Sharp polynomial decay rates for the damped wave equation with H\"older-like damping

Kiril Datchev; Perry Kleinhenz

Sharp polynomial decay rates for the damped wave equation with H\"older-like damping

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Pith's one-line read A power-law vanishing damping gives the sharp wave decay rate $t^{-(\beta+2)/(\beta+3)}$ on the torus.

desk verdict Sharp exponent for damped wave on the torus, proved with a Morawetz multiplier and dyadic decomposition; the argument is sound but the terse algebra will keep a referee busy. read the letter →

arxiv 1908.05631 v3 pith:5G5FKOLJ submitted 2019-08-15 math.AP math-phmath.MPmath.SP

Kiril Datchev , Perry Kleinhenz This is my paper · ORCID

classification math.APmath-phmath.MPmath.SP MSC 35B4035L0593D15

keywords dampedwaveequationenergydecayMorawetzmultipliertoruspolynomialpower-lawdampingHölder-likeresolventestimate

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 on the flat torus, when the damping $W=W(x)$ is bounded between two fixed positive multiples of the power $V(x)=(|x|-\sigma)^\beta$ near the edge of its support and vanishes on the strip $[-\sigma,\sigma]$, the energy of every solution to the damped wave equation decays at least like $t^{-(\beta+2)/(\beta+3)}$. This rate is sharp in the case $W=V$ near the edge, where earlier work of the second author rules out any faster polynomial decay. The proof reduces the time-decay question to a resolvent estimate for the stationary damped operator on the circle, then proves that estimate with a Morawetz multiplier adapted to the power-law transition layer. The result settles, in a translation-invariant model, exactly how the Hölder exponent of a damping's vanishing determines the best possible decay rate, with the exponent rising from $2/3$ at $\beta=0$ toward $1$ as $\beta\to\infty$.

What carries the argument

The engine is a one-dimensional resolvent estimate. After expanding in the $y$ Fourier variable and invoking a resolvent-to-decay reduction, the problem reduces to proving $\|(-\Delta+iqW-q^2)^{-1}\|_{L^2\to L^2}\le C q^{1/(\beta+2)}$ for large $q$. The proof of this bound uses the Morawetz multiplier method: with $F(x)=|u'|^2+E|u|^2$, it integrates $(bF)'=0$ against a piecewise-linear multiplier $b$ whose derivative is $q^\delta$ on the transition layer $[\sigma,\sigma+q^{-\delta}]$ and equals $1$ elsewhere. Two auxiliary cutoffs carry the power-law structure: $\mu$, which equals $q^\delta$ on the layer and $1$ outside, and $\chi$, which ramps linearly from $0$ to $1$ across the layer so that $(V\chi)'$ is controlled by $q^\delta W$. The choice $\delta=1/(\beta+2)$ balances the error terms and yields the resolvent bound, hence the decay exponent $(\beta+2)/(\beta+3)$.

What would settle it

Take $W=V$ on the torus with a fixed $\sigma\in(0,\pi)$ and $\beta\ge0$, and for large $q$ compute the $L^2\to L^2$ norm of $(-\Delta+iqW-q^2)^{-1}$; if along any sequence the norm grows faster than $C q^{1/(\beta+2)}$, the theorem's resolvent estimate (6) is false.

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Extended reading notes

Core claim

The central claim is the theorem in Section 1: for any $C_0>0$, $\sigma\in(0,\pi)$, and $\beta\ge0$, if $W=W(x)$ satisfies $\frac{1}{C_0}V\le W\le C_0V$ with $V=0$ on $[0,\sigma]$ and $V=(|x|-\sigma)^\beta$ on $(\sigma,\pi]$, then every solution of $\partial_t^2 v+W\partial_t v-\Delta v=0$ obeys $E(t)^{1/2}\le C t^{-(\beta+2)/(\beta+3)}(\|v_0\|_{H^2}+\|\partial_t v_1\|_{H^1})$ for large $t$, with $C$ depending only on $C_0$, $\sigma$, and $\beta$. The same exponent is optimal: when $W=V$ near $|x|=\sigma$, the second author's earlier construction of quasimodes shows that no $\alpha>(\beta+2)/(\beta+3)$ can satisfy such a bound. Thus the paper identifies the sharp polynomial decay rate for power-law vanishing damping on the torus.

Load-bearing premise

The proof depends on the damping being, up to fixed multiplicative constants, exactly the pure power $(|x|-\sigma)^\beta$ at the edge of its support and exactly zero on the undamped strip; a non-power vanishing profile would break the identity that controls $(V\chi)'$ inside the transition layer.

Editorial extensions

If this is right

For every $\beta\ge0$, every damping in the comparison class of (4) yields polynomial energy decay with exponent $(\beta+2)/(\beta+3)$, with a constant that depends only on $C_0$, $\sigma$, and $\beta$.
When $W=V$ near the edge $|x|=\sigma$, the exponent is best possible: no faster algebraic decay rate than $t^{-(\beta+2)/(\beta+3)}$ can hold.
The same proof, with the same constants, gives the identical decay rate on any product manifold $(\mathbb R/2\pi\mathbb Z)_x\times\Sigma_y$ with $\Sigma$ a compact Riemannian manifold.
The $\beta=0$ case recovers the previously known sharp rate $t^{-2/3}$ for constant damping on a strip.
As $\sigma\to\pi$ the constants blow up, matching the undamped limit where no decay is possible, while as $\sigma\to0$ the constants remain bounded even though the proved rate is weaker than what is known in the fully damped $\sigma=0$ case.

Reading between the lines

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

Inference: the transition-layer width $q^{-1/(\beta+2)}$ chosen by the proof suggests that high-frequency modes feel the damping through an average over a frequency-dependent window; a numerical computation of damped-mode resonance widths for $W=V$ could test this effective-averaging picture directly.
Inference: if the damping vanishes like a power times a slowly varying factor rather than as a pure power, the lemma controlling $(V\chi)'$ could fail, so the decay exponent may depend on the finer profile; a two-term asymptotic expansion of the resolvent near the edge would settle this.
Inference: the opposite monotonicity of the exponent in $\beta$ between the $\sigma>0$ and $\sigma=0$ regimes suggests a phase-transition curve for the optimal decay exponent as the undamped set shrinks, a question the present method does not reach.

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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

Dear [Colleague],

This is a short note that actually settles the right exponent for the damped wave equation on the torus when the damping vanishes like a power of distance to the boundary of its support. The new upper bound matches the known lower bound, so the rate t^{-(β+2)/(β+3)} is sharp for W = V near |x|=σ. The proof is a Morawetz multiplier method with a q-dependent multiplier and a dyadic decomposition in the boundary layer; it is more than a parameter scan, because the dyadic choice is what balances the exponents.

I found the argument coherent. The reduction from decay to a resolvent estimate is the standard Borichev-Tomilov route, and the Morawetz lemmas are explicit. I checked the two most delicate steps the stress-test highlights: the bound |(Vχ)'| ≤ C q^δ W in the transition layer, and the absorption step with (12) in Lemma 3. Both work; the exponent algebra in the dyadic cases is consistent. The proof is self-contained for the upper bound; [Kle19] is cited only for the matching lower bound, so there is no circularity.

The soft spots are more about presentation than substance. The algebra is compressed in several places—'Add a multiple of (13) and (14)' and 'by (12)' hide a few lines, and the dyadic case analysis is terse. A referee will want the intermediate inequalities spelled out. The two-sided comparability assumption (4) is load-bearing: the proof uses the exact power-law identity (Vχ)' = (β+1)q^δ V inside the transition layer, and a damping that vanished like x^β times a slowly varying factor could break that step. That is a natural limitation, not a flaw, but it is worth stating.

Overall, this is a solid, useful note. It closes a gap in the literature, and the technique is clean enough to be worth reading by anyone working on resolvent estimates for damped waves. I would send it to a serious referee; it deserves a careful reading rather than a desk rejection.

Best, [Your name]

Referee Report

0 major / 5 minor

Summary. The paper proves a sharp polynomial decay rate for the energy of solutions to the damped wave equation on the two-dimensional torus, for dampings depending only on x and comparable to (|x|-sigma)^beta near the boundary of their support in a strip where the damping vanishes. The main theorem states E(t)^{1/2} <= C t^{-(beta+2)/(beta+3)} (||v0||_{H^2}+||d_t v1||_{H^1}), with C depending only on C0, sigma, and beta. The proof reduces the time-dependent problem, via a standard Borichev-Tomilov resolvent criterion, to a one-dimensional resolvent estimate for -u''+iqWu-Eu. That estimate is proved by Morawetz multiplier lemmas using a dyadic decomposition of the transition layer of width q^{-1/(beta+2)}. The rate is sharp when W=V near |x|=sigma by the lower bound of [Kle19], so the theorem supplies the matching upper bound.

Significance. If correct, the result is significant: it completes the sharp polynomial decay rate for Holder-like dampings in this geometry and improves the earlier upper bound (beta+2)/(beta+4) from [Kle19] to the optimal (beta+2)/(beta+3), with no fitted parameters. The proof is self-contained, the dyadic Morawetz argument is explicit, and the main claim has a sharpness certificate from the cited lower bound. The result also extends verbatim to product manifolds with a compact factor in the y direction. The paper is concise and the algebraic steps, while terse in places, are checkable.

minor comments (5)

[Lemma 2] In the proof of Lemma 2 the sentence 'Add a multiple of (13) and (14) to both sides, and apply (12)' is more compressed than the rest of the paper; in particular the negative part of b' on (tau,pi) must be absorbed using (13)-(14) before (12) is applied. Expanding these two lines would remove the only place where a reader has to reconstruct the algebra.
[Lemma 3] The key pointwise bound |(V chi)'| <~ q^delta W in the proof of Lemma 3 is not shown; it follows from the identity (V chi)'=(beta+1)q^delta V on the transition layer and |V'|<=beta q^delta V outside it. Stating this explicitly would help.
[Proof of (10)] The optimization of the dyadic exponents eta_k is presented as a recurrence without explaining the aim; the recurrence equalizes the exponents A_j=-1/2+3 beta eta_j/4 - beta eta_{j+1}/4, and the condition beta <= 6(3^{N+1}-1) is exactly A_N <= 2 delta, which is what is needed for (10). A sentence to this effect would make the proof substantially easier to follow.
[Lemma 1] In the proof of (9) the multiplier b is not exhibited; writing b=1+epsilon cos x with epsilon>0 sufficiently small would make the hypotheses b>0 and b''<0 near [-sigma,sigma] transparent, and the absorption of the undamped region by (13) and (14) could then be made explicit.
[Throughout] There are several typographical slips, for example 'interesect' in Section 1 and the attribution 'due to Nonnenmacher [AL14]' in the introduction, where the appendix by Nonnenmacher could be cited explicitly; none of these affects the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the upper-bound proof is self-contained and the only self-citation is used for a non-load-bearing sharpness remark.

full rationale

The paper's central theorem is a conditional upper bound for the energy decay rate of the damped wave equation, assuming W is comparable to the explicit power-law V(x)=(|x|-sigma)^beta as in (4). The proof does not invoke the target decay rate as an input: the exponent alpha=(beta+2)/(beta+3) emerges from optimizing powers of q in Lemmas 2 and 3, the dyadic decomposition in the proof of (10), and the choice delta=1/(beta+2). No parameter is fitted to data, and no quantity is defined in terms of the conclusion. The reduction from the decay estimate (3) to the resolvent estimate (6) is imported from the external results [BT10] and [AL14], and the Morawetz multiplier method is attributed to [Mor61], [CV02], and [CD17]. The only self-citation is [Kle19], used in the abstract and in Remark (1) to state that the rate is optimal when W=V near |x|=sigma; this lower bound is not used in the derivation of the upper bound, so it is not load-bearing for the theorem's proof. The technical estimates in Lemma 3 follow from the explicit definitions of V and chi, e.g. |(V chi)'| ≲ q^delta W is justified by the exact identity (V chi)'=(beta+1) q^delta V in the transition layer and by V'/(q^delta V)=beta/(q^delta(|x|-sigma)) ≤ beta outside it. The proof is terse, but no circular step is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No numbers are fitted to data; the constants C, E0, and q0 are generic proof constants. The central claim rests on standard semigroup theory, the theorem's power-law damping hypothesis, and the closed-torus geometry. No new entities are introduced.

assumptions (3)

standard math Borichev-Tomilov resolvent criterion (Theorem 2.4 of [BT10] as formulated in Proposition 2.4 of [AL14]) links the resolvent bound (6) to the energy decay (3).
This is the semigroup theorem that converts the stationary resolvent estimate into the time-decay statement; it is cited, not proved.
standard math Sobolev embedding H^2(R/2pi Z) into C^1 justifies the integration by parts in the Morawetz estimate for the piecewise linear multiplier b.
The proof treats F(x)=|u'|^2+E|u|^2 and integrates (bF)' over the periodic torus; u in H^2 makes u' continuous enough for this.
domain assumption The damping function W is assumed to obey the two-sided bound (4): (1/C0)V is at most W and W is at most C0 V for all x, with V vanishing exactly on [-sigma,sigma].
This is the theorem hypothesis; the proof uses V as a proxy for W, for example in bounding |(V chi)'| and in Lemma 3.

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Cite this review

Pith. "Pith review of Sharp polynomial decay rates for the damped wave equation with H\"older-like damping." pith.science (2026). https://pith.science/paper/5G5FKOLJ

@misc{pith2026190805631,
  author       = {Pith},
  title        = {Pith review of: Sharp polynomial decay rates for the damped wave equation with H\"older-like damping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5G5FKOLJ}},
  note         = {Machine review of arXiv:1908.05631}
}

abstract

We study decay rates for the energy of solutions of the damped wave equation on the torus. We consider dampings invariant in one direction and bounded above and below by multiples of $x^{\beta}$ near the boundary of the support and show decay at rate $1/t^{\frac{\beta+2}{\beta+3}}$. In the case where $W$ vanishes exactly like $x^{\beta}$ this result is optimal by work of the second author. The proof uses a version of the Morawetz multiplier method.

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Pith tools

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

[1] [1]

Anantharaman and M

N. Anantharaman and M. L \'e autaud. Sharp polynomial decay rates for the damped wave equation on the torus. Anal. PDE , 7(1):159--214, 2014. With an appendix by S. Nonnenmacher

work page 2014

[2] [2]

Burq and P

N. Burq and P. G\'erard. Condition n\'ecessaire et suffisante pour la contr\^olabilit\'e exacte des ondes. , C. R. Math. Acad. Sci. Paris , 325(7):749--752, 1997

work page 1997

[3] [3]

Stabilisation of wave equations on the torus with rough dampings

N. Burq and P. G \'e rard. Stabilisation of wave equations on the torus with rough dampings. arXiv:1801.00983 , 2018

work page Pith review arXiv 2018

[4] [4]

Burq and M

N. Burq and M. Hitrik. Energy decay for damped wave equations on partially rectangular domains. Mathematical Research Letters , 14(1):35--47, 2007

work page 2007

[5] [5]

Bardos, G

C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM Journal on Control and Optimization , 30(5):1024--1065, 1992

work page 1992

[6] [6]

Borichev and Y

A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Mathematische Annalen , 347(2):455--478, 2010

work page 2010

[7] [7]

N. Burq. D\'ecroissance de l'\'energie locale de l'\'equation des ondes pour le probl\`eme ext\'erieur et absence de r\'esonance au voisinage du r\'eel. Acta Math. , 180(1):1--29, 1998

work page 1998

[8] [8]

Burq and M

N. Burq and M. Zworski. Control for Schr\"odinger operators on tori . Mathematical Research Letters , 19(2):309--324, 2012

work page 2012