The Calder\'on problem for third order nonlocal wave equations with time-dependent nonlinearities and potentials

TL;DR

This paper investigates the Calderón problem for third order nonlocal wave equations with time-dependent nonlinearities and potentials, establishing uniqueness results for the inverse problem using linearization techniques.

Contribution

It introduces new uniqueness results for the inverse problem of nonlocal third order wave equations with time-dependent nonlinearities and potentials, employing higher and first order linearization methods.

Findings

Unique determination of potential q from DN map.

Unique recovery of nonlinearities g from DN map.

Applicability to time-dependent unknowns.

Abstract

In this article, we study the Calder\'on 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 $\Lambda_{q,g}$: (i) If $g$ is a polynomial-type nonlinearity whose $m$-th order derivative is bounded, then $\Lambda_{q,g}$ uniquely determines $q$ and $(\partial^{\ell}_\tau g(x,t,0))_{2\leq \ell \leq m}$. (ii) If $g$ is a polyhomogeneous nonlinearity of finite order $L$, then $\Lambda_{q,g}$ 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.