Inverse problems for heat equation and space-time fractional diffusion equation with one measurement

Abstract

Given a connected compact Riemannian manifold $(M,g)$ without boundary, $\dim M\ge 2$, we consider a space--time fractional diffusion equation with an interior source that is supported on an open subset $V$ of the manifold. The time-fractional part of the equation is given by the Caputo derivative of order $\alpha \in (0,1]$, and the space fractional part by $(-{\Delta }_g{)}^{\beta }$, where $\beta \in (0,1]$ and ${\Delta }_g$ is the Laplace--Beltrami operator on the manifold. The case $\alpha =\beta =1$, which corresponds to the standard heat equation on the manifold, is an important special case. We construct a specific source such that measuring the evolution of the corresponding solution on $V$ determines the manifold up to a Riemannian isometry.

arXiv comment34 pages, 1 figure

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