Global solvability of inverse coefficient problem for one fractional diffusion equation with initial non-local and integral overdetermination conditions

In this work, we consider an inverse problem of determining the coefficient at the lower term of a fractional diffusion equation. The direct problem is the initial-boundary problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the unknown coefficient, an overdetermination condition of the integral form is specified with respect to the solution of the direct problem. Using Green's function for an ordinary fractional differential equation with a non-local boundary condition and the Fourier method, the inverse problem is reduced to an equivalent problem. Further, by using the fixed-point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained.