{"id":"3e260d82-d2bb-42d9-b517-251498741c7e","arxiv_id":"1908.08492","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For structurally damped sigma-evolution equations, solutions approach either a heat-type profile or an oscillating wave profile with explicitly optimal L2 decay rates.","lead":"This paper finds the exact large-time shape of solutions to a family of damped wave-like equations, and proves the decay rates it reports are optimal. It matters for stability theory and for nonlinear problems, because it shows when damping makes the system behave like heat diffusion and when it keeps wave oscillations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's velocity profile (8) and the j=1 lower bound in (9) are stated but never proved; the proof only establishes the u-profile (7).","rationale":"The reader's verdict of CONDITIONAL is appropriate, but the identified weakest assumption (the extra-derivative hypothesis in Lemma 3.2) is not the most serious issue. That hypothesis is verifiable by direct scaling: for the profiles in Theorems 1.1 and 1.3, || |D|^{s+1} φ_j || decays with an additional power t^{-1/[2(σ-δ1)]}, and for Theorem 1.2 it decays with an additional t^{-1/(2δ2)} because of the exponential weight. The genuinely missing piece is the proof of the velocity asymptotic (8) and the corresponding lower bound in (9) for j=1. The proof of Theorem 1.2 simply does not address B_t u. Since the theorem's statement includes these estimates, the central claim is only conditionally supported. I do not recommend REJECT: the gap is a missing argument, not an identified contradiction, and the available estimates in Proposition 3.4 strongly suggest the argument can be completed. I also do not recommend ACCEPT while an entire stated estimate is unproved. Hence the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":26074,"tokens_out":10299,"duration_ms":88010,"concrete_test":"Write the missing part of the proof of Theorem 1.2 for j=1. Decompose B_t u = B_t K0^cos * u0 + B_t K0^sin * u0 + B_t K1 * u1. For the u0 terms, use Proposition 3.4 (60) and (61) to show they are o(t^{-n/(4δ2)-s/(2δ2)}). For B_t K1 * u1, subtract P1 F^{-1}[e^{-t|ξ|^{2δ2}/2} cos(t|ξ|^σ)] and estimate the difference via Lemma 3.2 with a=s and φ = F^{-1}[e^{-t|ξ|^{2δ2}/2} cos(t|ξ|^σ)], checking that || |D|^{s+1} φ || = O(t^{-n/(4δ2)-(s+1)/(2δ2)}). If the total error is o(t^{-n/(4δ2)-s/(2δ2)}), then (8) and the j=1 lower bound in (9) follow; if not, the theorem fails as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim for the hyperbolic-like damping case includes not only the u-profile (7) but also the velocity profile (8) and the two-sided optimal bounds (9) for j=0,1. The proof of Theorem 1.2 in Section 3.2, however, decomposes only || |D|^s (u - P1 F^{-1}[e^{-t|ξ|^{2δ2}/2} sin(t|ξ|^σ)/|ξ|^σ]) || into J1-J4, obtains the o(...) estimate for J4 via Lemma 3.2, and then derives the upper and lower bounds for || |D|^s u ||. No analogous decomposition is written for B_t |D|^s u, no profile error of the form (8) is estimated, and no lower bound for the velocity is derived. Thus (8) and the j=1 part of (9) are unproved statements in a stated theorem. This is more load-bearing than the extra-derivative check in Lemma 3.2, which is satisfied by direct scaling and only needs to be recorded. The gap is likely repairable: using Proposition 3.4 estimates (60)-(61), the u0 contributions decay with an extra factor t^{-(1-σ/(2δ2))}, and Lemma 3.2 with φ = F^{-1}[e^{-t|ξ|^{2δ2}/2} cos(t|ξ|^σ)] should handle the replacement of u1 by its total mass. But until that argument is written, the theorem as stated is not fully established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the large-time asymptotic behavior of solutions to the Cauchy problem for a structurally damped σ-evolution equation with two damping terms, u_tt + (-Δ)^σ u + a(-Δ)^{δ1} u_t + b(-Δ)^{δ2} u_t = 0, with a,b ∈ {0,1} and 0<δ1<σ/2<δ2<σ. Three regimes are considered: parabolic-like damping (a=1,b=0), σ-evolution-like (hyperbolic-like) damping (a=0,b=1), and double damping (a=1,b=1) under the condition δ1+δ2>σ. For each regime, the paper claims an asymptotic profile given by an explicit Fourier multiplier expression, together with a matching two-sided decay estimate for ||B_t^j |D|^s u||_{L2} when the total mass P1 of u1 does not vanish. The proofs use the partial Fourier representation, pointwise estimates for the characteristic roots, decay estimates for low and high frequencies, and a lemma (Lemma 3.2) that replaces the initial velocity u1 by P1 δ0 in the profile error.","tokens_in":26382,"tokens_out":7790,"duration_ms":69204,"significance":"If the stated theorems are fully established, the paper would be a useful contribution: it extends the known asymptotic profile results for structurally damped wave equations (σ=1) to general σ≥1, provides explicit profiles that distinguish parabolic and σ-evolution behavior, and proves optimality of decay rates. The methods are standard and the derivations are not fitted to the target estimates; no free parameters enter and there is no circularity. However, as it stands, Theorem 1.2 is incomplete: the velocity profile (8) and the j=1 lower bound in (9) are stated but not proved. The gap appears to be repairable with the already-developed estimates of Proposition 3.4, but it is load-bearing for the main claim.","major_comments":[{"comment":"The proof of Theorem 1.2 does not establish the velocity profile (8) or the j=1 case of the lower bound in (9). The decomposition into J1-J4 and the subsequent bounds concern only || |D|^s (u - P1 F^{-1}[e^{-1/2 t|ξ|^{2δ2}} sin(t|ξ|^σ)/|ξ|^σ]) ||_{L2}; from this the paper derives (7) and the j=0 bounds. No analogous expression for B_t |D|^s u minus the cosine profile is estimated, and no lower bound for ||B_t |D|^s u||_{L2} is derived. Since (8) and (9) are explicit claims of Theorem 1.2, this is a load-bearing gap. The missing argument appears to be obtainable from Proposition 3.4, estimates (60)-(61), combined with Lemma 3.2 applied to φ = F^{-1}[e^{-1/2 t|ξ|^{2δ2}} cos(t|ξ|^σ)], but it must be written out.","section":"Section 3.2, Theorem 1.2"},{"comment":"In the applications of Lemma 3.2 to I5 (Section 3.1), J4 (Section 3.2), and I5 (Section 3.3), the paper checks only the first hypothesis || |D|^a φ ||_{L2} ≤ t^{-α} (e.g., (44) and (67)) and does not verify the second hypothesis || |D|^{a+1} φ ||_{L2} ≤ t^{-α-β}. This extra-derivative bound follows from the same scaling computation (for the parabolic profile the extra derivative raises the decay rate by 1/(2(σ-δ1)); for the oscillatory profile the factor |sin(t|ξ|^σ)|≤1 gives the analogous gain), but it is a hypothesis of the cited lemma and should be stated explicitly.","section":"Sections 3.1, 3.2, 3.3; Lemma 3.2"}],"minor_comments":[{"comment":"There are several spelling errors: 'Summurizing' for 'Summarizing', 'auxilliary' for 'auxiliary', 'condtion' for 'condition', 're-wirte' for 'rewrite', and 'in oder' for 'in order'.","section":"Global"},{"comment":"In equation (64) the norm notation '||u01||' should be '||u0||', and in equation (65) the norm '||u0||' appears in an estimate for the term involving u1; the data index should be corrected.","section":"Proof of Proposition 3.4, equations (64)-(65)"},{"comment":"The term 'Sobolev solutions' is used without a definition in the statements of the main theorems; a one-sentence definition or a reference to the standard notion would improve readability.","section":"Theorems 1.1-1.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript contains a clear repairable gap in Theorem 1.2: the velocity-profile statement and the j=1 lower bound are not proved. I believe the intended proof can be completed with existing estimates in the paper, so I recommend major revision rather than rejection. The other missing verification (the second hypothesis of Lemma 3.2) is cosmetic and easily fixed. There is no indication of circularity or parameter fitting; the work is a standard extension of known profile results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper extends known asymptotic profile results for structurally damped σ-evolution equations to general σ ≥ 1, covering parabolic-like damping, σ-evolution-like damping, and the double-damping case with δ1 + δ2 > σ. Theorems 1.1 and 1.3 state natural diffusion-dominated profiles with optimal two-sided decay rates, and the underlying Fourier-based strategy is the right one. No parameters are fitted, and the reliance on the author's own preprint [5] is limited to a generic integral estimate, so self-citation is not a concern.\n\nThe soft spot is Theorem 1.2. As written, the proof decomposes only the L² error between u and its sine profile, obtains the o(·) estimate, and then derives the upper and lower bounds for || |D|^s u ||. It never writes the analogous decomposition for B_t |D|^s u minus the cosine profile. Consequently (8) is unproved, and the j = 1 lower bound in (9) cannot follow from (7). This is a genuine gap in a stated theorem, not just a missing detail. It looks repairable: the decay estimates in Proposition 3.4, especially (60)–(61), plus Lemma 3.2 applied with φ = e^{−t|ξ|^{2δ2}/2} cos(t|ξ|^σ), should close it, and the u0 contributions decay faster in the claimed regime. But the argument needs to actually be written.\n\nA smaller issue: Lemma 3.2's extra-derivative hypothesis is invoked in terms I5 and J4 without being verified. For the heat-kernel profile the missing bound follows immediately by scaling, gaining an extra t^{−1/(2(σ−δ1))}; for the sine and cosine profiles it should also hold. Routine, but it should be recorded. There are also minor typos (\"Summurizing\", some inconsistent exponents in the preliminary estimates).\n\nOverall: the method is sound, Theorems 1.1 and 1.3 look solid, and Theorem 1.2 is likely correct but incomplete as written. This paper deserves a serious referee, and the author should be asked to supply the missing velocity profile proof and the Lemma 3.2 checks. I would not cite it in its current form, but I would revisit after revision.","headline":"Useful extension of asymptotic profiles to σ-evolution equations, but Theorem 1.2's velocity profile is stated without proof; the rest is broadly sound and worth refereeing after revision.","tokens_in":26905,"tokens_out":3102,"would_cite":false,"duration_ms":33223,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B40","35L30","35G10","35M11"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives the exact asymptotic profiles of solutions to linearly damped $\\sigma$-evolution equations in three damping regimes and proves the decay rates are optimal.","keywords":["sigma-evolution equations","structural damping","asymptotic profile","optimal decay estimates","parabolic-like damping","sigma-evolution-like damping","double damping","L1-L2 estimates"],"falsifier":"Take the profile $\\varphi(t,x)=B_t^j F^{-1}[e^{-t|\\xi|^{2(\\sigma-\\delta_1)}}/|\\xi|^{2\\delta_1}]$ and compute $\\| |D|^{s+1}\\varphi(t)\\|_{L^2}$ by Parseval's formula; if it does not decay like $t^{-\\alpha-1/(2(\\sigma-\\delta_1))}$ with $\\alpha$ the exponent in (44), the $o(\\cdot)$ error in Theorem 1.1 fails, and similarly for the oscillating kernel in Theorem 1.2.","tokens_in":25865,"feed_emoji":"⏳","tokens_out":14356,"duration_ms":119095,"temperature":0.7,"pith_summary":"This paper establishes the exact large-time shape of solutions to linearly damped $\\sigma$-evolution equations and shows that previously known decay rates cannot be improved. In the parabolic-like damping regime the solution is shown to approach a heat-like kernel with a power singularity $|\\xi|^{-2\\delta_1}$, while in the $\\sigma$-evolution-like regime it approaches a damped oscillating kernel. The double-damping case combines the two: the parabolic profile dominates, but only when $\\delta_1+\\delta_2>\\sigma$. A reader should care because knowing the profile, not just the decay rate, is the finer information needed to understand diffusion versus wave behavior and to study nonlinear perturbations.","feed_headline":"Damped sigma-evolution equations get optimal long-time profiles","feed_subtitle":"Three damping regimes settle into explicit kernels, and matching lower bounds prove the decay is optimal.","key_machinery":"The machinery is the Fourier representation of the solution, whose characteristic roots $\\lambda_{1,2}(\\xi)$ are expanded at low and high frequencies. For small $|\\xi|$ the dominant root in the parabolic-like case behaves like $-|\\xi|^{2(\\sigma-\\delta_1)}$, which produces the kernel $F^{-1}[e^{-t|\\xi|^{2(\\sigma-\\delta_1)}}/|\\xi|^{2\\delta_1}]$; in the $\\sigma$-evolution-like case the roots behave like $-|\\xi|^{2\\delta_2}\\pm i|\\xi|^\\sigma$, producing the damped sinusoid $e^{-t|\\xi|^{2\\delta_2}/2}\\sin(t|\\xi|^\\sigma)/|\\xi|^\\sigma$. A key tool is Lemma 3.2, which replaces the initial velocity $u_1$ by its total mass $P_1$ times a delta function in the profile comparison, so the error reduces to decay of the kernel itself. The oscillating lower bound also uses a Riemann-Lebesgue type lemma to show that the sine integral does not cancel as $t\\to\\infty$.","core_discovery":"Stated in the paper's own terms, the discovery is that for data in $L^1\\cap H^s$ and dimensions $n>4\\delta_1$ (or $n>2\\sigma$), the solution to (1) is asymptotically indistinguishable from a constant multiple of an explicit kernel. In the parabolic-like case $a=1,b=0$, and in the double-damping case $a=1,b=1$ with $\\delta_1+\\delta_2>\\sigma$, that kernel is $P_1 F^{-1}[e^{-t|\\xi|^{2(\\sigma-\\delta_1)}}/|\\xi|^{2\\delta_1}]$, with error $o(t^{-n/(4(\\sigma-\\delta_1))-s/(2(\\sigma-\\delta_1))-j+\\delta_1/(\\sigma-\\delta_1)})$ under $B_t^j|D|^s$. In the $\\sigma$-evolution-like case $a=0,b=1$, the kernel is $P_1 F^{-1}[e^{-\\frac12 t|\\xi|^{2\\delta_2}}\\sin(t|\\xi|^\\sigma)/|\\xi|^\\sigma]$, with a cos-profile for $B_t u$, and error $o(t^{-n/(4\\delta_2)-s/(2\\delta_2)+\\sigma/(2\\delta_2)})$. Whenever $P_1\\neq 0$, the same power gives a two-sided bound on $\\|B_t^j|D|^s u(t,\\cdot)\\|_{L^2}$, showing the decay rates from earlier $L^1\\cap L^2$ estimates are optimal.","pith_inferences":["A direct calculation with Parseval's formula indicates that the extra-derivative condition missing from the proof is actually satisfied for both profile kernels, so the stated theorems would follow once that estimate is written down.","Because the proofs work at the level of Fourier multipliers, the same profile comparison should extend to $L^q$ norms and to data in $L^m$ for $m\\in[1,2]$, yielding $L^m\\cap L^q$ analogues of Theorems 1.1-1.3.","For the corresponding semi-linear equations with power nonlinearity, the two-sided decay rates suggest that the critical power for global existence of small solutions in the parabolic-like case is the usual $1+2/n$; testing whether the profile persists in the nonlinear problem would be a natural next step."],"forward_implications":["In the parabolic-like and double-damping cases, the solution converges to the solution of the anomalous diffusion equation $v_t + (-\\Delta)^{\\sigma-\\delta_1}v = 0$ with initial mass $P_1$; the initial position $u_0$ is forgotten except through its regularity.","In the $\\sigma$-evolution-like case the profile keeps a sinusoid, so there is no diffusive simplification; the oscillation adds a positive term $\\sigma/(2\\delta_2)$ to the decay exponent, slowing the decay.","The two-sided bounds with $P_1\\neq 0$ make the earlier $L^1\\cap L^2$ decay estimates sharp, so no choice of constants or weights can improve the exponent.","For double damping with $\\delta_1+\\delta_2>\\sigma$, the required data regularity is set by the $\\sigma$-evolution damping $H^{s+2j\\delta_2}$, showing that the second damping term controls smoothness even though the first controls the profile."],"supporting_citations":[{"why":"Supplies Lemma 3.2, the tool that replaces the initial velocity by its total mass times the profile kernel in the error estimate.","marker":"[11]"},{"why":"Supplies Lemma 3.1, the elementary Fourier multiplier bound used to convert pointwise estimates into $L^2$ decay rates.","marker":"[5]"},{"why":"Provides the classification of parabolic-like versus $\\sigma$-evolution-like damping that fixes the expected profile in each regime.","marker":"[2]"},{"why":"Gives the $L^1\\cap L^2$ to $L^2$ decay estimates for structurally damped $\\sigma$-evolution equations that the upper bounds refine.","marker":"[4]"},{"why":"It supplies the earlier asymptotic profile result for wave equations with frictional and viscoelastic damping that the double-damping case extends to $\\sigma$-evolution order.","marker":"[10]"},{"why":"Introduces the parabolic-like/hyperbolic-like separation for damped wave equations and the estimates that motivate the present classification.","marker":"[6]"}],"fun_headline_variants":["Explicit kernels settle optimal decay for damped sigma-evolution","Sharp long-time profiles for damped sigma-evolution","Optimal decay proven for damped sigma-evolution","Exact asymptotic kernels for damped sigma-evolution","Asymptotic profiles optimal for damped sigma-evolution"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on a lemma that needs the profile kernels to keep decaying when one extra derivative is applied, and the paper does not explicitly check that second condition for the kernels it uses.","fun_headline_variants_meta":{"raw":{"variants":["Explicit kernels settle optimal decay for damped sigma-evolution","Sharp long-time profiles for damped sigma-evolution","Optimal decay proven for damped sigma-evolution","Exact asymptotic kernels for damped sigma-evolution","Asymptotic profiles optimal for damped sigma-evolution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000535,"raw_usage":{"total_tokens":2589,"prompt_tokens":977,"completion_tokens":1612,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":593,"completion_tokens_details":{"reasoning_tokens":1533}},"tokens_in":593,"tokens_out":1612,"duration_ms":11729,"temperature":1.0,"reasoning_tokens":1533,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:38:03.809965+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the profile $\\varphi(t,x)=B_t^j F^{-1}[e^{-t|\\xi|^{2(\\sigma-\\delta_1)}}/|\\xi|^{2\\delta_1}]$ and compute $\\| |D|^{s+1}\\varphi(t)\\|_{L^2}$ by Parseval's formula; if it does not decay like $t^{-\\alpha-1/(2(\\sigma-\\delta_1))}$ with $\\alpha$ the exponent in (44), the $o(\\cdot)$ error in Theorem 1.1 fails, and similarly for the oscillating kernel in Theorem 1.2.","supporting_citations":[{"cited_title":"Ikehata, H","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3.2, the tool that replaces the initial velocity by its total mass times the profile kernel in the error estimate."},{"cited_title":"Study of semi-linear $\\sigma$-evolution equations with frictional and visco-elastic damping","cited_arxiv_id":"1906.04471","evidence_quote":"Supplies Lemma 3.1, the elementary Fourier multiplier bound used to convert pointwise estimates into $L^2$ decay rates."},{"cited_title":"D’Abbicco, M.R","cited_arxiv_id":null,"evidence_quote":"Provides the classification of parabolic-like versus $\\sigma$-evolution-like damping that fixes the expected profile in each regime."},{"cited_title":"Duong, M","cited_arxiv_id":null,"evidence_quote":"Gives the $L^1\\cap L^2$ to $L^2$ decay estimates for structurally damped $\\sigma$-evolution equations that the upper bounds refine."},{"cited_title":"Ikehata, A","cited_arxiv_id":null,"evidence_quote":"It supplies the earlier asymptotic profile result for wave equations with frictional and viscoelastic damping that the double-damping case extends to $\\sigma$-evolution order."},{"cited_title":"D’Abbicco, M","cited_arxiv_id":null,"evidence_quote":"Introduces the parabolic-like/hyperbolic-like separation for damped wave equations and the estimates that motivate the present classification."}],"review_version":1}