An Inverse Problem for the Subdiffusion Equation with a Non-Local in Time Condition

An inverse problem of determining the right-hand side of the abstract subdiffusion equation with a fractional Caputo derivative is considered in a Hilbert space H. For the forward problem, instead of the Cauchy condition, the non-local in time condition u(0)=u(T) is taken. The right-hand side of the equation has the form g(t)f with a given function g(t) and an unknown element f∈H. If the function g(t) preserves its sign, then under a over-determined condition u(t0)=ψ, t0∈(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 the well-posedness of the problem by choosing properly t0. Moreover, we show that for some g(t), for the existence of a solution to the inverse problem, certain orthogonality conditions must be satisfied, but in this case there is no uniqueness. To the best knowledge of authors, the inverse problem with the non-local condition u(0)=u(T) has been considered for the first time. Moreover, all the results obtained are new not only for the subdiffusion equation, but also for the classical diffusion equation.