Inverse coefficient problems for the heat equation with fractional Laplacian

TL;DR

This paper investigates inverse problems for fractional heat equations, demonstrating conditions for solution existence and formulas for source coefficients using minimal data, with extensions to non-local data and space-dependent sources.

Contribution

It introduces new methods for solving inverse problems in fractional heat equations, including minimal data requirements and formulas for source recovery.

Findings

Existence of weak solutions with single observation data.

Explicit formula for the time-dependent source coefficient with additional data.

Extension to inverse problems with non-local data and space-dependent sources.

Abstract

In the present paper we study inverse problems related to determining the time-dependent coefficient and unknown source function of fractional heat equations. Our approach shows that having just one set of data at an observation point ensures the existence of a weak solution for the inverse problem. Furthermore, if there is an additional datum at the observation point, it leads to a specific formula for the time-dependent source coefficient. Moreover, we investigate inverse problems involving non-local data and recovering the space-dependent source function of the fractional heat equation.