Uniqueness of Inverse Source Problems for Time-Fractional Diffusion Equations with Singular Functions in Time

We consider a fractional diffusion equation of order $$\alpha \in (0,1)$$ whose source term is singular in time: $$ (\partial _t^\alpha +A)u(\boldsymbol{x},t)=\mu (t)f(\boldsymbol{x}),\quad (\boldsymbol{x},t)\in \Omega \times (0,T), $$ where $$\mu $$ belongs to a Sobolev space of negative order. In inverse source problems of determining $$f|_\Omega $$ by the data $$u|_{\omega \times (0,T)}$$ with a given subdomain $$\omega \subset \Omega $$ and $$\mu |_{(0,T)}$$ by the data $$u|_{\{\boldsymbol{x}_0\}\times (0,T)}$$ with a given point $$\boldsymbol{x}_0\in \Omega $$ , we prove the uniqueness by reducing to the case $$\mu \in L^2(0,T)$$ . The key is a transformation of a solution to an initial-boundary value problem with a regular function in time.