Inverse problem of recovering a time-dependent nonlinearity appearing in third-order nonlinear acoustic equations

TL;DR

This paper addresses the inverse problem of uniquely recovering a time-dependent nonlinearity in the third-order Jordan-Moore-Gibson-Thompson acoustic equation, using higher order linearization and complex geometric optics solutions.

Contribution

It introduces a method combining higher order linearization and CGO solutions to establish uniqueness in recovering the nonlinearity in the J-M-G-T equation.

Findings

Proved well-posedness for small initial and boundary data.

Established uniqueness of the nonlinearity recovery.

Developed a new approach using CGO solutions for third-order equations.

Abstract

In this paper, we consider the inverse problem of recovering a time-dependent nonlinearity for a third order nonlinear acoustic equation, which is known as the Jordan-Moore-Gibson-Thompson equation (J-M-G-T equation for short). This third order in time equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained for the small initial and boundary data. By the higher order linearization to the nonlinear equation, and construction of complex geometric optics (CGO for short) solutions for the linearized equation, we derive the uniqueness of recovering the nonlinearity.