The Calderón problem for a space-time fractional parabolic equation

Ru Yu Lai, Yi Hsuan Lin, Angkana Rüland

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14 Scopus citations

Abstract

In this article we study an inverse problem for the space-time fractional parabolic operator (∂t - Δ)s + Q with 0 < s < 1 in any space dimension. We uniquely determine the unknown bounded potential Q from infinitely many exterior Dirichlet-to-Neumann type measurements. This relies on Runge approximation and the dual global weak unique continuation properties of the equation under consideration. In discussing weak unique continuation of our operator, a main feature of our argument relies on a new Carleman estimate for the associated degenerate parabolic Caffarelli- Silvestre extension. Furthermore, we also discuss constructive single measurement results based on the approximation and unique continuation properties of the equation.

Original languageEnglish (US)
Pages (from-to)2655-2688
Number of pages34
JournalSIAM Journal on Mathematical Analysis
Volume52
Issue number3
DOIs
StatePublished - 2020

Bibliographical note

Funding Information:
∗Received by the editors June 24, 2019; accepted for publication (in revised form) March 16, 2020; published electronically June 4, 2020. https://doi.org/10.1137/19M1270288 Funding: The work of the first author was partially supported by the National Science Foundation grant DMS-1714490. The work of the second author was supported by the Finnish Centre of Excellence in Inverse Modelling and Imaging, Academy of Finland grant 284715, by the Academy of Finland project 309963, 2018-2019, and by the Ministry of Science and Technology, Taiwan (MOST) under the Columbus Program grant MOST-109-2636-M-009-006, 2020-2025. †School of Mathematics, University of Minnesota, Minneapolis, MN, 55455 (rylai@umn.edu).

Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

Keywords

  • Carleman estimate
  • Degenerate parabolic equations
  • Fractional parabolic Calderón problem
  • Nonlocal
  • Runge approximation
  • Unique continuation property

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