Global uniqueness in an inverse problem for time fractional diffusion equations

Yavar Kian (1, 2, 3) , Lauri Oksanen (4) , Eric Soccorsi (2, 1, 3) , Masahiro Yamamoto (5)

1 UTLN - Université de Toulon
2 CPT - Centre de Physique Théorique - UMR 7332
3 CPT - E8 Dynamique quantique et analyse spectrale
4 UCL - University College of London [London]
5 UTokyo - The University of Tokyo

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d \geq 2$, with boundary $\partial M$, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T) \times M$, $T>0$, with time-fractional Caputo derivative of order $\alpha \in (0,1) \cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solution on a subset of $\partial M$ at fixed time. In the ``flat" case where $M$ is a compact subset of $\mathbb R^d$, two out the three coefficients $\rho$ (weight), $a$ (conductivity) and $q$ (potential) appearing in the equation $\rho \partial_t^\alpha u-\textrm{div}(a \nabla u)+ q u=0$ on $(0,T)\times \Omega$ are recovered simultaneously.

Mots clés: Inverse problems; fractional diffusion equation; partial data

Domaines: Equations aux dérivées partielles [math.AP]

Yavar Kian, Lauri Oksanen, Eric Soccorsi, Masahiro Yamamoto. Global uniqueness in an inverse problem for time fractional diffusion equations. Journal of Differential Equations, 2018, 264 (2), pp.1146-1170. ⟨10.1016/j.jde.2017.09.032⟩. ⟨hal-01250971⟩