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# Analysis

### International mathematical journal of analysis and its applications

CiteScore 2018: 0.72

SCImago Journal Rank (SJR) 2018: 0.363
Source Normalized Impact per Paper (SNIP) 2018: 0.530

Mathematical Citation Quotient (MCQ) 2018: 0.36

Online
ISSN
2196-6753
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Volume 36, Issue 1

# A variable exponent Sobolev theorem for fractional integrals on quasimetric measure spaces

Stefan Samko
• Corresponding author
• Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, 8005-139 Faro, Portugal
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Published Online: 2015-08-27 | DOI: https://doi.org/10.1515/anly-2015-5003

## Abstract

We prove that the coefficients ${a}_{ij}\left(x\right)$, q(x) and the domain Ω of a multidimensional fractional diffusion equation can be recovered uniquely from measurements $u\left(b,t\right)$, $t\in \left({t}_{0},{t}_{1}\right)$, at an arbitrary single point b inside a bounded domain $\Omega \subset {ℝ}^{n}$. From the measurements we first recover infinitely many spectral data $\left({\lambda }_{m},{\varphi }_{m}\left(x\right)\right)$ of the elliptic operator associated with the fractional diffusion equation. Then, the coefficients ${a}_{ij}\left(x\right)$ and q(x) are found from linear algebraic systems of the form $Ay=b$, where A is a generalized Wronskian of some set of eigenfunctions that can be shown to be nontrivial. The domain Ω is reconstructed using the first eigenfunction ${\varphi }_{1}\left(x\right)$.

MSC: 46E30; 26A33; 31A05

Accepted: 2015-07-09

Published Online: 2015-08-27

Published in Print: 2016-02-01

Funding Source: Russian Fund of Basic Research

Award identifier / Grant number: 15-01-02732

Citation Information: Analysis, Volume 36, Issue 1, Pages 27–37, ISSN (Online) 2196-6753, ISSN (Print) 0174-4747,

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