# Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations

## TL;DR

This paper addresses the inverse problem of reconstructing source terms and coefficients in diffusion equations, demonstrating well-posedness and stability results for boundary-based recovery in cylindrical domains.

## Contribution

It introduces methods for reconstructing space-independent source terms and coefficients in diffusion equations, proving well-posedness and Lipschitz stability from boundary data.

## Key findings

- Source terms independent of one spatial variable can be reconstructed from boundary measurements.
- The inverse problem is shown to be well-posed under certain conditions.
- Lipschitz stability estimates are established for the recovery process.

## Abstract

We consider the inverse source problem of determining a source term depending on both time and space variable for fractional and classical diffusion equations in a cylindrical domain from boundary measurements. With suitable boundary conditions we prove that some class of source terms which are independent of one space direction, can be reconstructed from boundary measurements. Actually, we prove that this inverse problem is well-posed. We establish also some results of Lipschitz stability for the recovery of source terms which we apply to the stable recostruction of time-dependent coefficients.