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

Given a connected compact Riemannian manifold (M,g) without boundary, dim⁡M≥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 α∈(...

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Bibliographic Details
Published in:Journal of Differential Equations 2020-10, Vol.269 (9), p.7498-7528
Main Authors: Helin, Tapio, Lassas, Matti, Ylinen, Lauri, Zhang, Zhidong
Format: Article
Language:English
Subjects:
Inverse problem
Regularity
Space–time fractional diffusion equation
Uniqueness
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Summary:Given a connected compact Riemannian manifold (M,g) without boundary, dim⁡M≥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 α∈(0,1], and the space fractional part by (−Δg)β, where β∈(0,1] and Δg is the Laplace–Beltrami operator on the manifold. The case α=β=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.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2020.05.022