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

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.