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The Calderón problem for third order nonlocal wave equations with time-dependent nonlinearities and potentials

Song-Ren Fu, Yongyi Yu, Philipp Zimmermann

TL;DR

The paper addresses the Calderón problem for third-order nonlocal wave equations arising in nonlinear acoustics, focusing on semilinear MGT and Westervelt-type JMGT models with time-dependent nonlinearities and potentials. It develops a nonlocal inverse-problem framework based on Dirichlet-to-Neumann maps, Runge approximation, and unique continuation for the fractional Laplacian, proving unique recovery of the potential $q$ and derivatives of the nonlinearity $g$ (polynomial-type and polyhomogeneous) and, for Westervelt-type nonlinearities, the corresponding coefficients. The authors implement higher-order linearization for polynomial nonlinearities and first-order linearization for polyhomogeneous and Westervelt-type cases, achieving recovery without dimension or order restrictions and allowing time dependence in unknowns. The results advance nonlocal Calderón-type inverse problems by handling third-order temporal dynamics, finite propagation speed, and nontrivial time-dependent coefficients, with potential implications for applications in nonlinear acoustics and related nonlocal PDEs.

Abstract

In this article, we study the Calderón problem for nonlocal generalizations of the semilinear Moore--Gibson--Thompson (MGT) equation and the Jordan--Moore--Gibson--Thompson (JMGT) equation of Westervelt-type. These partial differential equations are third order wave equations that appear in nonlinear acoustics, describe the propagation of high-intensity sound waves and exhibit finite speed of propagation. For semilinear MGT equations with nonlinearity $g$ and potential $q$, we show the following uniqueness properties of the Dirichlet to Neumann (DN) map $Λ_{q,g}$: (i) If $g$ is a polynomial-type nonlinearity whose $m$-th order derivative is bounded, then $Λ_{q,g}$ uniquely determines $q$ and $(\partial^{\ell}_τg(x,t,0))_{2\leq \ell \leq m}$. (ii) If $g$ is a polyhomogeneous nonlinearity of finite order $L$, then $Λ_{q,g}$ uniquely determines $q$ and $g$. The uniqueness proof for polynomial-type nonlinearities is based on a higher order linearization scheme, while the proof for polyhomogeneous nonlinearities only uses a first order linearization. Finally, we demonstrate that a first linearization suffices to uniquely determine Westervelt-type nonlinearities from the related DN maps. We also remark that all the unknowns, which we wish to recover from the DN data, are allowed to depend on time.

The Calderón problem for third order nonlocal wave equations with time-dependent nonlinearities and potentials

TL;DR

The paper addresses the Calderón problem for third-order nonlocal wave equations arising in nonlinear acoustics, focusing on semilinear MGT and Westervelt-type JMGT models with time-dependent nonlinearities and potentials. It develops a nonlocal inverse-problem framework based on Dirichlet-to-Neumann maps, Runge approximation, and unique continuation for the fractional Laplacian, proving unique recovery of the potential and derivatives of the nonlinearity (polynomial-type and polyhomogeneous) and, for Westervelt-type nonlinearities, the corresponding coefficients. The authors implement higher-order linearization for polynomial nonlinearities and first-order linearization for polyhomogeneous and Westervelt-type cases, achieving recovery without dimension or order restrictions and allowing time dependence in unknowns. The results advance nonlocal Calderón-type inverse problems by handling third-order temporal dynamics, finite propagation speed, and nontrivial time-dependent coefficients, with potential implications for applications in nonlinear acoustics and related nonlocal PDEs.

Abstract

In this article, we study the Calderón problem for nonlocal generalizations of the semilinear Moore--Gibson--Thompson (MGT) equation and the Jordan--Moore--Gibson--Thompson (JMGT) equation of Westervelt-type. These partial differential equations are third order wave equations that appear in nonlinear acoustics, describe the propagation of high-intensity sound waves and exhibit finite speed of propagation. For semilinear MGT equations with nonlinearity and potential , we show the following uniqueness properties of the Dirichlet to Neumann (DN) map : (i) If is a polynomial-type nonlinearity whose -th order derivative is bounded, then uniquely determines and . (ii) If is a polyhomogeneous nonlinearity of finite order , then uniquely determines and . The uniqueness proof for polynomial-type nonlinearities is based on a higher order linearization scheme, while the proof for polyhomogeneous nonlinearities only uses a first order linearization. Finally, we demonstrate that a first linearization suffices to uniquely determine Westervelt-type nonlinearities from the related DN maps. We also remark that all the unknowns, which we wish to recover from the DN data, are allowed to depend on time.

Paper Structure

This paper contains 32 sections, 24 theorems, 416 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background of acoustic waves
  3. Nonlocal MGT and JMGT equations
  4. Assumptions and main results
  5. Comparison of our main results with the literature
  6. Inverse problems for nonlocal elliptic and parabolic equations
  7. Inverse problems for nonlocal wave equations
  8. Discussion of main results
  9. Organization of the article
  10. Preliminaries
  11. Fractional Sobolev spaces
  12. Time dependent function spaces
  13. Well-posedness results for linear and nonlinear forward problems
  14. Well-posedness of the linear nonlocal MGT equation
  15. Well-posedness of semilinear nonlocal MGT equations
  16. ...and 17 more sections

Key Result

Theorem 1.3

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain, $T>0$ and $s\in \mathbb{R}_+\setminus{\mathbb N}$. Suppose that the potentials $q_1,q_2\in L^{\infty}(0,T;L^p(\Omega))$ are real valued, $p$ satisfies the condition conp and $q_2$ is time-reversal invariant. Furthermore, let the nonlinea for all sufficiently small exterior values $\varphi\in C_c^{\infty}((W_1)_T)$, then we have a.e. i

Theorems & Definitions (57)

  • Remark 1.1
  • Example 1.2: Polynomial-type nonlinearities
  • Theorem 1.3: Unique determination of $q$ and $\{\partial_\tau^{\ell}g(0)\}_{\ell=1}^m$
  • Theorem 1.4: Unique determination of $q$ and $g$ (or $\{\alpha_k\}_{k=1}^L$)
  • Remark 1.5
  • Theorem 1.6: Unique determination of $\kappa$ or $\beta$
  • Remark 1.7
  • Proposition 2.1: Properties of fractional Laplacian
  • Proposition 2.2
  • Lemma 3.1
  • ...and 47 more