Simultaneous recovery of various time-dependent coefficients in a hyperbolic equation with space-time nonlocal attenuation

Abstract

This article is devoted to the inverse problem of uniquely determining different classes of coefficients in a hyperbolic equation with space-time nonlocal attenuation from a local source-to-solution map. Our objective is to exploit the nonlocal attenuation mechanism to address inverse coefficient problems that remain open, or are even fundamentally intractable, for linear and nonlinear hyperbolic equations without attenuation. More precisely, by leveraging the interactions induced by nonlocal attenuation, we simultaneously determine general classes of leading and lower order coefficients depending on both space and time. In addition, we formulate our results using data collected on disjoint sets, thereby extending the existing theory and introducing the notion of data on sets disjoint in both space and time for hyperbolic equations. Our approach builds on key features of nonlocal operators, such as memory effects and the unique continuation principle, while integrating the interplay between local and nonlocal operators along with structural properties of hyperbolic equations.