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

TL;DR

This paper proves the uniqueness of inverse source problems for time-fractional diffusion equations with singular time-dependent sources, by transforming the problem to a case with regular functions in time.

Contribution

It introduces a method to establish uniqueness in inverse problems involving singular time-dependent sources in fractional diffusion equations.

Findings

Uniqueness is proven for inverse source problems with singular time functions.

Reduction to cases with regular time functions is achieved through a solution transformation.

The approach applies to both spatially localized and pointwise data.

Abstract

We consider a fractional diffusion equations of order $\alpha\in(0,1)$ whose source term is singular in time: $(\partial_t^\alpha+A)u(x,t)=\mu(t)f(x)$, $(x,t)\in\Omega\times(0,T)$, 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 $\omega\subset\Omega$ or $\mu|_{(0,T)}$ by the data $u|_{\{x_0\}\times(0,T)}$ with a given point $x_0\in\Omega$, 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.