twisted divergence identity (2.4) for the twisted energy-momentum tensor is accepted from Dafermos-Holzegel-Rodnianski-Taylor [10, Appendix A.2].
    Section 2.1 states the identity and refers to [10] for its proof; it is used to derive all energy and r^p estimates.
  • standard math Sobolev inequalities (2.13)-(2.16), H^2 embeddings on R^3 and S^2, the interpolation Lemma 2.5, and Grönwall's inequality Lemma 2.7.
    Section 2.4; these functional inequalities are used in Sections 3 and 4 to pass from energy estimates to pointwise decay.
  • domain assumption Initial data are compactly supported smooth on the partially null hypersurface Σ_{τ0}.
    Initial value problem (1.4), Section 1.1; compact support is used in Section 4 so boundary terms on v→∞ vanish and local Sobolev estimates apply.
  • domain assumption The bound rt^{q-1} ≤ C of Lemma 2.2 holds uniformly on the future domain of dependence.
    Lemma 2.2, Section 2.2; it controls mixed r-t error terms in Propositions 3.3, 3.5 and 3.6.

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Cite this review

Pith. "Pith review of Boundedness and decay of waves on spatially flat decelerated FLRW spacetimes." pith.science (2026). https://pith.science/paper/DL2SXKUO

@misc{pith2026250516794,
  author       = {Pith},
  title        = {Pith review of: Boundedness and decay of waves on spatially flat decelerated FLRW spacetimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DL2SXKUO}},
  note         = {Machine review of arXiv:2505.16794}
}
abstract

We study the linear wave equation on a class of spatially homogeneous and isotropic Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetimes in the decelerated regime with spatial topology $\mathbb{R}^3$. Employing twisted $t$-weighted multiplier vector fields, we establish uniform energy bounds and derive integrated local energy decay estimates across the entire range of the decelerated expansion regime. Furthermore, we obtain a hierarchy of $r^p$-weighted energy estimates \`a la the Dafermos-Rodnianski $r^p$-method, which leads to energy decay estimates. As a consequence, we demonstrate pointwise decay estimates for solutions and their derivatives. In the wave zone, this pointwise decay is optimal in the "radiation" and "sub-radiation" cases, and almost optimal around the radiation case.

Figures

Figures reproduced from arXiv: 2505.16794 by the authors.

Figure 1
Figure 1. Penrose diagram of FLRW spacetimes (1.1) for 0 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The future domain of dependence of the initial hypersurface Σ [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

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Reference graph

Works this paper leans on

45 extracted references · 41 canonical work pages

  1. [1]

    2169, 20140361

    Bilal Abbasi and Walter Craig,On the initial value problem for the wave equation in Friedmann–Robertson–Walker space–times, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences470(2014), no. 2169, 20140361. 43

    work page 2014

  2. [2]

    1, 261–298

    Lars Andersson and David Fajman,Nonlinear stability of the Milne model with matter, Communications in Mathematical Physics378(2020), no. 1, 261–298

    work page 2020

  3. [3]

    Lars Andersson and Vincent Moncrief,Einstein spaces as attractors for the Einstein flow, Journal of differential geometry 89(2011), no. 1, 1–47

    work page 2011

  4. [4]

    Yannis Angelopoulos, Stefanos Aretakis, and Dejan Gajic,Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes, Advances in Mathematics323(2018), 529–621

    work page 2018

  5. [5]

    ,A vector field approach to almost-sharp decay for the wave equation on spherically symmetric, stationary space- times, Annals of PDE4(2018), 1–120

    work page 2018

  6. [6]

    04, 587–634

    Hamed Barzegar and David Fajman,Stable cosmologies with collisionless charged matter, Journal of Hyperbolic Differential Equations19(2022), no. 04, 587–634

    work page 2022

  7. [7]

    Volker Branding, David Fajman, and Klaus Kr¨ oncke,Stable cosmological Kaluza–Klein spacetimes, Communications in Mathematical Physics368(2019), 1087–1120

    work page 2019

  8. [8]

    2, European Mathematical Society, 2007

    Demetrios Christodoulou,The formation of shocks in 3-dimensional fluids, Vol. 2, European Mathematical Society, 2007

    work page 2007

Show all 45 references
  1. [9]

    01, 35–58

    Jo˜ ao L Costa, Jos´ e Nat´ ario, and Pedro FC Oliveira,Decay of solutions of the wave equation in expanding cosmological spacetimes, Journal of Hyperbolic Differential Equations16(2019), no. 01, 35–58

    2019

  2. [10]

    Mihalis Dafermos, Gustav Holzegel, Igor Rodnianski, and Martin Taylor,Quasilinear wave equations on asymptotically flat spacetimes with applications to kerr black holes, arXiv preprint arXiv:2212.14093 (2022)

    2022 arXiv

  3. [11]

    Mihalis Dafermos and Igor Rodnianski,A new physical-space approach to decay for the wave equation with applications to black hole spacetimes, XVIth international congress on mathematical physics: (with dvd-rom), 2010, pp. 421–432

    2010

  4. [12]

    Maxime Van de Moortel,An extension of ther p method for wave equations with scale-critical potentials and first-order terms, arXiv preprint arXiv:2505.13711 (2025), available at2505.13711

    2025 arXiv

  5. [13]

    Geren, and Rainer K

    Jurgen Ehlers, P. Geren, and Rainer K. Sachs,Isotropic solutions of the Einstein–Liouville equations, Journal of Mathemat- ical Physics9(196809), no. 9, 1344–1349, available athttps://pubs.aip.org/aip/jmp/article-pdf/9/9/1344/19089486/ 1344_1_online.pdf

  6. [14]

    David Fajman, Maciej Maliborski, Maximilian Ofner, Todd Oliynyk, and Zoe Wyatt,Phase transition between shock formation and stability in cosmological fluids, arXiv preprint arXiv:2405.03431 (2024)

    2024 arXiv

  7. [15]

    David Fajman, Maximilian Ofner, Todd Oliynyk, and Zoe Wyatt,Stability of fluids in spacetimes with decelerated expan- sion, arXiv preprint arXiv:2501.12798 (2025)

    2025 arXiv

  8. [16]

    David Fajman, Maximilian Ofner, and Zoe Wyatt,Slowly expanding stable dust spacetimes, Archive for Rational Mechanics and Analysis248(2024), no. 5, 83

    2024

  9. [17]

    Grigorios Fournodavlos,Future dynamics of FLR W for the massless-scalar field system with positive cosmological constant, Journal of Mathematical Physics63(2022), no. 3

    2022

  10. [18]

    Anahit Galstian, Tamotu Kinoshita, and Karen Yagdjian,A note on wave equation in Einstein and de Sitter space-time, Journal of mathematical physics51(2010), no. 5

    2010

  11. [19]

    Anahit Galstian and Karen Yagdjian,Microlocal analysis for waves propagating in Einstein & de Sitter spacetime, Math- ematical Physics, Analysis and Geometry17(2014), 223–246

    2014

  12. [20]

    01, 87–188

    Mahir Hadˇ zi´ c and Jared Speck,The global future stability of the FLR W solutions to the dust-Einstein system with a positive cosmological constant, Journal of Hyperbolic Differential Equations12(2015), no. 01, 87–188

    2015

  13. [21]

    Hawking and George F

    Stephen W. Hawking and George F. R. Ellis,The large scale structure of space-time, Cambridge Monographs on Mathe- matical Physics, Cambridge University Press, 1973

    1973

  14. [22]

    Daoyin He, Yaqing Sun, and Kangqun Zhang,Global existence for small amplitude semilinear wave equations with time- dependent scale-invariant damping, arXiv preprint arXiv:2501.01670 (2025)

    2025 arXiv

  15. [23]

    Joseph Keir,The weak null condition and global existence using thep-weighted energy method, arXiv preprint arXiv:1808.09982 (2018)

    2018 arXiv

  16. [24]

    Section A, Physique Th´ eorique35(1981), no

    Sergiu Klainerman and Peter Sarnak,Explicit solutions of□u= 0on the Friedmann-Robertson-Walker space-times, Annales de l’institut Henri Poincar´ e. Section A, Physique Th´ eorique35(1981), no. 4, 253–257

    1981

  17. [25]

    Christian L¨ ubbe and Juan Antonio Valiente Kroon,A conformal approach for the analysis of the non-linear stability of radiation cosmologies, Annals of Physics328(2013), 1–25

    2013

  18. [26]

    Elliot Marshall,Instability of slowly expanding FLR W spacetimes, arXiv preprint arXiv:2502.08095 (2025), available at 2502.08095

    2025 arXiv

  19. [27]

    7, 232 –236

    Akitaka Matsumura,Energy decay of solutions of dissipative wave equations, Proceedings of the Japan Academy, Series A, Mathematical Sciences53(1977), no. 7, 232 –236

    1977

  20. [28]

    Georgios Moschidis,Ther p-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications, Annals of PDE2(2016), 1–194. 44

    2016

  21. [29]

    Makoto Nakamura and Takuma Yoshizumi,Finite time blow-up of semi-linear Klein-Gordon equations with positive initial energy in FLR W spacetimes, arXiv preprint arXiv:2505.09077 (2025), available at2505.09077

    2025 arXiv

  22. [30]

    Jos´ e Nat´ ario and Flavio Rossetti,Explicit formulas and decay rates for the solution of the wave equation in cosmological spacetimes, Journal of Mathematical Physics64(2023), no. 3

    2023

  23. [31]

    Todd A Oliynyk,Future stability of the FLR W fluid solutions in the presence of a positive cosmological constant, Com- munications in Mathematical Physics346(2016), 293–312

    2016

  24. [32]

    Hans Ringstr¨ om,On the topology and future stability of the universe, Oxford University Press, 2013

    2013

  25. [33]

    Robertson,On the foundations of relativistic cosmology, Proceedings of the National Academy of Sciences15 (1929), no

    Howard P. Robertson,On the foundations of relativistic cosmology, Proceedings of the National Academy of Sciences15 (1929), no. 11, 822–829, available athttps://www.pnas.org/doi/pdf/10.1073/pnas.15.11.822

    1929 doi

  26. [34]

    ,Relativistic cosmology, Reviews of Modern Physics5(1933Jan), 62–90

  27. [35]

    ,Kinematics and world-structure, Astrophysical Journal82(November 1935), 284

    1935

  28. [36]

    Igor Rodnianski and Jared Speck,The stability of the irrotational Euler–Einstein system with a positive cosmological constant, arXiv preprint arXiv:0911.5501 (2009)

    2009 arXiv

  29. [37]

    17, 175004

    Flavio Rossetti and Alex Va˜ n´ o-Vi˜ nuales,Decay of solutions of the wave equation in cosmological spacetimes—a numerical analysis, Classical and Quantum Gravity40(2023), no. 17, 175004

    2023

  30. [38]

    ,Hyperboloidal approach for linear and non-linear wave equations in FLR W spacetimes, arXiv preprint arXiv:2502.20520 (2025)

    2025 arXiv

  31. [39]

    3, 515–600

    Volker Schlue,Decay of linear waves on higher-dimensional Schwarzschild black holes, Analysis & PDE6(2013), no. 3, 515–600

    2013

  32. [40]

    3, 633–715

    Jared Speck,The nonlinear future stability of the FLR W family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Mathematica18(2012), no. 3, 633–715

    2012

  33. [41]

    ,The stabilizing effect of spacetime expansion on relativistic fluids with sharp results for the radiation equation of state, Archive for Rational Mechanics and Analysis210(2013), 535–579

    2013

  34. [42]

    Martin Taylor,Future stability of expanding spatially homogeneous FLR W solutions of the spherically symmetric Einstein– massless Vlasov system with spatial topologyR 3, Journal of Mathematical Physics65(2024), no. 2

    2024

  35. [43]

    Kimitoshi Tsutaya and Yuta Wakasugi,Existence of solutions of semilinear wave equations with time-dependent propaga- tion speed and time derivative nonlinearity, 2025

    2025

  36. [44]

    1, 57 –65

    Hiroshi Uesaka,The total energy decay of solutions for the wave equation with a dissipative term, Journal of Mathematics of Kyoto University20(1980), no. 1, 57 –65

    1980

  37. [45]

    1, 101–124

    Jens Wirth,Solution representations for a wave equation with weak dissipation, Mathematical methods in the applied sciences27(2004), no. 1, 101–124. 45

    2004

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