An inverse problem for the fractionally damped wave equation

TL;DR

This paper addresses an inverse problem for a nonlinear wave equation with fractional damping, demonstrating unique coefficient recovery using advanced linearization and spectral properties, relevant for nonlinear acoustic imaging.

Contribution

It introduces a novel method for uniquely determining a nonlinear coefficient in a fractional damped wave equation from source-to-solution data.

Findings

Proves local well-posedness of the forward problem.

Establishes unique determination of the nonlinear coefficient.

Utilizes second order linearization and spectral fractional Laplacian properties.

Abstract

We consider an inverse problem for a Westervelt type nonlinear wave equation with fractional damping. This equation arises in nonlinear acoustic imaging, and we show the forward problem is locally well-posed. We prove that the smooth coefficient of the nonlinearity can be uniquely determined, based on the knowledge of the source-to-solution map and a priori knowledge of the coefficient in an arbitrarily small subset of the domain. Our approach relies on a second order linearization as well as the unique continuation property of the spectral fractional Laplacian.