Semilinear wave equations on extremal Reissner-Nordstr\"om black holes revisited

Ryan Unger; Yannis Angelopoulos

REVIEW 2 major objections 5 minor 6 references

Semilinear wave equations on extremal Reissner-Nordstr\"om black holes revisited

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Global existence for small-data semilinear wave equations on extremal Reissner–Nordström follows from a weak hierarchy of weighted energy estimates that is compatible with horizon derivative growth.

desk verdict A genuinely new proof technique for a known global-existence theorem on extremal RN, with a clean limitation statement; the main risk is the sketched bootstrap closure in Section 6. read the letter →

arxiv 2502.00210 v2 pith:UDYBAU6J submitted 2025-01-31 math.AP gr-qcmath-phmath.DGmath.MP

Yannis Angelopoulos , Ryan Unger This is my paper · ORCID

classification math.APgr-qcmath-phmath.DGmath.MP MSC 35L7135B4083C57

keywords semilinearwaveequationsextremalReissner-Nordströmglobalexistencestrongnullconditionhorizoninstabilityenergydecaybootstrapargumentblackholestability

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 proves that small smooth solutions of semilinear wave equations on the extremal Reissner–Nordström black hole exist globally and decay polynomially, for nonlinearities satisfying a strong null condition at the horizon and at null infinity. The proof uses a strictly weaker set of weighted energy and pointwise estimates than earlier work: it commutes only with time translation and rotations, keeps the weights $r^p$ and $(r-M)^{-p}$ below $p=2-\delta$, and never attempts to control the transverse null derivative of the solution on the horizon. This makes the stability statement compatible with, but not dependent on, the known non-decay and polynomial growth of horizon derivatives. The authors also sketch how the same scheme applies to semilinear problems on spacetimes that settle down to extremal Reissner–Nordström.

What carries the argument

The load-bearing object is the master energy $X_{p,k}(\tau_1,\tau_2)$, which combines $r^p$-weighted outgoing fluxes, $(r-M)^{-p}$-weighted ingoing fluxes, and integrated spacetime energies with a degeneration at the photon sphere; the paper propagates this hierarchy only up to $p=2-\delta$ and $k=n$ commutations. Around that hierarchy sits a bootstrap using $L^1_vL^\infty_{u,\omega}$ and $L^2_vL^\infty_{u,\omega}$ estimates for lower-order derivatives, characteristic pointwise bounds, and the strong null condition to organize nonlinear error terms. The weighted Hardy and Morawetz estimates replace the redshift effect in the extremal setting, while trapping is removed by a nondegenerate estimate at one higher commutation order. The method of characteristics yields $(r-M)^{-q}$ pointwise bounds that interpolate between boundedness and growth, which is enough to close the error estimates without proving sharp horizon asymptotics.

What would settle it

Take the semilinear equation with nonlinearity containing $r^{-1}\Gamma_i\phi\,Y\phi$ near the horizon and all other terms satisfying the strong null condition. If for a sequence of characteristic data with $\lVert\mathring{\phi}\rVert_\star\to 0$ the solution fails to exist globally, or if the bootstrap quantity $(r-M)^{-2}\partial_u\phi^{n-6}$ grows faster than $\tau^{1/2+\delta/2}$, then the claimed boundary of the theorem would be falsified.

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Extended reading notes

Core claim

The central claim is that global existence for semilinear systems on extremal Reissner–Nordström does not require the sharp near-horizon estimates that were previously thought necessary. For any mass $M>0$, $\delta\in(0,1/100)$, and $n\ge 12$, if the nonlinearity satisfies the strong null condition of Definition 2.2 and the characteristic initial data satisfy $\lVert\mathring{\phi}\rVert_\star\le \varepsilon_0\le \varepsilon_{\mathrm{stab}}$, then the solution exists on the entire domain of outer communication and extends smoothly to the event horizon. The solution obeys energy bounds such as $X_{0,n-2}(\tau,\infty)\le C\varepsilon_0^2\tau^{-2+\delta}$ and $X_{2-\delta,n-2}(\tau,\infty)\le C\varepsilon_0^2$, while the top-order weighted energy $E_{1+\delta,n}(\tau)+\underline{E}_{1+\delta,n}(\tau)$ is allowed mild growth like $\tau^{4\delta}$. The proof never commutes with the transverse null vector field $Y$ or with $r^2\partial_v$, so it can tolerate horizon growth of order $|Y\phi|\lesssim v^{1/2+\delta/2}$, exactly the kind of growth expected in more unstable settings.

Load-bearing premise

The decisive assumption is the strong null condition at the event horizon: near $H^+$ every quadratic derivative term must pair one good transverse derivative with a tangent derivative, or two angular derivatives, so terms such as $r^{-1}\Gamma_i\phi\,Y\phi$ are excluded; if such a term is present, the bootstrap needs a bound on $Y\phi^{n-6}$ that the weak hierarchy cannot provide.

Editorial extensions

If this is right

Small-data global existence holds on extremal Reissner–Nordström for all semilinear systems satisfying the strong null condition, with polynomial energy decay in all quantities except possibly a mildly growing top-order weighted energy.
The stability proof is independent of the horizon instability mechanism: it neither assumes nor proves decay of transverse null derivatives, so it remains consistent with faster horizon growth in charged scalar field and extremal Kerr settings.
Once global existence is known, one can revisit the same solution under the stronger norm of earlier work and recover the horizon derivative instability as a separate, later step.
The same weak hierarchy is sketched for asymptotically extremal spacetimes, provided the nonlinearity satisfies the null condition everywhere, which covers wave map systems on those backgrounds.
The proof uses only rotation and time-translation commutations in the exact extremal setting, and only rotation commutations in the dynamical background sketch, suggesting the method does not depend on a global timelike Killing field.

Reading between the lines

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

The paper's decoupling of stability from horizon instability suggests a practical two-stage strategy for future nonlinear stability problems on extremal Kerr: first prove existence with weak norms, then quantify horizon growth separately.
A direct testable extension is to run the same weak hierarchy for the charged scalar field system on extremal Reissner–Nordström; if the hierarchy closes there, the claimed compatibility with faster horizon growth becomes a proven theorem rather than a heuristic.
The failure for terms like $r^{-1}\Gamma_i\phi\,Y\phi$ indicates that the boundary of stable semilinearities is set by a horizon null-structure condition, not merely by the classical null condition at infinity; a classification of all quadratic horizon terms admitting global stability would pin down that boundary.
The asymptotic extremal sketch relies only on angular commutations, so it may carry over to backgrounds without any timelike symmetry, which would be essential for non-stationary dynamical settings.

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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

The thing to know about this paper is that it delivers a simpler proof of a known global-existence result, and the simplification is real. Angelopoulos and Unger prove small-data global existence for semilinear wave equations on extremal Reissner-Nordström under a strong null condition, using a substantially weaker initial-data norm than the earlier AAG20b proof and dropping the Y and r^2∂v commutations. That is worth something on its own, and it is explicitly engineered to be compatible with the larger horizon instabilities expected for charged scalar fields and extremal Kerr. Section 1.3.1 also carefully identifies a concrete class of 'good·bad' null forms that the new method cannot handle, which is exactly the kind of limitation statement you want.

The detailed error estimates in Sections 5.3–5.8 are the meat of the paper, and they look credible: the L1L∞ and L2L∞ estimates, the pointwise hierarchy with the (r−M)^{-q} interpolation, and the dyadic summation arguments are all spelled out with enough precision that a patient reader can check them. The Morawetz estimate in Section 4.4, however, is only sketched, and Section 6 is a sketch as well. The proof of Proposition 5.3 is where the bootstrap closes, and it is three paragraphs with a citation to a 'standard pigeonhole argument.'

The sharpest concern, and the one I would want a referee to press on, is Proposition 6.1. As written, the master hierarchy has E_{p,k}(τ2) on the right-hand side, which makes it the wrong shape for a bootstrap: X is defined as a supremum over [τ1,τ2], so this is X ≲ X + data + error. The text says the standard pigeonhole argument takes care of it, but that is not demonstrated here. Either the estimate is misstated and the final boundary belongs on the left, or the absorption step is a genuine gap. Since the entire proof of Theorem 3.3 rests on this step, the paper must either give the argument or point to the exact place in AKU24 (or elsewhere) where the identical situation is resolved. I don't think this is a fatal flaw, but it is load-bearing and currently unverifiable from the text.

The theorem itself is a re-proof, not a first-principles discovery, but the new technique is genuinely new and the paper is honest about its scope. The citation pattern is appropriate—most reliance is on the authors' own prior work, which is exactly the relevant background. I would send this to a serious referee, with the expectation that the revision fill the bootstrap closure gap and expand the Morawetz sketch. For anyone working on stability of extremal black holes, this is a useful and timely paper.

Referee Report

2 major / 5 minor

Summary. The paper revisits small-data global existence for systems of semilinear wave equations □_g φ = N(x,φ,dφ) on extremal Reissner–Nordström spacetimes, assuming a strong null condition on N near the horizon and null infinity (Definition 2.2). The main theorem (Theorem 3.3) asserts that smooth n-admissible characteristic data with sufficiently small norm ∥φ̇∥⋆ (with n≥12) produce solutions on the full domain of outer communication, smoothly extending to H+, with weighted energy estimates such as X_{0,n−2}(τ,∞) ≤ C ε0² τ^{−2+δ}, X_{2−δ,n−2} ≤ C ε0², and E_{1+δ,n}+Ē_{1+δ,n} ≤ C ε0² τ^{4δ}. The proof is organized as a bootstrap combining pointwise estimates, r^p and (r−M)^{−p} energy hierarchies, integrated local energy decay, and L¹L∞/L²L∞ estimates for lower-order terms. An extension to asymptotically extremal dynamical backgrounds is sketched as Theorem 1.7.

Significance. If the proof is completed, the paper would provide a substantial methodological simplification of the earlier result of Angelopoulos–Aretakis–Gajic: it propagates significantly weaker estimates, avoids commuting with Y and r²∂v, and is therefore compatible with the expected stronger horizon instabilities in charged scalar field and extremal Kerr settings. The explicit nonlinear error estimates in Section 5, the use of interpolation, and the honest discussion in Section 1.3.1 of which null forms are not covered are valuable contributions. The central claim is not circular and involves no fitted parameters; the limitations of the method are clearly stated. However, the manuscript currently leaves two load-bearing parts as sketches: the final bootstrap-closure argument in Section 6 and the Morawetz estimate in Proposition 4.4. These prevent full verification of Theorem 3.3 as written.

major comments (2)

[Section 6, Proposition 6.1 and proof of Proposition 5.3] The master hierarchy (6.1) places E_{p,k}^{τf}(τ2) on the right-hand side, and this quantity is part of the left-hand side X_{p,k}^{τf}(τ1,τ2) through the supremum over τ∈[τ1,τ2]. The proof of Proposition 5.3 asserts that a standard pigeonhole argument over dyadic intervals (citing [AKU24, Section 7.1]) yields X_{p,k}^{τf}(τ1,τ2) ≤ C(ε0²+ε^{5/2}) times the stated powers of τ1 or τ2, but no argument is given. This is the decisive step where the bootstrap constants are improved, and it is not a cosmetic omission: for k=n and p=1+δ the bootstrap permits X ≲ ε² τ2^{4δ}, so the future boundary term has the same τ-growth as the desired bound, and a direct absorption of E(τ2) would only recover the assumed constant. I ask for a self-contained lemma that iterates (6.1) over dyadic intervals and shows how E(τ2) is converted into E(τ1) plus an integrated bulk that is ε^{5/2}-small with constants independent of A, or, failing that, a precise statement of the cited lemma with its hypotheses verified for the present energy norms.
[Section 4.4.1, Proposition 4.4] The Morawetz estimate is labeled 'Sketch of proof.' It is a foundational a priori estimate on which all nonlinear error estimates in Section 5 rely. The sketch introduces several multipliers (X2, h2, χnear, g) without defining them and concludes (4.8) after 'straightforwardly bounding' the bulk error by E_{T,k}+E_{Z,k}. If this estimate is taken verbatim from [Are11a]/[AAG20a]/[HMVR24], please give a precise reference and state it as an imported theorem; if the present proof contains new modifications, they need to be written out. Otherwise the derivation of the master hierarchy (6.1) is not fully verifiable.

minor comments (5)

[Section 3.2] There is a typo in Theorem 3.3: 'extemal' should be 'extremal'.
[Section 5.6] In the estimate for E(∂v, i, k, R), the term X_{1+δ} on the right-hand side is missing the subscript k; it should read X_{1+δ,k}^{τf}.
[Section 5.7] The interpolation step after the endpoint estimates for E_{p,k}(R) is not shown; please specify the interpolation parameters used in (5.46) for at least one of the three rows, so the stated powers in (5.47) can be checked directly.
[Section 1.3.2 and Appendix A] Theorem 1.7 is stated as a theorem in the introduction but only sketched in Appendix A. Since the abstract advertises a sketch, I suggest rephrasing it as a conditional result or clearly marking it as a sketch within the statement.
[Section 5.5] The section title 'intermediater region' should be 'intermediate region'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main result is derived from the wave equation by in-paper energy and pointwise estimates; self-citations supply background and a routine pigeonhole step, not the claimed result.

full rationale

Theorem 3.3 is proved by a bootstrap argument rather than by fitting or relabeling. The linear energy hierarchies (Propositions 4.3–4.7), nonlinear error estimates (Propositions 5.12–5.16), and pointwise estimates (Propositions 5.10–5.11) are derived inside the paper from the wave equation under explicit bootstrap assumptions; the bootstrap is then closed by improving constants, which is a standard continuity argument and not a definitional loop. No quantity is fitted to data and then renamed a prediction. The initial data norm appears on the right-hand side only in the harmless sense that estimates at later times must be controlled by the data at τ=1. Self-citations to [AAG20a, AAG20b, AKU24] occur, but the cited items are technical lemmas or routine dyadic summation steps, not the target theorem, and the paper also cites non-overlapping sources for the routine steps. The only point worth scrutiny is Proposition 6.1, whose right-hand side contains E(τ2) while X contains the supremum of E(τ); the paper dispatches the absorption with a 'standard pigeonhole argument' citing [AKU24, Section 7.1]. This is an omitted proof detail and a possible correctness gap, but not a circular reduction: the claimed inequality is not identical to its input by construction, and no prior or fitted result is being presented as the conclusion. Overall, the central derivation is self-contained apart from routine technical citations, so no significant circularity is present.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities or fitted constants. It relies on standard background results for linear waves on extremal Reissner-Nordström, Hardy inequalities, and local well-posedness, all from cited literature. The only hand-chosen numerical parameter is the auxiliary δ, which is arbitrary small and not fitted to data.

free parameters (1)

δ
Auxiliary small parameter, 0 < δ < 1/100, chosen by hand to define the weaker hierarchy (p up to 2-δ) and to absorb losses in pointwise estimates. The theorem holds for any such δ, and the constants depend on it, so it is not fitted to data.

assumptions (4)

standard math (r-M)^{-p} and r^p linear energy hierarchies on extremal Reissner-Nordström (from AAG20a, DR10)
Used as black boxes in Section 4 to derive nonlinear versions of (1.4)-(1.6); they are prior theorems from the cited literature.
standard math Morawetz estimate for the linear wave equation on extremal Reissner-Nordström (Are11a, refined in AAG20b, Ape23, HMVR24)
The sketch of proof of Proposition 4.4 relies on these prior results.
standard math Hardy inequalities (Lemmas 4.1, 4.2) from AKU24
Used repeatedly to control lower-order terms; proved in the authors' earlier work.
standard math Local well-posedness for the characteristic initial value problem (Luk12)
Invoked in Section 3.1.3 to assert existence of solutions and that the bootstrap set is nonempty and connected.

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Cite this review

Pith. "Pith review of Semilinear wave equations on extremal Reissner-Nordstr\"om black holes revisited." pith.science (2026). https://pith.science/paper/UDYBAU6J

@misc{pith2026250200210,
  author       = {Pith},
  title        = {Pith review of: Semilinear wave equations on extremal Reissner-Nordstr\"om black holes revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UDYBAU6J}},
  note         = {Machine review of arXiv:2502.00210}
}

read the original abstract

We revisit global existence and decay for small-data solutions of semilinear wave equations on extremal Reissner-Nordstr\"om black hole backgrounds satisfying the classical null condition, a problem which was previously addressed by the first author in joint work with Aretakis and Gajic (Ann. of PDE, 2020). In this paper, we develop a new approach based on propagating a significantly weaker set of estimates, which allows for a simpler and more streamlined proof. Our proof does not require tracking sharp estimates for the solution in the near-horizon region, which means that it is compatible with, but does not imply, the non-decay and growth hierarchy of derivatives of the solution along the event horizon expected from the Aretakis instability. In particular, this approach is in principle compatible with other settings where stronger horizon instabilities are expected, such as nonlinear charged scalar fields on extremal Reissner-Nordstr\"om, or nonlinear waves on extremal Kerr. We also sketch how our proof applies to semilinear problems on spacetimes settling down to extremal Reissner-Nordstr\"om, such as those constructed in our joint work with Kehle (arXiv:2410.16234, 2024).

Figures

Figures reproduced from arXiv: 2502.00210 by the authors.

**Figure 2.** A Penrose diagram showing the main regions, null hypersurfaces, and energies involved in our [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** A Penrose diagram depicting the region R and the hypersurfaces I–VI used in the energy estimates in this section. 4.2 Hardy inequalities To bound lower order terms in our estimates, we require the following Hardy inequalities. They can be proved by using Lemmas 6.5 and 6.20 in [AKU24] with α = p − 2 and f = χψk for appropriate choices of cutoff functions χ, integrating over ω ∈ S 2 , and also using the trivial estim… view at source ↗

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Works this paper leans on

6 extracted references · 3 canonical work pages

[1]

Global spherically symmetric solutions of non-linear wave equations with null condition on extremal Reissner-Nordström spacetimes

[Ang16] Y. Angelopoulos. “Global spherically symmetric solutions of non-linear wave equations with null condition on extremal Reissner-Nordström spacetimes”.IMRN 2016.11 (2016), pp. 3279–3355. [AAG20a] Y. Angelopoulos, S. Aretakis, and D. Gajic. “Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds”.Adv. Math.375 (2020), ...

work page 2016
[60]

The non-linear stability of the Schwarzschild family of black holes

[DHRT] M. Dafermos, G. Holzegel, I. Rodnianski, and M. Taylor. “The non-linear stability of the Schwarzschild family of black holes” (2021). arXiv:2104.08222. [DHRT22] M. Dafermos, G. Holzegel, I. Rodnianski, and M. Taylor. “Quasilinear wave equations on asymp- totically flat spacetimes with applications to Kerr black holes” (2022). arXiv:2212.14093. [DL1...

arXiv 2021
[124]

Nonlinear stability of extremal Reissner–Nordström black holes in spherical symmetry

[AKU24] Y. Angelopoulos, C. Kehle, and R. Unger. “Nonlinear stability of extremal Reissner–Nordström black holes in spherical symmetry” (2024). arXiv:2410.16234. [Ape23] M. A. Apetroaie. “Instability of gravitational and electromagnetic perturbations of extremal Reissner-Nordström spacetime”.Ann. PDE 9.2 (2023), Paper No. 22,

arXiv 2024
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The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Sta- bility and Instability Results

[Are10] S. Aretakis. “The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Sta- bility and Instability Results” (2010). arXiv:1006.0283. [Are11a] S. Aretakis. “Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations I”.Comm. Math. Phys.307.1 (2011), pp. 17–63. [Are11b] S. Aretakis. “S...

arXiv 2010
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NonlinearscalarperturbationsofextremalReissner- Nordström spacetimes

[AAG20b] Y.Angelopoulos,S.Aretakis,andD.Gajic.“NonlinearscalarperturbationsofextremalReissner- Nordström spacetimes”.Ann. PDE 6.2 (2020), Paper No. 12,

work page 2020
[2025]

A note on integrated local energy decay estimates for spherically symmetric black hole spacetimes

[HMVR24] G. Holzegel, G. Mavrogiannis, and R. Velozo Ruiz. “A note on integrated local energy decay estimates for spherically symmetric black hole spacetimes” (2024). arXiv:2403.02533. [Joh79] F.John.“Blow-upofsolutionsofnonlinearwaveequationsinthreespacedimensions”. Manuscripta mathematica 28.1 (1979), pp. 235–268. [KU24] C. Kehle and R. Unger. “Extremal...

work page Pith review arXiv 2024

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Global spherically symmetric solutions of non-linear wave equations with null condition on extremal Reissner-Nordström spacetimes

[Ang16] Y. Angelopoulos. “Global spherically symmetric solutions of non-linear wave equations with null condition on extremal Reissner-Nordström spacetimes”.IMRN 2016.11 (2016), pp. 3279–3355. [AAG20a] Y. Angelopoulos, S. Aretakis, and D. Gajic. “Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds”.Adv. Math.375 (2020), ...

work page 2016

[2] [60]

The non-linear stability of the Schwarzschild family of black holes

[DHRT] M. Dafermos, G. Holzegel, I. Rodnianski, and M. Taylor. “The non-linear stability of the Schwarzschild family of black holes” (2021). arXiv:2104.08222. [DHRT22] M. Dafermos, G. Holzegel, I. Rodnianski, and M. Taylor. “Quasilinear wave equations on asymp- totically flat spacetimes with applications to Kerr black holes” (2022). arXiv:2212.14093. [DL1...

arXiv 2021

[3] [124]

Nonlinear stability of extremal Reissner–Nordström black holes in spherical symmetry

[AKU24] Y. Angelopoulos, C. Kehle, and R. Unger. “Nonlinear stability of extremal Reissner–Nordström black holes in spherical symmetry” (2024). arXiv:2410.16234. [Ape23] M. A. Apetroaie. “Instability of gravitational and electromagnetic perturbations of extremal Reissner-Nordström spacetime”.Ann. PDE 9.2 (2023), Paper No. 22,

arXiv 2024

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The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Sta- bility and Instability Results

[Are10] S. Aretakis. “The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Sta- bility and Instability Results” (2010). arXiv:1006.0283. [Are11a] S. Aretakis. “Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations I”.Comm. Math. Phys.307.1 (2011), pp. 17–63. [Are11b] S. Aretakis. “S...

arXiv 2010

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NonlinearscalarperturbationsofextremalReissner- Nordström spacetimes

[AAG20b] Y.Angelopoulos,S.Aretakis,andD.Gajic.“NonlinearscalarperturbationsofextremalReissner- Nordström spacetimes”.Ann. PDE 6.2 (2020), Paper No. 12,

work page 2020

[6] [2025]

A note on integrated local energy decay estimates for spherically symmetric black hole spacetimes

[HMVR24] G. Holzegel, G. Mavrogiannis, and R. Velozo Ruiz. “A note on integrated local energy decay estimates for spherically symmetric black hole spacetimes” (2024). arXiv:2403.02533. [Joh79] F.John.“Blow-upofsolutionsofnonlinearwaveequationsinthreespacedimensions”. Manuscripta mathematica 28.1 (1979), pp. 235–268. [KU24] C. Kehle and R. Unger. “Extremal...

work page Pith review arXiv 2024