Uniqueness for an inverse problem of determining order and temporal factor of the source for time-fractional evolution equations

This paper addresses the inverse problem of simultaneously recovering the fractional order $\alpha \in (0,1)\cup (1,2)$ and the time-dependent source factor $p(t)$ in the Cauchy problem for an evolution equation with a general self-adjoint operator $A$ in a Hilbert space $X$. The overdetermination condition is given by the scalar product $( u(t), \psi)_X$ for $0 < t < T$, where $\psi \in D(A)$ is an arbitrary fixed element. Uniqueness of the fractional order $\alpha$ is established independently of the specific form of the elliptic operator $A$ and the source function $p(t)$. Furthermore, uniqueness of the factor $p(t)$ is proved not only under the trivial overdetermination $( u(t), \psi)_X = 0$ for all $t \in (0,T)$, but also when the function $t \mapsto ( u(t), \psi)_X$ possesses sufficient smoothness. The proof relies on a decomposition of the solution near $t=0$ into a least smooth component and a smoother remainder.