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arxiv: 2509.13962 · v3 · submitted 2025-09-17 · 🧮 math.AP

Reconstruction of degeneracy region and power for parabolic equations and systems

Piermarco Cannarsa , Veronica Danesi , Anna Doubova This is my paper

Pith reviewed 2026-05-18 16:25 UTC · model grok-4.3

classification 🧮 math.AP
keywords inverse problemsdegenerate parabolic equationsBessel functionsspectral analysisuniquenessstabilityone-point observation
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The pith

Sufficient initial data conditions guarantee unique and stable recovery of the degeneracy point from one boundary measurement in degenerate parabolic equations.

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

This paper addresses the inverse problem of recovering the degeneracy point in the diffusion coefficient of a one-dimensional parabolic equation by measuring the normal derivative at a single boundary point. It derives sufficient conditions on the initial data to ensure stability and uniqueness in the strongly degenerate case. Additionally, it provides broader uniqueness theorems that identify the initial data, the zero-order coefficient, and the degeneracy power from time-dependent measurements. The method uses spectral analysis with explicit Bessel function representations of the solution. These findings are relevant for understanding and solving inverse problems in systems with singular diffusion coefficients, which model various physical processes.

Core claim

By analyzing the spectral problem and using an explicit form of the solution in terms of Bessel functions, sufficient conditions on the initial data are derived that guarantee the stability and uniqueness of recovering the degeneracy point from a one-point measurement. More general uniqueness theorems are presented that also cover the identification of the initial data, the coefficient of the zero order term and the degeneracy power, using measurements taken over time. The investigation extends to real 1-D degenerate parabolic systems of equations with a specific structure, supported by numerical simulations.

What carries the argument

The explicit representation of the solution in terms of Bessel functions for the spectral problem of the degenerate diffusion operator, which enables the uniqueness and stability analysis.

If this is right

  • Under the derived conditions, the degeneracy location is uniquely determined by the one-point observation.
  • Time measurements allow simultaneous recovery of initial data, zero-order coefficient, and degeneracy power.
  • The approach applies to coupled systems of real degenerate parabolic equations.
  • Numerical experiments confirm the theoretical uniqueness and stability results.

Where Pith is reading between the lines

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

  • The Bessel function approach could be adapted for numerical reconstruction algorithms in practice.
  • This framework might extend to identifying degeneracy in higher-dimensional or nonlinear parabolic problems.
  • It suggests that minimal observations suffice for parameter identification in degenerate media, potentially reducing experimental costs.

Load-bearing premise

The solution to the parabolic equation admits an explicit representation in terms of Bessel functions for the associated spectral problem.

What would settle it

Finding two different degeneracy points that yield the same normal derivative measurement at the boundary point for a qualifying initial datum would falsify the uniqueness result.

Figures

Figures reproduced from arXiv: 2509.13962 by Anna Doubova, Piermarco Cannarsa, Veronica Danesi.

Figure 1
Figure 1. Figure 1: Lack of stability, T = 0.7 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between several compact intervals [τ, γ]. degeneracy power and the initial data. Unlike in the previous section, where we considered point￾wise measurements of ∂xw(1, t), here we require measurements distributed over a time interval. For the first inverse problem, in addition to distributed measurements of ∂xw(1, t), the uniqueness of the coefficient c also requires distributed measurements of ∂… view at source ↗
Figure 4
Figure 4. Figure 4: Test 1, θ = 1.5, t ∗ = 1.99, u0 = 1, v0 = 1. Iterations in the computation of a by trust-region-reflective algorithm, aini = 0.1. 1 2 3 4 5 6 7 8 9 10 Iterations 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Function value Current function values [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Test 2, θ = 1.5, u0 = 1, v0 = 1. Iterations in the computation of a by trust-region-reflective algorithm, aini = 0.1. 1 2 3 4 5 6 7 8 9 10 11 Iterates 0 50 100 150 200 250 Function value Current Function Values [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Test 3, θ = 1.5. Iterations in the compu￾tation of u01 and u02 by trust-region-reflective algorithm, aini = 0.1, u01ini = 0.5, u02ini = 1.5. 0 5 10 15 Iterates 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Test 3, θ = 1.5. Evolution of the cost in trust-region-reflective algorithm, aini = 0.1, u01ini = 0.5, u02ini = 1.5. Test 4 We will take θ = 1.5, α = 1, β = 1, T = 2, t1 = 0, t2 = T , u01ini = 0.5, u02ini = 1.8 and aini = 0.1 as initial guesses to recover the desired value of ad = 0.5, u01d = 1, u02d = 2 using the minimization algorithm. The numerical results can be seen in Figures 11, 12 and 13. In [PIT… view at source ↗
Figure 11
Figure 11. Figure 11: Test 4, θ = 1.5. Itera￾tions in the computation of u01 and u02 by trust-region-reflective algorithm, aini = 0.1, u01ini = 0.5, u02ini = 1.8. 0 5 10 15 Iterates 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: Test 4, θ = 1.5. Evolution of the cost in trust-region-reflective algorithm, aini = 0.1, u01ini = 0.5, u02ini = 1.8. the inverse problem is now as follows: ( Minimize K(a, u02), where a ∈ Ua ad and (u a,u02 , va,u02 ) satisfies (51), where U a ad is given by (52) and K : (a, u02) ∈ Ua ad × R 7→ R is defined as follows: K(a, u02) = 1 2 Z t2 t1 |η(t) − ∂xu a,u02 (1, t)| 2 dt + 1 2 Z t2 t1 |ζ(t) − ∂xv a,u02 … view at source ↗
Figure 14
Figure 14. Figure 14: Test 5, θ = 1.5. Iterations in the com￾putation of u02 by trust-region-reflective algo￾rithm, aini = 0.1, u02ini = 1.8. 0 1 2 3 4 5 6 7 8 9 Iterates 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: Test 5, θ = 1.5. Evolution of the cost in trust-region-reflective algorithm, aini = 0.1 and u02ini = 1.8. 8.4 Degeneracy power and initial data reconstruction with distributed measurements In this sub-section, we will give a numerical simulation in line with the results 6.3 and 7.4. In particular, we assume that u0 is of the form (54) and v0 = 0, and fix the initial data u01 = 1 in (0, a), reconstructing … view at source ↗
Figure 17
Figure 17. Figure 17: Test 6, a = 0.5. Iterations in the computation of θ by trust-region-reflective al￾gorithm, θini = 1.1, u02ini = 1.5. 0 2 4 6 8 10 12 14 Iterates 0 0.5 1 1.5 2 2.5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 19
Figure 19. Figure 19: Test 6, a = 0.5. Evolution of the cost in trust-region-reflective algorithm, θini = 1.1, u02ini = 1.5. A Proof of Lemma 3.1 The proof of properties a), b) can be found in [33]. For property c) see [1]. With regard to property d), we have Z jν,n 0 s ν+1Jν(s) ds = [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
read the original abstract

We address the inverse problem of recovering a degeneracy point within the diffusion coefficient of a one-dimensional complex parabolic equation by observing the normal derivative at one point of the boundary. The strongly degenerate case is analyzed. In particular, we derive sufficient conditions on the initial data that guarantee the stability and uniqueness of the solution obtained from a one-point measurement. Moreover, we present more general uniqueness theorems, which also cover the identification of the initial data, the coefficient of the zero order term and the degeneracy power, using measurements taken over time. Our method is based on a careful analysis of the spectral problem and relies on an explicit form of the solution in terms of Bessel functions. Our investigation also covers the case of real 1-D degenerate parabolic systems of equations coupled with a specific structure. Theoretical results are also supported by numerical simulations.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates the inverse problem of recovering the degeneracy point in the diffusion coefficient of a one-dimensional parabolic equation from the normal derivative measured at a single boundary point, including the strongly degenerate case. Sufficient conditions on the initial data are derived to ensure stability and uniqueness of the reconstruction. More general uniqueness results are presented for identifying the initial data, the zero-order term coefficient, and the degeneracy power from time-dependent measurements. The method relies on spectral analysis of the associated degenerate Sturm-Liouville problem and explicit solution formulas involving Bessel functions. The results are extended to certain real 1-D degenerate parabolic systems, and numerical simulations are provided.

Significance. Should the central claims be verified, the paper would advance the field of inverse problems for degenerate parabolic PDEs by providing explicit conditions and methods based on spectral theory. This is significant for applications involving degeneracy, such as in mathematical biology or fluid mechanics. The explicit use of Bessel functions offers a concrete way to handle the strong degeneracy, distinguishing it from more abstract approaches.

major comments (1)
  1. [Spectral problem and solution representation] The uniqueness and stability results are obtained by reducing the parabolic problem to a spectral expansion with coefficients expressed via Bessel functions of the degenerate Sturm-Liouville operator. Since the degeneracy location and power are unknown parameters, the eigenfunctions depend on these unknowns. The manuscript requires additional justification that the closed-form Bessel representation remains valid uniformly in a neighborhood of the unknown parameters, particularly for interior degeneracy points or strongly degenerate regimes. This is load-bearing for the central claims regarding one-point measurements and joint identification.
minor comments (1)
  1. [Numerical simulations] The numerical simulations could include more discussion on how the degeneracy point is approximated in the discretization to validate the theoretical stability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the major comment below and will revise the manuscript to incorporate additional justification where needed.

read point-by-point responses

  1. Referee: The uniqueness and stability results are obtained by reducing the parabolic problem to a spectral expansion with coefficients expressed via Bessel functions of the degenerate Sturm-Liouville operator. Since the degeneracy location and power are unknown parameters, the eigenfunctions depend on these unknowns. The manuscript requires additional justification that the closed-form Bessel representation remains valid uniformly in a neighborhood of the unknown parameters, particularly for interior degeneracy points or strongly degenerate regimes. This is load-bearing for the central claims regarding one-point measurements and joint identification.

    Authors: We appreciate the referee's observation concerning the parameter dependence in the eigenfunctions. The manuscript derives the explicit Bessel representations for fixed degeneracy parameters via the standard transformation to Bessel equations for the degenerate Sturm-Liouville operator. To strengthen the argument for the inverse results, we will add a dedicated subsection (or appendix) in the revised version establishing uniform validity in a neighborhood of the true parameters. This will rely on the analytic dependence of Bessel functions on their order and argument, combined with continuity estimates for the eigenvalues and eigenfunctions with respect to the degeneracy location and power. Separate arguments will be provided for interior versus boundary degeneracy and for the strongly degenerate regime. These additions will directly support the one-point measurement and joint identification claims. revision: yes

Circularity Check

0 steps flagged

Derivation relies on standard spectral analysis with explicit Bessel forms; no reduction to fitted inputs or self-referential definitions

full rationale

The paper's uniqueness and stability results are obtained by reducing the parabolic problem to a spectral expansion whose coefficients are expressed via Bessel functions of the degenerate Sturm-Liouville operator. This representation is invoked as a known property of the operator rather than derived from the inverse data or fitted parameters within the present work. No step equates a prediction to a fitted quantity by construction, nor does any load-bearing claim rest solely on a self-citation whose validity is unverified outside the paper. The central claims therefore retain independent mathematical content once the spectral assumption is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of Sturm-Liouville operators and Bessel functions for the spectral analysis of degenerate parabolic problems; no free parameters or new postulated entities are introduced.

axioms (1)
  • standard math The eigenfunctions of the degenerate spatial operator admit an explicit representation in terms of Bessel functions.
    Invoked to obtain the explicit form of the solution used for uniqueness proofs.

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