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

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.

arXiv comment15 pages

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Cited by in corpus (1)

• Inverse $t$-source problem and a strict positivity property for coupled subdiffusion systems