# An inverse problem for the Linear Boltzmann Equation with time-dependent coefficient

## TL;DR

This paper investigates the inverse problem of determining a time-dependent absorption coefficient in the linear Boltzmann equation from boundary data, establishing uniqueness and stability results in multiple regions.

## Contribution

It provides the first stability estimates and uniqueness results for the inverse problem of identifying time-dependent coefficients in the linear Boltzmann equation.

## Key findings

- Unique determination of the absorption coefficient in a subset of the domain.
- Logarithmic stability estimate for the coefficient from boundary data.
- Extension of the determination to larger regions with more data.

## Abstract

In this paper, we study the stability in the inverse problem of determining the time dependent absorption coefficient appearing in the linear Boltzmann equation, from boundary observations. We prove in dimension $n\geq 2$, that the absorption coefficient can be uniquely determined in a precise subset of the domain, from the albedo operator. We derive a logarithm type stability estimate in the determination of the absorption coefficient from the albedo operator, in a subset of our domain assuming that it is known outside this subset. Moreover, we prove that we can extend this result to the determination of the coefficient in a larger region, and then in the whole domain provided that we have much more data. We prove also an identification result for the scattering coefficient appearing in the linear Boltzmann equation.