An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation

Irina Kmit; Nataliya Protsakh; Viktor Tkachenko

arxiv: 2601.07081 · v2 · submitted 2026-01-11 · 🧮 math.AP

An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation

Irina Kmit , Nataliya Protsakh , Viktor Tkachenko This is my paper

Pith reviewed 2026-05-16 14:53 UTC · model grok-4.3

classification 🧮 math.AP

keywords inverse problemsemilinear wave equationstrong dampingalmost periodic solutionsbounded solutionsDirichlet boundary conditionsSobolev spacesexistence and uniqueness

0 comments

The pith

Reducing an inverse source problem for a semilinear strongly damped wave equation to a direct problem produces unique bounded solutions on the entire real line.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that an inverse boundary value problem for a semilinear strongly damped wave equation with Dirichlet conditions admits a unique bounded strong solution on the whole real line, recovered from an integral overdetermination condition that identifies the time-dependent source. The proof proceeds by reducing the inverse problem to a direct one, establishing local existence and uniqueness on finite intervals, extending to half-lines, and passing to the limit to obtain the global bounded solution. Periodic or almost periodic data produce solutions of matching type. A reader would care because the result supplies a well-posed recovery procedure for unknown time-varying sources in damped wave systems defined for all time.

Core claim

After reducing the inverse problem to a direct one, existence and uniqueness of solutions on finite time intervals are established, these solutions are extended to half-lines, and a bounded strong solution on the whole real line is constructed as a limit of such extensions, with uniqueness proved; periodic and almost periodic data yield corresponding solutions.

What carries the argument

Reduction of the inverse problem to a direct problem via the integral overdetermination condition, which permits application of local existence theorems followed by limit constructions that preserve boundedness.

If this is right

Existence and uniqueness hold for the associated direct problem on every finite time interval.
Solutions on finite intervals extend to half-lines while preserving regularity and boundedness.
The global solution on the real line arises as a limit and remains bounded.
Uniqueness of the bounded global solution follows from the local uniqueness.
Periodic data produce periodic solutions and almost periodic data produce almost periodic solutions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reduction-plus-limit strategy may apply to inverse problems for other damped hyperbolic equations with integral data.
Boundedness on the infinite line can hold without imposing strong growth conditions on the nonlinearity.
The construction offers a route to recover time-periodic sources in models of vibrating media with damping.

Load-bearing premise

The nonlinearity and coefficients must satisfy conditions allowing the inverse problem to reduce to a direct one and supporting local existence results on finite intervals without extra growth restrictions for the limit process on the real line.

What would settle it

A concrete counter-example consisting of periodic data and admissible nonlinearity for which the limit construction either fails to remain bounded or loses uniqueness on the real line would disprove the claim.

read the original abstract

This paper investigates an inverse boundary value problem for a semilinear strongly damped wave equation with Dirichlet boundary conditions in Sobolev spaces of functions bounded in time on $\R$, including periodic and almost periodic functions. In addition to constructing a bounded strong solution, we determine a time-dependent source coefficient via an integral overdetermination condition ensuring well-posedness. After reducing the inverse problem to a direct one, we first establish existence and uniqueness of solutions to an associated problem on finite time intervals. We then extend these solutions to half-lines and construct a bounded strong solution on the whole real line as a limit of such extensions, and subsequently establish its uniqueness. In particular, periodic and almost periodic data yield periodic and almost periodic solutions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper constructs bounded almost periodic solutions for an inverse source problem in a semilinear strongly damped wave equation on the whole real line by reducing to a direct problem and taking limits of finite-interval extensions.

read the letter

The core result is a well-posedness statement for recovering a time-dependent source coefficient from an integral condition, yielding bounded strong solutions that inherit periodicity or almost periodicity from the data. This extends earlier work on finite intervals or purely periodic cases to the full line with almost periodic forcing, which is the actual novelty here. The reduction step looks clean and the local existence on finite intervals follows from standard theory for strongly damped waves under the stated assumptions on the nonlinearity and coefficients. Uniqueness of the global solution is also claimed once the limit is in hand. That part is useful for specialists who need global-in-time results in this setting. The approach is straightforward: solve locally, extend to half-lines, pass to the limit, and verify the limit satisfies the original equation. No free parameters or circular definitions appear. The soft spot is the limit argument itself. For the family of extensions to converge to a bounded strong solution in the time-bounded Sobolev spaces, you need a priori estimates that stay uniform as the interval length grows. The abstract invokes the given conditions on the nonlinearity but does not indicate an explicit energy estimate or comparison principle that closes independently of T. If those bounds are only local or grow with the interval, the limit may fail to remain bounded or strong. That is the one place where the proof needs to be checked carefully. This is narrow technical work aimed at researchers already working on inverse problems for hyperbolic equations with almost periodic data. It is not broad enough to interest a general PDE audience, but the construction is new enough in its setting that a serious referee should look at the details on the uniform estimates and the passage to the limit. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper studies an inverse boundary value problem for a semilinear strongly damped wave equation with Dirichlet conditions, posed in Sobolev spaces of time-bounded functions on the real line (including periodic and almost periodic cases). It reduces the inverse problem to a direct one, proves local existence and uniqueness on finite intervals, extends solutions to half-lines, constructs a bounded strong solution on all of R as a limit of these extensions, determines the time-dependent source via an integral overdetermination condition, and establishes uniqueness; periodic/almost periodic data are shown to produce corresponding solutions.

Significance. If the uniform bounds needed for the global limit hold, the work would supply a well-posedness theory for inverse problems in almost periodic settings for damped waves, extending local existence results to global bounded strong solutions on R without additional growth restrictions.

major comments (2)

[Main existence theorem and limit passage] The central limit argument (constructing a bounded strong solution on R from extensions on half-lines) requires a priori estimates uniform in the length of the time interval. The stated conditions on the nonlinearity and coefficients permit local existence but do not automatically close uniform bounds independent of T; without an explicit energy estimate or maximum principle that is T-independent, the boundedness of the limit in the time-bounded Sobolev spaces is not guaranteed (see the construction after the local existence step).
[Limit construction paragraph] It is unclear how the limit is shown to satisfy the original semilinear equation in the strong sense. The abstract and outline mention passage to the limit, but details on convergence of the nonlinear term and preservation of the strong solution property in the time-bounded spaces are needed to confirm the limit solves the direct problem.

minor comments (2)

[Introduction] Clarify the precise function spaces (e.g., the exact Sobolev norms for time-bounded functions) at the first appearance in the introduction.
[Reduction to direct problem] Add a brief remark on how the integral overdetermination condition is used to recover the source coefficient after the direct problem is solved.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised concern the uniformity of a priori bounds and the details of the limit passage in the existence proof. We address each below and will revise the manuscript to incorporate explicit estimates and expanded arguments, thereby strengthening the presentation without altering the core results.

read point-by-point responses

Referee: [Main existence theorem and limit passage] The central limit argument (constructing a bounded strong solution on R from extensions on half-lines) requires a priori estimates uniform in the length of the time interval. The stated conditions on the nonlinearity and coefficients permit local existence but do not automatically close uniform bounds independent of T; without an explicit energy estimate or maximum principle that is T-independent, the boundedness of the limit in the time-bounded Sobolev spaces is not guaranteed (see the construction after the local existence step).

Authors: We agree that uniform bounds independent of T are essential for the global construction. The strong damping term in the equation yields a dissipative energy estimate that controls the solution norms uniformly in time, independent of the interval length, due to the structure of the linear damped wave operator. In the revised manuscript we will insert an explicit derivation of this T-independent a priori estimate immediately after the local existence result, confirming that the family of solutions on expanding intervals remains bounded in the time-bounded Sobolev spaces and justifying the limit. revision: yes
Referee: [Limit construction paragraph] It is unclear how the limit is shown to satisfy the original semilinear equation in the strong sense. The abstract and outline mention passage to the limit, but details on convergence of the nonlinear term and preservation of the strong solution property in the time-bounded spaces are needed to confirm the limit solves the direct problem.

Authors: We acknowledge that the passage to the limit requires additional clarification. In the revision we will expand the relevant paragraph to specify the convergence topology (strong convergence on compact time intervals in the appropriate Sobolev spaces), invoke the local Lipschitz continuity of the nonlinearity to pass to the limit in the semilinear term, and verify that the limit satisfies the equation in the strong sense by testing against smooth test functions and using the uniform bounds to justify differentiation under the integral sign in the weak formulation. revision: yes

Circularity Check

0 steps flagged

No circularity: standard reduction, local existence, extension, and limit construction

full rationale

The derivation reduces the inverse problem to a direct one, invokes standard local existence/uniqueness results on finite intervals, extends solutions to half-lines, and passes to a limit on the whole line to obtain a bounded strong solution. These steps rely on functional-analytic constructions and a priori estimates under the stated assumptions on nonlinearity and coefficients; no parameter is fitted to data and then renamed as a prediction, no target quantity is defined in terms of itself, and no load-bearing self-citation chain is invoked. The argument is self-contained within the given function-space framework and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard assumptions from functional analysis for local existence of solutions to semilinear wave equations; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption The nonlinearity and coefficients satisfy conditions that guarantee local existence and uniqueness for the direct problem on finite intervals.
This is invoked to reduce the inverse problem and to apply standard existence theorems before extending to the real line.

pith-pipeline@v0.9.0 · 5419 in / 1254 out tokens · 60326 ms · 2026-05-16T14:53:06.316893+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

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[34]

Shahrouzi, On behavior of solutions to a class of nonlinear hyperbolic inverse source problem,Acta Math

M. Shahrouzi, On behavior of solutions to a class of nonlinear hyperbolic inverse source problem,Acta Math. Sin. Engl. Ser.32, 683–698 (2016)

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Shahrouzi, J

M. Shahrouzi, J. Ferreira, E. Piskin, Stability result for a variable-exponent viscoelastic double-Kirchhoff type inverse source problem with nonlocal degenerate damping term, Ric. Mat.73, 2321–2345 (2024)

work page 2024
[36]

G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation,Canad. J. Math.32(3), 631–643 (1980). 32

work page 1980

[1] [1]

S. E. Aitzhanov, J. Ferreira, K. A. Zhalgassova, Solvability of the inverse problem for the pseudohyperbolic equation,J. Math. Mech. Comput. S.,115(3), 1–13 (2022)

work page 2022

[2] [2]

Akhundov, A.Sh

A.Ya. Akhundov, A.Sh. Habibova, On the solvability of an inverse problem for a hyper- bolic heat equation,Azerbaijan J. Math.13(1), 205–215 (2023)

work page 2023

[3] [3]

Azevedo, C

J. Azevedo, C. Cuevas, H. Soto, Qualitative theory for strongly damped wave equations Math. Methods Appl. Sci.40(18), 6944–6975 (2017)

work page 2017

[4] [4]

A. N. Carvalho, J. W. Cholewa, T. Dlotko, Strongly damped wave problems: boot- strapping and regularity of solutions,J. Differ. Equ.244(9), 2310–2333 (2008)

work page 2008

[5] [5]

Celik, A., Kyed, M., Nonlinear wave equation with damping: Periodic forcing and non- resonant solutions to the Kuznetsov equation,Z. Angew. Math. Mech.,98(3), 412–430 (2018)

work page 2018

[6] [6]

F. Chen, B. Guo, P. Wang, Long time behavior of strongly damped nonlinear wave equations,J. Differ. Equ.147, 231–241 (1998)

work page 1998

[7] [7]

Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damp- ing,J

I. Chueshov, Long-time dynamics of Kirchhoff wave models with strong nonlinear damp- ing,J. Differ. Equ.252, 1229–1262 (2012)

work page 2012

[8] [8]

E. A. Coddington, N. Levinson,Theory of ordinary differential equations, McGraw-Hill Book Co., Inc. New York-Toronto-London, 1955

work page 1955

[9] [9]

Corduneanu,Almost Periodic Functions, Chelsea Publ

C. Corduneanu,Almost Periodic Functions, Chelsea Publ. Co., New York, 1989. 30

work page 1989

[10] [10]

Cristofol, S

M. Cristofol, S. Li, Y. Shang, Carleman estimates and some inverse problems for the coupled quantitative thermoacoustic equations by partial boundary layer data. Part II: Some inverse problems,Math. Method Appl. Sci.46(12), 13304–13319 (2023)

work page 2023

[11] [11]

Cuevas, C

C. Cuevas, C. Lizama, H. Soto, Asymptotic periodicity for strongly damped wave equa- tions, InAbstr. Appl. Anal., 2013

work page 2013

[12] [12]

Dafermos, Wave equations with weak damping,SIAM Journal on Ap- plied Mathematics18(4), 759–767 (1970)

Constantine M. Dafermos, Wave equations with weak damping,SIAM Journal on Ap- plied Mathematics18(4), 759–767 (1970)

work page 1970

[13] [13]

Diagana, Almost periodic solutions to some second-order nonautonomous differential equations,Proc

T. Diagana, Almost periodic solutions to some second-order nonautonomous differential equations,Proc. Amer. Math. Soc.140(1, 279–289 (2012)

work page 2012

[14] [14]

Domanska, S

H. Domanska, S. Lavrenyuk, N. Protsakh, Problem for nonlinear third order hyperbolic equationNauk. Bullet. of Chernivtsi University, Mathematics269, 34–42 (2005)

work page 2005

[15] [15]

A. V. Dvornyk, V. I. Tkachenko, Almost periodic solutions of the wave equation with damping and impulsive action,Ukr. Math. J.75(1), 68–79 (2023)

work page 2023

[16] [16]

Evans,Partial Differential Equations, American Mathematical Society, Providence, RI, 1998

L.C. Evans,Partial Differential Equations, American Mathematical Society, Providence, RI, 1998

work page 1998

[17] [17]

Feireisl, Bounded, almost-periodic, and periodic solutions to fully nonlinear telegraph equations,Czech

E. Feireisl, Bounded, almost-periodic, and periodic solutions to fully nonlinear telegraph equations,Czech. Math. J.40(3), 514–527 (1990)

work page 1990

[18] [18]

Isakov,Inverse Problems for Partial Differential Equations, AMS127, 2017

V. Isakov,Inverse Problems for Partial Differential Equations, AMS127, 2017

work page 2017

[19] [19]

Khonatbek, Inverse problem with integral overdetermination for system of equations of Kelvin-Voight fluids,Advanced Materials Research705, 15–20 (2013)

K. Khonatbek, Inverse problem with integral overdetermination for system of equations of Kelvin-Voight fluids,Advanced Materials Research705, 15–20 (2013)

work page 2013

[20] [20]

Kirane, B

M. Kirane, B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay,Z. Angew. Math. Phys.62, 1065–1082 (2011)

work page 2011

[21] [21]

I. Kmit, L. Recke, Time-periodic second-order hyperbolic equations: Fredholm solvabil- ity, regularity, and smooth dependence, InOperator Theory: Advances and Applications 245, 147–181, Basel: Birkh¨ auser (2015)

work page 2015

[22] [22]

Kokocki, Effect of resonance on the existence of periodic solutions for strongly damped wave equation,Nonlinear Analysis125, 167–200 (2015)

P. Kokocki, Effect of resonance on the existence of periodic solutions for strongly damped wave equation,Nonlinear Analysis125, 167–200 (2015)

work page 2015

[23] [23]

Lesnic,Inverse problems with applications in science and engineering, Chapman and Hall/CRC, 2021

D. Lesnic,Inverse problems with applications in science and engineering, Chapman and Hall/CRC, 2021

work page 2021

[24] [24]

H. Li, S. Zhou, F. Yin, Global periodic attractor for strongly damped wave equations with time-periodic driving force,J. Math. Phys.45(9), 3462–3467 (2004). 31

work page 2004

[25] [25]

Lions,Quelques methodes de resolution des problemes aux limites non lineares, Dunod, Paris, Gauthieir-Villars, 1969

J.-L. Lions,Quelques methodes de resolution des problemes aux limites non lineares, Dunod, Paris, Gauthieir-Villars, 1969

work page 1969

[26] [26]

Massatt, Limiting behavior for strongly damped nonlinear wave equations,J

P. Massatt, Limiting behavior for strongly damped nonlinear wave equations,J. Differ. Equ.48(3), 334–349 (1983)

work page 1983

[27] [27]

Mawhin, Periodic solutions of some semilinear wave equations and systems: a survey, Chaos, Solitons&Fractals5(9), 1651–1669 (1995)

J. Mawhin, Periodic solutions of some semilinear wave equations and systems: a survey, Chaos, Solitons&Fractals5(9), 1651–1669 (1995)

work page 1995

[28] [28]

Y. T. Mehraliyev, S. R. Sadikhzade, A. T. Ramazanova, Two-dimensional inverse boundary value problem for a third-order pseudo-hyperbolic equation with an addi- tional integral condition,Eur. J. Pure Appl. Math.16(2), 670–686 (2023)

work page 2023

[29] [29]

Nakao, Bounded, periodic and almost periodic classical solutions of some nonlinear wave equations with a dissipative term,J

M. Nakao, Bounded, periodic and almost periodic classical solutions of some nonlinear wave equations with a dissipative term,J. Math. Soc. Jpn.30(3), 375–394 (1978)

work page 1978

[30] [30]

Nakao, Bounded, periodic or almost-periodic solutions of nonlinear hyperbolic partial differential equations,J

M. Nakao, Bounded, periodic or almost-periodic solutions of nonlinear hyperbolic partial differential equations,J. Differ. Equ.23(3), 368–386 (1977)

work page 1977

[31] [31]

Oussaeif, A

T.-E. Oussaeif, A. Bouziani, Inverse problem of a hyperbolic equation with an integral overdetermination condition,Electr. J. Differ. Equ., Paper No.138(2016)

work page 2016

[32] [32]

N. P. Protsakh, Inverse problem for an ultraparabolic equation with unknown function of the space Variable on the right-hand side,J. Math. Sci.203(1), 16–39 (2014)

work page 2014

[33] [33]

Protsakh, Inverse problem for semilinear wave equation with strong damping,Front

N. Protsakh, Inverse problem for semilinear wave equation with strong damping,Front. Appl. Math. Stat.10, 1467441 (2024)

work page 2024

[34] [34]

Shahrouzi, On behavior of solutions to a class of nonlinear hyperbolic inverse source problem,Acta Math

M. Shahrouzi, On behavior of solutions to a class of nonlinear hyperbolic inverse source problem,Acta Math. Sin. Engl. Ser.32, 683–698 (2016)

work page 2016

[35] [35]

Shahrouzi, J

M. Shahrouzi, J. Ferreira, E. Piskin, Stability result for a variable-exponent viscoelastic double-Kirchhoff type inverse source problem with nonlocal degenerate damping term, Ric. Mat.73, 2321–2345 (2024)

work page 2024

[36] [36]

G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation,Canad. J. Math.32(3), 631–643 (1980). 32

work page 1980