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

Abstract

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.