Some remarks on the asymptotic profile of solutions to structurally damped $\sigma$-evolution equations

Tuan Anh Dao

REVIEW 2 major objections 3 minor 11 references

Some remarks on the asymptotic profile of solutions to structurally damped $\sigma$-evolution equations

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.08492 v1 pith:KF5PN4GL submitted 2019-08-22 math.AP

Tuan Anh Dao This is my paper · ORCID

classification math.AP MSC 35B4035L3035G1035M11

keywords sigma-evolutionequationsstructuraldampingasymptoticprofileoptimaldecayestimatesparabolic-likesigma-evolution-likedoubleL1-L2

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

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.

What carries the argument

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 $t\to\infty$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

A 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).

Overall: 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.

Referee Report

2 major / 3 minor

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.

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 (2)

[Section 3.2, Theorem 1.2] 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.
[Sections 3.1, 3.2, 3.3; Lemma 3.2] 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.

minor comments (3)

[Global] There are several spelling errors: 'Summurizing' for 'Summarizing', 'auxilliary' for 'auxiliary', 'condtion' for 'condition', 're-wirte' for 'rewrite', and 'in oder' for 'in order'.
[Proof of Proposition 3.4, equations (64)-(65)] 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.
[Theorems 1.1-1.3] 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.

Circularity Check

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No significant circularity: the asymptotic profiles are derived from the Fourier representation and explicit root expansions, with only a minor non-load-bearing self-citation for a standard integral estimate.

full rationale

The derivation is not circular. The paper starts from the explicit Fourier representation of the Cauchy problem (Section 2.1), expands the characteristic roots for small and large frequencies (Section 2.2), and then compares the solution kernels with model profiles such as F^{-1}[e^{-t|ξ|^{2(σ−δ1)}}/|ξ|^{2δ1}] and F^{-1}[e^{-t|ξ|^{2δ2}/2} sin(t|ξ|^σ)/|ξ|^σ] using mean value theorem expansions (Propositions 3.1, 3.3, 3.5). The amplitude P1 = ∫ u1 dx is not fitted; it arises from Lemma 3.2, cited to Ikehata–Takeda [11], as the natural leading moment after replacing u1 by P1 δ. No target estimate is assumed. The only self-citation is Lemma 3.1, attributed to the author's own preprint [5]; that estimate is a standard scaling bound for ∫ |ξ|^β e^{-c|ξ|^α t} dξ and is used only as a routine multiplier estimate inside decay proofs, not as the source of the profile or the optimal rates. Hence the self-citation is not load-bearing. I also note, as a non-circular correctness concern flagged by the proof structure, that in Theorem 1.2 the proof estimates only the u-profile (7) via the decomposition into J1–J4; the velocity profile (8) and the j = 1 part of (9) are stated in the theorem but no corresponding decomposition for B_t |D|^s u is written in Section 3.2. Similarly, Lemma 3.2 requires the extra-derivative bound || |D|^{a+1} φ ||_{L2} = O(t^{-α−β}), and the text verifies only the first bound, for example in (44) and (67). These are potential proof gaps, likely repairable by scaling and Proposition 3.4 estimates, but they are not circularity: they do not reduce a claimed output to an input.

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

The central claim is built entirely from the Fourier representation of the equation and standard estimates; there are no fitted numbers and no new physical or mathematical entities. The only inputs beyond standard analysis are the listed lemmas, the nonzero-mass condition, and the dimension restrictions stated in the theorems.

assumptions (5)

standard math Lemma 3.1: for n>=1, c>0, alpha>0, beta real with n+beta>0, the integrals of |ξ|^beta e^{-c|ξ|^alpha t} over low and high frequencies decay polynomially with rates (1+t)^{-(n+beta)/alpha} and t^{-(n+beta)/alpha}.
Cited to [5]; used throughout to convert multiplier weights into polynomial time decay in Lm0 norms, e.g. in Lemma 2.4.
standard math Lemma 3.2: if v is in L1 and the profile phi satisfies || |D|^a phi ||_{L2}=O(t^{-alpha}) and || |D|^{a+1} phi ||_{L2}=O(t^{-alpha-beta}), then || |D|^a (phi*v - (∫v)phi) ||_{L2}=o(t^{-alpha}).
Cited to [11]; replaces u1 by P1 delta_0 in the error terms I5, J4 and the analogous term in Theorem 1.3. The extra-derivative half of the hypothesis is not verified in the paper.
standard math Lemma 3.3: a variant Riemann-Lebesgue lemma for ∫_0^∞ f(r) e^{-zr} dr tending to zero as |z| goes to infinity in the half-plane Re z >= 0.
Used in the proof of Theorem 1.2 to show that the cosine oscillation term in the sin-profile norm tends to zero, yielding a positive lower constant.
domain assumption P1 = ∫_{R^n} u1(x) dx != 0.
Explicitly assumed in Theorems 1.1-1.3 for the lower-bound (optimality) parts; if P1=0, the leading asymptotic term vanishes and the stated two-sided bounds do not apply.
domain assumption Dimension-frequency conditions n>4δ1 (Theorems 1.1 and 1.3) and n>2σ (Theorem 1.2).
Stated in the theorems; they make the constants in (44) and (67) finite and the L1 data admissible.

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Pith. "Pith review of Some remarks on the asymptotic profile of solutions to structurally damped $\sigma$-evolution equations." pith.science (2026). https://pith.science/paper/KF5PN4GL

@misc{pith2026190808492,
  author       = {Pith},
  title        = {Pith review of: Some remarks on the asymptotic profile of solutions to structurally damped $\sigma$-evolution equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KF5PN4GL}},
  note         = {Machine review of arXiv:1908.08492}
}

abstract

In this paper, we are interested in analyzing the asymptotic profiles of solutions to the Cauchy problem for linear structurally damped $\sigma$-evolution equations in $L^2$-sense. Depending on the parameters $\sigma$ and $\delta$ we would like to not only indicate approximation formula of solutions but also recognize the optimality of their decay rates as well in the distinct cases of parabolic like damping and $\sigma$-evolution like damping. Moreover, such results are also discussed when we mix these two kinds of damping terms in a $\sigma$-evolution equation to investigate how each of them affects the asymptotic profile of solutions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[5]

T.A. Dao, H. Michihisa, Study of semi-linear σ -evolution equations with frictional and visco-elastic da mping, preprint, arXiv:1906.04471, 2019

work page Pith review arXiv 1906
[1]

Duong, Some results on the global solvability for st ructurally damped models with special nonlinearity, Ukrainian Math

P.T. Duong, Some results on the global solvability for st ructurally damped models with special nonlinearity, Ukrainian Math. J. , 70 (2019), 1395–1418

work page 2019
[2]

D’Abbicco, M.R

M. D’Abbicco, M.R. Ebert, A classiﬁation of structural d issipations for evolution operators, Math. Methods Appl. Sci., 39 (2016), 2558–2582

work page 2016
[3]

D’Abbicco, M.R

M. D’Abbicco, M.R. Ebert, A new phenomenon in the critica l exponent for structurally damped semi-linear evolution equations, Nonlinear Anal. , 149 (2017), 1–40

work page 2017
[4]

Duong, M

P.T. Duong, M. Kainane Mezadek, and M. Reissig, Global existence for semi-linear structurally damped σ - evolution models , J. Math. Anal. Appl., 431 (2015), 569–596

work page 2015
[6]

D’Abbicco, M

M. D’Abbicco, M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci. , 37 (2014), 1570–1592

work page 2014
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T.A. Dao, M. Reissig, An application of L1 estimates for oscillating integrals to parabolic like semi -linear struc- turally damped σ -evolution models , J. Math. Anal. Appl., 476 (2019), 426–463

work page 2019
[8]

T.A. Dao, M. Reissig, L1 estimates for oscillating integrals and their application s to semi-linear models with σ -evolution like structural damping , Discrete Contin. Dyn. Syst. A, 39 (2019), 5431–5463

work page 2019

Show all 11 references

[9]

Ikehata, H

R. Ikehata, H. Michihisa, Moment conditions and lower bo unds in expanding solutions of wave equations with double damping terms, Asymptot. Anal. , 114 (2019), 19–36

2019
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Ikehata, A

R. Ikehata, A. Sawada, Asymptotic proﬁle of solutions f or wave equations with frictional and viscoelastic damping terms, Asymptot. Anal. , 98 (2016), 59–77

2016
[11]

Ikehata, H

R. Ikehata, H. Takeda, Asymptotic proﬁles of solutions for structural damped wave equations, J. Dyn. Diﬀer. Equ., 31 (2019), 537–571. Tuan Anh Dao School of Applied Mathematics and Informatics, Hanoi Unive rsity of Science and Technology, No.1 Dai Co Viet road, Hanoi, Vietnam ...

2019

Pith tools

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

[1] [5]

T.A. Dao, H. Michihisa, Study of semi-linear σ -evolution equations with frictional and visco-elastic da mping, preprint, arXiv:1906.04471, 2019

work page Pith review arXiv 1906

[2] [1]

Duong, Some results on the global solvability for st ructurally damped models with special nonlinearity, Ukrainian Math

P.T. Duong, Some results on the global solvability for st ructurally damped models with special nonlinearity, Ukrainian Math. J. , 70 (2019), 1395–1418

work page 2019

[3] [2]

D’Abbicco, M.R

M. D’Abbicco, M.R. Ebert, A classiﬁation of structural d issipations for evolution operators, Math. Methods Appl. Sci., 39 (2016), 2558–2582

work page 2016

[4] [3]

D’Abbicco, M.R

M. D’Abbicco, M.R. Ebert, A new phenomenon in the critica l exponent for structurally damped semi-linear evolution equations, Nonlinear Anal. , 149 (2017), 1–40

work page 2017

[5] [4]

Duong, M

P.T. Duong, M. Kainane Mezadek, and M. Reissig, Global existence for semi-linear structurally damped σ - evolution models , J. Math. Anal. Appl., 431 (2015), 569–596

work page 2015

[6] [6]

D’Abbicco, M

M. D’Abbicco, M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci. , 37 (2014), 1570–1592

work page 2014

[7] [7]

T.A. Dao, M. Reissig, An application of L1 estimates for oscillating integrals to parabolic like semi -linear struc- turally damped σ -evolution models , J. Math. Anal. Appl., 476 (2019), 426–463

work page 2019

[8] [8]

T.A. Dao, M. Reissig, L1 estimates for oscillating integrals and their application s to semi-linear models with σ -evolution like structural damping , Discrete Contin. Dyn. Syst. A, 39 (2019), 5431–5463

work page 2019