Space- and Time-Dependent Source Identification Problem with Integral Overdetermination Condition

TL;DR

This paper investigates an inverse problem for a subdiffusion equation with a Caputo derivative, focusing on determining a space- and time-dependent source term from integral overdetermination data, establishing existence and uniqueness of solutions.

Contribution

It introduces the first study of fractional inverse problems with integral overdetermination conditions, proving existence and uniqueness of solutions for such problems.

Findings

Existence of a weak solution is proven.

Uniqueness of the solution is established.

Results are new for both fractional and classical parabolic equations.

Abstract

This paper is devoted to the study of the inverse problem of determining the right-hand side of the subdiffusion equation with the Caputo derivative with respect to time. In our case, the inverse problem consists in restoring the coefficient of the right-hand side, which depends on both the time and the spatial variable, when measured in integral form. Previously, similar inverse problems were studied for hyperbolic and parabolic equations with a different overdetermination condition, and in some works the existence and uniqueness of generalized solutions was established, while in others, the uniqueness of classical solutions was established. However, similar inverse problems for fractional equations with an integral overdetermination condition have not been considered before this work. The existence and uniqueness of a weak solution to the inverse problem under consideration is established. It is noteworthy that the results obtained are new for parabolic equations as well.