An inverse problem of finding two time-dependent coefficients in second order hyperbolic equations from Dirichlet to Neumann map

TL;DR

This paper addresses the inverse problem of simultaneously recovering a time-dependent velocity field and electric potential in a hyperbolic PDE from boundary measurements, providing stability estimates and extending the recoverability region.

Contribution

It introduces a method to stably determine both coefficients from boundary data in higher dimensions, expanding the known regions where recovery is possible.

Findings

Established stability estimates for the inverse problem.

Extended the recoverability region to include the entire domain.

Reduced the problem to a classic inverse problem for electromagnetic waves.

Abstract

In the present paper, we consider a non self adjoint hyperbolic operator with a vector field and an electric potential that depend not only on the space variable but also on the time variable. More precisely, we attempt to stably and simultaneously retrieve the real valued velocity field and the real valued potential from the knowledge of Neumann measurements performed on the whole boundary of the domain. We establish in dimension n greater than two, stability estimates for the problem under consideration. Thereafter, by enlarging the set of data we show that the unknown terms can be stably retrieved in larger regions including the whole domain. The proof of the main results are mainly based on the reduction of the inverse problem under investigation to an equivalent and classic inverse problem for an electro-magnetic wave equation.