Uniqueness of inverse source problems for time-fractional diffusion equations with singular functions in time

Abstract

We consider a fractional diffusion equations of order $α\in(0,1)$ whose source term is singular in time: $(\partial_t^α+A)u(x,t)=μ(t)f(x)$, $(x,t)\inΩ\times(0,T)$, where $μ$ belongs to a Sobolev space of negative order. In inverse source problems of determining $f|_Ω$ by the data $u|_{ω\times(0,T)}$ with a given subdomain $ω\subsetΩ$ or $μ|_{(0,T)}$ by the data $u|_{\{x_0\}\times(0,T)}$ with a given point $x_0\inΩ$, we prove the uniqueness by reducing to the case $μ\in L^2(0,T)$. The key is a transformation of a solution to an initial-boundary value problem with a regular function in time.