Inverse problem of recovering the time-dependent damping and nonlinear terms for wave equations

TL;DR

This paper addresses the inverse problem of determining time-dependent damping and nonlinear terms in wave equations on Riemannian manifolds, using Carleman estimates, Gaussian beams, and higher order linearization to establish uniqueness.

Contribution

It introduces a novel approach combining Carleman estimates, Gaussian beams, and higher order linearization for unique recovery of coefficients in semilinear wave equations.

Findings

Proved uniqueness of recovering damping and nonlinear terms.

Developed a method applicable to wave equations on Riemannian manifolds.

Enhanced inverse problem techniques with combined analytical tools.

Abstract

In this paper, we consider the inverse boundary problems of recovering the time-dependent nonlinearity and damping term for a semilinear wave equation on a Riemannian manifold. The Carleman estimate and the construction of Gaussian beams together with the higher order linearization are respectively used to derive the uniqueness results of recovering the coefficients.