Inverse problem for the subdiffusion equation with non-local in time condition

TL;DR

This paper investigates the inverse problem of determining a spatial function in a subdiffusion equation with fractional derivatives, establishing conditions for existence, uniqueness, and non-uniqueness of solutions under various assumptions.

Contribution

It provides new theoretical results on the inverse problem for subdiffusion equations with non-local time conditions, including criteria for uniqueness and existence based on properties of the function g(t).

Findings

Unique solution when g(t) does not change sign and conditions are met.

Non-uniqueness can occur for sign-changing g(t) without additional constraints.

Well-posedness can be achieved for certain g(t) by choosing the point t_0.

Abstract

In the Hilbert space $H$, the inverse problem of determining the right-hand side of the abstract subdiffusion equation with the fractional Caputo derivative is considered. For the forward problem, a non-local in time condition $u(0)=u(T)$ is taken. The right-hand side of the equation has the form $fg(t)$, and the unknown element is $f\in H$. If function $g(t)$ does not change sign, then under a over-determination condition $ u (t_0)= \psi $, $t_0\in (0, T)$, it is proved that the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution for some sign-changing functions $g(t)$. For such functions $g(t)$, under certain conditions on this function, one can achieve well-posedness of the problem by choosing $t_0$. And for some $g(t)$, for the existence of a solution to the inverse problem, certain orthogonality conditions must be satisfied and in this case there is no uniqueness. All the results obtained are also new for the classical diffusion equations.