{"id":"22ba28f0-2c23-46f7-af77-43d79fe1196c","arxiv_id":"1908.05631","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For translation-invariant damping on the torus that vanishes like x to the beta power at the support boundary, the damped wave energy decays exactly like t to the minus (beta+2)/(beta+3).","lead":"A mathematical proof shows that waves damped by a power-law friction on a torus lose energy at an exact polynomial rate, with the exponent determined by the Hölder exponent. The rate matches a known lower bound, settling the sharp decay exponent for this model.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the dyadic Morawetz estimates and the resolvent-to-decay reduction are internally consistent.","rationale":"The reader's verdict of ACCEPT with moderate confidence is appropriate. The paper's central new contribution is the upper bound matching the [Kle19] lower bound, and the proof is a coherent Morawetz-multiplier argument with a dyadic decomposition in x. I searched specifically for places where the argument could be circular or where a claimed estimate fails under the stated assumptions. The derivative bound on V chi holds, the bootstrap in Lemma 3 is not circular, the piecewise-linear multiplier produces no boundary terms on the torus, and the final dyadic optimization in the proof of (10) is consistent (the N-condition beta ≤ 6(3^(N+1)-1) indeed follows when the remaining q-power is required to be no worse than q^(2 delta), as needed for (10)). The only mild concerns are presentational: several applications of (12) are compressed, and constants are not tracked explicitly. These do not affect correctness. The reader's identified weakest assumption, the exact power-law comparison (4), is central, but it is not a vulnerability in the proof: the estimates use only comparability W ~ V, not the exact power-law identity, and the derivative bound survives non-power factors because the proof only needs W >= c V. Thus the verdict remains UNCHANGED.","tokens_in":8278,"tokens_out":39165,"duration_ms":347583,"concrete_test":"Independently re-derive the absorption step in Lemma 3 for a fixed beta, say beta = 1, writing out the application of inequality (12) term by term and verifying that the final coefficient (1 + E^(-1) q^(2 delta)) and the intermediate exponents q^(1-delta beta - delta) and q^(2 delta) are reproduced exactly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the central claim as the upper bound E(t)^(1/2) ≤ C t^(-(beta+2)/(beta+3)) under the two-sided comparability assumption (4), with sharpness inherited from [Kle19]. For this to hold, the resolvent estimate (8) must follow from the Morawetz lemmas. I checked the two most delicate points. First, the pointwise bound |(V chi)'| ≲ q^delta W in Lemma 3 is valid: inside the transition layer (V chi)' = (beta+1) q^delta V, and outside one has V'/(q^delta V) = beta/(q^delta (|x|-sigma)) ≤ beta since |x|-sigma ≥ q^(-delta). Second, the absorption step using (12) in Lemma 3 and in the proof of (10) balances the powers of q correctly; for example the condition beta ≤ 6(3^(N+1)-1) in case 4 comes from requiring q^(-1/2+3 beta eta_N/4) ≤ q^(2 delta), not ≤ 1, and the exponent algebra gives exactly that bound. I also found no missing boundary term from the piecewise-linear multiplier b, because b is continuous and the torus has no boundary. The proof is terse and would benefit from more intermediate algebra, but I did not find a load-bearing error.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":8542,"tokens_out":22725,"duration_ms":201393,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Lemma 2"},{"comment":"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.","section":"Lemma 3"},{"comment":"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.","section":"Proof of (10)"},{"comment":"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.","section":"Lemma 1"},{"comment":"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.","section":"Throughout"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThis 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.\n\nI 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.\n\nThe 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.\n\nOverall, 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.\n\nBest,\n[Your name]","headline":"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.","tokens_in":9074,"tokens_out":2830,"would_cite":true,"duration_ms":25668,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B40","35L05","93D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A power-law vanishing damping gives the sharp wave decay rate $t^{-(\\beta+2)/(\\beta+3)}$ on the torus.","keywords":["damped wave equation","energy decay","Morawetz multiplier","torus","polynomial decay","power-law damping","Hölder-like damping","resolvent estimate"],"falsifier":"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.","tokens_in":8082,"feed_emoji":"🌊","tokens_out":16162,"duration_ms":129213,"temperature":0.7,"pith_summary":"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 $\\beta\\to\\infty$.","feed_headline":"Damped waves on a torus decay like t^{-(β+2)/(β+3)}","feed_subtitle":"Bound holds for any damping within a fixed factor of (|x|-σ)^β near the edge, and is sharp for the pure power.","key_machinery":"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 $(\\beta+2)/(\\beta+3)$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the resolvent-to-decay theorem used to convert the time-decay estimate into the stationary resolvent bound.","marker":"[BT10]"},{"why":"Provides the formulation of that resolvent criterion and the earlier sharp torus decay results this paper refines.","marker":"[AL14]"},{"why":"Proves the matching lower bound, showing the exponent cannot be improved when W=V near the edge.","marker":"[Kle19]"},{"why":"Introduced the Morawetz multiplier identity for wave decay that the proof adapts.","marker":"[Mor61]"},{"why":"Gives the multiplier-method framework for resolvent estimates in cylindrical geometry.","marker":"[CV02]"},{"why":"Adapts that method to asymptotically cylindrical manifolds, the form used in the one-dimensional reduction.","marker":"[CD17]"}],"fun_headline_variants":["Sharp decay t^{-(β+2)/(β+3)} for damped torus waves","Optimal decay rate found for Hölder-like damped waves","Damped waves on torus: exact polynomial decay exponent","Power-law damping yields best possible energy decay"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sharp decay t^{-(β+2)/(β+3)} for damped torus waves","Optimal decay rate found for Hölder-like damped waves","Damped waves on torus: exact polynomial decay exponent","Power-law damping yields best possible energy decay"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000849,"raw_usage":{"total_tokens":3667,"prompt_tokens":894,"completion_tokens":2773,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":2700}},"tokens_in":510,"tokens_out":2773,"duration_ms":20238,"temperature":1.0,"reasoning_tokens":2700,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:10:29.818223+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}