Source Identification Problem for a Nonlinear Subdiffusion Equation

TL;DR

This paper addresses the inverse problem of identifying a nonlinear, space- and time-dependent source term in a subdiffusion equation with Caputo derivatives, establishing existence and uniqueness of solutions.

Contribution

It introduces a novel approach to reconstructing nonlinear source terms in subdiffusion equations with Caputo derivatives, extending previous results to more complex dependencies.

Findings

Proved existence and uniqueness of solutions for the inverse problem.

Established a priori estimates for the nonlinear subdiffusion equation.

Extended inverse problem analysis to equations with nonlinear right-hand sides.

Abstract

The work is devoted to the study of the inverse problem of determining the right-hand side of a nonlinear subdiffusion equation with a Caputo derivative with respect to time. Nonlinearity of the equation means that the right-hand side of the equation depends nonlinearly on the solution of the equation. The inverse problem consists of reconstructing the coefficient of the right-hand side, which depends on both time and spatial variables, under a measurement in an integral form. Similar inverse problems were previously studied in the case when the right-hand side depends only on time or on a spatial variable. A weak solution is sought by the Galerkin method. A priori estimates are proved, and with their help, the existence and uniqueness of a solution to the inverse problem under consideration are established. It is noteworthy that the results obtained are new for diffusion equations as well.