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Reconstruction of degeneracy region and power for parabolic equations and systems

Piermarco Cannarsa, Veronica Danesi, Anna Doubova

TL;DR

The paper studies the inverse problem of locating an interior degeneracy in a strongly degenerate 1-D parabolic equation from boundary flux data, using a spectral approach based on Bessel functions. It establishes well-posedness of the direct problem in weighted Sobolev spaces, derives explicit eigenfunctions and eigenvalues, and provides explicit formulas for the boundary normal derivative. It proves Lipschitz stability with one-point measurements under suitable nondegeneracy conditions, and develops general uniqueness results for several inverse problems with distributed time measurements, including extensions to real 1-D coupled systems and monotonicity in the degeneracy power. Numerical experiments validate the theoretical results, showing accurate recovery of the degeneracy position, initial data, and degeneracy power using both one-point and distributed data, and highlighting the role of measurement configuration. Overall, the work advances the theory and computation of inverse problems for degenerate diffusion with practical implications for materials with conductivity failures and related applications.

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.

Reconstruction of degeneracy region and power for parabolic equations and systems

TL;DR

The paper studies the inverse problem of locating an interior degeneracy in a strongly degenerate 1-D parabolic equation from boundary flux data, using a spectral approach based on Bessel functions. It establishes well-posedness of the direct problem in weighted Sobolev spaces, derives explicit eigenfunctions and eigenvalues, and provides explicit formulas for the boundary normal derivative. It proves Lipschitz stability with one-point measurements under suitable nondegeneracy conditions, and develops general uniqueness results for several inverse problems with distributed time measurements, including extensions to real 1-D coupled systems and monotonicity in the degeneracy power. Numerical experiments validate the theoretical results, showing accurate recovery of the degeneracy position, initial data, and degeneracy power using both one-point and distributed data, and highlighting the role of measurement configuration. Overall, the work advances the theory and computation of inverse problems for degenerate diffusion with practical implications for materials with conductivity failures and related applications.

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.

Paper Structure

This paper contains 17 sections, 14 theorems, 126 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Functional setting and well-posedness
  3. The eigenvalue problem
  4. Computation of the normal derivative
  5. Lipschitz stability with one point measurement
  6. Example of admissible initial data for stability estimates
  7. Uniqueness results for “ distributed” measurements
  8. Monotonicity of the first eigenvalue with respect to $\theta$
  9. Real systems of 1-D coupled degenerate parabolic equations
  10. Numerical results
  11. Degeneracy reconstruction with one-point measurements
  12. Degeneracy reconstruction with distributed measurements
  13. Degeneracy and initial data reconstruction with distributed measurements
  14. Piecewise-constant initial data, two side measurements
  15. Piecewise-constant initial data, one side measurements
  16. ...and 2 more sections

Key Result

Proposition 2.1

Given $\theta \in [1,2)$, we have: a) $H^1_{\theta} (0,1;\mathbb{C})$ is a Hilbert space. b) $\mathbb{A}: D(\mathbb{A}) \subset X \to X$ is a dissipative self-adjoint operator with dense domain. Therefore, $\mathbb{A}$ is the infinitesimal generator of an analytic semigroup of contractions $e^{t\mat

Figures (19)

  • Figure 1: Lack of stability, $T = 0.7$.
  • Figure 2: Stability for $t$ large, $T = 1.5$.
  • Figure 3: Comparison between several compact intervals $[\tau,\gamma]$.
  • Figure 4: Test \ref{['test1']}, $\theta = 1.5$, $t^\ast = 1.99$, $u_0=1$, $v_0=1$. Iterations in the computation of $a$ by trust-region-reflective algorithm, $aini=0.1$.
  • Figure 5: Test \ref{['test1']}, $\theta = 1.5$, $t^\ast = 1.99$, $u_0=1$, $v_0=1$. Evolution of the cost in trust-region-reflective algorithm, $aini=0.1$.
  • ...and 14 more figures

Theorems & Definitions (19)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.1
  • Proposition 3.1
  • Lemma 3.1: Properties of Bessel functions
  • Theorem 4.1
  • Theorem 5.1
  • Remark 5.1
  • Theorem 5.2
  • Theorem 6.1
  • ...and 9 more