Global uniqueness in an inverse problem for time fractional diffusion equations

Abstract

International audienceGiven $(M,g)$, 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

Initial-boundary value problem for distributed order time-fractional diffusion equations

“…This statement is of great interest in the analysis of inverse coefficient problems associated with timefractional diffusion equations, see e.g. [10,11,17]. Theorem 1.4.…”

Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations

“…For diffusion equations corresponding to the case α = 1 with time independent source terms, several authors investigated the conditional stability (e.g. [5,37,38] [15,16,17,19,23,31] where some inverse coefficient problems and some related results have been considered.…”

Inverse problems of determining coefficients of the fractional partial differential equations

“…Theorem 8 (Kian, Oksanen, Soccorsi and Yamamoto [25]). We assume that either of the following conditions is fulfilled.…”