# Inverse source problems for positive operators. I. Hypoelliptic diffusion and subdiffusion equations

## TL;DR

This paper investigates inverse problems for positive operators in parabolic and fractional diffusion equations, establishing existence and uniqueness results, and applies these findings to various hypoelliptic and subelliptic models on Lie groups.

## Contribution

It provides new theoretical results on inverse problems for a broad class of positive operators, including hypoelliptic and subelliptic cases, with applications to diverse diffusion models.

## Key findings

- Proved existence and uniqueness of solutions for inverse problems in fractional diffusion.
- Applied results to models on graded Lie groups, including Heisenberg group.
- Numerical example demonstrating the cooling problem with involution.

## Abstract

A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous left-invariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm-Liouville problems, differential models with involution, fractional Sturm-Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a "cooling function", and how the involution normally slows down the cooling speed of the rod.