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Inverse boundary value problems of determining nonlinear coefficients for the JMGT equation

Dong Qiu, Xiang Xu, Yeqiong Ye, Ting Zhou

Abstract

We consider inverse boundary value problems for the Jordan-Moore-Gibson-Thompson (JMGT) equation in nonlinear acoustics with quadratic nonlinearities of Kuznetsov-type and Westervelt-type. We show that the associated boundary Dirichlet-to-Neumann map uniquely determines the nonlinear coefficients $β$ in the Westervelt-type model, and the pair $(β,κ)$ in the Kuznetsov-type model, provided that the observation time is greater than the maximal boundary-to-boundary geodesic travel time. The results are obtained in both the Euclidean setting and on compact Riemannian manifolds with proper geometric assumptions. The proof is based on the idea of second order linearization combined with the construction of geometric optics and Gaussian beam solutions, reducing the inverse problem of uniqueness to the injectivity of associated geodesic ray transforms.

Inverse boundary value problems of determining nonlinear coefficients for the JMGT equation

Abstract

We consider inverse boundary value problems for the Jordan-Moore-Gibson-Thompson (JMGT) equation in nonlinear acoustics with quadratic nonlinearities of Kuznetsov-type and Westervelt-type. We show that the associated boundary Dirichlet-to-Neumann map uniquely determines the nonlinear coefficients in the Westervelt-type model, and the pair in the Kuznetsov-type model, provided that the observation time is greater than the maximal boundary-to-boundary geodesic travel time. The results are obtained in both the Euclidean setting and on compact Riemannian manifolds with proper geometric assumptions. The proof is based on the idea of second order linearization combined with the construction of geometric optics and Gaussian beam solutions, reducing the inverse problem of uniqueness to the injectivity of associated geodesic ray transforms.
Paper Structure (18 sections, 9 theorems, 169 equations)

This paper contains 18 sections, 9 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Statement of the Problem
  3. Main Results
  4. Previous Literature and Related Works
  5. Local Well-Posedness of JMGT equation
  6. The Euclidean Case
  7. Second Order Linearization
  8. Construction of GO solutions
  9. Proof of Theorem \ref{['thm1']}
  10. Gaussian Beam Solutions
  11. Fermi coordinates
  12. Eikonal and transport equations
  13. Construction of the phase function and the amplitude function
  14. Estimate of the Remainder term
  15. The Geometric Case
  16. ...and 3 more sections

Key Result

Proposition 1.1

Fix $\beta \in C^m(\overline M)$ and $\kappa \in C^m( \overline M)$. We assume that $\beta=0, \ \kappa=0$ on $\partial M$. There exists a constant $\delta$ such that if the boundary and initial values $(v_0,v_1,v_2,f)$ belong to the set then the nonlinear system JMGT admits a unique solution $u \in E^{m+2}$ and there exists a positive constant $C$ such that

Theorems & Definitions (18)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • proof
  • Remark 2.3
  • ...and 8 more