On overdetermination conditions in order determination inverse problems for fractional diffusion equations

The fractional diffusion equation in an infinite layer is considered, and an integral identity is proved that relates the solution of the Cauchy problem for the equation under consideration with entire harmonic functions. Based on this identity, overdetermination conditions are proposed in inverse problems of determining the order for the fractional diffusion equation. In these inverse problems, in addition to the desired solution, the order of fractional differentiation is unknown and must be determined. The proposed overdetermination conditions are spatially oriented: these conditions use information about the desired solution at fixed points in time. Theorems on the uniqueness and solvability of inverse problems with the proposed overdetermination conditions are proved.